Theses for computation and recursion on concrete and abstract structures

Abstract

The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the Church-Turing Thesis (abbreviated CT here) must be restricted to concrete structures whose objects are finite symbolic configurations of one sort or another. Informal and principled arguments for CT on concrete structures are reviewed. Next, it is argued that proposed generalizations of notions of computation to abstract structures must be considered instead under the general notion of algorithm. However, there is no clear general thesis in sight for that comparable to CT, though there are certain wide classes of algorithms for which plausible theses can be stated. The article concludes with a proposed thesis RT for recursion on abstract structures.