Converting a Non-Deterministic Finite Automaton (NFA) to a Deterministic Finite Automaton (DFA) involves creating a new DFA where each state corresponds to a set of NFA states reachable under certain conditions. Below is a simple Python program to perform this conversion. I'll explain each line of code:
```python
from collections import defaultdict
def epsilon_closure(states, transitions, epsilon):
closure = set(states)
stack = list(states)
while stack:
state = stack.pop()
for next_state in transitions[state].get(epsilon, []):
if next_state not in closure:
closure.add(next_state)
stack.append(next_state)
return closure
def nfa_to_dfa(nfa_states, nfa_transitions, alphabet, start_state, nfa_final_states):
dfa_states = []
dfa_transitions = defaultdict(dict)
dfa_start_state = epsilon_closure({start_state}, nfa_transitions, 'epsilon')
dfa_states.append(dfa_start_state)
stack = [dfa_start_state]
while stack:
current_state = stack.pop()
for symbol in alphabet:
next_states = set()
for nfa_state in current_state:
next_states.update(nfa_transitions[nfa_state].get(symbol, []))
epsilon_closures = epsilon_closure(next_states, nfa_transitions, 'epsilon')
if epsilon_closures not in dfa_states:
dfa_states.append(epsilon_closures)
stack.append(epsilon_closures)
dfa_transitions[current_state][symbol] = epsilon_closures
dfa_final_states = [state for state in dfa_states if any(nfa_final in state for nfa_final in nfa_final_states)]
return dfa_states, dfa_transitions, dfa_start_state, dfa_final_states
# Example NFA
nfa_states = {'q0', 'q1', 'q2'}
nfa_alphabet = {'0', '1'}
nfa_transitions = {
'q0': {'epsilon': {'q1'}},
'q1': {'0': {'q1'}, 'epsilon': {'q2'}},
'q2': {'1': {'q2'}}
}
nfa_start_state = 'q0'
nfa_final_states = {'q2'}
# Convert NFA to DFA
dfa_states, dfa_transitions, dfa_start_state, dfa_final_states = nfa_to_dfa(
nfa_states, nfa_transitions, nfa_alphabet, nfa_start_state, nfa_final_states
)
# Print DFA states and transitions
print("DFA States:", dfa_states)
print("DFA Transitions:", dfa_transitions)
print("DFA Start State:", dfa_start_state)
print("DFA Final States:", dfa_final_states)
```
**Explanation of the Code**:
1. Import the `defaultdict` class to create dictionaries with default values for non-existent keys.
2. Define a function `epsilon_closure(states, transitions, epsilon)` that calculates the ε-closure of a set of states in the NFA. It uses a stack-based approach to explore states reachable via ε-transitions.
3. Define a function `nfa_to_dfa(nfa_states, nfa_transitions, alphabet, start_state, nfa_final_states)` that converts an NFA to a DFA. It initializes the DFA's start state as the ε-closure of the NFA's start state.
4. The function uses a stack to iterate through DFA states and construct transitions.
5. Iterate over each symbol in the alphabet. For each symbol, calculate the ε-closure of states reachable via symbol transitions from the current DFA state.
6. If the calculated ε-closure is not already a DFA state, add it to the list of DFA states and the stack for further processing.
7. Identify DFA final states by checking if any NFA final state is included in the calculated DFA state.
8. An example NFA is defined with states, alphabet, transitions, start state, and final states.
9. The `nfa_to_dfa` function is called with the example NFA information to convert the NFA to a DFA.
10. Finally, the resulting DFA states, transitions, start state, and final states are printed.
This program demonstrates the process of converting an NFA to a DFA using the ε-closure concept and subset construction. Please note that this is a simplified example and might require further refinement for more complex NFAs.
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