Model-Theoretic Characterization of Boolean and Arithmetic Circuit Classes of Small Depth

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC1, SAC1 and AC1 as well as their arithmetic counterparts #NC1, #SAC1 and #AC1. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC0 = FO, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulae characterizing languages in the corresponding Boolean class.

[1]  Claude Laflamme,et al.  An Algebra and a Logic for NC¹ , 1990, Inf. Comput..

[2]  Heribert Vollmer,et al.  Descriptive Complexity of #AC0 Functions , 2016, CSL.

[3]  Thomas Schwentick,et al.  The Descriptive Complexity Approach to LOGCFL , 1998, J. Comput. Syst. Sci..

[4]  David A. Mix Barrington,et al.  Some results on uniform arithmetic circuit complexity , 1994, Mathematical systems theory.

[5]  Heribert Vollmer,et al.  Introduction to Circuit Complexity: A Uniform Approach , 2010 .

[6]  Heribert Vollmer,et al.  A Model-Theoretic Characterization ofźConstant-Depth Arithmetic Circuits , 2016 .

[7]  Neil Immerman,et al.  Expressibility and Parallel Complexity , 1989, SIAM J. Comput..

[8]  K. V. Subrahmanyam,et al.  Descriptive Complexity of #P Functions , 1995, J. Comput. Syst. Sci..

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

[10]  Heribert Vollmer,et al.  A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits , 2016, WoLLIC.

[11]  Leonid Libkin,et al.  Elements of Finite Model Theory , 2004, Texts in Theoretical Computer Science.

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

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

[14]  Neeraj Kayal,et al.  A Selection of Lower Bounds for Arithmetic Circuits , 2014 .

[15]  Heribert Vollmer,et al.  A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits , 2016, Workshop on Logic, Language, Information and Computation.

[16]  Martin Tompa,et al.  The complexity of short two-person games , 1990, Discret. Appl. Math..

[17]  Neil Immerman,et al.  Time, hardware, and uniformity , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[18]  Vikraman Arvind,et al.  Perspectives in Computational Complexity: The Somenath Biswas Anniversary Volume , 2014 .

[19]  Mikolás Janota,et al.  Digital Object Identifier (DOI): , 2000 .

[20]  Marcelo Arenas,et al.  Descriptive Complexity for counting complexity classes , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[21]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[22]  Meena Mahajan,et al.  Algebraic Complexity Classes , 2013, ArXiv.

[23]  Guillaume Malod,et al.  Characterizing Valiant's algebraic complexity classes , 2008, J. Complex..