A Theory of Computation Based on Quantum Logic

The (meta)logic underlying classical theory of computation is Boolean (twovalued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics about 70 years ago. It is currently understood as a logic whose truth values are taken from an orthomodular lattice. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. Finite automata and pushdown automata are two classes of the simplest mathematical models of computation. The present Chapter is a systematic exposition of automata theory based on quantum logic. We introduce the notions of orthomodular lattice-valued (quantum) finite and pushdown automaton. The classes of languages accepted by them are defined. Various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis, including equivalence between finite automata and regular expressions (the Kleene theorem) and equivalence between pushdown automata and context-free grammars. It is found that the universal validity of many important properties (for example, the Kleene theorem) of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties do not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between classical automata theory and automata theory based on quantum logic. ∗This work was partly supported by the National Foundation of Natural Sciences of China (Grant No: 60321002, 60496321) and the Key Grant Project of Chinese Ministry of Education (Grant No: 10403)