Algebraic Structures of Programs: First Steps to Algebraic Programming

Basic algebraic notions are introduced which can be used to describe and to transform program-data complexes for traditional languages, restricted computation, non-numeric computation and modeling. General algebraic program systems (GAPS) are free of assumptions what computing (programming) systems have as their particular constructs. First of all we argue why current technique is sometimes not adequate. Then we show how to interpret notions of abstract algebra (groupoid structure) as super-von Neumann computational structure and how to express many useful structures and notions in an algebra. Then GAPS are introduced and basic mathematical results are stated including the precise criterion when a given system of actions over programs can be added to given programming language. Various examples of GAPS are given. And at last we show possible primitives of ‘structured algebraic modeling’ (programming).