Bridges between Algebraic Automata Theory and Complexity Theory

The algebraic theory of finite automata has been one of the most successful tools to study and classify regular languages. These very same tools can in fact be used to understand more powerful models of computation and we discuss here the impact that semigroup theory can have in computational complexity.

[1]  Michael E. Saks,et al.  Non-Deterministic Communication Complexity with Few Witnesses , 1994, J. Comput. Syst. Sci..

[2]  Richard J. Lipton,et al.  Unbounded Fan-In Circuits and Associative Functions , 1985, J. Comput. Syst. Sci..

[3]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[4]  Pascal Weil,et al.  Polynomial closure and unambiguous product , 1995, Theory of Computing Systems.

[5]  Howard Straubing,et al.  Non-Uniform Automata Over Groups , 1990, Inf. Comput..

[6]  Denis Thérien,et al.  Computational complexity questions related to finite monoids and semigroups , 2003 .

[7]  Johan Håstad,et al.  On the power of small-depth threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[8]  Ran Raz,et al.  Separation of the Monotone NC Hierarchy , 1999, Comb..

[9]  Diederich Hinrichsen,et al.  Mathematical Systems Theory I , 2006, IEEE Transactions on Automatic Control.

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

[11]  Denis Thérien,et al.  Non-Uniform Automata Over Groups , 1987, Inf. Comput..

[12]  Howard Straubing.  Finite Automata, Formal Logic, and Circuit Complexity , 1994, Progress in Theoretical Computer Science.

[13]  Howard Straubing,et al.  On Logical Descriptions of Regular Languages , 2002, LATIN.

[14]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[15]  Ingo Wegener,et al.  Branching Programs and Binary Decision Diagrams , 1987 .

[16]  Toniann Pitassi,et al.  Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity , 2005, ICALP.

[17]  Denis Thérien,et al.  Logspace and logtime leaf languages , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[18]  Denis Thérien,et al.  Languages with Bounded Multiparty Communication Complexity , 2007, STACS.

[19]  Denis Thérien,et al.  DIAMONDS ARE FOREVER: THE VARIETY DA , 2002 .

[20]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[21]  Pavel Pudlák,et al.  Bounded-depth circuits: separating wires from gates , 2005, STOC '05.

[22]  Mario Szegedy.  Functions with Bounded Symmetric Communication Complexity, Programs over Commutative Monoids, and ACC , 1993, J. Comput. Syst. Sci..

[23]  Kristoffer Arnsfelt Hansen,et al.  Circuits on cylinders , 2006, computational complexity.

[24]  Pavel Pudlák.  An Application of Hindman's Theorem to a Problem on Communication Complexity , 2003, Comb. Probab. Comput..

[25]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[26]  Howard Straubing.  Languages Defined with Modular Counting Quantifiers , 2001, Inf. Comput..

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

[28]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[29]  Howard Straubing,et al.  Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables , 2001, STACS.

[30]  Denis Thérien,et al.  Monoids and Computations , 2004, Int. J. Algebra Comput..

[31]  Noam Nisan,et al.  Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs , 1992, J. Comput. Syst. Sci..

[32]  Jean-Éric Pin,et al.  Syntactic Semigroups , 1997, Handbook of Formal Languages.

[33]  Dominique Perrin,et al.  On the Expressive Power of Temporal Logic , 1993, J. Comput. Syst. Sci..

[34]  Denis Thérien,et al.  Algebraic Characterizations of Small Classes of Boolean Functions , 2003, STACS.

[35]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[36]  Pascal Weil,et al.  Closure of Varieties of Languages under Products with Counter , 1992, J. Comput. Syst. Sci..

[37]  Jean-Eric Pin,et al.  Profinite Semigroups, Mal'cev Products, and Identities☆ , 1996 .

[38]  Howard Straubing,et al.  Regular Languages Defined by Generalized First-Order Formulas with a Bounded Number of Bound Variables , 2002, Theory of Computing Systems.

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

[40]  Klaus-Jörn Lange,et al.  Characterizing TC0 in Terms of Infinite Groups , 2006, Theory of Computing Systems.

[41]  Howard Straubing,et al.  regular Languages Defined with Generalized Quantifiers , 1988, ICALP.

[42]  Jorge Almeida,et al.  Finite Semigroups and Universal Algebra , 1995 .

[43]  Denis Thérien,et al.  An Algebraic Approach to Communication Complexity , 1998, ICALP.

[44]  Denis Thérien,et al.  Complete Classifications for the Communication Complexity of Regular Languages , 2005, Theory of Computing Systems.

[45]  A BarringtonDavid.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1989 .

[46]  Dana Angluin,et al.  Learning Regular Sets from Queries and Counterexamples , 1987, Inf. Comput..

[47]  Imre Simon,et al.  Piecewise testable events , 1975, Automata Theory and Formal Languages.

[48]  Christoph Meinel,et al.  On relations between counting communication complexity classes , 2004, J. Comput. Syst. Sci..

[49]  Vince Grolmusz.  Separating the Communication Complexities of MOD m and MOD p Circuits , 1995, J. Comput. Syst. Sci..

[50]  Zoltán Ésik,et al.  Algebraic recognizability of regular tree languages , 2005, Theor. Comput. Sci..

[51]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[52]  Howard Straubing,et al.  Characterizations of regular languages in low level complexity classes , 2001, Bull. EATCS.

[53]  Jean-Camille Birget,et al.  Algorithmic problems in groups and semigroups , 2000 .

[54]  Vince Grolmusz,et al.  The BNS Lower Bound for Multi-Party Protocols in Nearly Optimal , 1994, Inf. Comput..

[55]  Denis Thérien.  SUBWORD COUNTING AND NILPOTENT GROUPS , 1983 .

[56]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[57]  N. Nisan.  The communication complexity of threshold gates , 1993 .

[58]  Francesco Bergadano,et al.  Learning Behaviors of Automata from Multiplicity and Equivalence Queries , 1994, SIAM J. Comput..

[59]  Howard Straubing,et al.  Superlinear Lower Bounds for Bounded-Width Branching Programs , 1995, J. Comput. Syst. Sci..

[60]  Howard Straubing,et al.  Regular Languages in NC¹ , 1992, J. Comput. Syst. Sci..

[61]  Cristopher Moore,et al.  Circuits and expressions with non-associative gates , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[62]  Thomas Wilke,et al.  Nesting Until and Since in Linear Temporal Logic , 2003, Theory of Computing Systems.

[63]  Denis Thérien,et al.  On the Languages Recognized by Nilpotent Groups (a translation of "Sur les Langages Reconnus par des Groupes Nilpotents") , 2001, Electron. Colloquium Comput. Complex..

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

[65]  Franco P. Preparata,et al.  Characterization of Associative Operations with Prefix Circuits of Constant Depth and Linear Size , 1990, SIAM J. Comput..

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

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

[68]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

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

[70]  Denis Thérien,et al.  NC1: The automata-theoretic viewpoint , 1991, computational complexity.

[71]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[72]  Kristoffer Arnsfelt Hansen.  Constant Width Planar Computation Characterizes ACC0 , 2005, Theory of Computing Systems.

[73]  Denis Thérien,et al.  Learning expressions and programs over monoids , 2006, Inf. Comput..

[74]  Denis Thérien,et al.  Restricted Two-Variable Sentences, Circuits and Communication Complexity , 2005, ICALP.

[75]  Noga Alon,et al.  Regular languages are testable with a constant number of queries , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[76]  Bernd Borchert.  On the Acceptance Power of Regular Languages , 1994, STACS.