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 VNC1 (corresponding to the “NC1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in VNC1. 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 Pudlák [9].

Highlights

  • Expander graphs have become one of the most useful combinatorial objects in theoretical computer science, with many beautiful applications in computer science and mathematics [19], and responsible for several breakthroughs in computational complexity [37, 17]

  • Our main contribution is the analysis of one of the iterative expander constructions from [38], which we show to be formalizable in the bounded arithmetic system VNC1

  • As our main application, building on Jeřábek [24] and Atserias, Galesi and Pudlák [7], we show that every proof in Gentzen’s sequent calculus LK of a monotone sequent can be simulated by a monotone LK (MLK) proof with only polynomial blowup in size

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Summary

Introduction

Expander graphs have become one of the most useful combinatorial objects in theoretical computer science, with many beautiful applications in computer science and mathematics [19], and responsible for several breakthroughs in computational complexity [37, 17]. Schwartz, and Shapira [4] gave a different construction of expanders, which combines algebraically constructed expanders of Alon and Roichman [3] with only two applications of a certain graph operation (replacement product), to obtain a constant-degree expander of arbitrary size They gave a fully combinatorial analysis of the replacement product operation they used in the second stage of the construction. 31:3 our expander construction to argue that any Gentzen’s sequent calculus LK proof (of a monotone sequent) can be simulated by a monotone LK (MLK) proof, with only polynomial blowup in proof size, improving upon the quasipolynomial simulation shown by Atserias, Galesi and Pudlák [7], and answering a question of Pudlák and Buss [36] This simulation result follows by the work of Jeřábek [24] who proved the result under the assumption that a certain expander graph family can be proved to exist within a system of NC1 reasoning. Our paper proves a strengthening of the assumption needed by Jeřábek

Our results
Expander constructions
Bounded arithmetic
Remainder of the paper
Notation
Expanders
Bounded arithmetic theory VNC1
LK and MLK proof systems
Constructing edge expanders
Graph operations
Effect of graph operations on edge expansion
Construction
Constructing bipartite vertex expanders
Getting a vertex expander from an edge expander
Getting a bipartite vertex expander
Formalizing the construction in bounded arithmetic
Defining NC1 functions within VNC1
A modified tree recursion
A conservation result
Expressing expander graph properties in VNC1
Formalizing edge expansion properties in VNC1
Application to monotone sequent calculus
Conclusions and open problems
Full Text
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