Abstract
AbstractThe theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.
Highlights
An important topic in unification theory (Siekmann, 1989; Baader and Siekmann, 1994; Baader and Snyder, 2001) is investigating the complexity of deciding solvability of unification problems w.r.t. an equational theory E, that is, for a fixed equational theory E one considers as input a set of equations between terms of the form = {s1 =? t1, . . . , sk =? tk}, and asks whether there exists a substitution σ such that σ =E σ holds for i = 1, . . . , k
4.3 General ACUIG-unification General ACUIG-unification problems differ from the ones we have considered until now in that the terms used in may contain free function symbols, that is, function symbols not occurring in the identities of ACUIG
We have considered approximate ACUI-unification w.r.t. three different ways of measuring the degree to which the equations of the unification problem are violated by a given substitution that is not a unifier
Summary
The valuation that sets p(a, x1), p(a, x2), p(a, x3), p(c, x3), and good(3) to false and all other variables to true satisfies all but the last clause This corresponds to the substitution σ with σ (x1) = σ (x2) = a and σ (x3) = a + c, which satisfies the first two equations and violates the third one. We will show that there is a 1–1-relationship between valuations satisfying certain clauses and substitutions satisfying the corresponding equations It is worth noting, that in Max-HSAT the number of satisfied clauses is maximized, whereas in MinVEq-ACUI the number of violated equations is minimized. The NP-hardness result holds even for ACUIunification problems with only one free constant
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