Logic, arithmetic, and automata

This paper is a summary of recent work in the application of mathematical logic to finite automata, and especially of mathematical logic beyond propositional calculus. To begin with a sketch of the history of the matter, let us recall that application of the "algebra of logic", i.e., elementary Boolean algebra, to the analysis of switching circuits was first suggested by Ehrenfest [A]. Nothing came of Ehrenfest's remark for many years, and it seems to have remained wholly unknown outside of Russia. Yanovskaya [G] says that details of the suggested application were worked out by Shestakoff in 1934-35. However, Shestakoff's candidate's dissertation, embodying the material, was presented to the University of Moscow in 1938, and the earliest publications by Shesta-koff are [D] and [E] in 1941. Meanwhile the same idea had occurred independently to Nakasima and Hanzawa [B] and Shannon [C]. For some time the development of the idea proceeded independently in Russia, in Japan, and in the United States, the three lines of development having had at first no influence on one another. This use of Boolean algebra is now widely familiar, and therefore requires no elaboration here. It is usually taken to be a Boolean algebra of cardinal number 2 that is used, although the character of the application would more naturally suggest propositional calculus. Use of the Boolean algebra and of propositional calculus are equivalent in a way that is well known. The choice of Boolean algebra is advantageous if algebraic methods and results are to be employed. But otherwise there is a certain artificiality in allowing only equations and inequalities to be asserted. And for further application of mathematical logic, the choice of propositional calculus provides a better basis. Mathematical logic beyond propositional calculus is first applied to autom-ata theory in the paper of McCulloch and Pitts [16], in which the context is biological. The authors are concerned with analyzing the behavior of a net of neurons and with the question of the existence of, and of finding, a neural net having some specified behavior. But their hypotheses about the behavior and the interaction of neurons are such that these questions become entirely similar to coiresponding questions about electronic digital computing circuits. The relevance of the ideas of McCulloch and Pitts to the theory of digital computing circuits was noticed by John von Neumann, and it was evidently this that led him to suggest application of mathematical …