On finite automata with limited nondeterminism

Abstract. We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by $2^{n} - 2$ where $n$ is the number of states. If the finite automaton is over a one-letter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than $n^{2}$. In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACE-hard. Since the problem is already known to be in PSPACE, it is therefore PSPACE-complete.

[1]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[2]  Kosaburo Hashiguchi.  New upper bounds to the limitedness of distance automata , 2000, Theor. Comput. Sci..

[3]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[4]  Jonathan Goldstine,et al.  On the Relation between Ambiguity and Nondeterminism in Finite Automata , 1992, Inf. Comput..

[5]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[6]  Detlef Wotschke,et al.  Amounts of nondeterminism in finite automata , 1980, Acta Informatica.

[7]  Philippe Gohon.  Automates de Coût Borné Sur Un Alphabet A Une Lettre , 1985, RAIRO Theor. Informatics Appl..

[8]  A. R. Meyer,et al.  Economy of Description by Automata, Grammars, and Formal Systems , 1971, SWAT.

[9]  Imre Simon,et al.  On Semigroups of Matrices over the Tropical Semiring , 1994, RAIRO Theor. Informatics Appl..

[10]  Andreas Weber.  Finite-Valued Distance Automata , 1994, Theor. Comput. Sci..

[11]  Jonathan Goldstine,et al.  On Measuring Nondeterminism in Regular Languages , 1990, Inf. Comput..

[12]  Hing Leung,et al.  On the topological structure of a finitely generated semigroup of matrices , 1988 .

[13]  Hing Leung.  Limitedness Theorem on Finite Automata with Distance Functions: An Algebraic Proof , 1991, Theor. Comput. Sci..

[14]  Kosaburo Hashiguchi,et al.  Improved Limitedness Theorems on Finite Automata with Distance Functions , 1990, Theor. Comput. Sci..

[15]  FRANK R. MOORE,et al.  On the Bounds for State-Set Size in the Proofs of Equivalence Between Deterministic, Nondeterministic, and Two-Way Finite Automata , 1971, IEEE Transactions on Computers.

[16]  Kosaburo Hashiguchi,et al.  Limitedness Theorem on Finite Automata with Distance Functions , 1982, J. Comput. Syst. Sci..

[17]  Hing Leung.  On Finite Automata with Limited Nondeterminism , 1992, MFCS.

[18]  Imre Simon,et al.  Recognizable Sets with Multiplicities in the Tropical Semiring , 1988, MFCS.

[19]  Imre Simon.  The Nondeterministic Complexity of a Finite Automaton , 1987 .