Expander construction in VNC1

Abstract We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44] , and show that this analysis can be formalized in the bounded arithmetic system VNC 1 (corresponding to the “ NC 1 reasoning”). As a corollary, we prove the assumption made by Jeřabek [28] that a construction of certain bipartite expander graphs can be formalized in VNC 1 . This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak [9] .

[1]  Toniann Pitassi,et al.  A new proof of the weak pigeonhole principle , 2000, STOC '00.

[2]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[3]  Stephen A. Cook,et al.  Log Depth Circuits for Division and Related Problems , 1986, SIAM J. Comput..

[4]  Milena Mihail,et al.  Conductance and convergence of Markov chains-a combinatorial treatment of expanders , 1989, 30th Annual Symposium on Foundations of Computer Science.

[5]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[6]  Emil Jerábek,et al.  A sorting network in bounded arithmetic , 2011, Ann. Pure Appl. Log..

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

[8]  S. Buss,et al.  Collapsing modular counting in bounded arithmetic and constant depth propositional proofs , 2015 .

[9]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[10]  Salil P. Vadhan,et al.  Derandomized Squaring of Graphs , 2005, APPROX-RANDOM.

[11]  Jeff B. Paris,et al.  Provability of the Pigeonhole Principle and the Existence of Infinitely Many Primes , 1988, J. Symb. Log..

[12]  Emil Jerábek.  Approximate counting by hashing in bounded arithmetic , 2009, J. Symb. Log..

[13]  M. Murty.  Ramanujan Graphs , 1965 .

[14]  Emil Jerábek.  Approximate counting in bounded arithmetic , 2007, J. Symb. Log..

[15]  A. Wigderson,et al.  ENTROPY WAVES, THE ZIG-ZAG GRAPH PRODUCT, AND NEW CONSTANT-DEGREE , 2004, math/0406038.

[16]  Noga Alon,et al.  An elementary construction of constant-degree expanders , 2007, SODA '07.

[17]  Eric Allender,et al.  Uniform constant-depth threshold circuits for division and iterated multiplication , 2002, J. Comput. Syst. Sci..

[18]  Samuel R. Buss,et al.  How to Lie Without Being (Easily) Convicted and the Length of Proofs in Propositional Calculus , 1994, CSL.

[19]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[20]  P. Pudlák.  Sets and Proofs: On the Complexity of the Propositional Calculus , 1999 .

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

[22]  Emil Jerábek,et al.  Substitution Frege and extended Frege proof systems in non-classical logics , 2009, Ann. Pure Appl. Log..

[23]  Joel I. Seiferas,et al.  Sorting Networks of Logarithmic Depth, Further Simplified , 2009, Algorithmica.

[24]  Richard J. Lipton,et al.  Model theoretic aspects of computational complexity , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[25]  S. Cook,et al.  Logical Foundations of Proof Complexity: INDEX , 2010 .

[26]  Samuel R. Buss.  Polynomial Size Proofs of the Propositional Pigeonhole Principle , 1987, J. Symb. Log..

[27]  Toshiyasu Arai,et al.  A Bounded Arithmetic AID for Frege Systems , 2000, Ann. Pure Appl. Log..

[28]  Ján Pich,et al.  Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic , 2014, Log. Methods Comput. Sci..

[29]  Pavel Pudlák,et al.  Monotone simulations of non-monotone proofs , 2001, J. Comput. Syst. Sci..

[30]  Noga Alon,et al.  Explicit construction of linear sized tolerant networks , 1988, Discret. Math..

[31]  Alexander A. Razborov,et al.  Proof Complexity of Pigeonhole Principles , 2001, Developments in Language Theory.

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

[33]  Stephen A. Cook,et al.  Quantified propositional calculus and a second-order theory for NC1 , 2005, Arch. Math. Log..

[34]  Pavel Pudlák,et al.  Ramsey's Theorem in Bounded Arithmetic , 1990, CSL.

[35]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[36]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[37]  Noga Alon,et al.  Random Cayley Graphs and Expanders , 1994, Random Struct. Algorithms.

[38]  Ricard Gavaldà,et al.  Monotone Proofs of the Pigeon Hole Principle , 2000, Math. Log. Q..