diff --git a/.github/workflows/run_tests.yaml b/.github/workflows/run_tests.yaml index b551861..ed0222b 100644 --- a/.github/workflows/run_tests.yaml +++ b/.github/workflows/run_tests.yaml @@ -10,11 +10,11 @@ run_all_tests: runs-on: 'ubuntu-latest' steps: - - uses: actions/checkout@v2 + - uses: actions/checkout@v4 - name: Install Python 3 - uses: actions/setup-python@v1 + uses: actions/setup-python@v5 with: - python-version: 3.9 + python-version: 3.11 - name: Install dependencies run: | python -m pip install --upgrade pip diff --git a/README.md b/README.md index 1575e03..ab8b7b4 100644 --- a/README.md +++ b/README.md @@ -2,7 +2,13 @@ Working with infinite data structures in Python. ## Introduction -Have you ever wondered what would happen if you could input an infinite structure (e.g., an infinite graph) into your algorithm instead of a finite one? With AutStr, you can do exactly that for certain infinite structures. AutStr provides an easy-to-use interface for defining relational structures over predefined infinite base structures. Currently, AutStr offers built-in support for a robust extension of linear integer arithmetic, while additional base structures over arbitrary countable domains can be defined via the low-level API. +What if your algorithms could process infinite structures — like complete infinite graphs or the entire set of natural numbers — with the same ease as manipulating a data frame? AutStr provides an intuitive interface for defining and manipulating exact representations of infinite relational structures. With AutStr, infinite structures become first-class citizens in Python. + +Targeted at researchers in algorithmic model theory and curious practitioners alike, AutStr offers: +- **Symbolic representation** of infinite sets and relations +- **Automata-based computation** for efficient manipulation +- **First-order logic interface** for formal queries +- **Visualization tools** for insight into infinite structures ### Installation with `uv` Here's how to install AutStr using `uv`, the high-performance Python package installer: @@ -222,31 +228,14 @@ VisualDFA(remaining.evaluate()).show_diagram() # width=2 (easier): narrow = "∃x.∃y.E(x,y) ∧ (∃z.E(y,z))" ``` - - Depth scales often better due to incremental minimization - - Width causes exponential alphabet growth: $|\Sigma|^k$ + - Depth can cause non-elementary(!) statespace explosion but scales often much better due to incremental minimization + - Width causes exponential alphabet growth: $|\Sigma|^k$ but SparseDFAs can avoid explicit enumeration of entire alphabet in many cases. 2. **Complexity Boundaries** | Parameter | Best Case | Worst Case | |------------------|-----------------|---------------------| | **Quantifier Depth** | Constant state space | Non-elementary state space | - | **Free Variables** | Exponential alphabet size | Exponential alphabet size | - -4. **Optimization Strategies** - - **Minimize aggressively**: - ```python - candidates = (candidates - multiples).minify() # Force state reduction - ``` - - **Avoid high-arity**: - ```python - # Decompose wide relations: - R1 = project(relation, ['x','y']) - R2 = project(relation, ['z']) - ``` - - **Use bounded quantification**: - ```python - # Instead of ∀x.φ(x), use: - bounded = ∀(x, x<1000).φ(x) # Finite domain restriction - ``` + | **Free Variables** | constant number of exceptions | Exponential alphabet size | #### Theoretical Insight While this infinite sieve beautifully demonstrates symbolic algorithm design, state complexity grows rapidly for sieved primes @@ -268,17 +257,53 @@ This implementation provides a complete toolkit for working with automatic struc ### Final Note AutStr began as a summer passion project in 2022—a practical exploration of the automatic structures I'd studied theoretically during my PhD. This library represents the intersection of academic curiosity and hands-on implementation, born from a desire to make abstract model theory concepts tangible. -Released in June 2025 as-is, the library remains fundamentally unchanged from its original vision except for: +Released in July 2025 following a major update, the library now includes significant new features beyond its original vision. This update was developed through a vibe coding session using DeepSeek, with extensive human testing and supervision throughout the process. + +Highlights of the update: +- Newly implemented `sparse_dfa` backend for efficient automata operations +- Serialization support for automata and structures +- MSO0 as finite powerset structure over natural numbers (e.g. index sets) - Modernized packaging (`pyproject.toml`) - Dependency version updates - Expanded documentation -While not actively maintained, AutStr stands as: -1. A functional implementation of basic automatic structure theory -2. A testament to the elegance of infinite-state computation -3. An invitation to explore algorithmic model theory hands-on +#### Performance and Maintenance +Recent updates have removed obvious performance bottlenecks through: + +- JAX-accelerated automata operations +- Sparse DFA representations +- Efficient serialization, which allows storing precompiled results + +However, significant optimization opportunities remain: +- Vectorization: Many sequential operations could still be parallelized +- Query Optimization: Advanced planning for first-order queries + +As this is a passion project developed outside my primary research, active maintenance will be limited. That said: + +- Bug reports are welcome and will be prioritized +- Performance contributions are especially appreciated +- Research collaborations involving AutStr are encouraged + + +## References on Automatic Structures +1. **Abu Zaid, F.** + *Algorithmic Solutions via Model Theoretic Interpretations.* + Dissertation, RWTH Aachen University, 2016. + DOI: [10.18154/RWTH-2017-07663](https://doi.org/10.18154/RWTH-2017-07663) + +2. **Blumensath, A., & Grädel, E.** + *Automatic Structures.* + Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS 2000). + Pages: 51–62. + URL: [LICS 2000 Proceedings](https://lics.siglog.org/2000/Grdel-AutomaticStructures.html) -> "Some things are worth building not because they scale, but because they reveal." -> — Faried Abu Zaid, June 2025 +3. **Khoussainov, B., & Nerode, A.** + *Automatic presentations of structures.* + In D. Leivant (Ed.), Logic and Computational Complexity (LCC 1994). Lecture Notes in Computer Science, vol 960. + Springer. DOI: [10.1007/3-540-60178-3_93](https://doi.org/10.1007/3-540-60178-3_93) -For researchers and enthusiasts: May this implementation spark new insights into the beautiful complexity of infinite structures. For practical applications, consider pairing with finite approximations or domain-specific abstractions. +4. **Khoussainov, B., Rubin, S., & Stephan, F.** + *Automatic Structures: Richness and Limitations.* + Logical Methods in Computer Science, Volume 3, Issue 2 (2007). + arXiv: [cs/0703064](https://arxiv.org/abs/cs/0703064) + DOI: [10.2168/LMCS-3(2:2)2007](https://doi.org/10.2168/LMCS-3%282%3A2%292007) diff --git a/autstr/arithmetic.py b/autstr/arithmetic.py index b7446a6..94646d7 100644 --- a/autstr/arithmetic.py +++ b/autstr/arithmetic.py @@ -2,22 +2,22 @@ from abc import ABC, abstractmethod from copy import deepcopy, copy -from typing import List, Union, Tuple, Dict, Set +from typing import List, Union, Tuple, Dict import math -from automata.fa.dfa import DFA - from autstr.buildin.automata import k_longer_automaton +from autstr.sparse_automata import SparseDFA from autstr.utils.automata_tools import iterate_language, lsbf_Z_automaton -from autstr.buildin.presentations import buechi_arithmetic_Z +from autstr.buildin.presentations import BuechiArithmeticZ from autstr.utils.misc import get_unique_id +from autstr.utils.misc import encode_symbol, decode_symbol class Term(ABC): """ Abstract class representing a term over the (base 2) Büchi arithmetic over the integers :math:`\\mathbb{Z}` """ - arithmetic = buechi_arithmetic_Z() + arithmetic = BuechiArithmeticZ() def __init__(self): self.presentation = None @@ -32,7 +32,7 @@ def update_presentation(self, recursive=True) -> None: """ raise NotImplementedError - def evaluate(self) -> DFA: + def evaluate(self) -> SparseDFA: """ Returns automatic presentation of the relation. @@ -119,7 +119,7 @@ def __contains__(self, item): words[i] = w + ('*' * difference) input_word = [tuple(w[i] for w in words) for i in range(l_max)] - return self.evaluate().accepts_input(input_word) + return self.evaluate().accepts(input_word) def drop(self, variables: List[Union[str, VariableETerm]]) -> DropRATerm: """ @@ -157,7 +157,7 @@ def isempty(self) -> bool: if self.presentation is None: self.update_presentation() - return self.presentation.isempty() + return self.presentation.is_empty() def isfinite(self) -> bool: """ @@ -168,7 +168,7 @@ def isfinite(self) -> bool: if self.presentation is None: self.update_presentation() - return self.presentation.isfinite() + return self.presentation.is_finite() def __iter__(self): """ @@ -180,7 +180,7 @@ def __iter__(self): if self.presentation is None: self.update_presentation() - for t in iterate_language(self.presentation, backward=True): + for t in iterate_language(self.presentation, backward=True, padding_symbol='*'): yield tuple( int( n.replace('*', '')[:-1], base=2 @@ -201,8 +201,8 @@ def update_presentation(self, recursive=True) -> None: self.subterm.update_presentation(recursive) sub_presentation = self.subterm.evaluate() - k_distance = len(sub_presentation.states) + 1 - inf_witness = k_longer_automaton(k_distance, len(self.subterm.get_variables()) - 1, self.arithmetic.sigma) + k_distance = sub_presentation.num_states + 1 + inf_witness = k_longer_automaton(k_distance, len(self.subterm.get_variables()) - 1, self.arithmetic.sigma, self.arithmetic.padding_symbol) arithmetic = deepcopy(self.arithmetic) T, L = get_unique_id(arithmetic.get_relation_symbols(), 2) arithmetic.update(**{T: self.subterm.evaluate(), L: inf_witness}) @@ -504,9 +504,9 @@ def gt(self, other) -> BaseRATerm: :param other: The term on the lhs :return: """ - return BaseRATerm('Lt', [other, self]) + return BaseRATerm('Gt', [self, other]) - def evaluate(self) -> DFA: + def evaluate(self) -> SparseDFA: if self.presentation is None: self.update_presentation() diff --git a/autstr/buildin/automata.py b/autstr/buildin/automata.py index 5e17a1d..e1cd7b9 100644 --- a/autstr/buildin/automata.py +++ b/autstr/buildin/automata.py @@ -1,107 +1,138 @@ -from typing import Set import itertools as it +from typing import Optional, Set +from autstr.sparse_automata import SparseDFA +from jax import numpy as jnp -from automata.fa.dfa import DFA +from autstr.utils.misc import encode_symbol -def length_automaton(n, sigma) -> DFA: +def length_automaton(n: int, base_alphabet: Set[int]) -> SparseDFA: """ - Creates an automaton that recognizes all words over sigma with length n + Creates an automaton that recognizes all words over base_alphabet with length exactly n. - :param n: The length - :param sigma: The input symbols - :return: + :param n: The exact word length + :param base_alphabet: Set of integer symbols + :return: SparseDFA recognizing words of length n """ - states = {f'{i}' for i in range(n + 1)} - transitions = {q: {a: str(int(q) + 1) if int(q) < n else q for a in sigma} for q in states} - - dfa = DFA( - states=states, - input_symbols=sigma, - initial_state='0', - transitions=transitions, - final_states={f'{n - 1}'} + # States: 0 (start), 1, 2, ..., n (accepting), n+1 (dead) + num_states = n + 2 + # Default transitions: move to next state or dead state + default_states = jnp.array([i + 1 for i in range(n)] + [n + 1, n + 1]) + # No exceptions needed (same behavior for all symbols) + exception_symbols = jnp.full((num_states, 0), -1) + exception_states = jnp.full((num_states, 0), -1) + # Only state n is accepting + is_accepting = jnp.array([False] * n + [True, False]) + start_state = 0 + + return SparseDFA( + num_states=num_states, + default_states=default_states, + exception_symbols=exception_symbols, + exception_states=exception_states, + is_accepting=is_accepting, + start_state=start_state, + symbol_arity=1, + base_alphabet=base_alphabet ) - return dfa -def k_longer_automaton(k: int, r: int, sigma: Set[str], padding_symbol='*') -> DFA: +def k_longer_automaton(k: int, r: int, base_alphabet: Set[int], padding_symbol: int) -> SparseDFA: """ - Creates an automaton that recognizes exactly the :math:`(r+1)`-tuple where the :math:`(r+1)`-th word is at least - :math:`k` letters longer than all other words in the tuple. - :param k: Minimal length distance to the other words. - :param r: Number of other words - :param sigma: The base alphabet - :return: The k-longer-automaton + Creates an automaton recognizing (r+1)-tuples where the last word is at least k letters + longer than the other r words. + + :param k: Minimal length difference + :param r: Number of reference words + :param base_alphabet: Set of integer symbols + :param padding_symbol: Padding symbol integer + :return: SparseDFA for the k-longer condition """ - assert padding_symbol in sigma - - states = set(range(k+1)) - states.add(-1) - input_symbols = set(it.product(sigma, repeat=r + 1)) - initial_state = 0 - - def get_nex_state(i, t): - if i >= 0: - if all([a == padding_symbol for a in t[:-1]]) and t[-1] != padding_symbol: - return min([i + 1, k]) # measure distance - elif t[-1] == padding_symbol: - return -1 # reject - elif i == 0: - return 0 # wait for all other words to end + # State mapping: [-1, 0, 1, ..., k] -> [0, 1, 2, ..., k+1] + state_mapping = {s: i for i, s in enumerate(range(-1, k+1))} + num_states = len(state_mapping) + sorted_alphabet = sorted(base_alphabet) + arity = r + 1 + + # Precompute all symbol tuples and their encodings + symbol_tuples = list(it.product(sorted_alphabet, repeat=arity)) + symbol_encodings = [encode_symbol(t, base_alphabet) for t in symbol_tuples] + + # Initialize DFA components + default_states = jnp.full(num_states, state_mapping[-1]) # Default to dead state + exception_list = [[] for _ in range(num_states)] + + # Build transitions + for state in range(-1, k+1): + state_idx = state_mapping[state] + for t, enc in zip(symbol_tuples, symbol_encodings): + # Compute next state + if state == -1: + next_state = -1 # Stay in dead state else: - return -1 # reject - else: - return -1 - - transitions = { - i: { - t: get_nex_state(i, t) for t in input_symbols - } for i in range(-1, k+1) - } - - final_states = {k} - - dfa = DFA( - states=states, - input_symbols=input_symbols, - initial_state=initial_state, - transitions=transitions, - final_states=final_states - ) - - return dfa - - -def zero() -> DFA: - """ - Creates an automaton over the empty symbol set that recognizes the empty language - - :return: - """ - result = DFA( - states={'0'}, - input_symbols=set(), - initial_state='0', - transitions={'0': {}}, - final_states=set() + if all(x == padding_symbol for x in t[:-1]) and t[-1] != padding_symbol: + next_state = min(state + 1, k) # Count extra length + elif t[-1] == padding_symbol: + next_state = -1 # Reject if padding the last word + elif state == 0: + next_state = 0 # Wait for other words to end + else: + next_state = -1 # Reject otherwise + + next_state_idx = state_mapping[next_state] + if next_state_idx != state_mapping[-1]: + exception_list[state_idx].append((enc, next_state_idx)) + + # Find max exceptions needed + max_exceptions = max(len(ex_list) for ex_list in exception_list) if exception_list else 0 + + # Build exception arrays + exception_symbols = jnp.full((num_states, max_exceptions), -1) + exception_states = jnp.full((num_states, max_exceptions), -1) + + for i, ex_list in enumerate(exception_list): + if ex_list: + syms, states = zip(*ex_list) + exception_symbols = exception_symbols.at[i, :len(syms)].set(jnp.array(syms)) + exception_states = exception_states.at[i, :len(states)].set(jnp.array(states)) + + # Final states: state k (meaning we've counted k extra symbols) + is_accepting = jnp.array([i == state_mapping[k] for i in range(num_states)]) + + return SparseDFA( + num_states=num_states, + default_states=default_states, + exception_symbols=exception_symbols, + exception_states=exception_states, + is_accepting=is_accepting, + start_state=state_mapping[0], + symbol_arity=arity, + base_alphabet=base_alphabet ) - return result - - -def one() -> DFA: - """ - Creates an automaton over the empty symbol set that recognizes language that contains only the empty word. - :return: - """ - result = DFA( - states={'0'}, - input_symbols=set(), - initial_state='0', - transitions={ - '0': {}, - }, - final_states={'0'} +def zero(symbol_arity: int = 1, base_alphabet: Optional[Set[int]] = None) -> SparseDFA: + """Automaton that rejects all inputs.""" + base_alphabet = base_alphabet or {0} + return SparseDFA( + num_states=1, + default_states=jnp.array([0]), + exception_symbols=jnp.full((1, 0), -1), + exception_states=jnp.full((1, 0), -1), + is_accepting=jnp.array([False]), + start_state=0, + symbol_arity=symbol_arity, + base_alphabet=base_alphabet ) - return result +def one(symbol_arity: int = 1, base_alphabet: Optional[Set[int]] = None) -> SparseDFA: + """Automaton that accepts all inputs.""" + base_alphabet = base_alphabet or {0} + return SparseDFA( + num_states=1, + default_states=jnp.array([0]), + exception_symbols=jnp.full((1, 0), -1), + exception_states=jnp.full((1, 0), -1), + is_accepting=jnp.array([True]), + start_state=0, + symbol_arity=symbol_arity, + base_alphabet=base_alphabet + ) \ No newline at end of file diff --git a/autstr/buildin/bin/buechi.autstr b/autstr/buildin/bin/buechi.autstr new file mode 100644 index 0000000..24326b3 Binary files /dev/null and b/autstr/buildin/bin/buechi.autstr differ diff --git a/autstr/buildin/bin/buechiZ.autstr b/autstr/buildin/bin/buechiZ.autstr new file mode 100644 index 0000000..0ad4405 Binary files /dev/null and b/autstr/buildin/bin/buechiZ.autstr differ diff --git a/autstr/buildin/bin/mso0.autstr b/autstr/buildin/bin/mso0.autstr new file mode 100644 index 0000000..4ab5e17 Binary files /dev/null and b/autstr/buildin/bin/mso0.autstr differ diff --git a/autstr/buildin/presentations.py b/autstr/buildin/presentations.py index d7b09f1..2efa036 100644 --- a/autstr/buildin/presentations.py +++ b/autstr/buildin/presentations.py @@ -1,13 +1,100 @@ -from automata.fa.dfa import DFA +from pathlib import Path import itertools as it +from typing import Dict, List, Set, Tuple +import jax.numpy as jnp + +from autstr.sparse_automata import SparseDFA from autstr.presentations import AutomaticPresentation +# Helper function to convert a symbol tuple to an integer encoding +def encode_symbol(tup: Tuple[str], base_alphabet: Set[str]) -> int: + """Encode a symbol tuple into an integer using base conversion.""" + base = len(base_alphabet) + mapping = {sym: idx for idx, sym in enumerate(sorted(base_alphabet))} + enc = 0 + for char in tup: + enc = enc * base + mapping[char] + return enc + +# Helper function to create a SparseDFA from a traditional DFA description +def create_sparse_dfa(states: List[str], input_symbols: Set[Tuple[str]], + transitions: Dict[str, Dict[Tuple[str], str]], + initial_state: str, final_states: Set[str]) -> SparseDFA: + """Convert a traditional DFA description to a SparseDFA.""" + # Map states to integers + state_to_index = {s: i for i, s in enumerate(states)} + num_states = len(states) + + # Determine base alphabet and arity + base_alphabet = set() + for sym in input_symbols: + for char in sym: + base_alphabet.add(char) + arity = len(next(iter(input_symbols))) if input_symbols else 0 + + # Create state arrays + default_states = [] + exception_symbols = [] + exception_states = [] + is_accepting = jnp.array([state in final_states for state in states], dtype=bool) + + # Build symbol mapping + symbol_map = {} + for symbol in input_symbols: + symbol_map[symbol] = encode_symbol(symbol, base_alphabet) + + # Process each state + for state in states: + # Find most common transition + next_states = [transitions[state][sym] for sym in input_symbols] + default_target = max(set(next_states), key=next_states.count) + default_states.append(state_to_index[default_target]) + + # Collect exceptions + exceptions = [] + for symbol, next_state in transitions[state].items(): + if next_state != default_target: + sym_enc = symbol_map[symbol] + next_idx = state_to_index[next_state] + exceptions.append((sym_enc, next_idx)) + + # Sort exceptions by symbol for consistency + exceptions.sort(key=lambda x: x[0]) + exception_symbols.append([e[0] for e in exceptions]) + exception_states.append([e[1] for e in exceptions]) + + # Pad exceptions + max_exceptions = max(len(e) for e in exception_symbols) if exception_symbols else 0 + padded_ex_syms = jnp.full((num_states, max_exceptions), -1, dtype=jnp.int32) + padded_ex_states = jnp.full((num_states, max_exceptions), -1, dtype=jnp.int32) + + for i in range(num_states): + if exception_symbols[i]: + padded_ex_syms = padded_ex_syms.at[i, :len(exception_symbols[i])].set(jnp.array(exception_symbols[i], dtype=jnp.int32)) + padded_ex_states = padded_ex_states.at[i, :len(exception_states[i])].set(jnp.array(exception_states[i], dtype=jnp.int32)) + + return SparseDFA( + num_states=num_states, + default_states=jnp.array(default_states, dtype=jnp.int32), + exception_symbols=padded_ex_syms, + exception_states=padded_ex_states, + is_accepting=is_accepting, + start_state=state_to_index[initial_state], + symbol_arity=arity, + base_alphabet=base_alphabet + ).minimize() + +def BuechiArithmetic() -> AutomaticPresentation: + """Load the serialized Büchi arithmetic presentation.""" + current_dir = Path(__file__).parent + bin_path = current_dir / 'bin' / 'buechi.autstr' + return AutomaticPresentation.automatic_presentation_from_file(str(bin_path)) + def buechi_arithmetic() -> AutomaticPresentation: - """ - create a presentation of Büchi arithmetic over the natural numbers :math:`\\mathbb{N}`. - """ - universe = DFA( + """Sparse version of Büchi arithmetic over natural numbers. Compiles definable relation from scratch.""" + # Universe automaton - only accepts valid binary numbers (no '*') + universe = create_sparse_dfa( states={'i', '0', '0+', '1', '*'}, input_symbols={('0',), ('1',), ('*',)}, transitions={ @@ -21,7 +108,7 @@ def buechi_arithmetic() -> AutomaticPresentation: final_states={'0', '1'} ) - addition = DFA( + addition = create_sparse_dfa( states={0, 1, 2}, input_symbols={('0', '0', '0'), ('0', '0', '1'), ('0', '0', '*'), ('0', '1', '0'), ('0', '1', '1'), ('0', '1', '*'), ('0', '*', '0'), ('0', '*', '1'), ('0', '*', '*'), @@ -61,7 +148,7 @@ def buechi_arithmetic() -> AutomaticPresentation: ) input_symbols = {a for a in it.product(['0', '1', '*'], repeat=2)} - weak_div = DFA( + weak_div = create_sparse_dfa( states={'0', '1', 'e'}, input_symbols=input_symbols, transitions={ @@ -76,110 +163,112 @@ def buechi_arithmetic() -> AutomaticPresentation: initial_state='0', final_states={'1'} ) + + # Create presentation presentation = AutomaticPresentation({'U': universe, 'A': addition, 'B': weak_div}) + + # Add bootstrap remaining relations presentation.update(Z='A(x,x,x)') presentation.update(Eq='exists z.(Z(z) and A(x,z,y))') presentation.update(Pt='B(x,x)') presentation.update(Lt='exists z.(not Z(z) and A(x, z, y))') - + presentation.update(Gt='exists z.(not Z(z) and A(y, z, x))') + return presentation +def BuechiArithmeticZ() -> AutomaticPresentation: + """Load the serialized Büchi arithmetic presentation.""" + current_dir = Path(__file__).parent + bin_path = current_dir / 'bin' / 'buechiZ.autstr' + return AutomaticPresentation.automatic_presentation_from_file(str(bin_path)) def buechi_arithmetic_Z() -> AutomaticPresentation: - """ - Creates a presentation of Büchi arithmetic over :math:`\\mathbb{Z}`. - """ - - base_input_symbols = {'0', '1', '*'} - - universe = DFA( - states={'-1', 'i+', 'i', '0', '0+', '1', '*'}, - input_symbols={(a,) for a in base_input_symbols}, - transitions={ - '-1': {('0',): 'i', ('1',): 'i+', ('*',): '*'}, - 'i+': {('0',): '0+', ('1',): '1', ('*',): '*'}, - 'i': {('0',): '0', ('1',): '1', ('*',): '*'}, - '0': {('0',): '0+', ('1',): '1', ('*',): '*'}, - '0+': {('0',): '0+', ('1',): '1', ('*',): '*'}, - '1': {('0',): '0+', ('1',): '1', ('*',): '*'}, - '*': {('0',): '*', ('1',): '*', ('*',): '*'}, - }, - initial_state='-1', - final_states={'0', '1'} + """Sparse version of Büchi arithmetic over integers.""" + # Universe automaton + universe_states = ['-1', 'i+', 'i', '0', '0+', '1', '*'] + universe_symbols = {('0',), ('1',), ('*',)} + universe_trans = { + '-1': {('0',): 'i', ('1',): 'i+', ('*',): '*'}, + 'i+': {('0',): '0+', ('1',): '1', ('*',): '*'}, + 'i': {('0',): '0', ('1',): '1', ('*',): '*'}, + '0': {('0',): '0+', ('1',): '1', ('*',): '*'}, + '0+': {('0',): '0+', ('1',): '1', ('*',): '*'}, + '1': {('0',): '0+', ('1',): '1', ('*',): '*'}, + '*': {('0',): '*', ('1',): '*', ('*',): '*'}, + } + universe = create_sparse_dfa( + universe_states, universe_symbols, universe_trans, '-1', {'0', '1'} ) - - addition_input_symbols = set(it.product(base_input_symbols, repeat=3)) - - addition_intermediate = DFA( - states={-1, 0, 1, 2}, - input_symbols=addition_input_symbols, - transitions={ - -1: {a: 0 if '*' not in a else 2 for a in addition_input_symbols}, - 0: { - ('0', '0', '0'): 0, ('0', '0', '1'): 2, ('0', '0', '*'): 2, ('0', '1', '0'): 2, ('0', '1', '1'): 0, - ('0', '1', '*'): 2, ('0', '*', '0'): 0, ('0', '*', '1'): 2, ('0', '*', '*'): 2, - ('1', '0', '0'): 2, ('1', '0', '1'): 0, ('1', '0', '*'): 2, ('1', '1', '0'): 1, ('1', '1', '1'): 2, - ('1', '1', '*'): 2, ('1', '*', '0'): 2, ('1', '*', '1'): 0, ('1', '*', '*'): 2, - ('*', '0', '0'): 0, ('*', '0', '1'): 2, ('*', '0', '*'): 2, ('*', '1', '0'): 2, ('*', '1', '1'): 0, - ('*', '1', '*'): 2, ('*', '*', '0'): 2, ('*', '*', '1'): 2, ('*', '*', '*'): 2, - }, - 1: { - ('0', '0', '0'): 2, ('0', '0', '1'): 0, ('0', '0', '*'): 2, ('0', '1', '0'): 1, ('0', '1', '1'): 2, - ('0', '1', '*'): 2, ('0', '*', '0'): 2, ('0', '*', '1'): 0, ('0', '*', '*'): 2, - ('1', '0', '0'): 1, ('1', '0', '1'): 2, ('1', '0', '*'): 2, ('1', '1', '0'): 2, ('1', '1', '1'): 1, - ('1', '1', '*'): 2, ('1', '*', '0'): 1, ('1', '*', '1'): 2, ('1', '*', '*'): 2, - ('*', '0', '0'): 2, ('*', '0', '1'): 0, ('*', '0', '*'): 2, ('*', '1', '0'): 1, ('*', '1', '1'): 2, - ('*', '1', '*'): 2, ('*', '*', '0'): 2, ('*', '*', '1'): 0, ('*', '*', '*'): 2, - }, - 2: { - ('0', '0', '0'): 2, ('0', '0', '1'): 2, ('0', '0', '*'): 2, ('0', '1', '0'): 2, ('0', '1', '1'): 2, - ('0', '1', '*'): 2, ('0', '*', '0'): 2, ('0', '*', '1'): 2, ('0', '*', '*'): 2, - ('1', '0', '0'): 2, ('1', '0', '1'): 2, ('1', '0', '*'): 2, ('1', '1', '0'): 2, ('1', '1', '1'): 2, - ('1', '1', '*'): 2, ('1', '*', '0'): 2, ('1', '*', '1'): 2, ('1', '*', '*'): 2, - ('*', '0', '0'): 2, ('*', '0', '1'): 2, ('*', '0', '*'): 2, ('*', '1', '0'): 2, ('*', '1', '1'): 2, - ('*', '1', '*'): 2, ('*', '*', '0'): 2, ('*', '*', '1'): 2, ('*', '*', '*'): 2, - }, + + # Addition automaton (intermediate) + add_states = [-1, 0, 1, 2] + add_symbols = set(it.product(['0', '1', '*'], repeat=3)) + add_trans = { + -1: {a: 0 if '*' not in a else 2 for a in add_symbols}, + 0: { + ('0','0','0'): 0, ('0','0','1'): 2, ('0','0','*'): 2, + ('0','1','0'): 2, ('0','1','1'): 0, ('0','1','*'): 2, + ('0','*','0'): 0, ('0','*','1'): 2, ('0','*','*'): 2, + ('1','0','0'): 2, ('1','0','1'): 0, ('1','0','*'): 2, + ('1','1','0'): 1, ('1','1','1'): 2, ('1','1','*'): 2, + ('1','*','0'): 2, ('1','*','1'): 0, ('1','*','*'): 2, + ('*','0','0'): 0, ('*','0','1'): 2, ('*','0','*'): 2, + ('*','1','0'): 2, ('*','1','1'): 0, ('*','1','*'): 2, + ('*','*','0'): 2, ('*','*','1'): 2, ('*','*','*'): 2, }, - initial_state=-1, - final_states={0} - ) - - weak_div_input_symbols = {a for a in it.product(['0', '1', '*'], repeat=2)} - weak_div = DFA( - states={'-1', '0', '1', 'e'}, - input_symbols=weak_div_input_symbols, - transitions={ - '-1': {a: '0' if a[1] == '0' else 'e' for a in weak_div_input_symbols}, - '0': { - a: '0' if a == ('0', '0') else '1' if a == ('0', '1') or a == ('1', '1') else 'e' for a in weak_div_input_symbols - }, - '1': { - a: 'e' if a[1] != '*' else '1' for a in weak_div_input_symbols - }, - 'e': {a: 'e' for a in weak_div_input_symbols} + 1: { + ('0','0','0'): 2, ('0','0','1'): 0, ('0','0','*'): 2, + ('0','1','0'): 1, ('0','1','1'): 2, ('0','1','*'): 2, + ('0','*','0'): 2, ('0','*','1'): 0, ('0','*','*'): 2, + ('1','0','0'): 1, ('1','0','1'): 2, ('1','0','*'): 2, + ('1','1','0'): 2, ('1','1','1'): 1, ('1','1','*'): 2, + ('1','*','0'): 1, ('1','*','1'): 2, ('1','*','*'): 2, + ('*','0','0'): 2, ('*','0','1'): 0, ('*','0','*'): 2, + ('*','1','0'): 1, ('*','1','1'): 2, ('*','1','*'): 2, + ('*','*','0'): 2, ('*','*','1'): 0, ('*','*','*'): 2, }, - initial_state='-1', - final_states={'1'} - ) - - N0_input_symbols = {(a,) for a in base_input_symbols} - - N0 = DFA( - states={-1, 0, 1}, - input_symbols=N0_input_symbols, - initial_state=-1, - transitions={ - -1: {a: 1 if a == ("0",) else 0 for a in N0_input_symbols}, - 0: {a: 0 for a in N0_input_symbols}, - 1: {a: 1 for a in N0_input_symbols} + 2: {s: 2 for s in add_symbols} + } + addition_intermediate = create_sparse_dfa(add_states, add_symbols, add_trans, -1, {0}) + + # Weak division automaton + div_states = ['-1', '0', '1', 'e'] + div_symbols = set(it.product(['0', '1', '*'], repeat=2)) + div_trans = { + '-1': {a: '0' if a[1] == '0' else 'e' for a in div_symbols}, + '0': { + a: '0' if a == ('0','0') + else '1' if a in {('0','1'), ('1','1')} + else 'e' for a in div_symbols }, - final_states={1} - ) - - presentation = AutomaticPresentation({'U': universe, 'A0': addition_intermediate, 'B': weak_div, 'N0': N0}) + '1': { + a: 'e' if a[1] != '*' else '1' for a in div_symbols + }, + 'e': {a: 'e' for a in div_symbols} + } + weak_div = create_sparse_dfa(div_states, div_symbols, div_trans, '-1', {'1'}) + + # N0 automaton + n0_states = [-1, 0, 1] + n0_symbols = {('0',), ('1',), ('*',)} + n0_trans = { + -1: {('0',): 1, ('1',): 0, ('*',): 0}, + 0: {s: 0 for s in n0_symbols}, + 1: {s: 1 for s in n0_symbols} + } + N0 = create_sparse_dfa(n0_states, n0_symbols, n0_trans, -1, {1}) + + # Create presentation + presentation = AutomaticPresentation({ + 'U': universe, + 'A0': addition_intermediate, + 'B': weak_div, + 'N0': N0 + }) presentation.update(Z='A0(x,x,x)') - + + # Define addition formula c000 = '(N0(x) and N0(y) and N0(z) and A0(x, y, z))' c001 = '(N0(x) and N0(y) and not N0(z) and exists a z0.(Z(z0) and A0(x,y,a) and A0(a,z,z0)))' c010 = '(N0(x) and not N0(y) and N0(z) and A0(z, y, x))' @@ -190,13 +279,244 @@ def buechi_arithmetic_Z() -> AutomaticPresentation: c111 = '(not N0(x) and not N0(y) and not N0(z) and A0(x,y,z))' phi_A = ' or '.join([c000, c001, c010, c011, c100, c101, c110, c111]) presentation.update(A=phi_A) - + presentation.update(Eq='exists z.(Z(z) and A(x,z,y))') presentation.update(Pt='B(x,x) and N0(x)') presentation.update(Lt='exists z.(N0(z) and not Z(z) and A(x, z, y))') + presentation.update(Gt='exists z.(N0(z) and not Z(z) and A(y, z, x))') presentation.update(Neg='exists z.(Z(z) and A(x,y,z))') - - # Delete auxiliary relations + + # Delete auxiliary relation del presentation.automata['A0'] + + return presentation + +def MSO0() -> AutomaticPresentation: + """Load the serialized MSO0 presentation.""" + current_dir = Path(__file__).parent + bin_path = current_dir / 'bin' / 'mso0.autstr' + return AutomaticPresentation.automatic_presentation_from_file(str(bin_path)) + +def finite_powerset() -> AutomaticPresentation: + # Base alphabet for the presentation + base_alphabet = {'0', '1', '*'} + + # 1. Universe Automaton (U) - Includes empty set + universe = create_sparse_dfa( + states={'start', 'empty', 'zero', 'one', 'pad_one', 'pad_zero', 'reject'}, + input_symbols={'0', '1', '*'}, + transitions={ + 'start': { + '*': 'empty', + '0': 'zero', + '1': 'one' + }, + 'empty': { + '*': 'empty', + '0': 'reject', + '1': 'reject' + }, + 'zero': { + '*': 'pad_zero', + '0': 'zero', + '1': 'one' + }, + 'one': { + '*': 'pad_one', + '0': 'zero', + '1': 'one' + }, + 'pad_zero': { + '*': 'pad_zero', + '0': 'reject', + '1': 'reject' + }, + 'pad_one': { + '*': 'pad_one', + '0': 'reject', + '1': 'reject' + }, + 'reject': { + '0': 'reject', + '1': 'reject', + '*': 'reject' + } + }, + initial_state='start', + final_states={'start', 'one','empty', 'pad_one'} # Accepts empty set and non-empty sets + ) + + # 2. Subset Automaton (Subset) - Empty set is subset of all sets + subset = create_sparse_dfa( + states={'start', 'error'}, + input_symbols=set(it.product(base_alphabet, repeat=2)), + transitions={ + 'start': { + ('0','0'): 'start', + ('0','1'): 'start', + ('0','*'): 'start', + ('1','1'): 'start', + ('*','0'): 'start', + ('*','1'): 'start', + ('*','*'): 'start', + ('1','0'): 'error', + ('1','*'): 'error' + }, + 'error': {k: 'error' for k in it.product(base_alphabet, repeat=2)} + }, + initial_state='start', + final_states={'start'} + ) + + # 3. Singleton Automaton (Sing) - Empty set is not a singleton + sing = create_sparse_dfa( + states={'start', 'one', 'one_pad', 'many', 'reject'}, + input_symbols=base_alphabet, + transitions={ + 'start': { + '0': 'start', + '1': 'one', + '*': 'reject' # Reject empty set immediately + }, + 'one': { + '0': 'reject', + '1': 'many', + '*': 'one_pad' + }, + 'one_pad': { + '0': 'many', + '1': 'many', + '*': 'one_pad' + }, + 'many': { + '0': 'many', + '1': 'many', + '*': 'many' + }, + 'reject': { + '0': 'reject', + '1': 'reject', + '*': 'reject' + } + }, + initial_state='start', + final_states={'one','one_pad'} # Only non-empty singletons + ) + + # 4. Successor Automaton (Succ) - Empty set has no successor + succ = create_sparse_dfa( + states={'start', 'after_x', 'after_y', 'error'}, + input_symbols=set(it.product(base_alphabet, repeat=2)), + transitions={ + 'start': { + ('0','0'): 'start', + ('1','0'): 'after_x', + ('0','1'): 'error', + ('1','1'): 'error', + ('*','*'): 'error', # Reject empty set + ('0','*'): 'error', + ('1','*'): 'error', + ('*','0'): 'error', + ('*','1'): 'error' + }, + 'after_x': { + ('*','1'): 'after_y', + ('0','0'): 'error', + ('0','1'): 'error', + ('1','0'): 'error', + ('1','1'): 'error', + ('*','0'): 'error', + ('0','*'): 'error', + ('1','*'): 'error', + ('*','*'): 'error' + }, + 'after_y': { + ('*','*'): 'after_y', + ('0','0'): 'error', + ('0','1'): 'error', + ('1','0'): 'error', + ('1','1'): 'error', + ('*','0'): 'error', + ('*','1'): 'error', + ('0','*'): 'error', + ('1','*'): 'error' + }, + 'error': {k: 'error' for k in it.product(base_alphabet, repeat=2)} + }, + initial_state='start', + final_states={'after_y'} # Only consecutive singletons + ) + + # 5. Less-Than on Singletons Automaton (Lt_sing) - Empty set not involved + lt_sing = create_sparse_dfa( + states={'init', 'x_first', 'x_first_accept', 'error'}, + input_symbols=set(it.product(base_alphabet, repeat=2)), + transitions={ + 'init': { + ('0','0'): 'init', + ('0','1'): 'error', + ('1','0'): 'x_first', + ('1','1'): 'error', + ('*','*'): 'error', + ('0','*'): 'error', + ('*','0'): 'error', + ('1','*'): 'error', + ('*','1'): 'error' + }, + 'x_first': { + ('0','0'): 'x_first', + ('0','1'): 'x_first_accept', + ('1','0'): 'error', + ('1','1'): 'error', + ('*','*'): 'error', + ('0','*'): 'error', + ('*','0'): 'x_first', + ('1','*'): 'error', + ('*','1'): 'x_first_accept' + }, + 'x_first_accept': { + ('*','*'): 'x_first_accept', + ('0','0'): 'error', + ('0','1'): 'error', + ('1','0'): 'error', + ('1','1'): 'error', + ('*','0'): 'error', + ('*','1'): 'error', + ('0','*'): 'error', + ('1','*'): 'error' + }, + 'error': {k: 'error' for k in it.product(base_alphabet, repeat=2)} + }, + initial_state='init', + final_states={'x_first_accept'} # Only when first singleton < second + ) + + # Create the presentation with base automata + presentation = AutomaticPresentation({ + 'U': universe, + 'Subset': subset, + 'Sing': sing, + 'Succ': succ, + 'Lt_sing': lt_sing + }) + + # Define additional relations using formulas + presentation.update( + In="Sing(x) and Subset(x, y)", + Eq_set="Subset(x, y) and Subset(y, x)", + Leq_sing="Lt_sing(x, y) or (Eq_set(x, y) and Sing(x) and Sing(y))", + Gt_sing="not Leq_sing(x, y) and Sing(x) and Sing(y)", + Min="(forall z. not Subset(z, x)) or " # Empty set case + "(Sing(y) and Subset(y, x) and forall z. (-(Sing(z) and Subset(z, x)) or Leq_sing(y, z)))", + Max="Sing(y) and Subset(y, x) and forall z. (-(Sing(z) and Subset(z, x)) or Leq_sing(z, y))", + Intersect="forall a. (-Sing(a) or ((Subset(a, z) and (Subset(a, x) and Subset(a, y))) or (-Subset(a, z) and -(Subset(a, x) and Subset(a, y)))))", + Union="forall a. (-Sing(a) or ((Subset(a, z) and (Subset(a, x) or Subset(a, y))) or (-Subset(a, z) and -(Subset(a, x) or Subset(a, y)))))", + SetMinus="forall a. (-Sing(a) or ((Subset(a, z) and (Subset(a, x) and not Subset(a, y))) or (-Subset(a, z) and -(Subset(a, x) and not Subset(a, y)))))" + ) + presentation.update( + Geq_sing="not Lt_sing(x, y) and Sing(x) and Sing(y)", + ) + return presentation + diff --git a/autstr/presentations.py b/autstr/presentations.py index b163db4..1432ba2 100644 --- a/autstr/presentations.py +++ b/autstr/presentations.py @@ -1,33 +1,147 @@ - from nltk.sem import logic -from automata.fa.dfa import DFA from typing import Dict, Optional, Union, List -from autstr.buildin.automata import zero, one -from autstr.utils.automata_tools import stringlify_states, product, projection, expand, unpad, pad -from autstr.utils.logic import get_free_elementary_vars - +from autstr.sparse_automata import SparseDFA, SparseDFASerializer +from autstr.utils.automata_tools import pad, unpad, product, projection, expand, stack +from autstr.buildin.automata import zero, one +from autstr.utils.logic import get_free_elementary_vars, optimize_query + +import json +import struct +import zlib +from typing import Dict + +class AutomaticPresentationSerializer: + MAGIC = b'APRS' + VERSION = 1 + HEADER_FORMAT = "4sB3sII" # Magic, version, reserved, checksum, payload_size + HEADER_SIZE = struct.calcsize(HEADER_FORMAT) + + @classmethod + def serialize(cls, presentation, filename: str) -> None: + """Serialize AutomaticPresentation to binary file""" + # Prepare payload components + payload = cls._create_payload(presentation) + + # Create header + checksum = zlib.crc32(payload) + header = struct.pack( + cls.HEADER_FORMAT, + cls.MAGIC, + cls.VERSION, + b'\0\0\0', # Reserved + checksum, + len(payload) + ) + + # Write to file + with open(filename, 'wb') as f: + f.write(header) + f.write(payload) + + @classmethod + def deserialize(cls, filename: str): + """Deserialize AutomaticPresentation from binary file""" + with open(filename, 'rb') as f: + # Read and validate header + header = f.read(cls.HEADER_SIZE) + magic, version, _, checksum, payload_size = struct.unpack(cls.HEADER_FORMAT, header) + + if magic != cls.MAGIC: + raise ValueError("Invalid file format (bad magic number)") + if version != cls.VERSION: + raise ValueError(f"Unsupported version: {version}") + + # Read and validate payload + payload = f.read(payload_size) + if zlib.crc32(payload) != checksum: + raise ValueError("Data corruption detected (checksum mismatch)") + + return cls._parse_payload(payload) + + @classmethod + def _create_payload(cls, presentation) -> bytes: + """Create binary payload from AutomaticPresentation""" + # Serialize metadata + metadata = { + "padding_symbol": presentation.padding_symbol, + "enforce_consistency": True # Not used in deserialization + } + metadata_json = json.dumps(metadata).encode('utf-8') + metadata_len = len(metadata_json) + + # Serialize automata dictionary + automata_data = {} + for name, dfa in presentation.automata.items(): + # Use SparseDFA serialization to bytes + dfa_bytes = SparseDFASerializer.to_bytes(dfa) + automata_data[name] = list(dfa_bytes) # Convert to list for JSON + + automata_json = json.dumps(automata_data).encode('utf-8') + automata_len = len(automata_json) + + # Pack components + return struct.pack("II", metadata_len, automata_len) + metadata_json + automata_json + + @classmethod + def _parse_payload(cls, payload: bytes): + """Parse binary payload into AutomaticPresentation""" + # Read lengths + metadata_len, automata_len = struct.unpack("II", payload[:8]) + offset = 8 + + # Decode metadata + metadata_json = payload[offset:offset+metadata_len] + metadata = json.loads(metadata_json.decode('utf-8')) + offset += metadata_len + + # Decode automata + automata_json = payload[offset:offset+automata_len] + automata_data = json.loads(automata_json.decode('utf-8')) + + # Convert back to SparseDFA instances + automata = {} + for name, dfa_bytes_list in automata_data.items(): + dfa_bytes = bytes(dfa_bytes_list) + automata[name] = SparseDFASerializer.from_bytes(dfa_bytes) + + # Reconstruct presentation + return AutomaticPresentation( + automata, + padding_symbol=metadata["padding_symbol"], + enforce_consistency=False + ) class AutomaticPresentation: """ A presentation of a possibly infinite structure by finite state machines. """ - def __init__(self, automata: Dict[str, DFA]): + def __init__(self, automata: Dict[str, SparseDFA], padding_symbol: Optional[any] = "*", enforce_consistency: bool = True) -> None: """ :param automata: dictionary containing the automata that recognize the domain and the relations of the structure. 'U' is reserved key for the universe. All other keys are assumed to recognize relations over L(U)^k. They can be addressed by their keys in first-order queries. """ - universe = pad(automata['U']).minify() + self.padding_symbol = padding_symbol + universe = pad(automata['U'], padding_symbol=self.padding_symbol).minimize() self.automata = {'U': universe} + # Prepare relation automata for R in automata: - if R != 'U': - arity = len(list(automata[R].input_symbols)[0]) - domain = product(universe, arity) - self.automata[R] = pad(stringlify_states(automata[R])).intersection(domain).minify() + if R != 'U': + if enforce_consistency: + self.automata[R] = self._prepare_automaton(automata[R]) + else: + self.automata[R] = pad(automata[R]).minimize() + + self.sigma = automata['U'].base_alphabet # Use base_alphabet for sigma + + def automatic_presentation_to_file(self, filename: str) -> None: + AutomaticPresentationSerializer.serialize(self, filename) - self.sigma = {t[0] for t in self.automata['U'].input_symbols} + @classmethod + def automatic_presentation_from_file(cls, filename: str): + return AutomaticPresentationSerializer.deserialize(filename) def get_relation_symbols(self) -> List[str]: """ @@ -39,27 +153,34 @@ def get_relation_symbols(self) -> List[str]: def update(self, **kwargs) -> None: for key in kwargs: - if isinstance(kwargs[key], DFA): + if isinstance(kwargs[key], SparseDFA): self.automata[key] = self._prepare_automaton(kwargs[key]) elif isinstance(kwargs[key], str): - self.automata[key] = self._prepare_automaton(self._build_automaton(kwargs[key])) + query = optimize_query(logic.Expression.fromstring(kwargs[key])) + self.automata[key] = self._prepare_automaton(self._build_automaton(query)) - def _prepare_automaton(self, dfa: DFA) -> DFA: + def _prepare_automaton(self, dfa: SparseDFA) -> SparseDFA: """Applies restriction to the universe and padding to the automaton""" - arity = len(list(dfa.input_symbols)[0]) - domain = product(self.automata['U'], arity) - return pad(stringlify_states(dfa)).intersection(domain).minify() - - def check(self, phi: logic.Expression) -> bool: + arity = dfa.symbol_arity # Get arity from symbol_arity attribute + domain = self.automata['U'] + for i in range(arity-1): + domain_i = expand(domain, new_arity=arity, pos=[i]).minimize() + dfa = dfa.intersection(domain_i).minimize() + return pad(dfa, self.padding_symbol).minimize() + + def check(self, phi: logic.Expression | str) -> bool: """Checks if a given first-order formula holds on the presented structure. Free variables are assumed be implicitly existentially quantified. :param phi: the first order formula :returns: the truth value of the formula, if the formula where all free variables are existentially quantified. """ - return not self._build_automaton(phi).isempty() + if isinstance(phi, str): + phi = logic.Expression.fromstring(phi) + phi = phi.simplify() + return not self._build_automaton(phi).is_empty() - def evaluate(self, phi: Union[str, logic.Expression], updates: Optional[Dict[str, Union[DFA, str]]] = None) -> DFA: + def evaluate(self, phi: Union[str, logic.Expression], updates: Optional[Dict[str, Union[SparseDFA, str]]] = None) -> SparseDFA: """Evaluates a given first-order query on the presented structure. Returns a presentation of the set of all satisfying assignments. @@ -69,25 +190,28 @@ def evaluate(self, phi: Union[str, logic.Expression], updates: Optional[Dict[str """ if isinstance(phi, str): phi = logic.Expression.fromstring(phi) + phi = optimize_query(phi) if updates is not None: for key in updates: if isinstance(updates[key], str): - updates[key] = self._prepare_automaton(self._build_automaton(updates[key])) + query = optimize_query(logic.Expression.fromstring(updates[key])) + updates[key] = self._prepare_automaton(self._build_automaton(query)) else: updates[key] = self._prepare_automaton(updates[key]) automata_backup = self.automata self.automata = dict(self.automata, **updates) - dfa_phi = self._build_automaton(phi) if len(get_free_elementary_vars(phi)) > 0: - dfa_phi = unpad(self._build_automaton(phi)) + dfa_phi = unpad(self._build_automaton(phi), self.padding_symbol).minimize() + else: + dfa_phi = self._build_automaton(phi) if updates is not None: self.automata = automata_backup return dfa_phi - def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: + def _build_automaton(self, phi: logic.Expression, verbose=False, init=True) -> SparseDFA: """ Creates a padded presentation of the satisfying assignments of phi. @@ -96,6 +220,11 @@ def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: len(free_vars)-ary. The dictionary maps each variable to it's position in the tuple. :return: Padded presentation of the satisfying assignments of phi """ + if init: + if verbose: + print(f'Building automaton for {str(phi)}') + return self._build_automaton(phi, verbose=verbose, init=False) + if isinstance(phi, str): phi = logic.Expression.fromstring(phi) @@ -105,53 +234,53 @@ def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: free_vars = get_free_elementary_vars(phi.term) if variable not in free_vars: - return self._build_automaton(phi.term) + return self._build_automaton(phi.term, verbose=verbose, init=False) else: psi = (phi.term.negate()).simplify() - dfa_rec = stringlify_states(self._build_automaton(psi).minify()) + dfa_rec = self._build_automaton(psi, verbose=verbose, init=False).minimize() pos = free_vars.index(variable) if len(free_vars) > 1: - domain = stringlify_states(product(self.automata['U'], len(free_vars) - 1)) + domain = product(self.automata['U'], len(free_vars) - 1) result = projection( dfa_rec, pos - ).minify().complement().minify().intersection(domain).minify() + ).minimize().complement().minimize().intersection(domain).minimize() else: - if dfa_rec.isempty(): + if dfa_rec.is_empty(): result = one() else: result = zero() if verbose: - print(f'{str(phi)}: {len(result.states)} states') + print(f'{str(phi)}: {result.num_states} states') return result elif isinstance(phi, logic.ExistsExpression): psi = phi.term variable = str(phi.variable) free_vars = get_free_elementary_vars(psi) - dfa_rec = stringlify_states(self._build_automaton(psi)) + dfa_rec = self._build_automaton(psi, verbose=verbose, init=False) if variable in free_vars: pos = free_vars.index(variable) if len(free_vars) > 1: - result = projection(dfa_rec, pos).minify() + result = projection(dfa_rec, pos).minimize() else: if verbose: print(f'{str(phi)}: 1 state') - if dfa_rec.isempty(): + if dfa_rec.is_empty(): return zero() else: return one() else: - result = stringlify_states(self._build_automaton(psi)) + result = self._build_automaton(psi, verbose=verbose, init=False) - result = pad(unpad(result)) + result = pad(unpad(result, self.padding_symbol).minimize(), self.padding_symbol).minimize() if verbose: - print(f'{str(phi)}: {len(result.states)} states') + print(f'{str(phi)}: {result.num_states} states') return result elif isinstance(phi, logic.AndExpression): left = phi.first @@ -161,25 +290,29 @@ def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: free_l = get_free_elementary_vars(left) free_r = get_free_elementary_vars(right) - dfa_l = expand(self._build_automaton(left), len(free_vars), pos=[free_vars.index(v) for v in free_l]) - dfa_r = expand(self._build_automaton(right), len(free_vars), pos=[free_vars.index(v) for v in free_r]) + dfa_l = expand(self._build_automaton(left, verbose=verbose, init=False), len(free_vars), pos=[free_vars.index(v) for v in free_l]) + dfa_r = expand(self._build_automaton(right, verbose=verbose, init=False), len(free_vars), pos=[free_vars.index(v) for v in free_r]) - result = dfa_l.intersection(dfa_r).minify() + result = dfa_l.intersection(dfa_r).minimize() if verbose: - print(f'{str(phi)}: {len(result.states)} states') + print(f'{str(phi)}: {result.num_states} states') return result elif isinstance(phi, logic.NegatedExpression): psi = phi.term - free_vars = get_free_elementary_vars(phi) + # Skip double negation + if isinstance(psi, logic.NegatedExpression): + return self._build_automaton(psi.term, verbose=verbose, init=False) + else: + free_vars = get_free_elementary_vars(phi) - domain = product(self.automata['U'], len(free_vars)) + domain = product(self.automata['U'], len(free_vars)) - result = self._build_automaton(psi).complement() - result = result.intersection(domain).minify() - if verbose: - print(f'{str(phi)}: {len(result.states)} states') - return result + result = self._build_automaton(psi, verbose=verbose, init=False).complement() + result = result.intersection(domain).minimize() + if verbose: + print(f'{str(phi)}: {result.num_states} states') + return result elif isinstance(phi, logic.OrExpression): left = phi.first right = phi.second @@ -188,12 +321,12 @@ def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: free_l = get_free_elementary_vars(left) free_r = get_free_elementary_vars(right) - dfa_l = expand(self._build_automaton(left), len(free_vars), pos=[free_vars.index(v) for v in free_l]) - dfa_r = expand(self._build_automaton(right), len(free_vars), pos=[free_vars.index(v) for v in free_r]) + dfa_l = expand(self._build_automaton(left, verbose=verbose, init=False), len(free_vars), pos=[free_vars.index(v) for v in free_l]) + dfa_r = expand(self._build_automaton(right, verbose=verbose, init=False), len(free_vars), pos=[free_vars.index(v) for v in free_r]) - result = dfa_l.union(dfa_r).minify() + result = dfa_l.union(dfa_r).minimize() if verbose: - print(f'{str(phi)}: {len(result.states)} states') + print(f'{str(phi)}: {result.num_states} states') return result elif isinstance(phi, logic.ApplicationExpression): R = str(phi.pred) @@ -203,10 +336,8 @@ def _build_automaton(self, phi: logic.Expression, verbose=False) -> DFA: self.automata[R], len(variables), [variables.index(v) for v in [str(v) for v in phi.args]] - ).minify() + ).minimize() if verbose: - print(f'{str(phi)}: {len(result.states)} states') - return result - - + print(f'{str(phi)}: {result.num_states} states') + return result \ No newline at end of file diff --git a/autstr/sparse_automata.py b/autstr/sparse_automata.py new file mode 100644 index 0000000..c5f7734 --- /dev/null +++ b/autstr/sparse_automata.py @@ -0,0 +1,1412 @@ +import json +import jax +import jax.numpy as jnp +import numpy as np +from collections import deque, defaultdict +from typing import Tuple, Optional, Callable, List, Set +import graphviz + +import struct +import zlib + + +from autstr.utils.misc import decode_symbol +from autstr.utils.misc import encode_symbol, complement + + + +# File format structure: +# [Header (16 bytes)] +# - Magic number: 4 bytes ('SDFA') +# - Version: 1 byte +# - Reserved: 3 bytes (0) +# - Checksum: 4 bytes (CRC32 of payload) +# - Payload size: 4 bytes +# [Payload] +# - Metadata (20 bytes) +# - Base alphabet +# - Default states +# - Exception symbols +# - Exception states +# - Acceptance array + +class SparseDFASerializer: + VERSION = 2 # Bump version for new format + HEADER_FORMAT = "4sB3sII" + HEADER_SIZE = struct.calcsize(HEADER_FORMAT) + METADATA_FORMAT = "IIIII" + METADATA_SIZE = struct.calcsize(METADATA_FORMAT) + + @classmethod + def serialize(cls, dfa: 'SparseDFA', filename: str) -> None: + """Serialize SparseDFA to binary file""" + # Prepare payload components + payload = cls._create_payload(dfa) + + # Create header + checksum = zlib.crc32(payload) + header = struct.pack( + cls.HEADER_FORMAT, + b'SDFA', # Magic number + cls.VERSION, # Format version + b'\0\0\0', # Reserved bytes + checksum, # CRC32 checksum + len(payload) # Payload size + ) + + # Write to file + with open(filename, 'wb') as f: + f.write(header) + f.write(payload) + + @classmethod + def deserialize(cls, filename: str) -> 'SparseDFA': + """Deserialize SparseDFA from binary file""" + with open(filename, 'rb') as f: + # Read and validate header + header = f.read(cls.HEADER_SIZE) + magic, version, _, checksum, payload_size = struct.unpack(cls.HEADER_FORMAT, header) + + if magic != b'SDFA': + raise ValueError("Invalid file format (bad magic number)") + if version != cls.VERSION: + raise ValueError(f"Unsupported version: {version}") + + # Read and validate payload + payload = f.read(payload_size) + if zlib.crc32(payload) != checksum: + raise ValueError("Data corruption detected (checksum mismatch)") + + return cls._parse_payload(payload) + + @classmethod + def _create_payload(cls, dfa: 'SparseDFA') -> bytes: + """Create binary payload from SparseDFA""" + # Convert arrays to numpy for efficient serialization + default_states = np.array(dfa.default_states, dtype=np.uint32) + exception_symbols = np.array(dfa.exception_symbols, dtype=np.int32) + exception_states = np.array(dfa.exception_states, dtype=np.int32) + is_accepting = np.array(dfa.is_accepting, dtype=np.uint8) + + # Serialize base alphabet as JSON + base_alphabet_json = json.dumps(sorted(dfa.base_alphabet)).encode('utf-8') + base_alphabet_len = len(base_alphabet_json) + + # Pack metadata + metadata = struct.pack( + cls.METADATA_FORMAT, + dfa.num_states, + dfa.max_exceptions, + dfa.start_state, + dfa.symbol_arity, + base_alphabet_len + ) + + # Pack components + components = [ + metadata, + base_alphabet_json, + default_states.tobytes(), + exception_symbols.tobytes(), + exception_states.tobytes(), + is_accepting.tobytes() + ] + + return b''.join(components) + + @classmethod + def _parse_payload(cls, payload: bytes) -> 'SparseDFA': + """Parse binary payload into SparseDFA""" + # Unpack metadata + meta = struct.unpack( + cls.METADATA_FORMAT, + payload[:cls.METADATA_SIZE] + ) + num_states, max_exceptions, start_state, symbol_arity, alpha_len = meta + + # Calculate offsets + offset = cls.METADATA_SIZE + base_alphabet_json = payload[offset:offset+alpha_len] + base_alphabet = set(json.loads(base_alphabet_json.decode('utf-8'))) + offset += alpha_len + + # Calculate array sizes + states_bytes = num_states * 4 + exceptions_bytes = num_states * max_exceptions * 4 + accepting_bytes = num_states + + # Extract arrays + default_states = jnp.array(np.frombuffer( + payload[offset:offset+states_bytes], + dtype=np.uint32 + )) + offset += states_bytes + + exception_symbols = jnp.array(np.frombuffer( + payload[offset:offset+exceptions_bytes], + dtype=np.int32 + ).reshape(num_states, max_exceptions)) + offset += exceptions_bytes + + exception_states = jnp.array(np.frombuffer( + payload[offset:offset+exceptions_bytes], + dtype=np.int32 + ).reshape(num_states, max_exceptions)) + offset += exceptions_bytes + + is_accepting = jnp.array(np.frombuffer( + payload[offset:offset+accepting_bytes], + dtype=np.uint8 + ).astype(bool)) + + return SparseDFA( + num_states=num_states, + default_states=default_states, + exception_symbols=exception_symbols, + exception_states=exception_states, + is_accepting=is_accepting, + start_state=start_state, + symbol_arity=symbol_arity, + base_alphabet=base_alphabet + ) + + @classmethod + def to_bytes(cls, dfa: 'SparseDFA') -> bytes: + """Serialize SparseDFA to bytes object""" + payload = cls._create_payload(dfa) + header = struct.pack( + cls.HEADER_FORMAT, + b'SDFA', + cls.VERSION, + b'\0\0\0', + zlib.crc32(payload), + len(payload) + ) + return header + payload + + @classmethod + def from_bytes(cls, data: bytes) -> 'SparseDFA': + """Deserialize SparseDFA from bytes object""" + if len(data) < cls.HEADER_SIZE: + raise ValueError("Data too short for header") + + header = data[:cls.HEADER_SIZE] + magic, version, _, checksum, payload_size = struct.unpack(cls.HEADER_FORMAT, header) + + if magic != b'SDFA': + raise ValueError("Invalid SparseDFA format") + if version != cls.VERSION: + raise ValueError(f"Unsupported SparseDFA version: {version}") + + payload = data[cls.HEADER_SIZE:cls.HEADER_SIZE+payload_size] + if len(payload) != payload_size: + raise ValueError("Payload size mismatch") + if zlib.crc32(payload) != checksum: + raise ValueError("SparseDFA data corruption detected") + + return cls._parse_payload(payload) + + +class SparseDFA: + def __init__(self, num_states: int, default_states: jnp.ndarray, + exception_symbols: jnp.ndarray, exception_states: jnp.ndarray, + is_accepting: jnp.ndarray, start_state: int, + symbol_arity: int = 1, base_alphabet: Optional[Set[int]] = None): + self.num_states = num_states + self.default_states = default_states + self.exception_symbols = exception_symbols + self.exception_states = exception_states + self.is_accepting = is_accepting + self.start_state = start_state + self.max_exceptions = exception_symbols.shape[1] + self.symbol_arity = symbol_arity + self.base_alphabet = base_alphabet or self._infer_base_alphabet() + self.base_alphabet_frozen = frozenset(self.base_alphabet) + + def _infer_base_alphabet(self) -> Set[int]: + """Infer base alphabet from exception symbols.""" + symbols = set(np.unique(self.exception_symbols)) + symbols.discard(-1) + return symbols or {0} + + def encode_symbol(self, symbol_tuple: Tuple[int]) -> int: + return encode_symbol(symbol_tuple, self.base_alphabet_frozen) + + def decode_symbol(self, symbol_enc: int) -> Tuple[int]: + return decode_symbol(symbol_enc, self.symbol_arity, self.base_alphabet_frozen) + + def transition(self, state: int, symbol: int) -> int: + """Get the next state for a given symbol.""" + + # Check if exception exists for this automaton + if self.exception_symbols.shape[1] == 0: + # No exceptions, use default state + return self.default_states[state] + else: + target = jnp.where(self.exception_symbols[state] == symbol, self.exception_states[state], -1).max() + target = jnp.where(target == -1, self.default_states[state], target) + + return target + + def compute(self, word: jnp.ndarray) -> int: + def step(state, symbol): + return self.transition(state, symbol), None + final_state, _ = jax.lax.scan(step, self.start_state, word) + return final_state + + def accepts(self, word: jnp.ndarray) -> jnp.ndarray: + # encode word for internal representation + word = jnp.array([encode_symbol(s, self.base_alphabet_frozen) for s in word]) + final_state = self.compute(word) + return self.is_accepting[final_state] + + def is_empty(self) -> bool: + """Check if the language is empty""" + # BFS to find any accepting state + visited = np.zeros(self.num_states, dtype=bool) + queue = deque([self.start_state]) + + while queue: + state = queue.popleft() + if self.is_accepting[state]: + return False + if visited[state]: + continue + visited[state] = True + + # Check all transitions + next_states = set() + next_states.add(int(self.default_states[state])) + for i in range(self.max_exceptions): + sym = self.exception_symbols[state, i] + if sym != -1: + next_states.add(int(self.exception_states[state, i])) + + for ns in next_states: + if not visited[ns]: + queue.append(ns) + + return True + + def is_finite(self) -> bool: + + E = jnp.array([[jnp.any(self.exception_states[i] == j)or self.default_states[i] == j for j in range(self.num_states)] for i in range(self.num_states)]) + + + + # Step 1: Find reachable states (forward BFS) + reachable = set() + queue = deque([self.start_state]) + while queue: + state = queue.popleft() + if state in reachable: + continue + reachable.add(state) + # Default transition + default_target = int(self.default_states[state]) + if default_target not in reachable: + queue.append(default_target) + # Exception transitions + for i in range(self.max_exceptions): + sym = int(self.exception_symbols[state, i]) + if sym == -1: + continue + target = int(self.exception_states[state, i]) + if target not in reachable: + queue.append(target) + + # Step 2: Find co-reachable states (backward BFS from accepting states) + # Precompute reverse transition map + rev_map = defaultdict(set) + for u in range(self.num_states): + # Default transitions + v_def = int(self.default_states[u]) + rev_map[v_def].add(u) + # Exception transitions + for i in range(self.max_exceptions): + sym = int(self.exception_symbols[u, i]) + if sym != -1: + v_ex = int(self.exception_states[u, i]) + rev_map[v_ex].add(u) + + co_reachable = set() + # Initialize queue with all accepting states + queue = deque([state for state in range(self.num_states) if self.is_accepting[state]]) + while queue: + state = queue.popleft() + if state in co_reachable: + continue + co_reachable.add(state) + for pred in rev_map[state]: + if pred not in co_reachable: + queue.append(pred) + + # Step 3: Useful states (reachable and co-reachable) + useful_states = reachable & co_reachable + + # If no useful states, language is empty -> finite + if not useful_states: + return True + + # Step 4: Build graph for useful states + graph = {} + for u in useful_states: + neighbors = set() + # Default transition + v_def = int(self.default_states[u]) + if v_def in useful_states: + neighbors.add(v_def) + # Exception transitions + for i in range(self.max_exceptions): + sym = int(self.exception_symbols[u, i]) + if sym != -1: + v_ex = int(self.exception_states[u, i]) + if v_ex in useful_states: + neighbors.add(v_ex) + graph[u] = neighbors + + # Step 5: Cycle detection with iterative DFS + color = {state: 0 for state in useful_states} # 0: white, 1: gray, 2: black + for state in useful_states: + if color[state] == 0: + stack = [state] + while stack: + u = stack.pop() + if color[u] == 0: + color[u] = 1 # Mark as gray + stack.append(u) # Push back for backtracking + for v in graph[u]: + if color[v] == 0: + stack.append(v) + elif color[v] == 1: + return False # Cycle found -> infinite language + else: + color[u] = 2 # Mark as black + return True # No cycles -> finite language + + def complement(self) -> 'SparseDFA': + new_accepting = ~self.is_accepting + return SparseDFA( + self.num_states, + self.default_states, + self.exception_symbols, + self.exception_states, + new_accepting, + self.start_state, + self.symbol_arity, + self.base_alphabet + ) + + def intersection(self, other: 'SparseDFA') -> 'SparseDFA': + return self._product(other, combine_accept=lambda a, b: a & b) + + + def union(self, other: 'SparseDFA') -> 'SparseDFA': + return self._product(other, combine_accept=lambda a, b: a | b) + + def _vectorized_transition(self, state: int, symbols: jnp.ndarray) -> jnp.ndarray: + """Vectorized transition lookup for a batch of symbols.""" + if symbols.size == 0: + return jnp.array([], dtype=jnp.int32) + + default_next = self.default_states[state] + result = jnp.full(symbols.shape, default_next, dtype=jnp.int32) + + # Process exceptions + ex_syms = self.exception_symbols[state] + ex_states = self.exception_states[state] + valid_mask = ex_syms != -1 + ex_syms_valid = ex_syms[valid_mask] + ex_states_valid = ex_states[valid_mask] + + for i in range(len(ex_syms_valid)): + mask = symbols == ex_syms_valid[i] + result = jnp.where(mask, ex_states_valid[i], result) + + return result + + def reverse_transition(self, state: int, symbol: int) -> jnp.ndarray: + """Get the states that transition to the given state on the given symbol.""" + default_mask = (self.default_states == state) & jnp.all(self.exception_symbols != symbol, axis=1) + ex_mask = (self.exception_states == state) & (self.exception_symbols == symbol) + + return jnp.arange(self.num_states)[default_mask | ex_mask] + + def successors(self, state: int) -> jnp.ndarray: + """Get all successor states from a given state.""" + default_succ = self.default_states[state] + ex_succ = self.exception_states[state, self.exception_symbols[state] != -1] + return jnp.unique(jnp.concatenate([jnp.array([default_succ]), ex_succ])) + + def _product(self, other: 'SparseDFA', combine_accept: Callable[[bool, bool], bool]) -> 'SparseDFA': + if self.symbol_arity != other.symbol_arity: + raise ValueError("Product requires same symbol arity") + + start_pair = (self.start_state, other.start_state) + queue = deque([start_pair]) + state_map = {start_pair: 0} + + new_default_states_list = [] + new_exception_symbols_list = [] + new_exception_states_list = [] + new_is_accepting_list = [] + + while queue: + current_pair = queue.popleft() + i, j = current_pair + current_idx = state_map[current_pair] + + new_is_accepting_list.append(combine_accept( + bool(self.is_accepting[i]), + bool(other.is_accepting[j])) + ) + + # Get relevant symbols + ex_i_mask = self.exception_symbols[i] != -1 + ex_j_mask = other.exception_symbols[j] != -1 + ex_i_symbols = self.exception_symbols[i][ex_i_mask] + ex_j_symbols = other.exception_symbols[j][ex_j_mask] + all_relevant_symbols = jnp.unique(jnp.concatenate([ex_i_symbols, ex_j_symbols])) + + # Default transition + def_i = int(self.default_states[i]) + def_j = int(other.default_states[j]) + default_target_pair = (def_i, def_j) + + if default_target_pair not in state_map: + state_map[default_target_pair] = len(state_map) + queue.append(default_target_pair) + new_default_states_list.append(state_map[default_target_pair]) + + # Process exceptions in batch + current_exceptions_symbols = [] + current_exceptions_states = [] + + if all_relevant_symbols.size > 0: + # Vectorized transition lookups + next_i_vec = self._vectorized_transition(i, all_relevant_symbols) + next_j_vec = other._vectorized_transition(j, all_relevant_symbols) + + # Identify non-default transitions + non_default_mask = (next_i_vec != def_i) | (next_j_vec != def_j) + non_default_symbols = all_relevant_symbols[non_default_mask] + non_default_i = next_i_vec[non_default_mask] + non_default_j = next_j_vec[non_default_mask] + + # Process non-default transitions + for idx in range(non_default_symbols.shape[0]): + sym = int(non_default_symbols[idx]) + next_pair = (int(non_default_i[idx]), int(non_default_j[idx])) + + if next_pair not in state_map: + state_map[next_pair] = len(state_map) + queue.append(next_pair) + current_exceptions_symbols.append(sym) + current_exceptions_states.append(state_map[next_pair]) + + new_exception_symbols_list.append(current_exceptions_symbols) + new_exception_states_list.append(current_exceptions_states) + + # Convert results to JAX arrays + num_new_states = len(state_map) + max_exceptions = max(len(ex) for ex in new_exception_symbols_list) if new_exception_symbols_list else 0 + + padded_ex_syms = jnp.full((num_new_states, max_exceptions), -1, dtype=jnp.int32) + padded_ex_states = jnp.full((num_new_states, max_exceptions), -1, dtype=jnp.int32) + + for i, (syms, states) in enumerate(zip(new_exception_symbols_list, new_exception_states_list)): + if syms: + padded_ex_syms = padded_ex_syms.at[i, :len(syms)].set(jnp.array(syms, dtype=jnp.int32)) + padded_ex_states = padded_ex_states.at[i, :len(states)].set(jnp.array(states, dtype=jnp.int32)) + + return SparseDFA( + num_new_states, + jnp.array(new_default_states_list, dtype=jnp.int32), + padded_ex_syms, + padded_ex_states, + jnp.array(new_is_accepting_list, dtype=bool), + 0, + self.symbol_arity, + self.base_alphabet.union(other.base_alphabet) + ) + + def alphabet_projection(self, projection_map: jnp.ndarray) -> 'SparseNFA': + new_ex_symbols = jnp.where( + self.exception_symbols != -1, + projection_map[self.exception_symbols], + -1 + ) + return SparseNFA( + num_states=self.num_states, + base_state=self.default_states, + exception_symbols=new_ex_symbols, + exception_states=self.exception_states, + is_accepting=self.is_accepting, + start_state=self.start_state, + symbol_arity=self.symbol_arity, + base_alphabet=self.base_alphabet + ) + + def intersect_subtapes(self, other: 'SparseDFA', tapes: List[int]) -> 'SparseDFA': + """ + Intersects two automata on specified tapes of the first automaton. + + Args: + other: Second automaton with arity = len(tapes) + tapes: List of tape indices from self to project to other + + Returns: + SparseDFA recognizing {x in L(self) | (x[tapes]) in L(other)} + """ + # TODO: Buggy (not all exception symbols induced by other are considered) + # Validate inputs + k = self.symbol_arity + l = other.symbol_arity + if len(tapes) != l: + raise ValueError(f"Tapes length ({len(tapes)}) must match other.arity ({l})") + if not all(0 <= t < k for t in tapes): + raise ValueError("All tape indices must be in [0, self.arity-1]") + if self.base_alphabet != other.base_alphabet: + raise ValueError("Automata must have the same base alphabet") + + base_alphabet = self.base_alphabet + n1 = self.num_states + n2 = other.num_states + num_states = n1 * n2 + + # Initialize arrays + default_states = np.zeros(num_states, dtype=np.int32) + exception_symbols = [[] for _ in range(num_states)] + exception_states = [[] for _ in range(num_states)] + is_accepting = np.zeros(num_states, dtype=bool) + + # Helper to project a full symbol to subtapes + def project_symbol(full_enc: int) -> int: + """Project full symbol to specified tapes""" + full_tuple = decode_symbol(full_enc, k, base_alphabet) + proj_tuple = tuple(full_tuple[t] for t in tapes) + return encode_symbol(proj_tuple, base_alphabet) + + # Build product automaton + for i in range(n1): + for j in range(n2): + idx = i * n2 + j + + # Default transitions for both automata + default_i = int(self.default_states[i]) + default_j = int(other.default_states[j]) + default_states[idx] = default_i * n2 + default_j + + # Set acceptance + is_accepting[idx] = self.is_accepting[i] and other.is_accepting[j] + + # Collect symbols that would cause exceptions + exception_symbols_set = set() + + # Add exception symbols from self + for pos1 in range(self.max_exceptions): + s_enc = int(self.exception_symbols[i, pos1]) + if s_enc == -1: + continue + exception_symbols_set.add(s_enc) + + # Add exception symbols from other (via projection) + for pos2 in range(other.max_exceptions): + p_enc = int(other.exception_symbols[j, pos2]) + if p_enc == -1: + continue + # Create a representative symbol by: + # 1. Decoding the projected symbol + # 2. Creating a full symbol with default values + # 3. Setting the specified tapes + base0 = sorted(base_alphabet)[0] # default symbol + full_tuple = [base0] * k + proj_tuple = decode_symbol(p_enc, l, base_alphabet) + for idx, t in enumerate(tapes): + full_tuple[t] = proj_tuple[idx] + full_enc = encode_symbol(tuple(full_tuple), base_alphabet) + exception_symbols_set.add(full_enc) + + # Process exception symbols + for full_enc in exception_symbols_set: + # Get transition in self + next_i = self.transition(i, full_enc) + + # Get projection and transition in other + proj_enc = project_symbol(full_enc) + next_j = other.transition(j, proj_enc) + + next_state = next_i * n2 + next_j + + # Only store if different from default + if next_state != default_states[idx]: + exception_symbols[idx].append(full_enc) + exception_states[idx].append(next_state) + + # Pad exceptions to uniform length + max_exceptions = max(len(ex) for ex in exception_symbols) if exception_symbols else 0 + padded_ex_syms = np.full((num_states, max_exceptions), -1, dtype=np.int32) + padded_ex_states = np.full((num_states, max_exceptions), -1, dtype=np.int32) + + for i, syms in enumerate(exception_symbols): + if syms: + padded_ex_syms[i, :len(syms)] = syms + padded_ex_states[i, :len(syms)] = exception_states[i] + + # Start state + start_state = self.start_state * n2 + other.start_state + + return SparseDFA( + num_states=num_states, + default_states=jnp.array(default_states), + exception_symbols=jnp.array(padded_ex_syms), + exception_states=jnp.array(padded_ex_states), + is_accepting=jnp.array(is_accepting), + start_state=start_state, + symbol_arity=k, + base_alphabet=base_alphabet + ) + + def regular_right_quotient(self, other: 'SparseDFA') -> 'SparseDFA': + nA, nB = self.num_states, other.num_states + + # Initialize reachability matrix + reachable = jnp.zeros((nA, nB), dtype=bool) + for i in range(nA): + for j in range(nB): + reachable = reachable.at[i, j].set( + bool(self.is_accepting[i]) & bool(other.is_accepting[j]) + ) + + # Backward propagation + changed = True + while changed: + new_reachable = reachable.copy() + changed = False + + for i in range(nA): + for j in range(nB): + if reachable[i, j]: + continue + + # Get unique symbols from both states + ex_i = self.exception_symbols[i][self.exception_symbols[i] != -1] + ex_j = other.exception_symbols[j][other.exception_symbols[j] != -1] + all_symbols = jnp.unique(jnp.concatenate([ex_i, ex_j])) + + # Check if any symbol leads to a reachable state + for sym in all_symbols: + next_i = self.transition(i, sym) + next_j = other.transition(j, sym) + if reachable[next_i, next_j]: + new_reachable = new_reachable.at[i, j].set(True) + changed = True + break + + reachable = new_reachable + + # New acceptance: state i is accepting if (i, other.start_state) is reachable + new_accept = jnp.array([ + reachable[i, other.start_state] for i in range(nA) + ]) + + return SparseDFA( + nA, + self.default_states, + self.exception_symbols, + self.exception_states, + new_accept, + self.start_state, + self.symbol_arity, + self.base_alphabet + ) + + def fill_defaults(self) -> 'SparseDFA': + """Fills in default transitions for all states. If default state is currently -1, it will be set to the most common exception state.""" + # check for unused default states (default_state == -1) + ex_states = jnp.where( + self.exception_states == -1, + self.num_states, + self.exception_states + ) + + n_symbols = len(self.base_alphabet_frozen)**self.symbol_arity + + counts = jnp.apply_along_axis( + jnp.bincount, axis=-1, arr=ex_states, length=self.num_states + 1 + )[:, :-1] + + if (self.default_states.min() < 0) or jnp.any(counts.sum(axis=-1) == n_symbols): + + change_mask = (self.default_states == -1) | (counts.sum(axis=-1) == n_symbols) + self.default_states = jnp.where( + change_mask, jnp.argmax(counts, axis=-1), self.default_states + ) + + # delete exceptions to new default states + new_ex_symbols = jnp.full((self.num_states, self.max_exceptions), -1, dtype=jnp.int32) + change_mask_expanded = jnp.expand_dims(change_mask, axis=1) + change_mask_expanded = jnp.repeat(change_mask_expanded, self.max_exceptions, axis=-1) + new_defaults_expanded = jnp.expand_dims(self.default_states, axis=1) + new_defaults_expanded = jnp.repeat(new_defaults_expanded, self.max_exceptions, axis=-1) + self.exception_symbols = jnp.where( + (self.exception_states == new_defaults_expanded) & change_mask_expanded, new_ex_symbols, self.exception_symbols + ) + self.exception_states = jnp.where( + (self.exception_states == new_defaults_expanded) & change_mask_expanded, jnp.full_like(self.exception_states, -1), self.exception_states + ) + + return self + + + def minimize(self) -> 'SparseDFA': + """Minimizes the DFA using Hopcroft's algorithm with sparse optimizations.""" + self.fill_defaults() + + # Convert to numpy for processing + default_states_np = np.array(self.default_states) + exception_symbols_np = np.array(self.exception_symbols) + exception_states_np = np.array(self.exception_states) + is_accepting_np = np.array(self.is_accepting) + + n_symbols = len(self.base_alphabet_frozen)**self.symbol_arity + + # Step 1: Precompute all symbols used + all_symbols = set(exception_symbols_np.flatten()) + all_symbols.discard(-1) + all_symbols = sorted(all_symbols) + + # Step 2: Compute reachable states + reachable = set() + queue = deque([self.start_state]) + while queue: + state = queue.popleft() + if state in reachable: + continue + reachable.add(state) + + # Default transition exists? + if (exception_symbols_np[state] != -1).sum() < n_symbols: + # Add default state + default_target = default_states_np[state] + if default_target not in reachable: + queue.append(default_target) + + # Add exception states + for i in range(self.max_exceptions): + sym = exception_symbols_np[state, i] + if sym == -1: + continue + target = exception_states_np[state, i] + if target not in reachable: + queue.append(target) + + # Step 3: Initial partition - accepting and non-accepting states + accepting_states = set() + non_accepting_states = set() + for state in reachable: + if is_accepting_np[state]: + accepting_states.add(state) + else: + non_accepting_states.add(state) + + if not accepting_states: + partitions = [non_accepting_states] + elif not non_accepting_states: + partitions = [accepting_states] + else: + partitions = [accepting_states, non_accepting_states] + + # Precompute state to partition index mapping + state_to_part_idx = {} + for part_idx, part in enumerate(partitions): + for state in part: + state_to_part_idx[state] = part_idx + + # Step 4: Partition refinement + changed = True + while changed: + changed = False + new_partitions = [] + # Precompute state to partition index for current partitions + state_to_part_idx = {} + for part_idx, part in enumerate(partitions): + for state in part: + state_to_part_idx[state] = part_idx + + for part in partitions: + # Split partition based on transitions + split_dict = defaultdict(list) + for state in part: + # Create key based on transitions + key = [] + + # Add default transition partition + default_next = default_states_np[state] + if default_next in reachable: + key.append(state_to_part_idx[default_next]) + + # Add exception transitions + for symbol in all_symbols: + next_state = self._get_transition( + state, symbol, + default_states_np, + exception_symbols_np, + exception_states_np + ) + key.append(state_to_part_idx[next_state]) + + split_dict[tuple(key)].append(state) + + # Add new partitions from splits + for subgroup in split_dict.values(): + new_partitions.append(set(subgroup)) + if len(subgroup) < len(part): + changed = True + + if changed: + partitions = new_partitions + + # Create mapping from old states to new state indices + state_to_part = {} + for part_idx, part in enumerate(partitions): + for state in part: + state_to_part[state] = part_idx + + # Build new DFA + num_new_states = len(partitions) + new_accepting = np.zeros(num_new_states, dtype=bool) + new_start = state_to_part[self.start_state] + + # Initialize new transition components + new_defaults = np.zeros(num_new_states, dtype=np.int32) + new_ex_syms = [] + new_ex_states = [] + + # Precompute state to partition index for transition processing + state_to_part_idx = {} + for part_idx, part in enumerate(partitions): + for state in part: + state_to_part_idx[state] = part_idx + + # Build transitions for new states + for part_idx, part in enumerate(partitions): + # Use representative state for transition information + rep_state = next(iter(part)) + + # Default transition + default_target = default_states_np[rep_state] + if default_target in reachable: + new_defaults[part_idx] = state_to_part_idx[default_target] + else: + new_defaults[part_idx] = -1 + + # Collect exceptions + exceptions = [] + for i in range(self.max_exceptions): + sym = exception_symbols_np[rep_state, i] + if sym == -1: + continue + target = exception_states_np[rep_state, i] + new_target = state_to_part_idx[target] + # Only add exception if different from default + if new_target != new_defaults[part_idx]: + exceptions.append((sym, new_target)) + + # Store exceptions + new_ex_syms.append([sym for sym, _ in exceptions]) + new_ex_states.append([state for _, state in exceptions]) + + # Set acceptance + new_accepting[part_idx] = is_accepting_np[rep_state] + + # Pad exception arrays + max_exceptions = max(len(ex) for ex in new_ex_syms) if new_ex_syms else 0 + padded_ex_syms = np.full((num_new_states, max_exceptions), -1) + padded_ex_states = np.full((num_new_states, max_exceptions), -1) + + for i in range(num_new_states): + syms = new_ex_syms[i] + [-1] * (max_exceptions - len(new_ex_syms[i])) + states = new_ex_states[i] + [-1] * (max_exceptions - len(new_ex_states[i])) + padded_ex_syms[i] = syms + padded_ex_states[i] = states + + # Convert back to JAX arrays + new_defaults_jax = jnp.array(new_defaults) + padded_ex_syms_jax = jnp.array(padded_ex_syms) + padded_ex_states_jax = jnp.array(padded_ex_states) + new_accepting_jax = jnp.array(new_accepting) + + return SparseDFA( + num_new_states, + new_defaults_jax, + padded_ex_syms_jax, + padded_ex_states_jax, + new_accepting_jax, + new_start, + self.symbol_arity, + self.base_alphabet + ).sparsify() + + def sparsify(self) -> 'SparseDFA': + """Computes the sparsest possible equivalent DFA by choosing optimal default states. + + + Returns: + SparseDFA with optimized default states and exceptions. A default state is changed if the most common exception state is more frequent than the default state. + """ + ex_states = jnp.where( + self.exception_states == -1, + self.num_states, + self.exception_states + ) + + counts = jnp.apply_along_axis( + jnp.bincount, axis=-1, arr=ex_states, length=self.num_states + 1 + ) + + counts = counts[:, :-1] # Ignore padding (num_states) + n_symbols = len(self.base_alphabet_frozen) + change_mask = counts.max(axis=-1) > (n_symbols ** self.symbol_arity) - counts.sum(axis=-1) + + # get new default states + maxfreq_states = jnp.argmax(counts, axis=-1) + new_defaults = jnp.where(change_mask, maxfreq_states, self.default_states) + + + # delete exceptions to new default states + new_ex_symbols = jnp.full((self.num_states, self.max_exceptions), -1, dtype=jnp.int32) + change_mask_expanded = jnp.expand_dims(change_mask, axis=1) + change_mask_expanded = jnp.repeat(change_mask_expanded, self.max_exceptions, axis=-1) + new_defaults_expanded = jnp.expand_dims(new_defaults, axis=1) + new_defaults_expanded = jnp.repeat(new_defaults_expanded, self.max_exceptions, axis=-1) + new_ex_symbols = jnp.where( + (self.exception_states == new_defaults_expanded) & change_mask_expanded, new_ex_symbols, self.exception_symbols + ) + new_ex_states = jnp.full((self.num_states, self.max_exceptions), -1, dtype=jnp.int32) + new_ex_states = jnp.where( + (self.exception_states == new_defaults_expanded) & change_mask_expanded, new_ex_states, self.exception_states + ) + + # For each state that changed, fill in the exceptions with the old default trasitions + for state in jnp.arange(self.num_states)[change_mask & (self.default_states != -1)]: + old_def_ex_syms = complement( + self.exception_symbols[state][self.exception_symbols[state] != -1], 0, n_symbols**self.symbol_arity - 1 + ) + + free_places = jnp.arange(new_ex_symbols.shape[1])[new_ex_symbols[state] == -1][:len(old_def_ex_syms)] + new_ex_symbols = new_ex_symbols.at[state, free_places].set(old_def_ex_syms) + new_ex_states = new_ex_states.at[state, free_places].set(self.default_states[state]) + + # Reorder exception symbols and states to minimize max_exceptions + # Find the minimum number of -1 entries in any row + num_minus_ones = jnp.sum(new_ex_symbols == -1, axis=1) + min_minus_ones_per_row = jnp.min(num_minus_ones) + + if min_minus_ones_per_row > 0: + # Create new padded arrays with reduced max_exceptions + new_max_exceptions = self.max_exceptions - min_minus_ones_per_row + reordered_ex_symbols = jnp.full((self.num_states, new_max_exceptions), -1, dtype=jnp.int32) + reordered_ex_states = jnp.full((self.num_states, new_max_exceptions), -1, dtype=jnp.int32) + + for i in range(self.num_states): + # Get non -1 entries + valid_symbols = new_ex_symbols[i][new_ex_symbols[i] != -1] + valid_states = new_ex_states[i][new_ex_states[i] != -1] + + # Fill into the reordered arrays + reordered_ex_symbols = reordered_ex_symbols.at[i, :len(valid_symbols)].set(valid_symbols) + reordered_ex_states = reordered_ex_states.at[i, :len(valid_states)].set(valid_states) + + new_ex_symbols = reordered_ex_symbols + new_ex_states = reordered_ex_states + + return SparseDFA( + self.num_states, + new_defaults, + new_ex_symbols, + new_ex_states, + self.is_accepting, + self.start_state, + self.symbol_arity, + self.base_alphabet + ) + + def _get_transition(self, state, symbol, default_states, exception_symbols, exception_states): + """Helper to get transition using numpy arrays""" + for i in range(self.max_exceptions): + if exception_symbols[state, i] == symbol: + return exception_states[state, i] + return default_states[state] + + def __str__(self) -> str: + lines = [] + lines.append(f"SparseDFA with {self.num_states} states (arity={self.symbol_arity})") + lines.append(f"Start state: {self.start_state}") + + # List accepting states + accepting_states = [i for i in range(self.num_states) if self.is_accepting[i]] + lines.append(f"Accepting states: {accepting_states}") + + # Add transitions header + lines.append("\nTransitions:") + lines.append("State | Default | Exceptions") + lines.append("------|---------|-----------") + + # Process each state + for state in range(self.num_states): + default = self.default_states[state] + + # Collect exception transitions + exceptions = [] + for i in range(self.max_exceptions): + sym = self.exception_symbols[state, i] + target = self.exception_states[state, i] + if sym != -1 and target != -1: + sym = decode_symbol(sym, self.symbol_arity, self.base_alphabet_frozen) + exceptions.append(f"{sym}→{target}") + + # Format exceptions or show none + exceptions_str = ", ".join(exceptions) if exceptions else "None" + + # Format state row + state_str = f"{state}{'*' if self.is_accepting[state] else ''}" + lines.append(f"{state_str:<5} | {default:<7} | {exceptions_str}") + + return "\n".join(lines) + + def show_diagram(self, filename: str = "automaton", format: str = "png", view: bool = True) -> graphviz.Digraph: + """ + Visualize the automaton using Graphviz, showing both default and exception transitions. + This version ensures all transitions are properly displayed. + """ + dot = graphviz.Digraph(engine='dot') + dot.attr(rankdir='LR') + + # Add nodes + for state in range(self.num_states): + if self.is_accepting[state]: + dot.node(str(state), shape='doublecircle') + else: + dot.node(str(state), shape='circle') + + # Add start arrow + dot.node('__start__', '', shape='none', width='0', height='0') + dot.edge('__start__', str(self.start_state)) + + # Collect all transitions + all_transitions = {} + + # First, add exception transitions + for state in range(self.num_states): + for i in range(self.max_exceptions): + symbol = int(self.exception_symbols[state, i]) + if symbol == -1: # Skip padding + continue + target = int(self.exception_states[state, i]) + key = (state, target) + + # Decode symbol + symbol_tuple = decode_symbol(symbol, self.symbol_arity, self.base_alphabet) + symbol_str = str(symbol_tuple) if self.symbol_arity > 1 else str(symbol_tuple[0]) + + if key not in all_transitions: + all_transitions[key] = [] + all_transitions[key].append(symbol_str) + + # Then add default transitions + for state in range(self.num_states): + default_target = int(self.default_states[state]) + key = (state, default_target) + + # Only add default if not already covered by exceptions + if key not in all_transitions: + all_transitions[key] = [] + + # Add "default" label to the list + all_transitions[key].append("default") + + # Create edges with all labels + for (from_state, to_state), symbols in all_transitions.items(): + # Combine all symbols for this edge + label = ", ".join(sorted(set(symbols))) + dot.edge(str(from_state), str(to_state), label=label) + + # Render and view + dot.render(filename=filename, format=format, view=view) + return dot + + def sparse_dfa_to_file(self, filename: str) -> None: + SparseDFASerializer.serialize(self, filename) + + @classmethod + def sparse_dfa_from_file(cls, filename: str) -> 'SparseDFA': + return SparseDFASerializer.deserialize(filename) + + +class SparseNFA: + def __init__(self, num_states: int, base_state: jnp.ndarray, + exception_symbols: jnp.ndarray, exception_states: jnp.ndarray, + is_accepting: jnp.ndarray, start_state: int, + symbol_arity: int = 1, base_alphabet: Optional[Set[int]] = None): + self.num_states = num_states + self.base_state = base_state + self.exception_symbols = exception_symbols + self.exception_states = exception_states + self.is_accepting = is_accepting + self.start_state = start_state + self.max_exceptions = exception_symbols.shape[1] + self.symbol_arity = symbol_arity + self.base_alphabet = base_alphabet if base_alphabet is not None else self._infer_base_alphabet() + + def _infer_base_alphabet(self) -> Set[int]: + """Infer base alphabet from exception symbols""" + symbols = set(np.unique(self.exception_symbols)) + symbols.discard(-1) + return symbols + + def _step(self, current_set: jnp.ndarray, symbol: int) -> Tuple[jnp.ndarray, None]: + """JAX-compatible single step transition""" + # Base transitions + base_next = jnp.zeros(self.num_states, dtype=bool) + base_next = base_next.at[self.base_state].set( + base_next[self.base_state] | current_set + ) + + # Exception transitions + exception_next = jnp.zeros(self.num_states, dtype=bool) + symbol_mask = (self.exception_symbols == symbol) & (self.exception_symbols != -1) + + # Process each exception slot + for i in range(self.max_exceptions): + # Get mask for this slot + slot_mask = symbol_mask[:, i] + slot_targets = self.exception_states[:, i] + + # Find states that are active and have matching exception + valid_transitions = current_set & slot_mask + + # Update exception_next for this slot + exception_next = exception_next.at[slot_targets].set( + exception_next[slot_targets] | valid_transitions + ) + + # Combine base and exception transitions + next_set = base_next | exception_next + return next_set, None + + def compute(self, word: jnp.ndarray) -> jnp.ndarray: + """Returns the set of states after processing the word""" + init = jnp.zeros(self.num_states, dtype=bool) + init = init.at[self.start_state].set(True) + final_set, _ = jax.lax.scan(self._step, init, word) + return final_set + + def accepts(self, word: jnp.ndarray) -> jnp.ndarray: + """Checks if the NFA accepts the word""" + final_set = self.compute(word) + return jnp.any(jnp.logical_and(final_set, self.is_accepting)) + + def determinize(self) -> SparseDFA: + """Converts NFA to DFA using subset construction""" + # Start state is epsilon closure (but no epsilons) + start_set = frozenset([self.start_state]) + state_queue = deque([start_set]) + state_to_id = {start_set: 0} + id_to_state = [start_set] + + # DFA components + dfa_defaults = [] + dfa_ex_syms = [] # List of exception symbols per state + dfa_ex_states = [] # List of exception states per state + dfa_accepting = [any(self.is_accepting[q] for q in start_set)] + + while state_queue: + current = state_queue.popleft() + current_id = state_to_id[current] + + # Compute base states and exception map + base_states = set() + exception_map = {} + has_exception = {} + + for q in current: + base_states.add(int(self.base_state[q])) + + for i in range(self.max_exceptions): + sym = self.exception_symbols[q, i] + if sym == -1: + continue + target = self.exception_states[q, i] + if sym not in exception_map: + exception_map[sym] = set() + has_exception[sym] = set() + + exception_map[sym].add(int(target)) + has_exception[sym].add(int(q)) + + # Default transition: T0 = base_states + T0 = frozenset(base_states) + if T0 not in state_to_id: + new_id = len(id_to_state) + state_to_id[T0] = new_id + id_to_state.append(T0) + dfa_accepting.append(any(self.is_accepting[q] for q in T0)) + state_queue.append(T0) + default_target = new_id + else: + default_target = state_to_id[T0] + dfa_defaults.append(default_target) + + # Process exception symbols + ex_symbols = [] + ex_states = [] + for sym, targets in exception_map.items(): + # Next state = base_states ∪ exception targets + targets_from_base = set([self.base_state[q] for q in current if q not in has_exception[sym]]) + T_sym = frozenset(targets_from_base | targets) + if T_sym not in state_to_id: + new_id = len(id_to_state) + state_to_id[T_sym] = new_id + id_to_state.append(T_sym) + dfa_accepting.append(any(self.is_accepting[q] for q in T_sym)) + state_queue.append(T_sym) + ex_states.append(new_id) + else: + ex_states.append(state_to_id[T_sym]) + ex_symbols.append(sym) + + dfa_ex_syms.append(ex_symbols) + dfa_ex_states.append(ex_states) + + # Pad exception arrays + max_ex = max(len(syms) for syms in dfa_ex_syms) if dfa_ex_syms else 0 + padded_ex_syms = jnp.full((len(id_to_state), max_ex), -1) + padded_ex_states = jnp.full((len(id_to_state), max_ex), -1) + + for i in range(len(id_to_state)): + if i < len(dfa_ex_syms): + syms = dfa_ex_syms[i] + [-1] * (max_ex - len(dfa_ex_syms[i])) + states = dfa_ex_states[i] + [-1] * (max_ex - len(dfa_ex_states[i])) + padded_ex_syms = padded_ex_syms.at[i].set(jnp.array(syms)) + padded_ex_states = padded_ex_states.at[i].set(jnp.array(states)) + + return SparseDFA( + num_states=len(id_to_state), + default_states=jnp.array(dfa_defaults, dtype=jnp.int32), + exception_symbols=padded_ex_syms, + exception_states=padded_ex_states, + is_accepting=jnp.array(dfa_accepting, dtype=bool), + start_state=0, + symbol_arity=self.symbol_arity, + base_alphabet=self.base_alphabet + ) + + def __str__(self) -> str: + """Returns a string representation of the NFA""" + lines = [] + lines.append(f"SparseNFA with {self.num_states} states (arity={self.symbol_arity})") + lines.append(f"Start state: {self.start_state}") + + # List accepting states + accepting_states = [i for i in range(self.num_states) if self.is_accepting[i]] + lines.append(f"Accepting states: {accepting_states}") + + # Add transitions header + lines.append("\nTransitions:") + lines.append("State | Base | Exceptions") + lines.append("------|------|-----------") + + # Helper to decode symbols + def symbol_to_str(sym: int) -> str: + if sym == -1: + return "ε" + tup = decode_symbol(sym, self.symbol_arity, self.base_alphabet) + if self.symbol_arity == 1: + return str(tup[0]) + return str(tup) + + # Process each state + for state in range(self.num_states): + base = self.base_state[state] + + # Collect exception transitions + exceptions = [] + for i in range(self.max_exceptions): + sym = int(self.exception_symbols[state, i]) + if sym == -1: + continue + target = int(self.exception_states[state, i]) + symbol_str = symbol_to_str(sym) + exceptions.append(f"{symbol_str}→{target}") + + # Format exceptions or show none + exceptions_str = ", ".join(exceptions) if exceptions else "None" + + # Format state row + state_str = f"{state}{'*' if self.is_accepting[state] else ''}" + lines.append(f"{state_str:<5} | {base:<4} | {exceptions_str}") + + return "\n".join(lines) + + def show_diagram(self, filename: str = "nfa", format: str = "png", view: bool = True) -> graphviz.Digraph: + """Visualizes the NFA using Graphviz""" + try: + import graphviz + except ImportError: + raise ImportError("Graphviz is required for visualization. Install with 'pip install graphviz'") + + dot = graphviz.Digraph(engine='dot') + dot.attr(rankdir='LR') + + # Helper to decode symbols + def symbol_to_str(sym: int) -> str: + if sym == -1: + return "ε" + tup = decode_symbol(sym, self.symbol_arity, self.base_alphabet) + if self.symbol_arity == 1: + return str(tup[0]) + return str(tup) + + # Add nodes + for state in range(self.num_states): + if self.is_accepting[state]: + dot.node(str(state), shape='doublecircle') + else: + dot.node(str(state), shape='circle') + + # Add start arrow + dot.node('__start__', '', shape='none', width='0', height='0') + dot.edge('__start__', str(self.start_state)) + + # Collect transitions + base_transitions = {} + exception_transitions = defaultdict(lambda: defaultdict(set)) + + # Base transitions + for state in range(self.num_states): + base_target = int(self.base_state[state]) + base_transitions[(state, base_target)] = "default" + + # Exception transitions + for state in range(self.num_states): + for i in range(self.max_exceptions): + sym = int(self.exception_symbols[state, i]) + if sym == -1: + continue + target = int(self.exception_states[state, i]) + symbol_str = symbol_to_str(sym) + exception_transitions[(state, target)][symbol_str] = True + + # Add base transitions to graph + for (from_state, to_state), label in base_transitions.items(): + dot.edge(str(from_state), str(to_state), label=label, style='dashed', color='blue') + + # Add exception transitions to graph + for (from_state, to_state), symbols in exception_transitions.items(): + label = ", ".join(sorted(symbols.keys())) + dot.edge(str(from_state), str(to_state), label=label) + + # Render and view + dot.render(filename=filename, format=format, view=view) + return dot \ No newline at end of file diff --git a/autstr/utils/automata_tools.py b/autstr/utils/automata_tools.py index 91336e5..46a9b2e 100644 --- a/autstr/utils/automata_tools.py +++ b/autstr/utils/automata_tools.py @@ -1,376 +1,770 @@ -from automata.fa.dfa import DFA -from automata.fa.nfa import NFA +import heapq +import jax +import jax.numpy as jnp +import numpy as np +from collections import deque, defaultdict +from functools import partial +from typing import Generator, Optional, Callable, Dict, List, Set, Union import itertools as it -from typing import List, Tuple, Callable, Union +from autstr.utils.logic import get_free_elementary_vars +from autstr.sparse_automata import SparseDFA, SparseNFA from autstr.buildin.automata import one -from autstr.utils.misc import get_unique_id -from autstr.utils.misc import heapify_llex as heapify -from autstr.utils.misc import heappop_llex as heappop -from autstr.utils.misc import heappush_llex as heappush +from autstr.utils.misc import decode_symbol, encode_symbol, complement -def stringlify_states(dfa: Union[DFA, NFA], convert_orgnames: bool = True) -> DFA: - """ - Auxiliary function to turn the states of a finite automation into strings - - :param dfa: the automaton A - :param convert_orgnames: If True, converts the original state names into strings. Otherwise, an arbitrary numbering - is chosen. - :return: dfa B such that B is identical to A but type(B.states) = Set[str] - """ - mapping = {s: str(s) if convert_orgnames else str(i) for i, s in enumerate(dfa.states)} - states = {mapping[q] for q in dfa.states} - if isinstance(dfa, DFA): - transitions = {mapping[q]: {a: mapping[dfa.transitions[q][a]] for a in dfa.input_symbols} for q in dfa.states} - else: - transitions = { - mapping[q]: {a: {mapping[p] for p in dfa.transitions[q][a]} for a in dfa.transitions[q]} for q in dfa.states - } - initial_state = mapping[dfa.initial_state] - final_states = {mapping[q] for q in dfa.final_states} - - if isinstance(dfa, DFA): - dfa_str = DFA( - initial_state=initial_state, - states=states, - transitions=transitions, - input_symbols=dfa.input_symbols, - final_states=final_states - ) - else: - dfa_str = NFA( - initial_state=initial_state, - states=states, - transitions=transitions, - input_symbols=dfa.input_symbols, - final_states=final_states - ) - return dfa_str -def stringlify_input_symbols(dfa: DFA, convert_orgnames=True) -> DFA: +# ====== Helper Functions ====== +def pad(dfa: SparseDFA, padding_symbol: int = -1) -> SparseDFA: + """Pad the automaton to accept trailing padding symbols by: + 1. Creating a new sub-automaton for (pad_tuple)* + 2. Connecting original accepting states to the new sub-automaton + 3. Determinizing and minimizing the result """ - Auxiliary function to turn the input_symbols of a finite automation into strings. Note that the function - - :param dfa: the automaton A - :param convert_orgnames: If True, converts the original input symbol names into strings. Otherwise, - an arbitrary numbering is chosen. - :return: dfa B such that B is identical to A but type(B.input_symbols) = Set[str] - """ - mapping = {s: str(s) if convert_orgnames else str(i) for i, s in enumerate(dfa.input_symbols)} - input_symbols = {mapping[s] for s in dfa.input_symbols} - transitions = {q: {mapping[a]: dfa.transitions[q][a] for a in dfa.input_symbols} for q in dfa.states} - - dfa_str = DFA( - initial_state=dfa.initial_state, - states=dfa.states, - transitions=transitions, - input_symbols=input_symbols, - final_states=dfa.final_states + arity = dfa.symbol_arity + base_alphabet = dfa.base_alphabet + if padding_symbol == -1: + padding_symbol = sorted(base_alphabet)[0] # Default to first symbol + pad_tuple = (padding_symbol,) * arity + pad_enc = encode_symbol(pad_tuple, base_alphabet) + + # If automaton is empty, return it immediately + if dfa.is_empty(): + return dfa + + # Convert JAX arrays to native Python types + default_states_np = np.array(dfa.default_states) + exception_symbols_np = np.array(dfa.exception_symbols) + exception_states_np = np.array(dfa.exception_states) + is_accepting_np = np.array(dfa.is_accepting) + + # Step 1: Build NFA components + n_orig = dfa.num_states + n_pad = n_orig # State for padding loop + n_dead = n_orig + 1 # Dead state + num_states = n_orig + 2 + + # Base state array (using native Python types) + base_state_arr = default_states_np.tolist() + [n_dead, n_dead] + + # Acceptance array (original acceptors + pad state) + is_accepting_arr = is_accepting_np.tolist() + [True, False] + + # Calculate needed exception slots + extra_slots = 1 # For new pad transitions + new_max_exceptions = dfa.max_exceptions + extra_slots + + # Initialize exception arrays + exception_symbols_arr = np.full((num_states, new_max_exceptions), -1, dtype=np.int32) + exception_states_arr = np.full((num_states, new_max_exceptions), -1, dtype=np.int32) + + # Copy original exceptions + for i in range(n_orig): + for j in range(dfa.max_exceptions): + sym = exception_symbols_np[i, j] + if sym != -1: + exception_symbols_arr[i, j] = sym + exception_states_arr[i, j] = exception_states_np[i, j] + + # Add new transitions: + # 1. From original accepting states to pad state on padding symbol + for i in range(n_orig): + if is_accepting_np[i]: + # Find first available slot + slot = np.where(exception_symbols_arr[i] == -1)[0] + if slot.size > 0: + slot_idx = slot[0] + exception_symbols_arr[i, slot_idx] = pad_enc + exception_states_arr[i, slot_idx] = n_pad + + # 2. From pad state to itself on padding symbol + exception_symbols_arr[n_pad, 0] = pad_enc + exception_states_arr[n_pad, 0] = n_pad + + # Build NFA using native Python types + nfa = SparseNFA( + num_states=num_states, + base_state=base_state_arr, + exception_symbols=exception_symbols_arr, + exception_states=exception_states_arr, + is_accepting=is_accepting_arr, + start_state=dfa.start_state, + symbol_arity=arity, + base_alphabet=base_alphabet ) - return dfa_str - - -def projection(dfa: DFA, i: int) -> DFA: - """Takes an automaton that recognizes a k-ary relation R and a position i i for s in dfa.input_symbols]) - done = set() - input_symbols = {s[:i] + s[i + 1:] for s in dfa.input_symbols} - input_symbols_i = {s[i] for s in dfa.input_symbols} - todo = [frozenset({dfa.initial_state})] - transitions = {} - while len(todo) > 0: - state = todo.pop(0) - done.add(state) - state_trans = {s: frozenset() for s in input_symbols} - for q in state: - for s in input_symbols: - state_trans[s] = frozenset( - state_trans[s].union({dfa.transitions[q][s[:i] + (a,) + s[i:]] for a in input_symbols_i}) + # Convert to DFA and minimize + return nfa.determinize()#.minimize() + +def unpad(dfa: SparseDFA, padding_symbol: int = -1, remove_blank: bool = False) -> SparseDFA: + """Remove trailing padding symbols from accepted words.""" + arity = dfa.symbol_arity + base_alphabet = dfa.base_alphabet + if padding_symbol == -1: + padding_symbol = sorted(base_alphabet)[0] # Default to first symbol + pad_tuple = (padding_symbol,) * arity + pad_tuple_enc = encode_symbol(pad_tuple, base_alphabet) + + # Compute closure under padding symbols + closure = {} + for q in range(dfa.num_states): + closure[q] = set() + stack = [q] + while stack: + state = stack.pop() + # Convert state to native int for set operations + state_int = int(state) if hasattr(state, '__int__') else state + + if state_int not in closure[q]: + closure[q].add(state_int) + # Follow padding transitions + if pad_tuple_enc in dfa.exception_symbols[state]: + idx = jnp.where(dfa.exception_symbols[state] == pad_tuple_enc)[0] + if idx.size > 0: + next_state = dfa.exception_states[state, idx[0]] + # Convert to native int before adding to stack + stack.append(int(next_state)) + # Default transition for padding + else: + next_state = dfa.default_states[state] + # Avoid infinite loops from self-transitions + if int(next_state) != state_int: + # Convert to native int before adding to stack + stack.append(int(next_state)) + + # New acceptance: states that can reach a final state via padding + new_accepting = jnp.array([ + any(dfa.is_accepting[p] for p in closure[q]) for q in range(dfa.num_states) + ]) + + # Create new automaton + if remove_blank: + # Remove padding symbol from input symbols + new_base_alphabet = base_alphabet - {padding_symbol} + # Filter out padding transitions + new_exception_symbols = jnp.full_like(dfa.exception_symbols, -1) + new_exception_states = jnp.full_like(dfa.exception_states, -1) + + for q in range(dfa.num_states): + valid_mask = (dfa.exception_symbols[q] != pad_tuple_enc) & (dfa.exception_symbols[q] != -1) + valid_indices = jnp.where(valid_mask)[0] + num_valid = len(valid_indices) + + if num_valid > 0: + new_exception_symbols = new_exception_symbols.at[q, :num_valid].set( + dfa.exception_symbols[q, valid_indices] ) - transitions[state] = state_trans - for s in state_trans.values(): - if s not in done: - todo.append(s) - - final = {s for s in done if any([q in dfa.final_states for q in s])} - result = DFA( - states=frozenset(done), - input_symbols=input_symbols, - initial_state=frozenset({dfa.initial_state}), - transitions=transitions, - final_states=final - ).minify(retain_names=False) - - return stringlify_states(result) - - -def expand(dfa: DFA, n: int, pos: List) -> DFA: - """ - Takes an automaton that recognizes a k-ary presentation R and expands it to a n-ary presentation S, n>=k, with - S = {(x_1,..., x_n) | (x_{pos[0]},..., x_{pos[k-1]}) in R} - - :param dfa: The automaton - :param n: The arity of the result - :param pos: where to encode the original relation - :return: Automaton recognizing {(x_1,..., x_n) | (x_{pos[0]},..., x_{pos[k-1]}) in L(dfa)} - """ - assert max(pos) < n - assert len(list(dfa.input_symbols)[0]) == len(pos) - - base_symbols = {symbol[0] for symbol in dfa.input_symbols} - - return DFA( - initial_state=dfa.initial_state, - input_symbols=set(it.product(base_symbols, repeat=n)), - states=dfa.states, - transitions={ - x: { - symbol: dfa.transitions[x][tuple(symbol[i] for i in pos)] for symbol in - it.product(base_symbols, repeat=n) - } for x in dfa.states - }, - final_states=dfa.final_states - ).minify() - - -def pad(dfa: DFA, padding_symbol: Tuple[str] = ('*',)) -> DFA: - """ - Create an automaton that recognises :math:`L(\\textrm{dfa}){\\textrm{padding_symbol}}^\\ast`. - - :param dfa: The automaton - :param padding_symbol: - :return: Automaton that recognizes :math:`L(\\textrm{dfa}){\\textrm{paddingsymbol}}^\ast` - """ - # TODO: padding symbol should be of type string - base_symbols = {a[0] for a in dfa.input_symbols}.union({padding_symbol[0]}) - arity = len(list(dfa.input_symbols)[0]) - input_symbols = set(it.product(base_symbols, repeat=arity)) - good, bad = get_unique_id(dfa.states, 2) - states = dfa.states.union({good, bad}) - padding_transitions = { - q: {padding_symbol * arity: good if q in dfa.final_states else bad} for q in dfa.states - } - transitions = {q: dfa.transitions[q] | padding_transitions[q] for q in dfa.states} - transitions[good] = {a: good if a == padding_symbol * arity else bad for a in input_symbols} - transitions[bad] = {a: bad for a in input_symbols} - final_states = dfa.final_states.union({good}) - - padded_dfa = DFA( - states=states, - input_symbols=input_symbols, - initial_state=dfa.initial_state, - transitions=transitions, - final_states=final_states - ).minify() - - return padded_dfa - - -def unpad(dfa: DFA, padding_symbol: Tuple[str] = ('*',), remove_blank: bool = False) -> DFA: - """ - Creates an automaton that recognizes {w | exists x in (padding_symbol^k)^*: wx in L(dfa)} where k is the arity of - the relation that is recognized by dfa. - - :param remove_blank: If True, removes the blank symbol from the input symbols - :param dfa: The automaton - :param padding_symbol: The padding symbol. Needs to be a length one tuple - :return: Automaton that recognizes {w | exists x in padding_symbol^*: wx in L(dfa)} - """ - arity = len(list(dfa.input_symbols)[0]) - if not remove_blank: - input_symbols = dfa.input_symbols - sink = get_unique_id(dfa.states, 1) - states = dfa.states.union({sink}) - transitions = { - q: { - a: dfa.transitions[q][a] if padding_symbol * arity != a else sink for a in dfa.input_symbols - } for q in dfa.states - } - transitions[sink] = {a: sink for a in input_symbols} + new_exception_states = new_exception_states.at[q, :num_valid].set( + dfa.exception_states[q, valid_indices] + ) + + return SparseDFA( + num_states=dfa.num_states, + default_states=dfa.default_states, + exception_symbols=new_exception_symbols, + exception_states=new_exception_states, + is_accepting=new_accepting, + start_state=dfa.start_state, + symbol_arity=arity, + base_alphabet=new_base_alphabet + ).minimize() else: - input_symbols = {a for a in dfa.input_symbols if padding_symbol * arity != a} - states = dfa.states - transitions = { - q: { - a: dfa.transitions[q][a] for a in dfa.input_symbols if padding_symbol * arity != a - } for q in dfa.states - } - - final_states = dfa.final_states - last_fstates = None - while final_states != last_fstates: - last_fstates = final_states - final_states = final_states.union( - {q for q in dfa.transitions if dfa.transitions[q][padding_symbol * arity] in final_states} - ) - - unpadded_dfa = DFA( - states=states, - input_symbols=input_symbols, - initial_state=dfa.initial_state, - transitions=transitions, - final_states=final_states - ).minify() - - return unpadded_dfa - - -def product(dfa: DFA, n: int) -> DFA: - """ - Create an automaton that recognizes the n-fold cartesian product of L(dfa) - - :param dfa: The automaton - :param n: number of convolutions - :return: Automaton that recognizes the n-fold cartesian product of L(dfa) - """ + return SparseDFA( + num_states=dfa.num_states, + default_states=dfa.default_states, + exception_symbols=dfa.exception_symbols, + exception_states=dfa.exception_states, + is_accepting=new_accepting, + start_state=dfa.start_state, + symbol_arity=arity, + base_alphabet=base_alphabet + ).minimize() + +def product(dfa: SparseDFA, n: int) -> SparseDFA: + """Create the n-fold Cartesian product of the automaton's language.""" if n == 0: - result = one() + return one() + if n == 1: + return dfa + else: + result = dfa + for _ in range(n-1): + result = stack(result, dfa) return result - base_automata = [stringlify_states(expand(dfa, n, [i])) for i in range(n)] - result = base_automata[0] - for base in base_automata[1:]: - result = result.intersection(base).minify() - - return result - -def iterate_language(dfa: DFA, decoder: Callable = None, backward: bool = False, padding_symbol: str = '*'): +def stack(dfa1: SparseDFA, dfa2: SparseDFA) -> SparseDFA: """ - Generator over the language that is represented by a DFA. Provides functionality for decoding of words by user - defined decoder functions. Iterates through the words in length-lexicographic order. - - :param dfa: The automaton - :param decoder: Decoder function that maps (tuples of) words to python object - :param backward: If True, iterate over the elements of the reversed language - :param padding_symbol: The padding symbol - :return: + Creates a stacked automaton that recognizes the concatenation of two relations + without explicitly generating all possible symbols. + + The new automaton accepts tuples (x1,...,xk,y1,...,yl) where: + (x1,...,xk) is accepted by dfa1 and + (y1,...,yl) is accepted by dfa2 + + Args: + dfa1: First automaton of arity k + dfa2: Second automaton of arity l + + Returns: + SparseDFA of arity k+l recognizing the stacked relation """ - arity = len(list(dfa.input_symbols)[0]) - nfa = dfa - - nfa = stringlify_states(nfa) - - if backward: - nonempty = set() - for q in nfa.states: - nfa_q = DFA( - states=nfa.states, - input_symbols=nfa.input_symbols, - initial_state=nfa.initial_state, - final_states={q}, - transitions=nfa.transitions - ) - if not nfa_q.isempty(): - nonempty.add(q) + # Validate common base alphabet + if dfa1.base_alphabet != dfa2.base_alphabet: + raise ValueError("Automata must have the same base alphabet") + + dfa1 = pad(dfa1) + dfa2 = pad(dfa2) + + # Get arities + k = dfa1.symbol_arity + l = dfa2.symbol_arity + arity = k + l + base_alphabet = dfa1.base_alphabet + + # Create product states + n1 = dfa1.num_states + n2 = dfa2.num_states + num_states = n1 * n2 + + # On-the-fly construction: only generate reachable states + start_pair = (dfa1.start_state, dfa2.start_state) + queue = deque([start_pair]) + state_map = {start_pair: 0} + + new_default_states_list = [] + new_exception_symbols_list = [] + new_exception_states_list = [] + new_is_accepting_list = [] + + # Helper function to split symbol + def split_symbol(full_symbol_enc): + """Split encoded symbol into two components""" + full_tuple = decode_symbol(full_symbol_enc, arity, base_alphabet) + s1_tuple = full_tuple[:k] + s2_tuple = full_tuple[k:] + s1_enc = encode_symbol(s1_tuple, base_alphabet) + s2_enc = encode_symbol(s2_tuple, base_alphabet) + return s1_enc, s2_enc + + # Build product automaton + idx_counter = 0 + while queue: + current_pair = queue.popleft() + i, j = current_pair + current_idx = state_map[current_pair] + + # Add acceptance status + new_is_accepting_list.append(bool(dfa1.is_accepting[i]) and bool(dfa2.is_accepting[j])) + + # Collect all unique symbols that cause an exception in either DFA + # or are part of the full alphabet + all_relevant_symbols = set() + + # Add symbols that are exceptions in dfa1 + for pos1 in range(dfa1.max_exceptions): + s1_enc = int(dfa1.exception_symbols[i, pos1]) + if s1_enc == -1: + continue + # Generate corresponding symbols for the full arity + for symbol_char in base_alphabet: + base_tuple = decode_symbol(s1_enc, k, base_alphabet) + full_tuple = base_tuple + (symbol_char,) * l + full_enc = encode_symbol(full_tuple, base_alphabet) + all_relevant_symbols.add(full_enc) + + # Add symbols that are exceptions in dfa2 + for pos2 in range(dfa2.max_exceptions): + s2_enc = int(dfa2.exception_symbols[j, pos2]) + if s2_enc == -1: + continue + # Generate corresponding symbols for the full arity + for symbol_char in base_alphabet: + base_tuple = decode_symbol(s2_enc, l, base_alphabet) + full_tuple = (symbol_char,) * k + base_tuple + full_enc = encode_symbol(full_tuple, base_alphabet) + all_relevant_symbols.add(full_enc) + + # Add all symbols from the combined alphabet to ensure all transitions are considered + for symbol_tuple_chars in it.product(sorted(base_alphabet), repeat=arity): + all_relevant_symbols.add(encode_symbol(symbol_tuple_chars, base_alphabet)) + + # Determine default transition for the product state + # The default transition for the product automaton is formed by the default transitions + # of the individual automata. + def_i = int(dfa1.default_states[i]) + def_j = int(dfa2.default_states[j]) + default_target_pair = (def_i, def_j) + + if default_target_pair not in state_map: + state_map[default_target_pair] = len(state_map) + queue.append(default_target_pair) + new_default_states_list.append(state_map[default_target_pair]) + + # Process exceptions for the current product state + current_exceptions_symbols = [] + current_exceptions_states = [] + + for full_enc in sorted(list(all_relevant_symbols)): # Sort for deterministic output + s1_enc, s2_enc = split_symbol(full_enc) + + next_i = int(dfa1.transition(i, s1_enc)) + next_j = int(dfa2.transition(j, s2_enc)) + next_pair = (next_i, next_j) + + # Only add as an exception if it deviates from the default transition + if next_pair != default_target_pair: + if next_pair not in state_map: + state_map[next_pair] = len(state_map) + queue.append(next_pair) + current_exceptions_symbols.append(full_enc) + current_exceptions_states.append(state_map[next_pair]) + + new_exception_symbols_list.append(current_exceptions_symbols) + new_exception_states_list.append(current_exceptions_states) + + idx_counter += 1 + + # Pad exceptions to uniform length + num_new_states = len(state_map) + max_exceptions = max(len(ex) for ex in new_exception_symbols_list) if new_exception_symbols_list else 0 + padded_ex_syms = jnp.full((num_new_states, max_exceptions), -1, dtype=jnp.int32) + padded_ex_states = jnp.full((num_new_states, max_exceptions), -1, dtype=jnp.int32) + + for i in range(num_new_states): + syms = new_exception_symbols_list[i] + states = new_exception_states_list[i] + if syms: + padded_ex_syms = padded_ex_syms.at[i, :len(syms)].set(jnp.array(syms, dtype=jnp.int32)) + padded_ex_states = padded_ex_states.at[i, :len(states)].set(jnp.array(states, dtype=jnp.int32)) + + # Create and return the stacked automaton + return SparseDFA( + num_states=num_new_states, + default_states=jnp.array(new_default_states_list), + exception_symbols=jnp.array(padded_ex_syms), + exception_states=jnp.array(padded_ex_states), + is_accepting=jnp.array(new_is_accepting_list), + start_state=0, # Start state is always 0 in the new mapping + symbol_arity=arity, + base_alphabet=base_alphabet + ) - transitions = {q: {a: set() for a in nfa.input_symbols} for q in nfa.states} - for p in nfa.states: - for q in nfa.states: - for a in nfa.transitions[q]: - transitions[nfa.transitions[q][a]][a].add(q) +def projection(dfa: SparseDFA, i: int) -> SparseDFA: + """Project the automaton by existentially quantifying the i-th position.""" + arity = dfa.symbol_arity + base_alphabet = dfa.base_alphabet + new_arity = arity - 1 + + # Create new automaton via subset construction + start_state = frozenset([int(dfa.start_state)]) + state_queue = deque([start_state]) + state_to_id = {start_state: 0} + id_to_state = [start_state] + + # New DFA components + new_default_states = [] + new_exception_symbols = [] + new_exception_states = [] + new_accepting = [] + + # Generate all possible symbols for the new alphabet + new_alphabet = set(it.product(sorted(base_alphabet), repeat=new_arity)) + + while state_queue: + state_set = state_queue.popleft() + state_id = state_to_id[state_set] + + # For each possible symbol in new alphabet + trans_map = {} + for symbol_tuple in new_alphabet: + next_set = set() + for q in state_set: + # Try all possible values at position i + for a in base_alphabet: + full_symbol = symbol_tuple[:i] + (a,) + symbol_tuple[i:] + full_symbol_enc = encode_symbol(full_symbol, base_alphabet) + next_state = dfa.transition(q, full_symbol_enc) + next_set.add(int(next_state)) # Convert to native int + trans_map[symbol_tuple] = frozenset(next_set) + + # Handle case with no transitions + if not trans_map: + # Create a dead state if no transitions exist + dead_state = frozenset() + if dead_state not in state_to_id: + new_id = len(id_to_state) + state_to_id[dead_state] = new_id + id_to_state.append(dead_state) + # Dead state never accepts and transitions to itself + new_accepting.append(False) + new_default_states.append(new_id) + new_exception_symbols.append([]) + new_exception_states.append([]) + default_target = state_to_id[dead_state] + state_id_for_default = default_target + else: + # Find the most common transition + targets = list(trans_map.values()) + default_target_set = max(set(targets), key=targets.count) + + # Get or create state ID for default target + if default_target_set not in state_to_id: + new_id = len(id_to_state) + state_to_id[default_target_set] = new_id + id_to_state.append(default_target_set) + state_queue.append(default_target_set) + state_id_for_default = new_id + else: + state_id_for_default = state_to_id[default_target_set] + + # Create exceptions for deviations + exceptions = [] + for symbol_tuple, target_set in trans_map.items(): + if target_set != default_target_set: + # Get or create state ID for this target + if target_set not in state_to_id: + new_id = len(id_to_state) + state_to_id[target_set] = new_id + id_to_state.append(target_set) + state_queue.append(target_set) + target_id = new_id + else: + target_id = state_to_id[target_set] + + symbol_enc = encode_symbol(symbol_tuple, base_alphabet) + exceptions.append((symbol_enc, target_id)) + + new_exception_symbols.append([e[0] for e in exceptions]) + new_exception_states.append([e[1] for e in exceptions]) + + new_default_states.append(state_id_for_default) + new_accepting.append(any(dfa.is_accepting[q] for q in state_set)) + + # Pad exceptions + max_ex = max(len(ex) for ex in new_exception_symbols) if new_exception_symbols else 0 + padded_ex_syms = jnp.full((len(id_to_state), max_ex), -1) + padded_ex_states = jnp.full((len(id_to_state), max_ex), -1) + + for idx, (syms, states) in enumerate(zip(new_exception_symbols, new_exception_states)): + if syms: + padded_ex_syms = padded_ex_syms.at[idx, :len(syms)].set(jnp.array(syms, dtype=jnp.int32)) + padded_ex_states = padded_ex_states.at[idx, :len(states)].set(jnp.array(states, dtype=jnp.int32)) + + return SparseDFA( + num_states=len(id_to_state), + default_states=jnp.array(new_default_states, dtype=jnp.int32), + exception_symbols=padded_ex_syms, + exception_states=padded_ex_states, + is_accepting=jnp.array(new_accepting), + start_state=0, + symbol_arity=new_arity, + base_alphabet=base_alphabet + ) - final = {nfa.initial_state} - else: - nonempty = set() - for q in nfa.states: - nfa_q = DFA( - states=nfa.states, - input_symbols=nfa.input_symbols, - initial_state=q, - final_states=nfa.final_states, - transitions=nfa.transitions +def expand(dfa, new_arity: int, pos: List[int]): + # Input validation (unchanged) + original_arity = dfa.symbol_arity + base_alphabet = dfa.base_alphabet + sorted_alphabet = sorted(base_alphabet) + m = len(base_alphabet) + new_num_states = dfa.num_states + K = m ** (new_arity - len(set(pos))) # Account for duplicate positions + new_max_exceptions = dfa.max_exceptions * K + + # Precompute fixed positions and free indices + fixed_mask = jnp.zeros(new_arity, dtype=bool) + for idx in pos: + fixed_mask = fixed_mask.at[idx].set(True) + free_indices = jnp.where(~fixed_mask, size=new_arity, fill_value=-1)[0] + free_count = jnp.sum(~fixed_mask).item() + + # Precompute encoding powers + powers = m ** jnp.arange(new_arity-1, -1, -1, dtype=jnp.int64) + sorted_alphabet_arr = jnp.arange(len(sorted_alphabet)) + + # Prepare new DFA arrays + new_default_states = jnp.zeros(new_num_states, dtype=jnp.int32) + new_exception_symbols = jnp.full((new_num_states, new_max_exceptions), -1, dtype=jnp.int32) + new_exception_states = jnp.full((new_num_states, new_max_exceptions), -1, dtype=jnp.int32) + + # Process each state + for state in range(dfa.num_states): + # Calculate new default state via frequency counts + # The default state for the expanded DFA is simply the default state of the original DFA + # as the 'default' transition applies when no exception matches, regardless of arity. + new_default_states = new_default_states.at[state].set(int(dfa.default_states[state])) + + # Collect original exceptions + orig_exceptions = dfa.exception_symbols[state][dfa.exception_symbols[state] != -1] + orig_targets = dfa.exception_states[state][dfa.exception_symbols[state] != -1] + + # Process exception blocks + all_exception_symbols = [] + all_exception_states = [] + for sym_enc, target in zip(orig_exceptions, orig_targets): + # Decode and convert to indices + orig_tuple = decode_symbol(int(sym_enc), original_arity, base_alphabet) + + # Check consistency for duplicate positions + expected = {} + consistent = True + for orig_idx, new_idx in enumerate(pos): + val = orig_tuple[orig_idx] + if new_idx in expected and expected[new_idx] != val: + consistent = False + break + expected[new_idx] = val + if not consistent: + continue # Skip inconsistent symbol + + # Generate expanded symbols as a block + expanded_symbols = generate_expanded_block( + sym_enc, + sorted_alphabet_arr, + fixed_mask, + free_indices, + free_count, + powers, + original_arity, + m, + pos ) - if not nfa_q.isempty(): - nonempty.add(q) - - transitions = dfa.transitions - final = nfa.final_states + all_exception_symbols.append(expanded_symbols) + all_exception_states.append(jnp.full_like(expanded_symbols, target)) + + if all_exception_symbols: + # Flatten the lists and pad to max_exceptions + all_exception_symbols = jnp.concatenate(all_exception_symbols) + all_exception_states = jnp.concatenate(all_exception_states) + new_exception_symbols = new_exception_symbols.at[state, :len(all_exception_symbols)].set(all_exception_symbols) + new_exception_states = new_exception_states.at[state, :len(all_exception_states)].set(all_exception_states) + + return SparseDFA( + num_states=new_num_states, + default_states=new_default_states, + exception_symbols=new_exception_symbols, + exception_states=new_exception_states, + is_accepting=dfa.is_accepting, + start_state=dfa.start_state, + symbol_arity=new_arity, + base_alphabet=base_alphabet + ) - queue = [(('',) * arity, q) for q in nfa.final_states] - queue = heapify(queue) - while len(queue) > 0: - word, state = heappop(queue) - if state in final: - if decoder is None: - yield word - else: - yield decoder(word) - - for a in transitions[state]: - if a != (padding_symbol,) * arity: - for q in transitions[state][a]: - if q in nonempty: - heappush( - queue, - (tuple( - [f'{wordcomp}{b}' if b != padding_symbol else wordcomp for wordcomp, b in - zip(word, a)] - ), q) - ) - - -def lsbf_automaton(n: int) -> DFA: +def generate_expanded_block(sym_enc, sorted_alphabet_arr, fixed_mask, free_indices, + free_count, powers, original_arity, m, pos): + # Decode original symbol + powers_orig = m ** jnp.arange(original_arity-1, -1, -1, dtype=jnp.int64) + digits = (sym_enc // powers_orig) % m + orig_tuple = sorted_alphabet_arr[digits] + + # Create fixed values template + fixed_values = jnp.full(len(fixed_mask), -1, dtype=sorted_alphabet_arr.dtype) + for orig_idx, new_idx in enumerate(pos): + fixed_values = fixed_values.at[new_idx].set(orig_tuple[orig_idx]) + + # Generate free combinations + if free_count > 0: + n_comb = m ** free_count + grid = jnp.indices((m,)*free_count).reshape(free_count, -1).T + free_vals = sorted_alphabet_arr[grid] + symbol_tensors = jnp.tile(fixed_values, (n_comb, 1)) + symbol_tensors = symbol_tensors.at[:, free_indices[:free_count]].set(free_vals) + else: + symbol_tensors = fixed_values[None, :] + + # Encode symbols + indices = jnp.searchsorted(sorted_alphabet_arr, symbol_tensors) + return jnp.sum(indices * powers, axis=1, dtype=jnp.int32) + +# We'll define a custom heap structure for length-lexicographic ordering +class LengthLexHeap: + def __init__(self): + self.heap = [] + + def push(self, item): + # item: (word_tuple, state) + # word_tuple is tuple of strings + # Priority: 1. Total length (sum of lengths), 2. Lex order + total_length = max(len(comp) for comp in item[0]) + heapq.heappush(self.heap, (total_length, item[0], item[1])) + + def pop(self): + _, word, state = heapq.heappop(self.heap) + return (word, state) + + def __len__(self): + return len(self.heap) + +def iterate_language(dfa: SparseDFA, decoder: Callable = None, + backward: bool = False, padding_symbol: int = -1) -> Generator: """ - generates an automation that recognizes exactly the least-significant-bit-first binary encoding of n + Generator over the language of a SparseDFA. Yields words in length-lexicographic order. + Note: The algorithm assumes minimality and optimal sparsity of the automaton. + + :param dfa: Sparse automaton + :param decoder: Function to decode words to Python objects + :param backward: If True, generate words in reverse order + :param padding_symbol: Integer representing padding symbol + :return: Generator of words (or decoded objects) """ - bits = format(n, 'b')[::-1] - n_bits = len(bits) - - states = set(range(n_bits + 2)) - initial_state = 0 - final_states = {n_bits} - input_symbols = {('0',), ('1',), ('*',)} - - transitions = {i: {a: i + 1 if a == (bits[i],) else n_bits + 1 for a in input_symbols} for i in range(n_bits)} - transitions[n_bits] = {a: n_bits + 1 for a in input_symbols} - transitions[n_bits + 1] = {a: n_bits + 1 for a in input_symbols} - - result = DFA( - states=states, - input_symbols=input_symbols, - initial_state=initial_state, - transitions=transitions, - final_states=final_states - ) - - return stringlify_states(result) - -def lsbf_Z_automaton(n: int) -> DFA: + successors = {q: dfa.successors(q) for q in range(dfa.num_states)} + nonempty = {q for q in range(dfa.num_states) if len(successors[q]) > 0 or q not in successors[q]} + + arity = dfa.symbol_arity + + # Build reversed transitions: state -> symbol -> set of previous states + rev_transitions = {} + for state in range(dfa.num_states): + rev_transitions[state] = {} + + + start_set = {dfa.start_state} + final_set = set(jnp.where(dfa.is_accepting)[0].tolist()) + + # Initialize heap with starting states + heap = LengthLexHeap() + for state in start_set: + if state in nonempty: + # Represent words as tuple of empty strings + heap.push((tuple(["" for _ in range(arity)]), state)) + + def cat(word, symbol): + """Concatenate symbol to word based on direction.""" + if backward: + return str(symbol) + word + else: + return word + str(symbol) + + def push(heap, word_tuple, sym_enc, next_state): + """Push a new word onto the heap with the given extension symbol and next state.""" + if encode_symbol((padding_symbol,) * arity, dfa.base_alphabet) == sym_enc: + # Skip padding symbols + return + # Decode symbol + symbol_tuple = decode_symbol(sym_enc, arity, dfa.base_alphabet) + + # Create new word components + new_components = [] + for comp, sym in zip(word_tuple, symbol_tuple): + if sym == padding_symbol: + # Keep component unchanged + new_components.append(comp) + else: + # Prepend symbol to component + new_components.append(cat(comp, sym)) + + new_word_tuple = tuple(new_components) + + # Add to heap + heap.push((new_word_tuple, next_state)) + + # Main loop + visited_words = set() + while heap: + word_tuple, state = heap.pop() + + # Skip duplicates + word_key = (state, word_tuple) + if word_key in visited_words: + continue + visited_words.add(word_key) + + # Check if we've reached a final state + if state in final_set: + if decoder: + yield decoder(word_tuple) + else: + yield word_tuple + + # process transitions + ex_mask = dfa.exception_symbols[state] != -1 + ex_symbols = dfa.exception_symbols[state, ex_mask] + ex_states = dfa.exception_states[state, ex_mask] + for sym_enc, next_state in zip(ex_symbols, ex_states): + sym_enc, next_state = int(sym_enc), int(next_state) + if next_state not in nonempty: + continue + + push(heap, word_tuple, sym_enc, next_state) + + default = int(dfa.default_states[state]) + if default in nonempty: + # get all non-exception symbols + default_symbols = complement(ex_symbols, 0, len(dfa.base_alphabet)**dfa.symbol_arity - 1) + for sym_enc in default_symbols: + push(heap, word_tuple, sym_enc, default) + + + + + +def lsbf_Z_automaton(z: int) -> SparseDFA: """ - generates an automation that recognizes exactly the least-significant-bit-first binary encoding of n over Z + Creates a SparseDFA for LSB-first representation of integer z with sign bit and padding. + Alphabet encoding: + "*" = 0 + "0" = 1 + "1" = 2 """ - bits = format(abs(n), 'b')[::-1] - n_bits = len(bits) - - states = set(range(n_bits + 2)) - states.add(-1) - initial_state = -1 - final_states = {n_bits} - input_symbols = {('0',), ('1',), ('*',)} - - transitions = {i: {a: i + 1 if a == (bits[i],) else n_bits + 1 for a in input_symbols} for i in range(n_bits)} - transitions[-1] = { - a: 0 if (n >= 0 and a == ('0',)) or (n < 0 and a == ('1',)) else n_bits + 1 for a in input_symbols - } - transitions[n_bits] = {a: n_bits + 1 for a in input_symbols} - transitions[n_bits + 1] = {a: n_bits + 1 for a in input_symbols} - - result = DFA( - states=states, - input_symbols=input_symbols, - initial_state=initial_state, - transitions=transitions, - final_states=final_states - ) - - return stringlify_states(result) + # Handle special case for zero + if z == 0: + return SparseDFA( + num_states=4, + default_states=jnp.array([3, 3, 3, 3], dtype=jnp.int32), + exception_symbols=jnp.array([[1], [1], [0], [-1]], dtype=jnp.int32), # "0"=1, "*"=0 + exception_states=jnp.array([[1], [2], [2], [-1]], dtype=jnp.int32), + is_accepting=jnp.array([False, False, True, False]), + start_state=0, + symbol_arity=1, + base_alphabet={"*", "0", "1"} # "*"=0, "0"=1, "1"=2 + ) + + # Determine sign and magnitude + sign_symbol = 1 if z >= 0 else 2 # "0"=1 for positive, "1"=2 for negative + magnitude = abs(z) + + # Convert to LSB-first bits (without trailing zeros) + bits = [] + while magnitude: + bits.append(2 if magnitude & 1 else 1) # 1→"0"=1, 2→"1"=2 + magnitude >>= 1 + + # Create representation: [sign_symbol] + bits (LSB first) + rep = [sign_symbol] + bits + n = len(rep) + + # States: + # 0 to n-1: processing representation + # n: accepting state (after full representation) + # n+1: dead state + num_states = n + 2 + + # Create arrays with vectorized operations + default_states = jnp.full(num_states, n+1, dtype=jnp.int32) # Default to dead state + + # Exception symbols: rep for states 0..n-1, 0 ('*') for state n + exception_symbols = jnp.full((num_states, 1), -1, dtype=jnp.int32) + exception_symbols = exception_symbols.at[:n, 0].set(jnp.array(rep, dtype=jnp.int32)) + exception_symbols = exception_symbols.at[n, 0].set(0) # '*' for accepting state + + # Exception states: next state for representation, self for padding + exception_states = jnp.full((num_states, 1), -1, dtype=jnp.int32) + exception_states = exception_states.at[:n, 0].set(jnp.arange(1, n+1)) + exception_states = exception_states.at[n, 0].set(n) # loop in accepting state + + # Accepting state is state n + is_accepting = jnp.zeros(num_states, dtype=bool) + is_accepting = is_accepting.at[n].set(True) + + return SparseDFA( + num_states=num_states, + default_states=default_states, + exception_symbols=exception_symbols, + exception_states=exception_states, + is_accepting=is_accepting, + start_state=0, + symbol_arity=1, + base_alphabet={"*", "0", "1"} # "*"=0, "0"=1, "1"=2 + ) \ No newline at end of file diff --git a/autstr/utils/logic.py b/autstr/utils/logic.py index be145ec..e32c7e8 100644 --- a/autstr/utils/logic.py +++ b/autstr/utils/logic.py @@ -1,5 +1,5 @@ import nltk -from nltk.sem.logic import Expression +from nltk.sem.logic import Expression, AllExpression, ExistsExpression, NegatedExpression, ApplicationExpression, AndExpression, OrExpression from typing import List @@ -18,4 +18,32 @@ def get_free_elementary_vars(phi: Expression) -> List[str]: return free_vars +def optimize_query(query: Expression) -> Expression: + """ + Optimize a query for optimized automata construction. + :param query: The query expression + :return: Optimized query expression + """ + if isinstance(query, AllExpression): + query.term = optimize_query(query.term) + if query.variable.name in get_free_elementary_vars(query): + return query.term + else: + return query + elif isinstance(query, ExistsExpression): + query.term = optimize_query(query.term) + if query.variable.name in get_free_elementary_vars(query): + return query.term + else: + return query + elif isinstance(query, NegatedExpression): + if isinstance(query.term, AllExpression): + term = optimize_query(NegatedExpression(query.term)) + return ExistsExpression(query.variable, term) + elif isinstance(query.term, NegatedExpression): + return optimize_query(query.term.term) + else: + return NegatedExpression(optimize_query(query.term)) + else: + return query \ No newline at end of file diff --git a/autstr/utils/misc.py b/autstr/utils/misc.py index 1c91e20..b56d8c6 100644 --- a/autstr/utils/misc.py +++ b/autstr/utils/misc.py @@ -1,6 +1,8 @@ from functools import cmp_to_key from heapq import heapify, heappop, heappush -from typing import Set, List, Union +from typing import FrozenSet, Set, List, Tuple, Union + +from jax import numpy as jnp def cmp_llex(v: str, w: str) -> int: """ @@ -68,3 +70,58 @@ def get_unique_id(current_id: List[str], n: int = 1) -> Union[str, List[str]]: else: n_symbols = len(str(n)) return [f'{max_element}{i:{n_symbols}d}' for i in range(n)] + + +# ====== Symbol Encoding/Decoding ====== +def encode_symbol(tuple_symbol: Tuple[int], base_alphabet: FrozenSet[int]) -> int: + """Encode a symbol tuple into a single integer.""" + if not tuple_symbol: + return 0 + m = len(base_alphabet) + alphabet_sorted = sorted(base_alphabet) + mapping = {sym: idx for idx, sym in enumerate(alphabet_sorted)} + enc = 0 + for sym in tuple_symbol: + enc = enc * m + mapping[sym] + return enc + + +def decode_symbol(enc: int, arity: int, base_alphabet: FrozenSet[int]) -> Tuple[int]: + """Decode an integer into a symbol tuple.""" + if arity == 0: + return () + m = len(base_alphabet) + alphabet_sorted = sorted(base_alphabet) + symbols = [] + num = enc + for _ in range(arity): + num, r = divmod(num, m) + symbols.append(alphabet_sorted[r]) + return tuple(reversed(symbols)) + +def complement(values, min_val: int, max_val: int) -> jnp.ndarray: + """Find the complement of a set of values within a specified range. + + Args: + values (jnp.ndarray): The input array of values. + min (int): The minimum value of the range. + max (int): The maximum value of the range. + + Returns: + jnp.ndarray: The complement of the input array. + """ + + unique_idxs = jnp.unique(values) + if unique_idxs.size == 0: + result = jnp.array([], dtype=values.dtype) + else: + + # Create full range and mask for present values + full_range = jnp.arange(min_val, max_val + 1) + mask = jnp.zeros_like(full_range, dtype=bool) + indices_in_full = unique_idxs - min_val + mask = mask.at[indices_in_full].set(True) + + # Extract missing values + result = full_range[~mask] + return result \ No newline at end of file diff --git a/notebooks/arithmetic.ipynb b/notebooks/arithmetic.ipynb new file mode 100644 index 0000000..5f2910a --- /dev/null +++ b/notebooks/arithmetic.ipynb @@ -0,0 +1,155 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "id": "014fe7bd-e587-491b-9862-4dca24c0e69f", + "metadata": {}, + "outputs": [], + "source": [ + "from autstr.arithmetic import VariableETerm as Var\n", + "from time import time\n", + "x = Var('x')" + ] + }, + { + "cell_type": "markdown", + "id": "45dccf70-33f8-4617-b6e7-a5ffadac9cf2", + "metadata": {}, + "source": [ + "# Algorithmic Design with Infinite Sets: The Sieve of Eratosthenes\n", + "AutStr enables novel algorithm design using infinite sets as first-class citizens. This implementation of the Sieve of Eratosthenes maintains the infinite candidate prime set symbolically:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "966543c9-eac4-46f0-8502-c14fd3edd37d", + "metadata": {}, + "outputs": [], + "source": [ + "def infinite_sieve(steps):\n", + " \"\"\"Sieve of Eratosthenes over infinite integers\"\"\"\n", + " x = Var('x')\n", + " candidates = (x.gt(1)) # Initial infinite set: {2,3,4,...}\n", + " primes = []\n", + " \n", + " for _ in range(steps):\n", + " # Find smallest candidate (symbolic operation)\n", + " start = time()\n", + " for p in candidates: # Elements are listed in ascending order (absolute values) \n", + " primes.append(p[0])\n", + " print(primes)\n", + " break\n", + " end = time()\n", + " print(f\"computing next prime took: {end - start} seconds\")\n", + " \n", + " # Remove multiples: candidates = candidates \\ {k·p | k>1}\n", + " p = primes[-1]\n", + " y = Var(\"y\")\n", + " multiples = (x.eq(p * y)).drop(\"y\")\n", + " candidates = candidates & ~multiples \n", + " \n", + " return primes, candidates\n", + "\n", + "# Execute first 5 sieving steps\n", + "primes, remaining = infinite_sieve(steps=3)\n", + "print(f\"Primes found: {primes}\") # [2,3,5,7,11]\n", + "print(f\"Remaining infinite set:\") \n", + "VisualDFA(remaining.evaluate()).show_diagram()" + ] + }, + { + "cell_type": "markdown", + "id": "26fc46e1-b0bf-4b5f-8a49-3c7d1d70dd1b", + "metadata": {}, + "source": [ + "# Integer Linear Equation Systems\n", + "As we have seen, working with infinite structures is for sure compute intense. This is due to the fact that we can define (and in principle solve) extremly complext problems within our formalism. Solving linear equation systems with integer constraints is one such problem. In contrast to their continuous counterparts, integer linear equation systems are known to bw NP hard to solve. \n", + "\n", + "With the Arithmetic package, we can easily define and solve such systems." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "060cbf58-3d09-4196-bb3b-4e0baf0fabeb", + "metadata": {}, + "outputs": [], + "source": [ + "# Define variables\n", + "x,y, z = Var(\"x\"), Var(\"y\"), Var(\"z\")" + ] + }, + { + "cell_type": "markdown", + "id": "dc62f306-4dbc-4257-b62a-66760cf3acd5", + "metadata": {}, + "source": [ + "# Solvable" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "32c9b9fd-7e48-4db8-a4bf-7ee4079a26e0", + "metadata": {}, + "outputs": [], + "source": [ + "%%time\n", + "eq1 = (x + y).eq(6) \n", + "eq2 = (y + z).eq(12) \n", + "eq3 = (x + z).eq(10) \n", + "eqn_sys = eq1 & eq2 & eq3\n", + "\n", + "for s in eqn_sys:\n", + " print(s)" + ] + }, + { + "cell_type": "markdown", + "id": "510cc124-40a8-48e1-9a5c-8e73598620f9", + "metadata": {}, + "source": [ + "## Unsolvable" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "86cd7a58-eab3-4c6d-94ff-05b6ef614868", + "metadata": {}, + "outputs": [], + "source": [ + "%%time\n", + "eq1 = (3 * y + 5 * z).eq(1) \n", + "eq2 = (x + y).eq(5) \n", + "eq3 = (2*z + x).eq(17) \n", + "eqn_sys = eq1 & eq2 & eq3\n", + "\n", + "eqn_sys.isempty()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.0" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/notebooks/buechi.autstr b/notebooks/buechi.autstr new file mode 100644 index 0000000..6bb66a1 Binary files /dev/null and b/notebooks/buechi.autstr differ diff --git "a/notebooks/b\303\274chi.ipynb" "b/notebooks/b\303\274chi.ipynb" new file mode 100644 index 0000000..fd05b9e --- /dev/null +++ "b/notebooks/b\303\274chi.ipynb" @@ -0,0 +1,326 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from autstr.buildin.presentations import BuechiArithmetic" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Büchi Arithmetic\n", + "Since many predifined relations of the structure are dynamically generated upon creation, this might take a few seconds." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "ba = BuechiArithmetic()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Visualize Addition \n", + "We can explore the presentations visually with the ```show_diagram``` method." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ba.automata[\"A\"].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Define Zero\n", + "We will make use of the fact that $0$ is the only number that satisfies $x+x=x$ " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "ba.update(O='A(x,x,x)')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ba.automata['O'].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Define One\n", + "$1$ is the only number apart from $0$ that has only trivial additive decompositions." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "# Define one\n", + "ba.update(I='not O(x) and forall y z.(O(y) or O(z) or not A(y, z, x))')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You can also convert the automaton to a string to obtain a readable representation" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "print(ba.automata['I'])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Define Successor" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "ba.update(S='exists x.(I(x) and A(u, x, v))')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ba.automata['S'].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Define powers of $2$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "ba.update(P='B(x,x)')\n", + "ba.automata['P'].show_diagram()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "# Define linear order\n", + "ba.update(Smaller='exist z.(not O(z) and A(x, z, y))')\n", + "ba.automata['Smaller'].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Define Bit Predicate\n", + "The bit predicate lets us effective \"talk\" about the binary presentation of a number within formulas. Positions in the presentation are represented as powers of $2$:\n", + "$$\n", + " \\textrm{BitPredicate(x,y)} \\colon\\Leftrightarrow y=2^n \\land bin(x)[n] = 1\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "# Define bit predicate E(x, y) <=> y==2^n bin(x)[n] == 1\n", + "ba.update(\n", + " Theorem='exists x0 x1.(P(y) and Smaller(x0, y) and A(x0, x1, x) and B(x1, y) and not exists y1.(Smaller(y, y1) and B(x1, y1)))'\n", + ")\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ba.automata['Theorem'].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Proof the \"Sandwich Theorem\"\n", + "Finally, we will prove a small theorem: For every positive natural number $x$ there is a power of two in $[x, 2x]$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "jupyter": { + "outputs_hidden": false + }, + "pycharm": { + "name": "#%%\n" + } + }, + "outputs": [], + "source": [ + "%%time\n", + "# Evaluate 'For all x>0 there is a power of 2 in [x, 2x]'\n", + "a = ba.evaluate('forall x.(O(x) or exists y u v w.(A(x, x, y) and A(x, u, v) and P(v) and A(v, w, y)))')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Formulas without free variables (sentences), the result automaton will have exactly one state that is excepting iff the sentence is true. Alternatively, you can also call ```check``` instead of ```evaluate``` to directly obtain the truth value." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "a.show_diagram()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/notebooks/demo.ipynb b/notebooks/demo.ipynb deleted file mode 100644 index 3457a52..0000000 --- a/notebooks/demo.ipynb +++ /dev/null @@ -1,2149 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "from autstr.buildin.presentations import buechi_arithmetic\n", - "from visual_automata.fa.dfa import VisualDFA\n", - "from autstr.utils.automata_tools import iterate_language" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "jupyter": { - "outputs_hidden": false - }, - "pycharm": { - "name": "#%%\n" - } - }, - "outputs": [], - "source": [ - "ba = buechi_arithmetic()" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "jupyter": { - "outputs_hidden": false - }, - "pycharm": { - "name": "#%%\n" - } - }, - "outputs": [ - { - "data": { - "image/svg+xml": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Initial\n", - "\n", - "\n", - "\n", - "\n", - "1\n", - "\n", - "1\n", - "\n", - "\n", - "\n", - "Initial->1\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "0\n", - "\n", - "0\n", - "\n", - "\n", - "\n", - "0->0\n", - "\n", - "\n", - " ('0',) \n", - "\n", - "\n", - "\n", - "0->0\n", - "\n", - "\n", - " ('1',) \n", - "\n", - "\n", - "\n", - "0->0\n", - "\n", - "\n", - " ('*',) \n", - "\n", - "\n", - "\n", - "1->0\n", - "\n", - "\n", - " ('1',) \n", - "\n", - "\n", - "\n", - "1->0\n", - "\n", - "\n", - " ('*',) \n", - "\n", - "\n", - "\n", - "2\n", - "\n", - "\n", - "2\n", - "\n", - "\n", - "\n", - "1->2\n", - "\n", - "\n", - " ('0',) \n", - "\n", - "\n", - "\n", - "2->0\n", - "\n", - "\n", - " ('0',) \n", - "\n", - "\n", - "\n", - "2->0\n", - "\n", - "\n", - " ('1',) \n", - "\n", - "\n", - "\n", - "2->2\n", - "\n", - "\n", - " ('*',) \n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Define zero\n", - "ba.update(O='A(x,x,x)')\n", - "VisualDFA(ba.automata['O']).show_diagram()" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - 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"metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Define successor\n", - "ba.update(S='exists x.(I(x) and A(u, x, v))')\n", - "VisualDFA(ba.automata['S']).show_diagram()" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "jupyter": { - "outputs_hidden": false - }, - "pycharm": { - "name": "#%%\n" - } - }, - "outputs": [ - { - "data": { - "image/svg+xml": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Initial\n", - "\n", - "\n", - "\n", - "\n", - "1\n", - "\n", - "1\n", - "\n", - "\n", - "\n", - "Initial->1\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "0\n", - "\n", - "\n", - "0\n", - "\n", - "\n", - "\n", - "0->0\n", - "\n", - "\n", - " ('*',) \n", - "\n", - "\n", - "\n", - "2\n", - "\n", - "2\n", - "\n", - "\n", - "\n", - "0->2\n", - "\n", - "\n", - " ('0',) \n", - "\n", - "\n", - "\n", - "0->2\n", - "\n", - "\n", - " ('1',) \n", - "\n", - "\n", - "\n", - "1->0\n", - "\n", - "\n", - 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"\n", - "\n", - "\n", - "7->1\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "7->4\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "7->6\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "7->7\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "8->0\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "8->2\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "8->2\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "8->6\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "8->6\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "8->6\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "8->6\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "8->6\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "8->8\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Define linear order\n", - "ba.update(Smaller='exist z.(not O(z) and A(x, z, y))')\n", - "VisualDFA(ba.automata['Smaller']).show_diagram()" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false, - "jupyter": { - "outputs_hidden": false - }, - "pycharm": { - "name": "#%%\n" - } - }, - "outputs": [ - { - "data": { - "image/svg+xml": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Initial\n", - "\n", - "\n", - "\n", - "\n", - "0\n", - "\n", - "0\n", - "\n", - "\n", - "\n", - "Initial->0\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "1\n", - "\n", - "\n", - "1\n", - "\n", - "\n", - "\n", - "0->1\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "3\n", - "\n", - "\n", - "3\n", - "\n", - "\n", - "\n", - "0->3\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "4\n", - "\n", - "4\n", - "\n", - "\n", - "\n", - "0->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "0->4\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "0->4\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "0->4\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "0->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "5\n", - "\n", - "5\n", - "\n", - "\n", - "\n", - "0->5\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "0->5\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "1->1\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "2\n", - "\n", - "2\n", - "\n", - "\n", - "\n", - "1->2\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "1->3\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "1->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "2->1\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "2->2\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "2->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "3->3\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "3->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n", - "4->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "5->1\n", - "\n", - "\n", - " ('1', '1') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('*', '1') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('0', '1') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('1', '*') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('*', '*') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('0', '*') \n", - "\n", - "\n", - "\n", - "5->4\n", - "\n", - "\n", - " ('*', '0') \n", - "\n", - "\n", - "\n", - "5->5\n", - "\n", - "\n", - " ('0', '0') \n", - "\n", - "\n", - "\n", - "5->5\n", - "\n", - "\n", - " ('1', '0') \n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 8, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Define bit predicate E(x, y) <=> y==2^n bin(x)[n] == 1\n", - "ba.update(\n", - " E='exists x0 x1.(P(y) and Smaller(x0, y) and A(x0, x1, x) and B(x1, y) and not exists y1.(Smaller(y, y1) and B(x1, y1)))'\n", - ")\n", - "VisualDFA(ba.automata['E']).show_diagram()" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "collapsed": false, - "jupyter": { - "outputs_hidden": false - }, - "pycharm": { - "name": "#%%\n" - } - }, - "outputs": [ - { - "data": { - "image/svg+xml": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Initial\n", - "\n", - "\n", - "\n", - "\n", - "0\n", - "\n", - "\n", - "0\n", - "\n", - "\n", - "\n", - "Initial->0\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Evaluate 'between a number x>0 and its double lies a power of 2'\n", - "a = ba.evaluate('forall x.(O(x) or exists y u v w.(A(x, x, y) and A(x, u, v) and P(v) and A(v, w, y)))')\n", - "VisualDFA(a).show_diagram()" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.12.0" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/notebooks/mso0.ipynb b/notebooks/mso0.ipynb new file mode 100644 index 0000000..e85f5ab --- /dev/null +++ b/notebooks/mso0.ipynb @@ -0,0 +1,210 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "id": "e26c9d9b-3839-4742-ab56-e1a34dbbd8b9", + "metadata": {}, + "outputs": [], + "source": [ + "from autstr.buildin.presentations import MSO0" + ] + }, + { + "cell_type": "markdown", + "id": "9474b029-aacd-4780-b4c1-63ba2a0ddcd3", + "metadata": {}, + "source": [ + "# Finite Powerset Structure of (N, <) and its Connection to Büchi's MSO Logic\n", + "\n", + "## Introduction\n", + "\n", + "This notebook demonstrates an automatic structure for the finite powerset of natural numbers with order relations, a fundamental structure in mathematical logic and automata theory. The structure is given by:\n", + "\n", + "$$\\mathcal{P}_{fin}(\\mathbb{N}) = (P_{fin}(\\mathbb{N}), \\subseteq, \\text{Sing}, \\text{Succ}, <_{\\text{sing}}, \\text{min}, \\text{max})$$\n", + "\n", + "where:\n", + "- $P_{fin}(\\mathbb{N})$ is the set of all finite subsets of $\\mathbb{N}$\n", + "- $\\subseteq$ is the subset relation\n", + "- $\\text{Sing}$ is the set of singleton sets $\\{\\{n\\} \\mid n \\in \\mathbb{N}\\}$\n", + "- $\\text{Succ}$ is the successor relation on singletons $\\{(\\{n\\}, \\{n+1\\}) \\mid n \\in \\mathbb{N}\\}$\n", + "- $<_{\\text{sing}}$ is the less-than relation on singletons $\\{(\\{m\\}, \\{n\\}) \\mid m < n\\}$\n", + "- $\\text{min}$ and $\\text{max}$ are functions returning the smallest/largest element in a set\n", + "\n", + "This structure is particularly significant because it is equivalent to Büchi's monadic second-order logic (MSO) on the natural numbers with the successor relation (S1S). Büchi's seminal theorem shows that the sets definable in MSO over $(\\mathbb{N}, <)$ are exactly the regular sets.\n", + "\n", + "## Key Connections to Büchi's Work\n", + "\n", + "1. **Representation of Sets**: \n", + " - Finite subsets of $\\mathbb{N}$ are represented as finite words over $\\{0,1\\}$\n", + " - Each position in the word corresponds to a natural number\n", + " - '1' indicates membership in the set\n", + " - The empty set is represented by the empty word\n", + "\n", + "2. **Automata-Theoretic Foundation**:\n", + " - Büchi showed that MSO-definable sets correspond to regular languages\n", + " - Our implementation uses finite automata to recognize relations in the structure\n", + " - First-order definitions of operations (∩, ∪, \\) mirror MSO definability\n", + "\n", + "3. **Decidability**:\n", + " - Büchi's proof established the decidability of MSO over $(\\mathbb{N}, <)$\n", + " - Our automatic presentation allows deciding first-order queries using automata operations\n", + " - The structure is closed under first-order definitions\n", + "\n", + "## Structure Highlights\n", + "\n", + "Our implementation features automata for:\n", + "1. **Universe**: Recognizes valid set representations (empty set + sets ending with 1)\n", + "2. **Subset**: $X \\subseteq Y$ (empty set is subset of all)\n", + "3. **Singleton**: $\\{\\{n\\} \\mid n \\in \\mathbb{N}\\}$\n", + "4. **Successor**: $\\{(\\{n\\}, \\{n+1\\}) \\mid n \\in \\mathbb{N}\\}$\n", + "5. **Less-than on singletons**: $\\{(\\{m\\}, \\{n\\}) \\mid m < n\\}$\n", + "\n", + "With first-order definitions for:\n", + "- Set equality ($X = Y$)\n", + "- Intersection ($X \\cap Y$)\n", + "- Union ($X \\cup Y$)\n", + "- Set difference ($X \\setminus Y$)\n", + "- Minimum and maximum elements\n", + "\n", + "## Why This Matters\n", + "\n", + "This implementation provides a concrete computational foundation for Büchi's theoretical results, demonstrating how:\n", + "1. MSO formulas can be compiled to automata\n", + "2. Set operations correspond to automata manipulations\n", + "3. Decidability results can be made practical\n", + "4. Infinite structures can be finitely represented\n", + "\n", + "The connection between automata and logic established by Büchi remains foundational in computer science, influencing areas from formal verification to database theory." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2cacf418-2f8b-49cd-91c5-2717531d49d1", + "metadata": {}, + "outputs": [], + "source": [ + "%%time\n", + "mso0 = MSO0()\n", + "mso0.automata[\"Subset\"].show_diagram()" + ] + }, + { + "cell_type": "markdown", + "id": "3727b3b0-833a-4363-a18c-d452501fa677", + "metadata": {}, + "source": [ + "# Practical Application: Patient Selection in Clinical Trials using Automata\n", + "\n", + "## Index Sets as Patient Groups in Medical Studies\n", + "\n", + "Consider a clinical trial scenario where we need to select groups of patients that satisfy complex combinatorial constraints. This application demonstrates how our finite powerset structure of natural numbers (equivalent to Büchi's MSO logic) provides an elegant solution to this real-world problem.\n", + "\n", + "### The Problem: K-Partitions with Regular Constraints\n", + "Imagine we have:\n", + "- A patient set P = {p₀, p₁, p₂, ..., pₙ} indexed by natural numbers\n", + "- A need to partition patients into K groups with specific constraints:\n", + " 1. **Consecutive enrollment**: Groups must consist of patients enrolled consecutively\n", + " 2. **Risk-based exclusion**: Groups must avoid certain high-risk patients\n", + " 3. **Size constraints**: Groups must have 2-5 participants\n", + " 4. **Representation requirement**: Each group must include at least one high-risk patient\n", + "\n", + "### Our Automata-Theoretic Solution\n", + "Using our finite powerset structure, we can:\n", + "1. **Represent groups** as finite sets of indices\n", + "2. **Encode constraints** as automata-recognizable relations\n", + "3. **Compute valid partitions** through automata operations\n", + "\n", + "```python\n", + "def find_valid_groups(patient_set, excluded, high_risk):\n", + " # Build the automatic presentation\n", + " ap = finite_powerset()\n", + " \n", + " # Define constraints via automata operations\n", + " constraints = f\"\"\"\n", + " Interval(X) and \n", + " SizeConstraint(X) and \n", + " forall s: (Subset(s, X) -> not Subset(s, {excluded})) and \n", + " exists s: (Subset(s, X) and Subset(s, {high_risk}))\n", + " \"\"\"\n", + " \n", + " # Evaluate constraints to get valid groups\n", + " valid_groups_automaton = ap.evaluate(constraints)\n", + " return extract_valid_groups(valid_groups_automaton)\n", + "```\n", + "\n", + "### Key Advantages\n", + "1. **Regular Pattern Recognition**:\n", + " - Interval constraint: `∀a,b∈X, ∀c: (a < c < b) → c∈X`\n", + " - Encoded as automaton recognizing consecutive indices\n", + "\n", + "2. **Complex Constraints**:\n", + " ```python\n", + " SizeConstraint = \"\"\"\n", + " exists m, M: (min(X, m) and max(X, M) and \n", + " (Succ(m, M) or \n", + " exists s1: (Succ(m, s1) and Succ(s1, M)) or \n", + " exists s1,s2: (Succ(m, s1) and Succ(s1, s2) and Succ(s2, M)) or \n", + " exists s1,s2,s3: (Succ(m, s1) and Succ(s1, s2) and Succ(s2, s3) and Succ(s3, M))\n", + " )\n", + " \"\"\"\n", + " ```\n", + "\n", + "3. **Efficient Verification**:\n", + " - Automata can validate groups in linear time\n", + " - Precompiled constraints enable instant verification\n", + "\n", + "### Practical Example\n", + "**Input**:\n", + "- Excluded patients E = {3,5} (high-risk)\n", + "- Required high-risk patients H = {0,2,4}\n", + "\n", + "**Valid Groups**:\n", + "- {0,1}: Size 2, includes high-risk patient 0\n", + "- {1,2}: Size 2, includes high-risk patient 2\n", + "- {0,1,2}: Size 3, includes high-risk patients 0 and 2\n", + "- {4,5}: **Invalid** (contains excluded patient 5)\n", + "- {3,4}: **Invalid** (contains excluded patient 3)\n", + "\n", + "### Why This Matters\n", + "This application demonstrates how Büchi's seminal work on MSO logic:\n", + "1. Provides a foundation for formalizing real-world constraints\n", + "2. Enables efficient computation through automata theory\n", + "3. Solves practical problems in healthcare analytics\n", + "4. Offers a decision procedure for complex combinatorial constraints\n", + "\n", + "The connection between abstract automata theory and concrete applications like patient selection showcases the enduring power of Büchi's framework in computer science and beyond." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7f9a8a27-2ebd-4430-933b-7bbd19cf6a77", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.0" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/poetry.lock b/poetry.lock new file mode 100644 index 0000000..e3b2f44 --- /dev/null +++ b/poetry.lock @@ -0,0 +1,4068 @@ +# This 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