Lecture4 - 15-453 FORMAL LANGUAGES, AUTOMATA AND...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
MINIMIZING DFAs THURSDAY Jan 24
Background image of page 2
IS THIS MINIMAL? 1 1 1 1 0 0 0 0 NO
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
IS THIS MINIMAL? 0 1 0 1
Background image of page 4
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)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
NOT TRUE FOR NFAs 0 0 0 0
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ^ ^
Background image of page 8
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 ^ ^
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
0 0,1 0 0 1 1 1 q0 q1 q2 q3 ε distinguishes accept from non-accept states
Background image of page 10
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 ^ ^ ^
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 }
Background image of page 12
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 }
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1 1 1 1 0 0 0 0
Background image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 43

Lecture4 - 15-453 FORMAL LANGUAGES, AUTOMATA AND...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online