Extended canonical systems

Abstract

Let L be a (finite) alphabet containing K as a sub-alphabet; let (S) be a system, like a Post canonical system [1; 2], except that the production variables range over all strings in K (rather than all strings in L, as in Post). The symbols in L may, however, occur in the axioms and production statements of (S). Such a system we call an extended canonical system; more specifically an L-K system. We say that the system is in the alphabet L, but over the alphabet K. We have found such systems to be more wieldy than the Post systemns; fewer axioms are usually required, and the axioms are usually shorter.2 It is easy to show the equivalence of representability in an extended canonical system to representability in a canonical system. It is well known that if K contains only 1 symbol then not every recursively enumerable set of strings in K is representable in a canonical system in the alphabet K; only the recursive sets can be so represented. We raise the problem: if K contains only 1 symbol, is every r.e. set of strings in K representable in some extended canonical system over K? We answer this question affirmatively. To simplify our proof, somewhat, we shall appeal to Post's normal form theorem.3 We let K be an alphabet containing just one symbol; call this symbol 1. We shall identify a string of l's of length n with the positive integer n. Let A be a recursively enumerable set (of positive integers). Appealing to Post's normal form theorem, there is a normal canonical system (C) in the alphabet { 1, b } and a string a such that for every (positive) integer n, nfEA iff an is provable in (C). We let K2 be the alphabet { 1, b } and we let L be the 8-symbol alphabet { 1, b, N, C, To, P, Q, }. We shall construct an L-K system in which A is represented. Along the way, we will have to represent certain relations of numbers (strings in K). For any L-K system (S), any string ir in L, and any relation R(xi, * * *, x) of strings in K, the string r is said to represent R iff the following condition holds: