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State Minimization

State Minimization. Section 2.5 Fri, Oct 7, 2005. Minimizing a DFA. To minimize a DFA is to find an equivalent DFA that has the least possible number of states.

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State Minimization

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  1. State Minimization Section 2.5 Fri, Oct 7, 2005

  2. Minimizing a DFA • To minimize a DFA is to find an equivalent DFA that has the least possible number of states. • If the DFA is going to be used to write a program (e.g., a compiler) or to design hardware, then there may be a significant benefit in minimizing it before implementing it.

  3. Indistinguishable States • Let M be a DFA. Two states p and q are called indistinguishable if the DFA obtained by making p the initial state is equivalent to the DFA obtained by making q the initial state. • Write p ≡ q. • This is an equivalence relation on K. • Why?

  4. Distinguishing States • We say that a string wdistinguishes states p and q if • (p, w) M* (f, e), for some final state f, and • (q, w) M* (u, e) for some nonfinal state u, • or vice versa. • Therefore, two states are indistinguishable if there is no string that distinguishes them.

  5. Minimizing a DFA • The problem is to determine which states are distinguishable and which are indistinguishable. • To minimize a DFA, we want to identify its equivalence classes of indistinguishable states and replace them with single states.

  6. The Relation ≡n • Let p and q be states. Define the relation p ≡nq to mean that p and q are not distinguishable by any string w with |w| n. • That is, no sequence of up to n moves will distinguish p and q. • This, too, is an equivalence relation. • Clearly, p ≡0q if and only if p, qF or p, q F.

  7. Properties of ≡n • The following are obvious: • For all n 0, if p ≡ q, then p ≡nq. • For all n 0, if p ≡nq then p ≡ q. • For all n 1, if p ≡nq, then p ≡n – 1q. • But it is not the case that if p ≡n – 1q, then p ≡nq.

  8. Another Property of ≡n • Theorem: For any two states p and q and any n 1, p ≡nq if and only if • p ≡n – 1q, and • for all a Σ, δ(p, a) ≡n – 1 δ(q, a). • This theorem characterizes the relation ≡n entirely in terms of the relation ≡n – 1 and the function . • It leads to an algorithm to determine which states are indistinguishable.

  9. The Minimization Algorithm • Initialize the equivalence classes for ≡0 as F and K – F. • For n = 0, 1, 2, …, compute the equivalence classes of ≡n from ≡n – 1 and , as described in the previous theorem. • Repeat this until, for some n, the classes of ≡n are the same as the classes of ≡n – 1. • This must eventually happen. • Why?

  10. The Minimization Algorithm • At that point, the equivalence classes of ≡nare the equivalence classes of ≡. • Why? • Replace each equivalence class of states under ≡ by a single state.

  11. a b b a b 2 3 4 5 6 a a b a a b 1 a a a b b b 7 8 9 10 b a, b Example of Minimization • Minimize the following DFA.

  12. Example: The ≡0 Partition • The relation ≡0 partitions K into two classes. {1, 2, 3, 4, 5, 6, 7, 8, 9}, {10}

  13. Example: The ≡1 Partition • Apply the “a-transitions” to each class: {2, 6, 8, 5, 10, 10, 2, 5, 4}, {10} • Apply the “b-transitions” to each class: {7, 3, 9, 3, 7, 3, 9, 7, 10}, {10}

  14. Example: The ≡1 Partition • The a-transitions indicate that 5 and 6 are in a different class than 1, 2, 3, 4, 7, 8, and 9. • The b-transitions indicate that 9 is in a different class than 1, 2, 3, 4, 5, 6, 7, and 8. • The partition of ≡1 is {1, 2, 3, 4, 7, 8}, {5, 6}, {9}, {10}

  15. Example: The ≡2 Partition • Apply the a-transitions to these classes: {2, 6, 8, 5, 2, 5}, {10, 10}, {4}, {10} • Apply the b-transitions to these classes: {7, 3, 9, 3, 9, 7}, {7, 3}, {10}, {10} • The a-transitions split {1, 2, 3, 4, 7, 8} into {1, 3, 7} and {2, 4, 8}. • The b-transitions split {1, 2, 3, 4, 7, 8} into {1, 2, 4, 8} and {3, 7}.

  16. Example: The ≡2 Partition • Combining these, we get the ≡2partition {1}, {3, 7}, {2, 4, 8}, {5, 6}, {9}, {10}. • Apply the a-transitions to these classes: {2}, {8, 2}, {6, 5, 5}, {10, 10}, {4}, {10}. • Apply the b-transitions to these classes: {7}, {9, 9}, {3, 3, 7}, {7, 3}, {10}, {10}. • This produces no new classes.

  17. Example: The ≡3 Partition • Therefore, these are the classes of ≡, namely {1}, {2, 4, 8}, {3, 7}, {5, 6}, {9}, {10}.

  18. a b b a b 2 3 4 5 6 a a b a a b 1 a a a b b b 7 8 9 10 b a, b Example of Minimization • Combine states 2, 4, and 8 into one state.

  19. a b b a b 2 3 4 5 6 a a b a a b 1 a a a b b b 7 8 9 10 b a, b Example of Minimization • Combine states 2, 4, and 8 into one state.

  20. a b b a b 2 3 4 5 6 a a b a a b 1 a a a b b b 7 8 9 10 b a, b Example of Minimization • Combine states 3 and 7 into one state.

  21. a b b a b 2 3 4 5 6 a a b a a b 1 a a a b b b 7 8 9 10 b a, b Example of Minimization • Combine states 5 and 6 into one state.

  22. a 248 56 a, b a a b 1 b a 10 a b b 37 9 b Example of Minimization • Replace each of these classes with a single class.

  23. Example of Minimization • Now we can see that the language of this DFA is {w *w contains aaa or bbb}.

  24. Example • Find a minimal DFA that accepts the language {w* | w contains 010 and 101}.

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