notes05 Equivalence of DFA and NFA

notes05 Equivalence of DFA and NFA - CS 373: Theory of...

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CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1
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NFA vs. DFA Expressive Power of NFAs and DFAs Is there a language that is recognized by a DFA but not by any NFAs? No! Is there a language that is recognized by an NFA but not by any DFAs? No! Main Theorem Theorem 1. A language L is regular if and only if there is an NFA N such that L ( N ) = L . In other words: For any DFA D , there is an NFA N such that L ( N ) = L ( D ) , and for any NFA N , there is a DFA D such that L ( D ) = L ( N ) . 2 NFAs for Regular Languages Converting DFAs to NFAs Proposition 2. For any DFA D , there is an NFA N such that L ( N ) = L ( D ) . Proof. Is a DFA an NFA? Essentially yes! Syntactically, not quite. The formal definition of DFA has δ DFA : Q × Σ Q whereas δ NFA : Q × ∪ { ± } ) → P ( Q ). For DFA D = ( Q, Σ D ,q 0 ,F ), define an “equivalent” NFA N = ( Q, Σ N ,q 0 ,F ) that has the exact same set of states, initial state and final states. Only difference is in the transition function. δ N ( q,a ) = { δ D ( q,a ) } for a Σ and δ N ( q,± ) = for all q Q . 3 NFAs recognize Regular Languages 3.1 Simulating an NFA Simulating an NFA on Your Computer NFA Acceptance Problem Given an NFA N and an input string w , does N accept w ? How do we write a computer program to solve the NFA Acceptance problem?
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This note was uploaded on 10/04/2011 for the course CS 373 taught by Professor Viswanathan,m during the Fall '08 term at University of Illinois, Urbana Champaign.

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notes05 Equivalence of DFA and NFA - CS 373: Theory of...

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