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Ullman (TOC) Edition 3 Exercise 9.2 Question 1 (Page No. 390)
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Show that the halting problem, the set of $(M,w)$ pairs such that $M$ halts (with or without accepting) when given input $w$ is $RE$ but not recursive.$ ($See the box on "The Halting Problem" in Section $9.2.4)$
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Ullman (TOC) Edition 3 Exercise 9.2 Question 2 (Page No. 390 - 391)
In the box "Why 'Recursive'?" in Section $9.2.1$ we suggested that there was a notion of "recursive function" that competed with the Turing machine as a model for what can be computed. In this exercise, we shall explore ... : Evaluate $A(2,1).$ What function of $x$ is $A(x,2)$? Evaluate $A(4,3)$.
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Ullman (TOC) Edition 3 Exercise 9.1 Question 2 (Page No. 382)
Write one of the possible codes for the Turing machine of Fig.$8.9.$
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Ullman (TOC) Edition 3 Exercise 9.5 Question 1 (Page No. 418)
Let $L$ be the set of (codes for) context-free grammars $G$ such that $L(G)$ contains at least one palindrome. Show that $L$ is undecidable. Hint: Reduce PCP to $L$ by constructing, from each instance of PCP a grammar whose language contains a palindrome if and only if the PCP instance has a solution.
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Ullman (TOC) Edition 3 Exercise 9.4 Question 1 (Page No. 412)
Tell whether each of the following instances of $PCP$ has a solution. Each is presented as two lists $A$ and $B$, and the $i^{th}$ strings on the two lists correspond for each $i = 1,2,\cdot\cdot\cdot\cdot$ $A=(01,001,10); \ B = (011,10,00).$ $A=(01,001,10); \ B = (011,01,00).$ $A=(ab,a,bc,c); \ B = (bc,ab,ca,a).$
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