Automata theory based on quantum logic: some characterizations

Abstract

Automata theory based on quantum logic (abbr. l -valued automata theory) may be viewed as a logical approach of quantum computation. In this paper, we characterize some fundamental properties of l -valued automata theory, and discover that some properties of the truth-value lattices of the underlying logic are equivalent to certain properties of automata. More specifically (i) the transition relations of l -valued automata are extended to describe the transitions enabled by strings of input symbols, and particularly, these extensions depend on the distributivity of the truth-value lattices (Proposition 3.1); (ii) some properties of the l -valued successor and source operators and l -valued subautomata are demonstrated to be equivalent to a property of the truth-value lattices which is exactly equivalent to the distributive law (Proposition 4.3 and Corollary 4.4). This is a new characterization of Boolean algebras in the framework of l -valued automata theory; (iii) we verify that the intersection of two l -valued subautomata is still an l -valued subautomaton if and only if the multiplication (&) is distributive over the union in the truth-value lattices (Proposition 4.5), which is strictly weaker than the usual distributivity; (iv) we show that some topological characterizations in terms of the l -valued successor and source operators also rely on the distributivity of truth-value lattices (Theorem 5.6). Finally, we address some related topics for further study.