practice_exam_ch4.pdf - Theory of Computation Questions and...

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Theory of Computation: Questions and Solutions Chapter 4: Decidability A decider is a Turing machine that always halts. To show that a language A is decidable is to show that there is a decider M that decides it. In other words, given a string w , by simulating M on input w : M must accept w , if w A , and M must reject w , if w 6∈ A . To show that a Turing machine is a decider is to show that it always halts. In other words, every step of the Turing machine will take a finite amount of time (no chance for infinite loop). Theorem 4.1 Consider the problem of determining whether a DFA B accepts input string w . Convert this problem into a membership problem by converting it into a language and show that the language is decidable. Solution : The problem can be converted into a membership problem as follows: A DFA = {h B, w i | B is a DFA that accepts input string w } To show that A DFA is decidable, we need to show that there exists a Turing machine M that decides it. In other words, M must accept h B, w i if h B, w i ∈ A DFA and M must reject h B, w i if h B, w i 6∈ A DFA . Note that h B, w i ∈ A DFA if an only if B is a DFA that accepts input string w . Given a DFA B and a string w , we can check whether B accepts w by simply run B on input w until the last symbol of the string w has been process. If the current state after processing the last symbol of the input string w is an accept state, B accepts w . Otherwise, B rejects w . With this idea, the Turning machine M that decides A DFA can be constructed as follows: M =“On input h B, w i , where B is a DFA and w is a string: 1. Simulate B on input w 2. If the simulation ends in an accept state, accept . If it ends in a nonaccepting state, reject .” A DFA always halts after processing the last symbol of an input string. Simulating a DFA on an input will take a finite amount of time since an input string is a finite sequence of symbols. Checking whether a state is in a set of accept states takes a finite amount of time. Therefore, the Turing machine M is a decider. Next, we need to show that h B, w i ∈ A DFA iff M accepts h B, w i . Assume that h B, w i ∈ A DFA . Since h B, w i ∈ A DFA , by the definition of the language A DFA , B is a DFA and B accepts w . Since B accepts w , by simulating B on input w in step 1, B willl accept w which causes TM M to accept the input h B, w i .
CS 1502 — Formal Method in Computer Science Page 1
Theory of Computation: Questions and Solutions

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