Proof Systems Based on Structured Circuits

TL;DR

This paper explores advanced proof systems based on structured circuits like SDDs and d-SDNNFs, demonstrating their potential for more compact refutations of unsatisfiable CNFs and analyzing their relative strengths.

Contribution

It extends proof system research by comparing SDDs and d-SDNNFs to OBDDs, establishing size advantages, and analyzing rule-based separations with a novel sat-to-unsat lifting theorem.

Findings

SDDs and d-SDNNFs can produce smaller refutations than OBDDs.

The relative strength of proof systems varies with derivation rules allowed.

A sat-to-unsat lifting theorem links CNF complexity in different proof systems.

Abstract

Since their introduction by Atserias, Kolaitis, and Vardi in 2004, proof systems where each line is represented by an ordered binary decision diagram (OBDD) have been intensively studied as they allow to compactly represent Boolean functions. We extend this line of work by considering representation formats that can be even more succinct than OBDDs and have gained a lot of attention in the area of knowledge compilation: sentential decision diagrams (SDDs) and deterministic structured DNNF circuits (d-SDNNFs). We show that both variants can provide strictly smaller refutations of unsatisfiable CNFs than their OBDD counterparts. Furthermore, we investigate the relative strength of these systems depending on which of the three fundamental derivation rules join, reordering, and weakening are allowed. Here we obtain several separations and identify interesting open problems. To streamline our proofs we establish a sat-to-unsat lifting theorem that might be of independent interest: it turns satisfiable CNFs that are hard to represent by SDDs and d-SDNNFs into unsatisfiable CNFs that are hard to refute in the corresponding proof system.