القائمة
Study of abstract machines and computation models including finite automata, pushdown automata, Turing machines, regular languages, context-free languages, and computational limits
التثبيت باستخدام Codex أو Claude انسخ هذا Prompt والصقه في Codex أو Claude أو مساعد آخر ليراجع صفحة Skill ويثبّتها لك.
استنادا إلى تصنيف SOC المهني
| name | Automata Theory |
| description | Study of abstract machines and computation models including finite automata, pushdown automata, Turing machines, regular languages, context-free languages, and computational limits |
| license | MIT |
| compatibility | universal |
| audience | Theoretical Computer Scientists, Language Designers, Algorithm Engineers |
| category | Computer Science |
I specialize in automata theory—the study of abstract computational devices and the languages they recognize. My expertise spans finite automata (DFA, NFA), regular languages, context-free grammars and pushdown automata, context-sensitive grammars, linear bounded automata, and Turing machines. I apply this theoretical foundation to design lexical analyzers, parsers, pattern matching algorithms, and to understand the fundamental limits of computation.
# Deterministic Finite Automaton Implementation
from typing import Set, Dict, Tuple, Optional
from dataclasses import dataclass
from enum import Enum
class Symbol:
EPSILON = None
@dataclass
class DFA:
states: Set[str]
alphabet: Set[str]
transitions: Dict[Tuple[str, str], str]
start_state: str
accept_states: Set[str]
def accepts(self, input_string: str) -> bool:
"""Check if DFA accepts the input string."""
current = self.start_state
for symbol in input_string:
if (current, symbol) not in self.transitions:
return False
current = self.transitions[(current, symbol)]
return current in self.accept_states
def to_minimal_dfa(self) -> 'DFA':
"""Minimize DFA using Hopcroft's algorithm."""
# Initial partition: accept vs reject states
P = [self.accept_states.copy(),
(self.states - self.accept_states).copy()]
W = [self.accept_states.copy()]
while W:
A = W.pop(0)
for symbol in self.alphabet:
# Find states that transition on symbol into A
X = set()
for state in self.states:
if symbol in self.alphabet and (state, symbol) in self.transitions:
if self.transitions[(state, symbol)] in A:
X.add(state)
# Refine partition
new_P = []
for Y in P:
intersection = X & Y
difference = Y - X
if intersection and difference:
new_P.append(intersection)
new_P.append(difference)
if Y in W:
W.remove(Y)
W.append(intersection)
W.append(difference)
else:
if len(intersection) <= len(difference):
W.append(intersection)
else:
W.append(difference)
else:
new_P.append(Y)
P = new_P
# Build minimized DFA from partitions
partition_map = {}
for i, part in enumerate(P):
for state in part:
partition_map[state] = f"q{i}"
new_states = {partition_map[s] for s in self.states}
new_transitions = {}
for (state, symbol), next_state in self.transitions.items():
new_from = partition_map[state]
new_to = partition_map[next_state]
new_transitions[(new_from, symbol)] = new_to
new_start = partition_map[self.start_state]
new_accept = {partition_map[s] for s in self.accept_states}
return DFA(new_states, self.alphabet, new_transitions, new_start, new_accept)
# Example: DFA for binary strings divisible by 3
def create_div_by_3_dfa() -> DFA:
states = {'q0', 'q1', 'q2'} # Remainder states
alphabet = {'0', '1'}
transitions = {
('q0', '0'): 'q0',
('q0', '1'): 'q1',
('q1', '0'): 'q2',
('q1', '1'): 'q0',
('q2', '0'): 'q1',
('q2', '1'): 'q2',
}
return DFA(states, alphabet, transitions, 'q0', {'q0'})
# Test
dfa = create_div_by_3_dfa()
test_strings = ['0', '1', '11', '110', '1001', '10101']
for s in test_strings:
print(f"{s} divisible by 3: {dfa.accepts(s)}")
# NFA to DFA Conversion (Subset Construction)
from typing import Set, Dict, List, Tuple, FrozenSet
from collections import defaultdict
class NFA:
def __init__(self, states: Set[str], alphabet: Set[str],
transitions: Dict[Tuple[str, Optional[str]], Set[str]],
start_state: str, accept_states: Set[str]):
self.states = states
self.alphabet = alphabet
self.transitions = transitions
self.start_state = start_state
self.accept_states = accept_states
def epsilon_closure(self, states: Set[str]) -> Set[str]:
"""Compute epsilon closure of a set of states."""
stack = list(states)
closure = set(states)
while stack:
state = stack.pop()
key = (state, None) # epsilon transition
if key in self.transitions:
for next_state in self.transitions[key]:
if next_state not in closure:
closure.add(next_state)
stack.append(next_state)
return closure
def move(self, states: Set[str], symbol: str) -> Set[str]:
"""Find states reachable from states via symbol."""
result = set()
for state in states:
key = (state, symbol)
if key in self.transitions:
result.update(self.transitions[key])
return result
class DFACreator:
def __init__(self, nfa: NFA):
self.nfa = nfa
self.dfa_states: List[FrozenSet[str]] = []
self.dfa_transitions: Dict[Tuple[int, str], int] = {}
self.accept_states: Set[int] = set()
self._build_dfa()
def _build_dfa(self):
"""Build DFA using subset construction."""
start_closure = self.nfa.epsilon_closure({self.nfa.start_state})
start_set = frozenset(start_closure)
self.dfa_states.append(start_set)
if any(s in self.nfa.accept_states for s in start_set):
self.accept_states.add(0)
queue = [start_set]
state_index = {start_set: 0}
while queue:
current_set = queue.pop(0)
current_idx = state_index[current_set]
for symbol in self.nfa.alphabet:
# Move on symbol, then epsilon closure
move_result = self.nfa.move(current_set, symbol)
new_set = self.nfa.epsilon_closure(move_result)
new_frozen = frozenset(new_set)
if new_frozen not in state_index:
new_idx = len(self.dfa_states)
self.dfa_states.append(new_frozen)
state_index[new_frozen] = new_idx
if any(s in self.nfa.accept_states for s in new_set):
self.accept_states.add(new_idx)
queue.append(new_frozen)
self.dfa_transitions[(current_idx, symbol)] = state_index[new_frozen]
def get_dfa(self):
"""Return equivalent DFA."""
from .automata import DFA # Assume DFA class defined above
states = {f"q{i}" for i in range(len(self.dfa_states))}
start = "q0"
accept = {f"q{i}" for i in self.accept_states}
transitions = {}
for (state_idx, symbol), next_idx in self.dfa_transitions.items():
transitions[(f"q{state_idx}", symbol)] = f"q{next_idx}"
return DFA(states, self.nfa.alphabet, transitions, start, accept)
# Example: NFA for (a|b)*abb
def create_nfa_for_pattern() -> NFA:
states = {'q0', 'q1', 'q2', 'q3'}
alphabet = {'a', 'b'}
transitions = {
('q0', 'a'): {'q0', 'q1'},
('q0', 'b'): {'q0', 'q2'},
('q1', 'b'): {'q3'},
('q2', 'b'): {'q3'},
('q3', None): set(), # Accepting state
}
return NFA(states, alphabet, transitions, 'q0', {'q3'})
nfa = create_nfa_for_pattern()
creator = DFACreator(nfa)
dfa = creator.get_dfa()
# Test
test_strings = ['abb', 'aabb', 'ababb', 'babb', 'aaabbb']
for s in test_strings:
print(f"'{s}': {dfa.accepts(s)}")
# Pushdown Automaton for Balanced Parentheses
from typing import Set, Dict, Tuple, Optional, List
from dataclasses import dataclass
class PDA:
"""
Pushdown Automaton for balanced parentheses: (^n ^n)
"""
def __init__(self):
self.states = {'q0', 'q1', 'q2'}
self.alphabet = {'(', ')'}
self.stack_alphabet = {'(', '$'}
# Transition: (state, input, stack_top) -> [(new_state, stack_push)]
self.transitions: Dict[Tuple[str, Optional[str], str], List[Tuple[str, str]]] = {}
# q0: Start state
self._add_transition('q0', '(', '$', 'q0', '($')
self._add_transition('q0', '(', '(', 'q0', '((')
# q0 -> q1 on ')', pop matching '('
self._add_transition('q0', ')', '(', 'q1', '')
# q1: Still processing
self._add_transition('q1', ')', '(', 'q1', '')
self._add_transition('q1', None, '$', 'q2', '') # Accept on empty stack
def _add_transition(self, state: str, symbol: Optional[str],
stack_top: str, new_state: str, stack_push: str):
key = (state, symbol, stack_top)
self.transitions.setdefault(key, []).append((new_state, stack_push))
def accepts(self, input_string: str) -> bool:
"""Check if PDA accepts the string."""
# Each configuration: (state, stack_tuple)
from collections import deque
configurations = deque()
configurations.append(('q0', ('$',))) # Start with bottom of stack
while configurations:
state, stack = configurations.pop()
# If input consumed and in accept state
if not input_string and state == 'q2':
return True
symbol = input_string[0] if input_string else None
# Try transitions
key = (state, symbol, stack[0] if stack else None)
if key in self.transitions:
for new_state, stack_push in self.transitions[key]:
new_stack = stack_push[::-1] + stack[1:] if stack_push else stack[1:]
new_input = input_string[1:] if symbol else input_string
configurations.append((new_state, new_stack))
# Try epsilon transitions
key_eps = (state, None, stack[0] if stack else None)
if key_eps in self.transitions:
for new_state, stack_push in self.transitions[key_eps]:
new_stack = stack_push[::-1] + stack[1:] if stack_push else stack[1:]
configurations.append((new_state, new_stack))
return False
# Test PDA
pda = PDA()
test_strings = ['', '()', '(())', '(()())', '((( )))', ')(', '(()']
for s in test_strings:
print(f"'{s}': {pda.accepts(s)}")
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