Characterizing Tseitinformulas with short regular resolution refutations
Abstract
Tseitinformulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitinformulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs $G$ for that class is in $O(\logV(G))$. To do so, we show that any regular resolution refutation of an unsatisfiable Tseitinformula with graph $G$ of bounded degree has length $2^{\Omega(tw(G))}/V(G)$, thus essentially matching the known $2^{O(tw(G))}poly(V(G))$ upper bound up. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitinformulas to the size of representations of \textit{satisfiable} Tseitinformulas in decomposable negation normal form (DNNF). Then we prove that for every graph $G$ of bounded degree, every DNNFrepresentation of every satisfiable Tseitinformula with graph $G$ must have size $2^{\Omega(tw(G))}$ which yields our lower bound for regular resolution.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.09609
 arXiv:
 arXiv:2103.09609
 Bibcode:
 2021arXiv210309609D
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 20 pages including references