Abstract
First-order program schemata represent one of the most simple models of sequential imperative programs intended for solving verification and optimization problems. We consider the decidable relation of logical-thermal equivalence on these schemata and the problem of their size minimization while preserving logical-thermal equivalence. We prove that this problem is decidable. Further we show that the first-order program schemata supplied with logical-thermal equivalence and finite-state deterministic transducers operating over substitutions are mutually translated into each other. This relationship makes it possible to adapt equivalence checking and minimization algorithms developed in one of these models of computation to the solution of the same problems for the other model of computation. In addition, on the basis of the discovered relationship, we describe a subclass of first-order program schemata such that minimization of the program schemata from this class can be performed in polynomial time by means of known techniques for minimization of finite-state transducers operating over semigroups. Finally, we demonstrate that in the general case the minimization problem for finite state transducers over semigroups may have several non-isomorphic solutions.
Highlights
We prove that this problem is decidable
Further we show that the first-order program schemata supplied with logical-thermal equivalence and finite state deterministic transducers operating over substitutions are mutually translated into each other
On the basis of the discovered relationship, we have found a subclass of firstorder program schemata such that their minimization can be performed in polynomial time by means of known techniques for minimization of finite state transducers operating over semigroups
Summary
Для произвольных заданных множеств переменных X, функциональных символов F и предикатных символов P будем использовать запись T erm(X, F) для обозначения множества термов, построенных из переменных и функциональных символов, а запись Atom(X, F, P) для обозначения множества атомарных формул (атомов), построенных из предикатных символов и термов. Для каждого терма или атома T обозначим записью V arT множество переменных, входящих в выражение T. Множество всех таких подстановок условимся обозначать записью Subst(X, Y, F ). Xn} конечное множество переменных и θ(xi) = ti для всех i, 1 ≤ i ≤ n, то такая подстановка называется конечной и определяется списком пар {x1/t1, . Композиция η ◦ θ подстановок θ ∈ Subst(X, Y, F ) и η ∈ Subst(Y, Z, F) это такая подстановка из множества Subst(X, Z, F) (ее традиционно обозначают записью θη), которая определяется равенством θη(x) = θ(x)η для каждой переменной x, x ∈ X. Множество подстановок Subst(X, X, F) с операцией композиции образует полугруппу, в которой нейтральным элементом служит тождественная подстановка ε = {x1/x1, . Квазиупорядоченное множество (Subst(X, X, F ), ) образует квазирешетку, наименьшим элементом которой является тождественная подстановка ε. Более подробные сведения об алгебре подстановок могут быть почерпнуты из работы [15]
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