Bounded-width polynomial-size branching programs recognize exactly those languages in NC1

Abstract We show that any language recognized by an NC 1 circuit (fan-in 2, depth O (log n )) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC 1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC 1 . Further, following Ruzzo ( J. Comput. System Sci. 22 (1981), 365–383) and Cook ( Inform. and Control 64 (1985) 2–22) , if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n )-uniform NC 1 circuits, that is, those languages in ATIME(log n ). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC 1 .

[1]  H. Zassenhaus.  The theory of groups , 1949 .

[2]  C. Y. Lee.  Representation of switching circuits by binary-decision programs , 1959 .

[3]  M. Hall.  The Theory Of Groups , 1959 .

[4]  John E. Savage,et al.  Computational Work and Time on Finite Machines , 1972, JACM.

[5]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[6]  William Joseph Masek,et al.  A fast algorithm for the string editing problem and decision graph complexity , 1976 .

[7]  M. Fischer,et al.  Parallel Prefix Computation , 1980, J. ACM.

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

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

[10]  Leslie G. Valiant,et al.  A complexity theory based on Boolean algebra , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[11]  Leslie G. Valiant,et al.  A complexity theory based on Boolean algebra , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[12]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[13]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.

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

[15]  Andrew C. Yao,et al.  Lower bounds by probabilistic arguments , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[16]  Allan Borodin,et al.  Bounds for width two branching programs , 1983, SIAM J. Comput..

[17]  Stephen A. Cook,et al.  Log Depth Circuits for Division and Related Problems , 1984, SIAM J. Comput..

[18]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[19]  Pavel Pudlák.  A Lower Bound on Complexity of Branching Programs (Extended Abstract) , 1984, MFCS.

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

[21]  Ronald Fagin,et al.  Bounded-Depth, Polynomial-Size Circuits for Symmetric Functions , 1985, Theoretical Computer Science.

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

[23]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[24]  os AjtaiIBM.  Two lower bounds for branching programs , 1986 .

[25]  Vojtech Rödl,et al.  Two lower bounds for branching programs , 1986, STOC '86.

[26]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[27]  D. A. Barrington.  BOUNDED WIDTH BRANCHING PROGRAMS , 1986 .

[28]  Allan Borodin,et al.  Bounds for Width Two Branching Programs , 1986, SIAM J. Comput..

[29]  Noga Alon,et al.  Meanders, Ramsey theory and lower bounds for branching programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[30]  Stephen A. Cook,et al.  Problems Complete for Deterministic Logarithmic Space , 1987, J. Algorithms.

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

[32]  Denis Thérien,et al.  Finite monoids and the fine structure of NC1 , 1987, STOC.

[33]  Ian Parberry,et al.  Parallel complexity theory , 1987, Research notes in theoretical computer science.

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

[35]  Georg Schnitger,et al.  Parallel Computation with Threshold Functions , 1988, J. Comput. Syst. Sci..