A new approach for fuzzy topology (III)
Fuzzy Sets and Systems
Fuzzifying topology based on complete residuated lattice-valued logic (I)
Fuzzy Sets and Systems
A logic for approximate reasoning
Journal of Symbolic Logic
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Orthologic and quantum logic: models and computational elements
Journal of the ACM (JACM)
Quantum automata and quantum grammars
Theoretical Computer Science
Automata, Languages, and Machines
Automata, Languages, and Machines
Automata and Computability
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
On the power of quantum finite state automata
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Automata theory based on unsharp quantum logic†
Mathematical Structures in Computer Science
Myhill--Nerode type theory for fuzzy languages and automata
Fuzzy Sets and Systems
Determinization of weighted finite automata over strong bimonoids
Information Sciences: an International Journal
DLT'10 Proceedings of the 14th international conference on Developments in language theory
Finite automata theory with membership values in lattices
Information Sciences: an International Journal
An improved algorithm for determinization of weighted and fuzzy automata
Information Sciences: an International Journal
Weighted automata and multi-valued logics over arbitrary bounded lattices
Theoretical Computer Science
A theory of computation based on unsharp quantum logic: Finite state automata and pushdown automata
Theoretical Computer Science
Weighted nested word automata and logics over strong bimonoids
CIAA'12 Proceedings of the 17th international conference on Implementation and Application of Automata
Hi-index | 5.23 |
The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than 60 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. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. We introduce the notion of orthomodular lattice-valued (quantum) automaton. Various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis. We define the class of regular languages accepted by orthomodular lattice-valued automata. The acceptance abilities of orthomodular lattice-valued nondeterministic automata and their various modifications (such as deterministic automata and automata with ε-moves) are compared. The closure properties of orthomodular lattice-valued regular languages are derived. The Kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic. We also present a pumping lemma for orthomodular lattice-valued regular languages. It is found that the universal validity of many properties (for example, the Kleene theorem, the equivalence of deterministic and nondeterministic automata) of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does 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 the classical theory of computation and the computation theory based on quantum logic.