Pebble Games and Algebraic Proof Systems

2025-06-26

AI Summary

This paper establishes precise correspondences between three pebbling games—reversible, black, and black-white—on directed acyclic graphs (DAGs) and three algebraic proof systems: Nullstellensatz (NS), Monomial Calculus (MC), and Polynomial Calculus (PC). By modeling pebbling strategies as algebraic refutations, it achieves, for the first time, parallel characterizations across space–time and degree–size complexity measures, yielding tight matching bounds for variable-space complexity. Leveraging pebbling lower bounds, the paper proves strict separations among NS, MC, and PC with respect to degree, exposes a sharp size–strength trade-off for MC, and obtains separations in variable-space complexity across all three systems. This work unifies graph-theoretic pebbling and algebraic proof complexity, providing a novel structural framework for analyzing proof complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Analyzing refutations of the well known 0pebbling formulas Peb$(G)$ we prove some new strong connections between pebble games and algebraic proof system, showing that there is a parallelism between the reversible, black and black-white pebbling games on one side, and the three algebraic proof systems Nullstellensatz, Monomial Calculus and Polynomial Calculus on the other side. In particular we prove that for any DAG $G$ with a single sink, if there is a Monomial Calculus refutation for Peb$(G)$ having simultaneously degree $s$ and size $t$ then there is a black pebbling strategy on $G$ with space $s$ and time $t+s$. Also if there is a black pebbling strategy for $G$ with space $s$ and time $t$ it is possible to extract from it a MC refutation for Peb$(G)$ having simultaneously degree $s$ and size $ts$. These results are analogous to those proven in {deRezende et al.21} for the case of reversible pebbling and Nullstellensatz. Using them we prove degree separations between NS, MC and PC, as well as strong degree-size tradeoffs for MC. We also notice that for any directed acyclic graph $G$ the space needed in a pebbling strategy on $G$, for the three versions of the game, reversible, black and black-white, exactly matches the variable space complexity of a refutation of the corresponding pebbling formula Peb$(G)$ in each of the algebraic proof systems NS, MC and PC. Using known pebbling bounds on graphs, this connection implies separations between the corresponding variable space measures.

Problem

• Connects pebble games to algebraic proof systems
• Proves space-time tradeoffs in pebbling and proof systems
• Establishes variable space complexity equivalences

Innovation

• Links pebbling games to algebraic proof systems
• Proves space-time tradeoffs for pebbling strategies
• Matches pebbling space to variable space complexity