Unification Procedure for Terms with Sequence Variables and Sequence Functions

We study term equations with sequence variables and sequence function symbols. A sequence variable can be instantiated by any finite sequence of terms, including the empty sequence. A sequence function abbreviates a finite sequence of functions all having the same argument lists. An instance of such a function is IntegerDivision(x,y) that abbreviates the sequence Quotient(x, y),Remainder(x, y). Bringing sequence functions in the language naturally allows Skolemization over sequence variables: Let x, y be individual variables, x be a sequence variable, and p be a flexible arity predicate symbol. Then ∀x∀y∃x.p(x, y, x) Skolemizes to ∀x∀y.p(x, y, f(x, y)), where f is a binary Skolem sequence function symbol. Another example, ∀y∃x.p(y, x), where y is a sequence variable, after Skolemization introduces a flexible arity sequence function symbol g: ∀y.p(y, g(y)). In this paper we introduce a unification procedure for solving equations in the free theory with individual and sequence variables, and individual and sequence function symbols. Function symbols can have either fixed or flexible arity. Unification in such a theory is decidable. The procedure enumerates an almost minimal complete set of solutions, and terminates if the set is finite. This work is an extension and refinement of our previous results [12]. We implemented the procedure (without the decision algorithm) in Mathematica [25] on the base of a rule-based programming system ρLog [15]. Equation solving with sequence variables plays an important role in various applications in automated reasoning, artificial intelligence, and programming. Historically, probably the first attempt to implement unification with sequence variables (without sequence functions) was made in the system MVL [7]. It