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# What Are The Axioms And Rules of Inference of A Mathematical Science of Computation?

Ideally we would like a mathematical theory in which every true statement about procedures would have a proof, and preferably a proof that is easy to find, not too long, and easy to check. In 1931, Gödel proved a result, one of whose immediate consequences is that there is no complete mathematical theory of computation. Given any mathematical theory of computation there are true statements expressible in it which do not have proofs. Nevertheless, we can hope for a theory which is adequate for practical purposes, like proving that compilers work; the unprovable statements tend to be of a rather sweeping character, such as that the system itself is consistent.

It is almost possible to take over one of the systems of elementary number theory such as that given in Mostowski's book "Sentences Undecidable in Formalized Arithmetic" since the content of a theory of computation is quite similar. Unfortunately, this and similar systems were designed to make it easy to prove meta-theorems about the system, rather than to prove theorems in the system. As a result, the integers are given such a special role that the proofs of quite easy statements about simple procedures would be extremely long.

Therefore it is necessary to construct a new, though similar, theory in which neither the integers nor any other domain, (e.g. strings of symbols) are given a special role. Some partial results in this direction are described in this paper. Namely, an integer-free formalism for describing computations has been developed and shown to be adequate in the cases where it can be compared with other formalisms. Some methods of proof have been developed, but there is still a gap when it comes to methods of proving that a procedure terminates. The theory also requires extension in order to treat the properties of data spaces.

Next: What Important Results Have Up: Towards a Mathematical Science Previous: What Kinds of Facts

John McCarthy
Tue May 14 13:32:03 PDT 1996