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# l4-handout - Outline Finish-NFAs Lecture 4 Regular...

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Lecture 4: Regular Expressions David Dill Department of Computer Science 1 Outline Finish ± -NFAs Introduction to Regular expressions More Operations on Languages Recursive Definition of Syntax and Semantics Converting Regular Expressions to ± -NFAs Converting NFAs to regular expressions. 2 Definition of ECLOSE Def: S is the least set satisfying property P if whenever P ( S 0 ) holds, S S 0 . ECLOSE ( q ) is the least set satisfying: q ECLOSE ( q ) δ ( s, ± ) ECLOSE ( q ) for all s ECLOSE ( q ) Also, ECLOSE ( S ) = S { ECLOSE ( q ) | q S } (“Extend to sets” – this is overloading) 3 Acceptance by an ± -NFA We can define ˆ δ for an ± -NFA taking ECLOSE into account: Base: ˆ δ ( q, ± ) = ECLOSE ( q ) Induction: ˆ δ ( q, xa ) = ECLOSE ( S s ˆ δ ( q,x ) δ ( s, a )) (Same as ˆ δ for NFA, with ECLOSE wrapped around state sets.) Acceptance (same as before). x L ( E ) if ˆ δ ( q 0 , x ) F 6 = Example: ˆ δ ( q 0 , 001) = { q 1 , q 2 } – accepted because q 2 F . 4

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Converting an ± -NFA to an DFA Theorem: For every ± -NFA E , there exists an DFA D such that L ( D ) = L ( E ) . The proof uses a modified subset construction, where every subset is ± -closed. Given E = ( Q, Σ , q 0 , δ E , F E ) , we can define D = (2 Q , Σ , ECLOSE ( q 0 ) , δ D , F D ) as follows: F D = { S Q | S F 6 = ∅} δ D ( S, a ) = ECLOSE ( S s S δ E ( s, a )) Note: δ D has no ± transitions! 5 ± -elimination example q0 q1 1 q2 0 0 e 1 e converts to the DFA: {q0} {q1,q2} 0,1 0,1 6 ± -elimination Proof (not presented in lecture) Lemma: If D is the DFA obtained from the ± - NF A N by the ± -elimination, then ˆ δ D ( ECLOSE ( q 0 ) , x ) = ˆ δ E ( q 0 , x ) for all x Σ * . Proof: By induction on strings: Basis: ˆ δ D ( ECLOSE ( q 0 ) , ± ) = ECLOSE ( q 0 ) Def of ˆ δ for DFA = ˆ δ E ( q 0 , ± ) def of ˆ δ for ± -NFA Induction: ˆ δ D ( ECLOSE ( q 0 ) , xa ) = δ D ( ˆ δ D ( ECLOSE ( q 0 ) , x ) , a ) Def of ˆ δ for DFA = δ D ( ˆ δ E ( q 0 , x ) , a ) Ind. Hyp.
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l4-handout - Outline Finish-NFAs Lecture 4 Regular...

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