Conceptual confluence in 1936: Post and turing

Abstract

In 1936, Post and Turing independently proposed two models of computation that are virtually identical. Turing refers back to these models in his (The word problem in semi-groups with cancellation. Ann. Math. 52, 491–505) and calls them “the logical computing machines introduced by Post and the author”. The virtual identity is not to be viewed as a surprising coincidence, but rather as a natural consequence of the way in which Post and Turing conceived of the steps in mechanical procedures on finite strings. To support our view of the underlying conceptual confluence, we discuss the two 1936 papers, but explore also Post’s work in the 1920s and Turing’s paper (Solvable and unsolvable problems. Sci. News 31, 7–23). In addition, we consider their overlapping mathematical work on the word-problem for semigroups (with cancellation) in Post’s (Recursive unsolvability of a problem of Thue. J. Symb. Log. 12, 1–11) and Turing’s (The word problem in semi-groups with cancellation. Ann. Math. 52, 491–505). We argue that the unity of their approach is of deep significance for the theory of computability.