A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems

This paper presents a polynomial-time algorithm for equivalence under certain semiring congruences. These congruences arise when studying the isomorphism of state spaces for a class of hybrid systems. The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of 7nite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and 7nite mathematics are used in the studyof PL sy stems, in conjunction with linear algebra techniques, all in the context of a “PL algebra” formalism. PL systems are of interest as controllers as well as identi7cation models. Basic questions for any class of systems are those of equivalence, and, in particular, whether state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexitywas potentially exponential; here it is shown to be solvable in polynomial time. c 2001 Elsevier Science B.V. All rights reserved.