Towards a unified view of finite automata and semi-Markov flowgraph models

Abstract

Two disparate areas of knowledge are interrelated in this article: (a) Theory of Finite Automata and (b) semi-Markov Flowgraphs. These two areas, which at first sight do not have anything in common, are shown to have similar structures and to be governed by similar rules. It is shown that the renewal argument applies in both areas, resulting in equations marking at certain points in ‘time’ system regeneration. The corresponding sets of equations look formally the same but hold in entirely different domains, (a) language equations and (b) equations for Laplace transforms of waiting time distributions. The equations thus obtained can be solved iteratively, as their solutions are Fixed Points. There are established theories addressing this type of problem in each field, (a) Knaster–Tarski Fixed Point Theorem and (b) Contraction Mapping Principle. Even though the mathematical structures in both areas are different ((a): partially ordered set; (b): complete metric space), the formal steps of arriving at the equations, their structure and solution techniques used are remarkably similar. The paper has two goals. The first is its cross-disciplinary contribution, identifying structural similarities in two distant areas. This structural result is hoped to be the first step towards finding a single theoretical framework encompassing both fields. Furthermore, the analogy observed here may in the future allow known techniques from one field to be ‘carried over’ to the other, thus generating new results. The paper's second contribution is educational. It is instructive to draw learners' attention to analogous structures in another field as it reinforces the learning experience.