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Decidable Questions About Regular languages 1) Membership problem: “Given a specification of known type and a string w, is w in the language specified?” 2) Emptiness problem: “Given a specification of known type, does it specify the empty set?” 3) Finiteness problem: “Given a specification of known type, does it specify finite set?” 4) Equivalence problem: “Given two specifications of the same known type, do they specify the same language?” Algorithm: (Decision of membership problem for DFA) Input: A DFA M=(Q, , , q0, F) and w in *. Output: “YES” if w is in L(M), “NO’ otherwise. Method: If (q0, w) is in F, say “YES”; If not say “NO”. Algorithm: (Decision of emptiness problem for finite automata.) Input: A DFA M=(Q, , , q0, F). Output: “YES” if L(M) = , “NO’ otherwise. Method: Compute the set of states accessible from q0. {p |(q0, w) = p } If this contains NO final state, say “YES”; If not say “NO”. ^ Algorithm: (Decision of finiteness problem for finite automata.) Input: A DFA M=(Q, , , q0, F). Output: “YES” if L(M) = is a finite set, “NO’ otherwise. Method: Test if there are any cycles. Answer “YES” if there are no cycles, otherwise “NO”. Definition: let M = (Q, , , q0, F) be a DFA, and let q1 and q2 be two distinct states. We say that a string x in * distinguishes q1 from q2 if exactly one of (q1, x) and (q2, x) is in F. We say q1 and q2 are k-indistinguishable if and only if there is no x, with |x| k, which distinguishes q1 and q2. ^ ^ Definition: We say q1 and q2 are (q1 q2) indistinguishable if and only if they are k-indistinguishable for all k 0. q1 and q2 are equivalent states. 0 B 0 start 1 C 1 F 0 A 1 1 0 0 1 D E 1 0 0 G 1 Table filling algorithm (p157) Basis: If p is an accepting state and q is a nonaccepting state, then (p, q) is a distinguishable pair of states. Induction: Let p and q be states such that (p,a) = r and (q,a) = s for some input symbol a. If (r, s) is a known pair of distinguishable states, then (p,q) is also a pair of distinguishable states. Table of state inequivalences B C D E F G A B C D E F • Theorem 4.20: It two states are not distinguished by the table-filling algorithm, then the states are equivalent • Theorem4.24: If we create for each state q of a DFA a block consisting of q and all the states equivalent to q, then the different blocks of states form a partition of the set of states. • State minimization algorithm (p162) Algorithm: (Decision of equivalence problem for DFA) Input: Two DFA M1 = (Q1, 1, 1, q1, F1) and M2 = (Q2, 2, 2, q2, F2) such that Q1 Q2 = . Output: “YES” if L(M1) = L(M2), “NO” otherwise Method: Construct the FA M = (Q1 Q2 , 1 2, 1 2, q1, F1 F2). Determine if q1 q2. If so say “YES”; otherwise say “NO”. Observation: L(M1) = L(M2) if and only if (L(M1) L(M2)) (L(M1) L(M2)) = .