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Unformatted text preview: 1 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15453 For next time: Read 2.1 & 2.2 2 MINIMIZING DFAs 3 IS THIS MINIMAL? 1 1 1 1 NO 4 IS THIS MINIMAL? 1 1 5 THEOREM For every regular language L, there exists a unique (up to relabeling of the states) minimal DFA M such that L = L (M) 6 NOT TRUE FOR NFAs 7 EXTENDING Given DFA M = (Q, , , q , F) extend to : Q * Q as follows: (q, ) = (q, ) = (q, w 1 w k+1 ) = ( (q, w 1 w k ) , w k+1 ) ^ ^ ^ ^ ^ A string w * distinguishes states q 1 from q 2 if (q 1 , w) F (q 2 , w) F ^ ^ q (q, ) 8 0,1 1 1 1 q q 1 q 2 q 3 distinguishes accept from nonaccept states 9 Fix M = (Q, , , q , F) and let p, q, r Q Definition: p is distinguishable from q iff there is a w * that distinguishes p from q p is indistinguishable from q iff p is not distinguishable from q 10 Fix M = (Q, , , q , F) and let p, q, r Q Define relation ~ : p ~ q iff p is indistinguishable from q p ~ q iff p is distinguishable from q / Proposition: ~ is an equivalence relation p ~ p (reflexive) p ~ q q ~ p (symmetric) p ~ q and q ~ r p ~ r (transitive) p q r w w Suppose not: 11 so ~ partitions the set of states of M into disjoint equivalence classes Fix M = (Q, , , q , F) and let p, q, r Q q Q q [q] = { p  p ~ q } Proposition: ~ is an equivalence relation 12 1 1 1 1 13 Algorithm MINIMIZE Input: DFA M Output: DFA M MIN such that: M M MIN M MIN has no inaccessible states M MIN is irreducible states of M MIN are pairwise distinguishable  Theorem: M MIN is the unique minimum 14 Idea: States of M MIN will be blocks of equivalent states of M Theorem: M MIN is the unique minimum Must show 2 things: Suppose M is a DFA equiv M, then 1. (# states in M) (# states in M MIN ) 2. If (# states in M) = (# states in M MIN ) then M is isomorphic to M MIN (i.e., same up to relabeling of states) 15 Idea: States of M MIN will be blocks of equivalent states of M 16...
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 Spring '09
 EdmundM.Clarke

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