Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
A polynomial-time algorithm for the equivalence of probabilistic automata
SIAM Journal on Computing
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Quantum automata and quantum grammars
Theoretical Computer Science
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Characterizations of 1-Way Quantum Finite Automata
SIAM Journal on Computing
On the power of quantum finite state automata
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
1-way quantum finite automata: strengths, weaknesses and generalizations
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Introduction to probabilistic automata (Computer science and applied mathematics)
Introduction to probabilistic automata (Computer science and applied mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
A faster PSPACE algorithm for deciding the existential theory of the reals
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Determining the equivalence for one-way quantum finite automata
Theoretical Computer Science
Note: A note on quantum sequential machines
Theoretical Computer Science
Unbounded-error quantum computation with small space bounds
Information and Computation
Quantum Computation and Quantum Information: 10th Anniversary Edition
Quantum Computation and Quantum Information: 10th Anniversary Edition
Characterizations of one-way general quantum finite automata
Theoretical Computer Science
Hi-index | 0.00 |
Several types of automata, such as probabilistic and quantum automata, require to work with real and complex numbers. For such automata the acceptance of an input is quantified with a probability. There are plenty of results in the literature addressing the complexity of checking the equivalence of these automata, that is, checking whether two automata accept all inputs with the same probability. On the other hand, the critical problem of finding the minimal automata equivalent to a given one has been left open [C. Moore, J.P. Crutchfield, Quantum automata and quantum grammars, Theoret. Comput. Sci. 237 (2000) 275-306, see p. 304, Problem 5]. In this work, we reduce the minimization problem of probabilistic and quantum automata to finding a solution of a system of algebraic polynomial (in)equations. An EXPSPACE upper bound on the complexity of the minimization problem is derived by applying Renegar@?s algorithm. More specifically, we show that the state minimization of probabilistic automata, measure-once quantum automata, measure-many quantum automata, measure-once generalized quantum automata, and measure-many generalized quantum automata is decidable and in EXPSPACE. Finally, we also solve an open problem concerning minimal covering of stochastic sequential machines [A. Paz, Introduction to Probabilistic Automata, Academic Press, New York, 1971, p. 43].