An Introduction to the Theory of Computation by Eitan Gurari

By Eitan Gurari

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Configurations of Programs An execution of a program on a given input is a discrete process in which the input is consumed, an output is generated, the variables change their values, and the program traverses its instructions. Each stage in the process depends on the outcome of the previous stage, but not on the history of the stages. The outcome of each stage is a configuration of the program that indicates the instruction being reached, the values stored in the variables, the portion of the input left to be read, and the output that has been generated so far.

Emptiness problem for grammars Domain: { G | G is a grammar }. Question: Does the given grammar define an empty language? Ambiguity problem for grammars Domain: { G | G is a grammar }. Question: Does the given grammar G have two or more different derivation graphs for some string in L(G)? Equivalence problem for grammars, Domain: { (G1, G2) | G1 and G2 are grammars }. Question: Does the given pair of grammars generate the same language? 5 Reducibility among Problems A common approach in solving problems is to transform them to different problems, solve the new ones, and derive the solutions for the original problems from those for the new ones.

Question: Does the given program have two or more accepting computations that define the same output for some input? Single-valuedness problem for programs Domain: Set of all programs. Question: Does the given program define a function, that is, does the given program for each input have at most one output? Equivalence problem for programs Domain: { (P1, P2) | P1 and P2 are programs }. Question: Does the given pair (P1, P2) of programs define the same relation, that is, does R(P1) = R(P2)? 8 inputs.

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