# Where p s aa a aa bb b ab bc c ac ? a ba ba other

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), where P = { S aA A aA | bB B aB | bC C aC | λ } a + ba * ba * Other examples: Examples 3.29, 3.2.10, 3.2.11, 3.2.12

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6 6.2 Expression Graphs Claim(2) . Every language accepted by a FSA is regular . Defn. 6.2.1 An expression graph is a labeled digraph in which arcs are labeled by regular expressions . Paths in expression graphs generate regular expressions Algorithm 6.2.2 Construction of regular expression from a FSA. Produce an arbitrary expression graph by repeatedly removing nodes from the state diagram. Produce the language of the FSA by the union of the sets of strings whose processing successfully terminates in one of the accepting states Case 1.2.1 (i) (ii) w j , i w i , k k i j j k w j , i w i , k w i , i k j k j i w j , i w i , k w j , i ( w i , i )* w i , k
7 6.2 Expression Graphs Note. If there are multiple final states in the given FSA M , then for each accepting state F , we produce an expression for the strings accepted by F . The language accepted by M is the union of the regular expressions.

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8 6.2 Expression Graphs 6.2.2 j k i i j k cc cc
9 6.2 Expression Graphs i j = k There is no such i that is neither the start state nor a final state

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10 6.3 Regular Grammars and FA Given a regular grammar G , there exists a NFA M such that L ( G ) = L ( M ) Theorem. 6.3.1 Let G = ( V , , P , S ) be a regular grammar. Define the NFA M = ( Q , , δ , S , F ) as follows: a) V { Z }, where Z V , if P contains a rule A a V O.W. b) δ ( A , a) = B whenever A aB P δ ( A , a) = Z whenever A a P c) { A | A λ P } { Z }, if Z Q { A | A λ P } O.W. Then L ( M ) = L ( G ) Q = F =
11 6.3 Regular Grammars and FA Example 6.3.1 Given G = ( V , , P , S ), where V = { S , B }, = { a , b }, and P = { S aS | bB | a , B bB | λ }, the corresponding NFA M = ( Q , , δ , S , F ), where Q = { S , B , Z }, = { a , b }, δ ( S , a ) = S, δ ( S , b ) = B , δ ( S , a ) = Z , δ ( B , b ) = B, F = { B , Z } Theorem 6.3.2 Given an NFA M , there exists a regular grammar G . 220d . L ( M ) = L ( G ). (i) V = Q (ii) The transition δ ( A , a ) = B yields the rule A aB in G (iii) For each accepting state C , create the rule C λ in G Example 6.3.4 P : S bB , S aA ; A aS , A bC ; B bS , B aC , B λ ; C bA , C aB ;

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12 6.4 Regular Languages A language over an alphabet is regular if it is i) a regular set/expression over ii) accepted by DFA / NFA / NFA- λ iii) generated by a regular grammar Regular languages are closed under , , *, ¯ , and
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• Winter '12
• DennisNg
• Formal language, Regular expression, Regular language, Nondeterministic finite state machine

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