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# Lecture4x - 15-453 FORMAL LANGUAGES AUTOMATA AND...

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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453

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MINIMIZING DFAs THURSDAY Jan 24
IS THIS MINIMAL? 1 1 1 1 0 0 0 0 NO

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IS THIS MINIMAL? 0 1 0 1
THEOREM For every regular language L, there exists a UNIQUE (up to re-labeling of the states) minimal DFA M such that L = L(M)

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NOT TRUE FOR NFAs 0 0 0 0
EXTENDING δ Given DFA M = (Q, Σ, δ , q 0 , F) extend δ to δ : Q × Σ* → Q as follows: δ (q, ε) = δ (q, σ ) = δ (q, σ 1 σ k+1 ) = δ ( δ (q, σ 1 σ k ) , σ k+1 ) ^ ^ ^ ^ ^ String w Σ* distinguishes states q 1 and q 2 iff δ (q 1 , w) F δ (q 2 , w) F ^ ^ q δ (q, σ ) ^ Note: δ (q 0 , w) F M accepts w

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EXTENDING δ Given DFA M = (Q, Σ, δ , q 0 , F) extend δ to δ : Q × Σ* → Q as follows: δ (q, ε) = δ (q, σ ) = δ (q, σ 1 σ k+1 ) = δ ( δ (q, σ 1 σ k ) , σ k+1 ) ^ ^ ^ ^ ^ String w Σ* distinguishes states q 1 and q 2 iff q δ (q, σ ) ^ Note: δ (q 0 , w) F M accepts w exactly ONE of δ (q 1 , w), δ (q 2 , w) is a final state ^ ^
Fix M = (Q, Σ, δ , q 0 , F) and let p, q Q DEFINITION: p is distinguishable from q iff there is a w Σ* that distinguishes p and q p is indistinguishable from q iff p is not distinguishable from q iff for all w Σ*, δ (p, w) F δ (q, w) F ^ ^

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0 0,1 0 0 1 1 1 q 0 q 1 q 2 q 3 ε distinguishes accept from non-accept states
Fix M = (Q, Σ, δ , q 0 , 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) Proof (of transitivity): for all w, we have: δ (p, w) F δ (q, w) F δ (r, w) F ^ ^ ^

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Fix M = (Q, Σ, δ , q 0 , F) and let p, q, r Q Proposition: ~ is an equivalence relation so ~ partitions the set of states of M into disjoint equivalence classes q 0 Q q [q] = { p | p ~ q }
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) so ~ partitions the set of states of M into disjoint equivalence classes q 0 Q q [q] = { p | p ~ q }

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1 1 1 1 0 0 0 0
Algorithm MINIMIZE Input: DFA M Output: DFA M MIN such that: M M MIN (that is, L(M) = L(M MIN )) M MIN has no inaccessible states M MIN is irreducible all states of M MIN are pairwise distinguishable || Theorem: M MIN is the unique minimum DFA

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