A sorting network in bounded arithmetic

Abstract We formalize the construction of Paterson’s variant of the Ajtai–Komlos–Szemeredi sorting network of logarithmic depth in the bounded arithmetical theory VNC ∗ 1 (an extension of VNC 1 ), under the assumption of the existence of suitable expander graphs. We derive a conditional p-simulation of the propositional sequent calculus in the monotone sequent calculus MLK .

[1]  János Komlós,et al.  An 0(n log n) sorting network , 1983, STOC.

[2]  Pavel Pudlák,et al.  Monotone simulations of nonmonotone proofs , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[3]  E. Szemerédi,et al.  O(n LOG n) SORTING NETWORK. , 1983 .

[4]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[5]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[6]  Stephen A. Cook,et al.  A second-order theory for NL , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[7]  Mike Paterson,et al.  Improved sorting networks withO(logN) depth , 1990, Algorithmica.

[8]  Noam Nisan,et al.  Symmetric logspace is closed under complement , 1995, STOC '95.

[9]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.

[10]  L. Bonet.  The Complexity of Resource-Bounded Propositional Proofs , 2001 .

[11]  Emil Jerábek,et al.  On theories of bounded arithmetic for NC1 , 2011, Ann. Pure Appl. Log..

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

[13]  E. Szemerédi,et al.  Sorting inc logn parallel steps , 1983 .

[14]  Domenico Zambella.  Notes on Polynomially Bounded Arithmetic , 1996, J. Symb. Log..