On monotone planar circuits

In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC/sup 0/. This provides a striking contrast to the non-planar case, where exactly NC/sup 1/ is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outerface only, can be solved in LOGDCFL/spl sube/SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outerface only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outerface can be computed by a monotone planar circuit of width O(w) and depth w/sup O(1)/. Finally, we show that monotone planar read-once circuits, with inputs on the outerface only, can be efficiently learned using membership queries.

[1]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[2]  Vijaya Ramachandran,et al.  An Efficient Parallel Algorithm for the Layered Planar Monotone Circuit Value Problem , 1997, Algorithmica.

[3]  S. Rao Kosaraju,et al.  An NC Algorithm for Evaluating Monotone Planar Circuits , 1995, SIAM J. Comput..

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

[5]  Vijaya Ramachandran,et al.  An efficient parallel algorithm for the general planar monotone circuit value problem4 , 1994, SODA '94.

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

[7]  Sorin Istrail,et al.  Bounded-Width Polynomial-Size Boolean Formulas Compute Exactly Those Functions in AC^0 , 1994, Inf. Process. Lett..

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

[9]  Peter Bro Miltersen,et al.  Searching Constant Width Mazes Captures the AC0 Hierarchy , 1997, STACS.

[10]  Peter Bro Miltersen,et al.  Searching constant width mazes captures the AC 0 hierarchy , 1997 .

[11]  William F. McColl,et al.  On the Planar Monotone Computation of Threshold Functions , 1985, STACS.

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

[13]  David A. Mix Barrington,et al.  Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹ , 1989, J. Comput. Syst. Sci..

[14]  Leslie M. Goldschlager,et al.  A Space Efficient Algorithm for the Monotone Planar Circuit Value Problem , 1980, Information Processing Letters.

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

[16]  Jia-Wei Hong.  Computation - computability, similarity and duality , 1986, Research notes in theoretical computer science.

[17]  Stephen A. Cook,et al.  Complexity Theory of Parallel Time and Hardware , 1989, Inf. Comput..