Jean Gallier
Paliath Narendran
Abstract
In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (first-order) languages with equality. A decidable version of E-unification called rigid E-unification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid E-unification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of E-unification.
Problem Given E→ ={Ei| 1≤i≤n} a family of n finite sets of equations and S={〈ui,vi〉 |1≤i≤n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e., holding the variables in θ(Ei) “rigid”), θ(ui), and θ(vi) are provably equal from θ(Ei) for i=1,...,n?
Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i=1,...,n?
A substitution θ solving the above problem is called a rigid E→-unifier of S, and a pair 〈E→,S〉 such that S has some rigid E→-unifier is called an equational premating. It is shown that deciding whether a pair 〈 E→,S〉is an equational premating is an NP-complete problem.
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Index Terms
- Theorem proving using equational matings and rigid E-unification
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