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(r1,...,rn). The decidability of the satisfiability problem of programs consisting of such a rule, a fact and a goal - called - 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.
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