Halting problem of one binary Horn clause is undecidable

Abstract

This paper proposes a codification of the halting problem of any Turing machine in the form of only one right-linear binary Horn clause as follows: p(t) ← p(tt). where t (resp. tt) is any (resp. linear) term. Recursivity is well-known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.

Some direct consequences are presented here. For instance, there exists an explicitly constructible right-linear binary Horn clause for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the non-linear case is an unexpected extension.

The proof of the main result is based on the unpredictable iterations of periodically linear functions defined by J.H. Conway within number theory. Let us note that these new undecidability results are proved w.r.t. any type of resolution (bottom-up or top-down, depth-first or breadth-first, unification with or without occur-check).

This work has been partially supported by GRECO de Programmation of CNRS.