Abstract

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

Highlights

  • In this work, we obtain strong trade-offs in proof complexity by making a connection to pebble games played on graphs

  • This was strengthened in [29], which used the connection between designs and Nullstellensatz degree discussed above to establish that the degree needed to refute a pebbling formula exactly coincides with the reversible pebbling price of the corresponding directed acyclic graph (DAG)

  • In this paper we prove that size and degree of Nullstellensatz refutations in any field of pebbling formulas are exactly captured by time and space of the reversible pebble game on the underlying graph

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Summary

Introduction

We obtain strong trade-offs in proof complexity by making a connection to pebble games played on graphs. Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

Proof Complexity
Pebble Games
Our Contributions
Outline of This Paper
Preliminaries
Nullstellensatz
Reversible Pebbling and Pebbling Formulas
Reversible Pebblings and Nullstellensatz Refutations
From Pebbling to Refutation
From Refutation to Pebbling
Nullstellensatz Trade-offs from Reversible Pebbling
Findings
Concluding Remarks

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