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.