Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables

AbstractAbstract. We consider generalized first-order sentences over < using both ordinary and modular quantifiers. It is known that the languages definable by such sentences are exactly the regular languages whose syntactic monoids contain only solvable groups. We show that any sentence in this logic is equivalent to one using three variables only, and we prove that the languages expressible with two variables are those whose syntactic monoids belong to a particular pseudovariety of finite monoids, namely the wreath product of the pseudovariety DA (which corresponds to the languages definable by ordinary first-order two-variable sentences) with the pseudovariety of finite solvable groups. This generalizes earlier work of Thérien and Wilke on the expressive power of two-variable formulas in which only ordinary quantifiers are present. If all modular quantifiers in the sentence are of the same prime modulus, this provides an algorithm to decide if a regular language has such a two-variable definition.

[1]  John Rhodes,et al.  The kernel of monoid morphisms , 1989 .

[2]  Thomas Wilke,et al.  Classifying Discrete Temporal Properties , 1999, STACS.

[3]  Howard Straubing,et al.  Families of recognizable sets corresponding to certain varieties of finite monoids , 1979 .

[4]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

[5]  Howard Straubing,et al.  Locally trivial categories and unambiguous concatenation , 1988 .

[6]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[7]  Kousha Etessami,et al.  First-order logic with two variables and unary temporal logic , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[8]  Jean-Eric Pin.  BG = PG: A Success Story , 1995 .

[9]  M. Schützenberger,et al.  Sur Le Produit De Concatenation Non Ambigu , 1976 .

[10]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[11]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[12]  Danièle Beauquier,et al.  Factors of Words , 1989, ICALP.

[13]  Luis Ribes,et al.  The Pro-P Topology of a Free Group and Algorithmic Problems in Semigroups , 1994, Int. J. Algebra Comput..

[14]  Denis Thérien,et al.  Classification of Finite Monoids: The Language Approach , 1981, Theor. Comput. Sci..

[15]  Howard Straubing,et al.  When Can One Finite Monoid Simulate Another , 2000 .

[16]  Mark V. Sapir,et al.  Closed Subgroups in Pro-V Topologies and the Extension Problem for Inverse Automata , 2001, Int. J. Algebra Comput..

[17]  Bret Tilson,et al.  Categories as algebra: An essential ingredient in the theory of monoids , 1987 .

[18]  Raymond E. Miller,et al.  Varieties of Formal Languages , 1986 .

[19]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[20]  Thomas Wilke,et al.  Over words, two variables are as powerful as one quantifier alternation , 1998, STOC '98.

[21]  Berndt Farwer,et al.  ω-automata , 2002 .

[22]  Uwe Schöning.  Probabilistic Complexity Classes and Lowness , 1989, J. Comput. Syst. Sci..

[23]  Jorge Almeida.  A Syntactical Proof of Locality of da , 1996, Int. J. Algebra Comput..

[24]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[25]  Johan Anthory Willem Kamp,et al.  Tense logic and the theory of linear order , 1968 .

[26]  Howard Straubing,et al.  Regular Languages Defined with Generalized Quanifiers , 1995, Inf. Comput..

[27]  Benjamin Steinberg,et al.  Finite state automata: A geometric approach , 2001 .

[28]  Price E. Stiffler,et al.  Chapter 1. Extension of the fundamental theorem of finite semigroups , 1973 .