Multi-valued Logics, Automata, Simulations, and Games

Abstract

Multi-valued systems are systems in which the atomic propositions and the transitions are not Boolean and can take values from some set. Latticed systems, in which the elements in the set are partially ordered, are useful in abstraction, query checking, and reasoning about multiple view-points. For example, abstraction involves systems in which an atomic proposition can take values from {true, unknown, false}, and these values can be partially ordered according to a “being more true” order (true ≥ unknown ≥ false) or according to a “being more informative” order (true ≥ unknown and false ≥ unknown). For Boolean temporal logics, researchers have developed a rich and beautiful theory that is based on viewing formulas as descriptors of languages of infinite words or trees. This includes a relation between temporal-logic formulas and automata on infinite objects, a theory of simulation relation between systems, a theory of two-player games, and a study of the relations among these notions. The theory is very useful in practice, and is the key to almost all algorithms and tools we see today in verification.