Theory of Computation Lecture 5

Equivalence of NFAs and DFAs

• Deterministic Finite Automata (DFA) and Nondeterministic Finite Automata (NFA) recognize the same class of languages.
• NFAs may seem more powerful, yet they do not recognize more languages than DFAs.
• Describing an NFA for a language can often be easier than for a DFA.
• Two machines are considered equivalent if they recognize the same language.

Theorems

• Theorem 1.39: Every NFA has an equivalent DFA.
• Corollary 1.40: A language is regular if and only if some NFA recognizes it.

Steps to Convert NFA to DFA

• Step 1: Initialize Q' to the empty set (ϕ).
• Step 2: Add the start state (q0) of the NFA to Q' and determine transitions.
• Step 3: For each input symbol, find possible sets of states. Add new sets to Q' if they are not already present.
• Step 4: In the DFA, all states containing final states of the NFA will also be final states.

Example Conversions

• Construct a transition table from the given NFA to organize state transitions.
• Form δ' transitions for newly identified sets of states based on input symbols.

Example 1 NFA to DFA Conversion

• Initial state processing:
  • δ'([q0], 0) = [q0]
  • δ'([q0], 1) = [q1]
• State q1 yields a new state:
  • δ'([q1], 0) = [q1, q2]
  • δ'([q1], 1) = [q1]
• Final state processing involves states that include q2.
• Transition table and diagram reflect all derived transitions accurately, eliminating unreachable states.

Example 2: NFA with Null Conversion

• ε-closure method identifies the states based on epsilon transitions:
  • ε-closure(q0) = {q0, q1, q2}
• Determine transitions based on ε-closure outputs leading to combinations of states.
• Final state determination includes every state set containing final states (like q2).

Important Notes for NFA to DFA Conversion

• Transition tables must be drawn accurately to represent all possible states and their transitions.
• If the transition from the start state over an input symbol is null, assign it to a dead state in the DFA.
• Final states in the DFA are any state that contains a final state from the NFA.
• Continue adding states and transitions until no new states remain.

Explore the equivalence between nondeterministic finite automata (NFAs) and deterministic finite automata (DFAs) in this comprehensive quiz. Understand how both types of automata recognize the same class of languages and the surprising implications of this equivalence. Join us to deepen your knowledge in the Theory of Computation.