Lecture4

# Lecture4 - 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  Given DFA M = (Q, Σ, , q0, F) extend  to  : Q  Σ* → Q as follows: (q, ε) = (q, ) = (q, 1 …k+1 ) = ( (q, 1 …k ) , k+1 ) ^ ^ ^ ^ ^ String w  Σ* distinguishes states q1 and q2 iff (q1, w)  F (q2, w)  F ^ ^ q (q, ) ^ Note: (q0, w)  F M accepts w

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EXTENDING  Given DFA M = (Q, Σ, , q0, F) extend  to  : Q  Σ* → Q as follows: (q, ε) = (q, ) = (q, 1 …k+1 ) = ( (q, 1 …k ) , k+1 ) ^ ^ ^ ^ ^ String w  Σ* distinguishes states q1 and q2 iff q (q, ) ^ Note: (q0, w)  F M accepts w exactly ONE of (q1, w), (q2, w) is a final state ^ ^
Fix M = (Q, Σ, , q0, 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*, (p, w)  F  (q, w)  F ^ ^

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0 0,1 0 0 1 1 1 q0 q1 q2 q3 ε distinguishes accept from non-accept states
Fix M = (Q, Σ, , q0, 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: (p, w)  F  (q, w)  F  (r, w)  F ^ ^ ^

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Fix M = (Q, Σ, , q0, F) and let p, q, r  Q Proposition: ~ is an equivalence relation so ~ partitions the set of states of M into disjoint equivalence classes q0 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 q0 Q q [q] = { p | p ~ q }

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1 1 1 1 0 0 0 0
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Lecture4 - 15-453 FORMAL LANGUAGES AUTOMATA AND...

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