{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–17. Sign up to view the full content.

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

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

View Full Document Right Arrow Icon
MINIMIZING DFAs THURSDAY Jan 24
Image of page 2
IS THIS MINIMAL? 1 1 1 1 0 0 0 0 NO
Image of page 3

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

View Full Document Right Arrow Icon
IS THIS MINIMAL? 0 1 0 1
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)
Image of page 5

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

View Full Document Right Arrow Icon
NOT TRUE FOR NFAs 0 0 0 0
Image of page 6
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
Image of page 7

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

View Full Document Right Arrow Icon
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 ^ ^
Image of page 8
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 ^ ^
Image of page 9

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

View Full Document Right Arrow Icon
0 0,1 0 0 1 1 1 q 0 q 1 q 2 q 3 ε distinguishes accept from non-accept states
Image of page 10
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 ^ ^ ^
Image of page 11

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

View Full Document Right Arrow Icon
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 }
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 q 0 Q q [q] = { p | p ~ q }
Image of page 13

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

View Full Document Right Arrow Icon
1 1 1 1 0 0 0 0
Image of page 14
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
Image of page 15

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

View Full Document Right Arrow Icon
Idea:
Image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern