Towards a New Algebraic Foundation of Flowchart Scheme Theory

Abstract

We develope a formalism for the algebraic study of flowchart schemes and their behaviours, based on a new axiomatic looping operation, called feedback. This formalism is based on certain flownomial expressions. Such an expression is built up from two types of atomic schemes (i.e., elements in a double-ranked set X considered as unknown computation processes, and elements in a “theory” T considered as known computation processes) by using three operations: sum, composition, and feedback. Flownomial expressions are subject to certain rules of identification. The axiomatization of flowchart schemes is based on the fact that a flowchart scheme may be identified with a class of isomorphic flownomial expressions in normal form. The corresponding algebra for flowchart schemes is called biflow. This axiomatization is extended to certain types of behaviour. We present axiomatizations for accessible flowchart schemes, reduced flowchart schemes, minimal flowchart schemes with respect to the input behaviour, minimal flowchart schemes with respect to the input-output behaviour etc. Some results are new, others are simple translations in terms of feedback of previous results obtained by using Elgot’s iteration or Kleene’s repetition. The paper also contains some historical comments.