The Fixpoint Combinator in Combinatory Logic – A Step towards Autonomous Real-time Testing of Software?

Combinatory Logic is an elegant and powerful logical theory that is used in computer science as a theoretical model for computation. Its algebraic structure supports self-application and is Turing-complete. However, contrary to Lambda Calculus, it untangles the problem of substitution, because bound variables are eliminated by inserting specific terms called Combinators. It was introduced by Schönfinkel (1924) and Curry (1930). Combinatory Logic uses just one algebraic operation, namely combining two terms, yielding another valid term of Combinatory Logic. Terms in models of Combinatory Logic look like some sort of assembly language for mathematical logic. A Neural Algebra, modeling the way we think, constitutes an interesting model of Combinatory Logic. There are other models, also based on the Graph Model (Engeler 1981), such as software testing. This paper investigates what Combinatory Logic contributes to modern software testing. Keywords: combinatory logic, combinatory algebra, autonomous real-time testing, recursion, software testing, artificial intelligence