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Problem 19

Show that every infinite Turing-recognizable language has an infinite decidable subset.

Expert verified

We are given an infinite Turing-recognizable language L with Turing machine M that accepts L. To show that L has an infinite decidable subset S, we enumerate the strings in L and construct S by including each string in the sequence if M halts and accepts it. Then we construct a Turing machine M' that decides S by checking the existence of any input string w in S. M' halts and accepts on every string in S and halts and rejects on every string not in S, proving that S is an infinite decidable subset of L.

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Chapter 3

Let \(c_{1} x^{n}+c_{2} x^{n-1}+\cdots+c_{n} x+c_{n+1}\) be a polynomial with a root at \(x=x_{0}\). Let \(c_{\max }\) be the largest absolute value of a \(c_{i}\). Show that $$ \left|x_{0}\right|<(n+1) \frac{c_{\mathrm{max}}}{\left|c_{1}\right|} $$

Chapter 3

Examine the formal definition of a Turing machine to answer the following questions, and explain your reasoning. a. Can a Turing machine ever write the blank symbol \(\lrcorner\) on its tape? b. Can the tape alphabet \(\Gamma\) be the same as the input alphabet \(\Sigma\) ? c. Can a Turing machine's head ever be in the same location in two successive steps? d. Can a Turing machine contain just a single state?

Chapter 3

Show that the collection of Turing-recognizable languages is closed under the operation of \(A_{\text {a. union. }}\) d. intersection. b. concatenation. e. homomorphism. c. star.

Chapter 3

The language \(A\) is one of the two languages \(\\{0\\}\) or \(\\{1\\} .\) In either case, the language is finite and hence decidable. If you aren't able to determine which of these two languages is \(A\), you won't be able to describe the decider for \(A\). However, you can give two Turing machines, one of which is \(A\) 's decider.

Chapter 3

Show that a language is decidable iff some enumerator enumerates the language in the standard string order.

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