Minimization of DFAs

Transcription

1 CS 72: Computability and Complexity Minimization of DFAs Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: L.von Ahn, L. Blum, M. Blum, A. Sinclair

2 What is Minimization? Minimized DFA for language L = DFA with fewest states that recognizes L Also called minimal DFA S. A. Seshia 2

3 Why is Minimization Important? DFAs are how computers manipulate regular languages (expressions) DFA size determines space/time efficiency S. A. Seshia 3

4 IS THIS MINIMAL? NO S. A. Seshia 4

5 HOW ABOUT THIS? YES S. A. Seshia 5

6 Equivalent DFAs S. A. Seshia 6

7 Main Result of this Lecture For every regular language L, there exists a unique, minimal DFA that recognizes L uniqueness up to re-labeling of states S. A. Seshia 7

8 Words States Let DFA M = (Q, Σ, δ, q, F) Each word w in Σ * corresponds to a unique state in Q The ending state of M on w Given x, y Σ * x M y iff M ends in the same state on both x and y M is an equivalence relation (why?) How many equivalence classes are there? S. A. Seshia 8

9 Example: Is M? M? S. A. Seshia 9

10 Indistinguishable Words/Strings Let DFA M = (Q, Σ, δ, q, F) recognize L Given x, y Σ * x L y (x and y are indistinguishable) iff z Σ *, xz L iff yz L Compare with x M y iff M ends in the same state on both x and y S. A. Seshia

11 Example: What are indistinguishable words? S. A. Seshia

12 L and M Let DFA M = (Q, Σ, δ, q, F) recognize L Given x, y Σ * x L y (x and y are indistinguishable) iff z Σ *, xz L iff yz L x M y iff M ends in the same state on both x and y True or False: If x M y then x L y If x L y then x M y TRUE FALSE S. A. Seshia 2

13 Indistinguishable Words Let DFA M = (Q, Σ, δ, q, F) recognize L Given x, y Σ * x L y (x and y are indistinguishable) iff z Σ *, xz L iff yz L x M y iff M ends in the same state on both x and y Which has more equivalence classes -- M or L? S. A. Seshia 3

14 Myhill-Nerode Theorem (a version) The relation L defines a DFA M L for language L where the states of M L correspond to the equivalence classes of L M L is the unique, minimal DFA for L (up to isomorphism) S. A. Seshia 4

15 Proof of Myhill-Nerode Thm. S. A. Seshia 5

16 Next: Algorithm for DFA Minimization S. A. Seshia 6

17 Indistinguishable States Idea: Merge indistinguishable states Recall: States of DFA M map - to equivalence classes of M Each equivalence class of M is in some equivalence class of L States p and q are indistinguishable iff their corresponding M equivalence classes are in the same class of L We write p q p ~ q p and q are distinguishable S. A. Seshia 7

18 The Algorithm We Want Input: DFA M Output: DFA M L such that: M M L M L has no unreachable states M L is irreducible states of M L are pairwise distinguishable Theorem: M L is the unique minimum S. A. Seshia 8

19 DFA Minimization Algo.: Idea States of M L are equivalence classes of L Equivalence classes of L can be obtained by merging states of M Our algorithm works in reverse: Start by assuming all states as being merged together Identify pairs of distinguishable states Repeat until no new distinguishable state-pairs identified S. A. Seshia 9

20 TABLE-FILLING ALGORITHM Input: DFA M = (Q, Σ, δ, q, F) Output: Table: { (p,q) p,q Q and p ~ q } / States of M L = { [q] q Q } q q Base Case: p accepts and q rejects p ~ q / q i d d d d d q n d q q q i q n d Recursion: p σ p q σ ~/ q p ~ q / S. A. Seshia 2

21 q Example q q 2 d d d q 3 d d d q q q 2 q 3, q q q 2 q 3 S. A. Seshia 2

22 q q q q d q 3 q 2 q 2 d q 3 d d q q q 2 q 3 S. A. Seshia 22

23 q q q 4, q, q 5 q q 2 q 3 q 3 q 4 q 5 Do Try this at Home! q q q 3 q 4 q 5 S. A. Seshia 23

24 Correctness of Algorithm.If algorithm marks (p, q) as d, then p ~ q 2.If p ~ q, then algorithm marks (p, q) as d Proving () is easy. Use induction on the step at which (p, q) was marked d. S. A. Seshia 24

25 Part (2): If p ~ q, then the algorithm marks (p, q) as d Proof (by contradiction): Suppose p ~ q, but the algorithm does not mark (p, q) as d Since p ~ q there exists w such that: δ(p, ^ w) F and δ(q, ^ w) F Of all such bad pairs (p, q), let p, q be a pair with the smallest w S. A. Seshia 25

26 If p ~ q, then the algorithm marks (p, q) as d Proof (by contradiction): Suppose p ~ q, but the algorithm does not mark (p, q) as d δ(p, ^ w) F and δ(q, ^ w) F Of all such bad pairs (p, q), let p, q be a pair with the smallest w w = σw, where σ Σ (w is not ε, why?) Let p = δ(p,σ) and q = δ(q,σ) Then (p, q ) is also a bad pair Contradiction! (why?) S. A. Seshia 26

27 Complexity of Algorithm For DFA M, let Number of states of M be n Size of the input alphabet Σ be k Initialization of table: O(n 2 ) Rest of the algorithm: O(k n 2 ) S. A. Seshia 27

28 Minimal NFA is NOT Unique S. A. Seshia 28

29 Next Steps Read Sipser 2. in preparation for next lecture S. A. Seshia 29