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## Two-sided program unification and its application to program refactoring.

#### Abstract

It is generally accepted that to unify a pair of substitutions θ_1 and θ_2 means to find out a pair of substitutions η' and η'' such that the compositions θ_1 η' and θ_2 η'' are the same. Actually, unification is the problem of solving linear equations of the form θ_1 X=θ_2 Y in the semigroup of substitutions. But some other linear equations on substitutions may be also viewed as less common variants of unification problem. In this paper we introduce a two-sided unification as the process of bringing a given substitution θ_1 to another given substitution θ_2 from both sides by giving a solution to an equation Xθ_1 Y=θ_2. Two-sided unification finds some applications in software refactoring as a means for extracting instances of library subroutines in arbitrary pieces of program code. In this paper we study the complexity of two-sided unification for substitutions and programs. In Section 1 we discuss the concept of unification on substitutions and programs, outline the recent results on program equivalence checking that involve the concept of unification and anti-unification, and show that the problem of library subroutine extraction may be viewed as the problem of two-sided unification for programs. In Section 2 we formally define the concept of two-sided unification on substitutions and show that the problem of two-unifiability is NP-complete (in contrast to P-completeness of one-sided unifiability problem). NP-hardness of two-sided unifiability problem is established in Section 4 by reducing to it the bounded tiling problem which is known to be NP-complete; the latter problem is formally defined in Section 3. In Section 5 we define two-sided unification of sequential programs in the first-order model of programs supplied with strong equivalence (logic&term equivalence) of programs and proved that this problem is NP-complete. This is the main result of the paper.

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#### Edition

Proceedings of the Institute for System Programming, vol. 26, issue 2, 2014, pp. 245-268.

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

DOI: 10.15514/ISPRAS-2014-26(2)-11

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