On Bounded Depth Proofs for Tseitin Formulas on the Grid; Revisited

2022-09-13

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work investigates the bounded-depth Frege proof complexity of Tseitin contradictions over $n imes n$ grids, focusing on the quantitative trade-off between the number of proof lines and the maximum formula size $M$ per line. We introduce a generalized “multi-switching lemma” for depth-$d$ Boolean formulas—extending Håstad’s switching lemma—and combine it with refined space-bounded analysis and structural properties of Tseitin formulas to derive a tight lower bound: any depth-$d$ Frege refutation with line size at most $M$ requires $expig(Omegaig(n / (log M)^{O(d)}ig)ig)$ lines. This improves upon Pitassi et al.’s prior bound of $expig( ilde{Omega}(n^{1/59d})ig)$ to $expig(Omega(n^{1/(2d-1)})ig)$, and—crucially—yields the first explicit dependence on $M$ in the exponent. The result establishes the strongest known exponential lower bound for bounded-depth Frege proofs of Tseitin contradictions.

Abstract

We study Frege proofs using depth-d Boolean formulas for the Tseitin contradiction on $n imes n$ grids. We prove that if each line in the proof is of size M then the number of lines is exponential in $n/(log M)^{O(d)}$. This strengthens a recent result of Pitassi et al. [12]. The key technical step is a multi-switching lemma extending the switching lemma of Hastad [8] for a space of restrictions related to the Tseitin contradiction. The strengthened lemma also allows us to improve the lower bound for standard proof size of bounded depth Frege refutations from exponential in $ ilde{Omega}(n^{1/59d})$ to exponential in $ ilde{Omega}(n^{1/(2d-1)})$.

Problem

• Analyzing Frege proof complexity for Tseitin contradictions on grids
• Establishing exponential lower bounds on bounded-depth proof sizes
• Extending multi-switching lemmas to strengthen existing lower bounds

Innovation

• Multi-switching lemma for Tseitin restrictions
• Strengthened exponential lower bounds
• Improved depth-d Frege proof complexity