c318f00f - CSc 318 Final Examination Saturday 16 December...

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CSc 318 Final Examination Saturday 16 December 2000 8 AM >>>>>>>>>>>SUGGESTED ANSWERS<<<<<<<<<<<<<<<<<<<<<<< 1. Consider the set of DFA’s which can be constructed over some alphabet Σ . Prove that equivalence of DFA’s is an equivalence relation on this set. I show that the relation is reflexive, symmetric, and transitive. First, a DFA is clearly equivalent to itself, because if M is the DFA, L(M)=L(M). Second, if M is equivalent to M’ then L(M)=L(M’)==>L(M’)=L(M)==> M’ is equivalent to M. Third, if M is equivalent to M’ and M’ equivalent to M” then L(M)=L(M’)=L(M”) ==> M is equivalent to M”. 2. All semester you have been asked to write out definitions I have given you. Now I ask you to create the definition of what is called a “lazy NFA.” All of the FA’s we have defined this semester “chew up” ε or single symbols of an alphabet at each move. A lazy NFA “chews up” strings, that is, moving from one state to another consumes a string. Provide a formal definition for a lazy NFA, a formal definition of acceptance of a string by a lazy NFA, and a formal definition of the language of a lazy NFA. A lazy NFA, M, is a 5-tuple, M=(Q, Σ , s, F, ), where Q is a finite set of states, Σ is an alphabet, s Q is the start state, F Q is a set of final states, and ∆⊆ Qx Σ *xQ is a transition relation. A string w is accepted by M if we may write w=u 1 u 2 … u n and find states a 1 a 2 … a n such that a n F, (s, a 1 , u 1 ), (u i , a i+1 , u i+1 ) ∈∆ for i=1, 2, …, n-1. The language of M is the set of strings accepted by M.
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