A Special Case of Schematic Syntactic Unification

TL;DR

This paper introduces semiloop unification, a special case of schematic unification, providing a sufficient condition for unifiability based on initial segment structure, with implications for recursive term constructions.

Contribution

It defines semiloop unification, analyzes its properties, and offers a sufficient condition for unifiability, advancing understanding of recursive unification problems.

Findings

Provided a sufficient condition for semiloop unification unifiability.

Linked semiloop unification to narrowing and recursive grammars.

Identified open questions regarding necessity and extension of conditions.

Abstract

We present a unification problem based on first-order syntactic unification which ask whether every problem in a schematically-defined sequence of unification problems is unifiable, so called loop unification. Alternatively, our problem may be formulated as a recursive procedure calling first-order syntactic unification on certain bindings occurring in the solved form resulting from unification. Loop unification is closely related to Narrowing as the schematic constructions can be seen as a rewrite rule applied during unification, and primal grammars, as we deal with recursive term constructions. However, loop unification relaxes the restrictions put on variables as fresh as well as used extra variables may be introduced by rewriting. In this work we consider an important special case, so called semiloop unification. We provide a sufficient condition for unifiability of the entire sequence based on the structure of a sufficiently long initial segment. It remains an open question whether this condition is also necessary for semiloop unification and how it may be extended to loop unification.