Satisfiability of the smallest binary program

Abstract

Recursivity is well known to be a crucial and important concept in programming theory. The simplest scheme of recursion in the context of logic programming is the binary Horn clause P(l_{1},...,l_{n})leftarrow P(r_{1},...,r_{n}). The decidability of the satisfiability problem of programs consisting of such a rule, a fact and a goal -- called smallest binary program -- has been a goal of research for some time. In this paper the undecidability of the smallest binary program is shown by a simple reduction of the Post Correspondence Problem.