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## Polynomial time algorithm for checking strong equivalence of program.

#### Abstract

Strong (logic&term) equivalence of programs is the weakest decidable equivalence relation which approximates the functional equivalence of programs. In this paper we develop a new variant of the algorithm for checking strong equivalence of programs. A distinguished feature of our algorithm is that it relies completely on the algebra of finite substitutions which includes the operations of composition and antiunification. The paper begins with a short introduction to the theory of first-order substitutions. In the next section the formal first-order model of sequential programs and strong equivalence are defined. We also describe a preliminary variant of equivalence checking algorithm. This algorithm deals with composition and antiunification operations on finite substitutions. Since the complexity of these operations depends on data structures that represent substitutions, we consider in Section 4 graph-theoretic representations of substitutions as well as upper and lower bounds on the complexity of basic operations on substitutions. To reduce the complexity of equivalence checking algorithm for sequential programs we introduce in Section 5 a new class of truncated substitutions and study the basic algebraic properties of composition and antiunification operations on truncated substitutions. In Section 6 we showed that both operations on truncated substitutions can be performed in time O(n^2) . Finally, in Section 7 using the properties of truncated substitutions we introduced an improved version of the equivalence checking algorithm and proved that its time complexity is O(n^6 ) on the size of programs to be checked.

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#### Edition

Proceedings of the Institute for System Programming, vol. 22, 2012, pp.435-455.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2012-22-23