A theory of computation based on unsharp quantum logic: Finite state automata and pushdown automata

Abstract

When generalizing the projection-valued measurements to the positive operator-valued measurements, the notion of the quantum logic generalizes from the sharp quantum logic to the unsharp quantum logic. It is known that: (i) the distributive law is one of the main differences between the sharp quantum logic and the boolean logic, and the block or the center of the sharp quantum structures are boolean algebras; (ii) the unsharp quantum logic does not satisfy the non-contradiction law, which forces the block or the center of unsharp quantum structures to be multiple valued algebras, rather than boolean algebras. Multiple valued algebras, as special quantum structures, are the algebraic semantics of multiple valued logic. Interestingly, we recently discovered that the difference between some unsharp quantum structures and multiple valued algebras is also some kind of distributive law.Choosing an orthomodular lattice (an algebraic model of a sharp quantum logic) to be the truth valued lattice, Ying et al. have systematically developed automata theory based on sharp quantum logic. In this paper, choosing a lattice ordered quantum multiple valued algebra E (an extended lattice ordered effect algebra E, respectively) to be the truth valued lattice, we also systematically develop an automata theory based on unsharp quantum logic. We introduce E-valued finite-state automata and E-valued pushdown automata in the framework of unsharp quantum logic. We study the classes of languages accepted by these automata and re-examine their various properties in the framework of unsharp quantum logic. The study includes the equivalence between finite-state automata and regular expressions, as well as the equivalence between pushdown automata and context-free grammars. It is also demonstrated that the universal validity of some important properties (such as some closure properties of languages and Kleene theorem etc.) depends heavily on the aforementioned distributive law. More precisely, when the underlying model degenerates into an MV algebra, then all the counterparts of properties in classical automata are valid. This is the main difference between automata theory based on unsharp quantum logic and automata theory based on sharp quantum logic.