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TOC-grand test-Q9


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+1 vote

Consider the following problems concerning finite automata with alphabet \small \{0,1\}. Which of the following are DECIDABLE ?

S1 : on input a DFA D, determine whether D accepts infinitely many palindromes.

S2 : on input a DFA D; determine whether D accepts a pair of distinct strings u and v such that u is a prefix of v.

(A) Only S1

(B) Only S2

(C) Both S1 and S2

(D) Neither S1 or S2

asked Sep 7, 2019 by gbeditor-2 (72,840 points)
reshown Sep 8, 2019 by gbeditor-2

1 Answer

0 votes
 
Best answer

S1 : DECIDABLE,

Rephrasing, we need to determine whether L(D) \cap P is infinite, where P is the context-free language of palindromes. Construct a PDA for L(D) \cap P by applying the Cartesian-product construction to a PDA for P and the given DFA D. Then, convert the resulting PDA to a grammar and use the algorithm from class to check whether the grammar accepts infinitely many strings.

 

S2 : DECIDABLE ,

Use any graph search algorithm to identify the set F' of accept states of D that are reachable from the start state. Output “yes” if and only if there exists a pair of state q' , q'' \in F (not necessarily distinct) with a path of nonzero length from $q' to q'')

Hence Option C

 

 

 

answered Sep 11, 2019 by (91,540 points)
Answer:

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