l3 - Lecture 3: -NFAs and Regular Expressions David Dill...

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Unformatted text preview: Lecture 3: -NFAs and Regular Expressions David Dill Department of Computer Science 1 Outline -NFAs More operations on languages Regular expression syntax and semantics 2 -NFA Idea; Allow state changes that dont consume input symbols (silent moves). Choice of whether to take -transition is non-deterministic. q0 q1 1 q2 e 1 e (e in the drawing is ) The input 001 is accepted. The path followed is q q 2 q 1 q q 1 q 2 . 3 Definition of -NFA The only difference in the mathematical definition is in : : Q ( { } ) 2 Q 4 -closure Preliminary: the -closure of state q , ECLOSE ( q ) , is the set of all states reachable from q by silent moves. ECLOSE ( q ) = { q } ECLOSE ( q 1 ) = { q 1 ,q 2 } ECLOSE ( q 2 ) = { q 1 ,q 2 } 5 Definition of ECLOSE Def: S is the least set satisfying property P if whenever P ( S ) holds, S S . ECLOSE ( q ) is the least set satisfying: q ECLOSE ( q ) ( s, ) ECLOSE ( q ) for all s ECLOSE ( q ) Also, ECLOSE ( S ) = uniontext { ECLOSE ( q ) | q S } (Extend to sets this is overloading) 6 Acceptance by an -NFA We can define for an -NFA taking ECLOSE into account. Here is a recursive definition on the structure of strings: Base: w = : ( q, ) = ECLOSE ( q ) Induction: w = xa : ( q,xa ) = ECLOSE ( uniontext s ( q,x ) ( s,a )) (Same as for NFA, with...
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This note was uploaded on 01/12/2010 for the course CS 154 at Stanford.

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l3 - Lecture 3: -NFAs and Regular Expressions David Dill...

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