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What Important Results Have Been Obtained Relevant to A Mathematical Science of Computation?

In 1936 the notion of a computable function was clarified by Turing, and he showed the existence of universal computers that, with an appropriate program, could compute anything computed by any other computer. All our stored program computers, when provided with unlimited auxiliary storage, are universal in Turing's sense. In some subconscious sense even the sales departments of computer manufacturers are aware of this, and they do not advertise magic instructions that cannot be simulated on competitors machines, but only that their machines are faster, cheaper, have more memory, or are easier to program.

The second major result was the existence of classes of unsolvable problems. This keeps all but the most ignorant of us out of certain Quixotic enterprises such as trying to invent a debugging procedure that can infallibly tell if a program being examined will get into a loop.

Later in this paper we shall discuss the relevance of the results of mathematical logic on creative sets to the problem of whether it is possible for a machine to be as intelligent as a human. In my opinion it is very important to build a firmer bridge between logic and recursive function theory on the one side, and the practice of computation on the other.

Much of the work on the theory of finite automata has been motivated by the hope of applying it to computation. I think this hope is mostly in vain because the fact of finiteness is used to show that the automaton will eventually repeat a state. However, anyone who waits for an IBM 7090 to repeat a state, solely because it is a finite automaton, is in for a very long wait.

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