Elementary Algorithms and Their Implementations

In the sequence of articles [3, 4, 5, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a set-theoretic “definition” of algorithms, much like the “definition” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a probability space. The aim is to provide a traditional foundation for the theory of algorithms, a development of it within axiomatic set theory in the same way as analysis and probability theory are rigorously developed within set theory on the basis of the set theoretic modeling of their basic notions. A characteristic feature of this approach is the adoption of a very abstract notion of algorithm that takes recursion as a primitive operation and is so wide as to admit “non-implementable” algorithms: implementations are special, restricted algorithms (which include the familiar models of computation, e.g., Turing and random access machines), and an algorithm is implementable if it is reducible to an implementation.