A Note On Square-free Sequences and Anti-unification Type

TL;DR

This paper investigates anti-unification in term algebras with associative-idempotent functions, revealing that minimal complete solution sets may not exist and are structurally related to infinite square-free sequences.

Contribution

It demonstrates the nullarity of certain anti-unification problems and links their solution structure to infinite square-free sequences.

Findings

Complete solution sets contain infinite chains of comparable generalizations.

Generalization problems can lack minimal complete solutions (nullarity).

Solution structures are isomorphic to subsequences of infinite square-free sequences.

Abstract

Error: Peer-review process exposed an error in Theorem 1 that, unfourtunately, is not repairable. Idempotent semigroups are always finite. See Green and Rees [1952], Siekmann and Szab\'o [1981] for details Anti-unification is a fundamental operation used for inductive inference. It is abstractly defined as a process deriving from a set of symbolic expressions a new symbolic expression possessing certain commonalities shared between its members. We consider anti-unification over term algebras where some function symbols are interpreted as associative-idempotent $(f (x, f (y, z)) = f (f (x, y), z)$ and $f (x, x) = x$, respectively) and show that there exists generalization problems for which a minimal complete set of solutions does not exist (Nullary), that is every complete set must contain comparable elements with respect to the generality relation. In contrast to earlier techniques for showing the nullarity of a generalization problem, we exploit combinatorial properties of complete sets of solutions to show that comparable elements are not avoidable. We show that every complete set of solutions contains an infinite chain of comparable generalizations whose structure is isomorphic to a subsequence of an infinite square-free sequence over three symbols.