Is the Church-Turing Thesis True? CAROL E. CLELAND Department of Philosophy and Institute of Cognitive Science, University of Colorado, Boulder, CO, U.S.A. Abstract. The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930's. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn't be computed by a Turing machine but could be computed by means of some other kind of effective procedure. Since that time, however, other interpretations of the thesis have appeared in the literature. In this paper I identify three domains of application which have been claimed for the thesis: (1) the number theoretic functions; (2) all functions; (3) mental and/or physical phenomena. Subsequently, I provide an analysis of our intuitive concept of a procedure which, unlike Turing's, is based upon ordinary, everyday procedures such as recipes, directions and methods; I call them "mundane procedures." I argue that mundane procedures can be said to be effective in the same sense in which Turing machine procedures can be said to be effective. I also argue that mundane procedures differ from Turing machine procedures in a fundamental way, viz., the former, but not the latter, generate causal processes. I apply my analysis to all three of the above mentioned interpretations of the Church-Turing thesis, arguing that the thesis is (i) clearly false under interpretation (3), (ii) false in at least some possible worlds (perhaps even in the actual world) under interpretation (2), and (iii) very much open to question under interpretation (1). Key words. Church-Turing thesis, Turing machine, effective procedure, causal process, analog process The Church-Turing thesis is founded upon two independent analyses of the concept of an effective procedure. The first was based upon the lambda-calculus, a logical system developed by Alonzo Church (1934) in the mid-1930's as a foundation for standard mathematics. Church argued that ,~-definability repre- sented an intuitively plausible notion of what it is for a number theoretic function (a function defined on the natural numbers) to be effectively computable ("calcul- able"). When it was subsequently demonstrated (Kleene, 1936) that ,~-definability was extensionally equivalent (vis,~-vis the computation of number theoretic functions) to the other well known mathematical notion of effective computabili- ty, viz., Herbrand-Godel general recursiveness, Church postulated (1965) that ,~-definability represented an ultimate limit to the mathematical possibilities for computing a number theoretic function. That is, he proposed that there weren't any number theoretic functions which were effectively computable but not h-definable. This proposal became known as "Church's thesis." A few years later Alan Turing (1965), unaware of Church's work, proposed yet another analysis of the concept of effective computability. Turing based his analysis on the concept of a very simple abstract mechanism now known as a Minds and Machines 3: 283-312, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.