On Uniform Circuit Complexity

Abstract We argue that uniform circuit complexity introduced by Borodin is a reasonable model of parallel complexity. Three main results are presented. First, we show that alternating Turing machines are also a surprisingly good model of parallel complexity, by showing that simultaneous size/depth of uniform circuits is the same as space/time of alternating Turing machines, with depth and time within a constant factor and likewise log(size) and space. Second, we apply this to characterize NC, the class of polynomial size and polynomial-in-log depth circuits, in terms of tree-size bounded alternating TMs and other models. In particular, this enables us to show that context-free language recognition is in NC. Third, we investigate various definitions of uniform circuit complexity, showing that it is fairly insensitive to the choice of definition.

[1]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[2]  Stephen A. Cook,et al.  Hardware complexity and parallel computation , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[3]  Ivan Hal Sudborough.  Time and Tape Bounded Auxiliary Pushdown Automata , 1977, MFCS.

[4]  Allan Borodin,et al.  On Relating Time and Space to Size and Depth , 1977, SIAM J. Comput..

[5]  Juris Hartmanis,et al.  On the Power of Multiplication in Random Access Machines , 1974, SWAT.

[6]  I. H. Sudborough.  The complexity of the membership problem for some extensions of context-free languagest† , 1977 .

[7]  Nicholas Pippenger.  Pebbling with an Auxiliary Pushdown , 1981, J. Comput. Syst. Sci..

[8]  Leslie M. Goldschlager,et al.  A unified approach to models of synchronous parallel machines , 1978, STOC.

[9]  Stephen A. Cook,et al.  Deterministic CFL's are accepted simultaneously in polynomial time and log squared space , 1979, STOC.

[10]  Walter J. Savitch,et al.  Parallel and Nondeterministic Time Complexity Classes (Preliminary Report) , 1978, ICALP.

[11]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[12]  Nicholas Pippenger,et al.  On simultaneous resource bounds , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[13]  Michael J. Fischer,et al.  Relations Among Complexity Measures , 1979, JACM.

[14]  Walter L. Ruzzo,et al.  Tree-size bounded alternation(Extended Abstract) , 1979, J. Comput. Syst. Sci..

[15]  R. Ladner.  The circuit value problem is log space complete for P , 1975, SIGA.

[16]  Steven Fortune,et al.  Parallelism in random access machines , 1978, STOC.

[17]  Stephen A. Cook,et al.  Characterizations of Pushdown Machines in Terms of Time-Bounded Computers , 1971, J. ACM.

[18]  Charles J. Colbourn,et al.  Linear Time Automorphism Algorithms for Trees, Interval Graphs, and Planar Graphs , 1981, SIAM J. Comput..

[19]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[20]  Leslie Michael Goldschlager,et al.  Synchronous parallel computation. , 1978 .

[21]  Larry J. Stockmeyer,et al.  A characterization of the power of vector machines , 1974, STOC '74.

[22]  Larry J. Stockmeyer,et al.  A Characterization of the Power of Vector Machines , 1976, J. Comput. Syst. Sci..

[23]  Walter J. Savitch,et al.  Time Bounded Random Access Machines with Parallel Processing , 1979, JACM.