The class of primitive recursive functions may be defined as the closure of certain initial functions, namely the zero, successor and identity functions, under two schemes, namely composition (sometimes called “substitution”) and recursion. For a detailed definition the reader is referred to any standard work, for instance p. 219 of [2], by Kleene.

This paper resolves 3 problems left open by R. M. Robinson in [3].We recall that the set of primitive recursive functions is the closure under (i) substitution (or “composition”), and (ii) recursion, of the set P consisting of the zero, successor and projection functions (see any textbook, for instance p. 120 of [2]).

Our method, broadly speaking, is to take a decision problem of the required degree of unsolvability associated with a semi-Thue system, and translate from ssmi-Thue system to propositional calculus. An outline of the strategy of the proof is given in §4.

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