Combination of constraint solving techniques: An algebraic point of view

Abstract

In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to apply—namely that unification with so-called linear constant restrictions is decidable in the single theories—is equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras—called combinability—that clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipulations of terms (such as ordered rewriting, abstraction functions, etc.) With this proof method, the previous combination results for unification can easily be extended to the case of constraint solvers that also take relational constraints (such as ordering constraints) into account. Background information from universal algebra about free structures is given to clarify the algebraic meaning of our results.