Quasipolynomial size circuit classes

Circuit complexity theory has tried to understand which problems can be solved by 'small' circuits of constant depth. Normally 'small' has meant 'polynomial in the input size', but a number of recent results have dealt with circuits of size 2 to the log n/sup 0(1)/ power, or quasipolynomial size. The author summarizes the reasons for thinking about the complexity classes so introduced, surveys these results and gives an overview of these classes. He also shows that the Barrington-Immerman-Straubing uniformity definition for polynomial-size classes can easily be extended to quasipolynomial size as well, with most of the key results remaining true in the uniform setting.<<ETX>>

[1]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

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

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

[4]  Steven Lindell.  A purely logical characterization of circuit uniformity , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

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

[6]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[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]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[9]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[10]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[11]  Georg Schnitger,et al.  Parallel Computation with Threshold Functions , 1986, J. Comput. Syst. Sci..

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

[13]  Neil Immerman,et al.  Expressibility as a complexity measure: results and directions , 1987, SCT.

[14]  Jacobo Torán,et al.  The power of the middle bit , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[15]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[16]  Andrew Chi-Chih Yao,et al.  ON ACC and threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[17]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[18]  Pierre McKenzie,et al.  Extensions to Barrington's M-program model , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[19]  David A. Mix Barrington,et al.  Representing Boolean functions as polynomials modulo composite numbers , 1992, STOC '92.

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

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

[22]  Pavel Pudlák,et al.  Threshold circuits of bounded depth , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[23]  E. Allender,et al.  On strong separations from ACo , 1991 .

[24]  Denis Thérien,et al.  Automata Theory Meets Circuit Complexity , 1989, ICALP.

[25]  Eric Allender,et al.  A note on the power of threshold circuits , 1989, 30th Annual Symposium on Foundations of Computer Science.

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

[27]  Daniel A. Spielman,et al.  The perceptron strikes back , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.