ALGEBRAIC ISOMORPHISM INVARIANTS FOR GRAPHS OF AUTOMATA

Abstract

This chapter highlights algebraic isomorphism invariants for the graphs of automata. Finite automata, which are mathematical models of discrete-time finite-state systems, can be represented by a finite sequence of directed graphs called transition graphs. These graphs are studied from an algebraic point of view in terms of a natural representation of the graphs by linear transformations. The classical invariants of linear transformation similarity become invariants of graphical isomorphism. The principal objective of the investigation is to determine the extent to which these algebraic invariants specify the structure (isomorphism class) of an arbitrary transition graph. Based on the solution obtained for connected graphs, it is shown that the elementary divisors can be formulated in terms of the depths of the points of a forest. By combining these results and solving for the graphical invariants in terms of the elementary divisors, one is able to determine, for an arbitrary transition graph, the precise extent of the structural information conveyed by these algebraic isomorphism invariants.