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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 δ Given DFA M = (Q, Σ, δ , q , F) extend δ to δ : Q × Σ* → Q as follows: δ (q, ε) = δ (q, σ ) = δ (q, w 1 …w k+1 ) = δ ( δ (q, w 1 …w k ) , w k+1 ) ^ ^ ^ ^ ^ A string w ∈ Σ* distinguishes states q 1 from q 2 if δ (q 1 , w) ∈ F ⇔ δ (q 2 , w) ∉ F ^ ^ q δ (q, σ ) 8 0,1 1 1 1 q q 1 q 2 q 3 ε distinguishes accept from nonaccept states 9 Fix M = (Q, Σ, δ , q , F) and let p, q, r ∈ Q Definition: p is distinguishable from q iff there is a w ∈ Σ* that distinguishes p from q p is indistinguishable from q iff p is not distinguishable from q 10 Fix M = (Q, Σ, δ , 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, Σ, δ , 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
 Equivalence relation, base case, Mmin

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