An Introduction to the Theory of Computation by Eitan Gurari PDF

By Eitan Gurari

ISBN-10: 0716781824

ISBN-13: 9780716781820

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E1 + E2) and (E1 - E2) are polynomial expressions of degree max(d1, d2). ii. (E1 * E2) is a polynomial expression of degree d1 + d2. A polynomial is called a Diophantine polynomial if it can be is represented by a polynomial expression, and its variables are over the natural numbers. Hilbert's tenth problem is the problem of determining for any given Diophantine polynomial Q(x1, . . , xn) with variables x1, . . , xn whether or not there exist 1, . . , n such that Q( 1, . . , n) = 0. A LOOP program is a program that consists only of instructions of the form x 0, x y, x x + 1, and do x end.

Membership problem for programs Domain: { (P, x, y) | P is a program, and x and y are sequences of values from the domain of the variables of P }. Question: Is (x, y) in the relation R(P) for the given instance (P, x, y), that is, does P have an accepting computation on x with output y? Emptiness problem for programs Domain: Set of all programs. Question: Does the given program accept an empty language, that is, does the program accept no input? html Set of all programs. Question: Does the given program have two or more accepting computations that define the same output for some input?

3 Show that Hilbert's tenth problem is reducible to the nonemptiness problem for programs. html Show that the problem of determining the existence of solutions over the natural numbers for systems of Diophantine equations of the following form is reducible to Hilbert's tenth problem. Each Qi(x1, . . , xn) is assumed to be a Diophantine polynomial. 5 For each of the following cases show that K1 is reducible to K2. a. K1 is the emptiness problem for Type 0 grammars, and K2 is the equivalence problem for Type 0 grammars.

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An Introduction to the Theory of Computation by Eitan Gurari


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