Distance automata having large finite distance or finite ambiguity

A distance automaton is a (nondeterministic finite) automaton which is equipped with a nonnegative cost function on its transitions. The distance of a word recognized by such a machine quantifies the expenses associated with the recognition of this word. The distance of a distance automaton is the maximal distance of a word recognized by this machine or is infinite, depending on whether or not a maximum exists. We present distance automata havingn states and distance 2n − 2. As a by-product we obtain regular languages having exponential finite order. Given a finitely ambiguous distance automaton withn states, we show that either its distance is at most 3n − 1, or the growth of the distance in this machine is linear in the input length. The infinite distance problem for these distance automata is NP-hard and solvable in polynomial space. The infinite-order problem for regular languages is PSPACE-complete.

[1]  Imre Simon,et al.  Limited subsets of a free monoid , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

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

[3]  Kosaburo Hashiguchi,et al.  Algorithms for Determining Relative Star height and Star Height , 1988, IFIP Congress.

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

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

[6]  Andreas Weber.  Distance Automata Having Large Finite Distance or Finite Ambiguity , 1990, MFCS.

[7]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

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

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

[10]  K. Hashiguchi,et al.  Representation Theorems on Regular Languages , 1983, J. Comput. Syst. Sci..

[11]  Hing Leung.  An algebraic method for solving decision problems in finite automata theory , 1987 .

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

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

[14]  Arto Salomaa.  Jewels of formal language theory , 1981 .

[15]  Hing Leung.  On finite automata with limited nondeterminism , 1998, Acta Informatica.

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

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

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

[19]  Helmut Seidl,et al.  On the Degree of Ambiguity of Finite Automata , 1991, Theor. Comput. Sci..

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

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

[22]  Kosaburo Hashiguchi,et al.  A Decision Procedure for the Order of Regular Events , 1979, Theor. Comput. Sci..

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

[24]  Helmut Seidl,et al.  On finitely generated monoids of matrices with entries in N , 1991, RAIRO Theor. Informatics Appl..

[25]  Toshihide Ibaraki.  Finite Automata Having Cost Functions: Nondeterministic Models , 1978, Inf. Control..