A Lower Bound on DNNF Encodings of Pseudo-Boolean Constraints

TL;DR

This paper proves that certain pseudo-Boolean constraints inherently require exponentially large DNNF representations, highlighting fundamental size limitations in encoding these constraints for SAT solving.

Contribution

It establishes a lower bound on the size of DNNF encodings for some pseudo-Boolean constraints, demonstrating intrinsic complexity barriers.

Findings

Existence of PB-constraints with exponential DNNF size

DNNF encodings can be inherently large for certain constraints

Size limitations impact encoding strategies for pseudo-Boolean constraints

Abstract

Two major considerations when encoding pseudo-Boolean (PB) constraints into SAT are the size of the encoding and its propagation strength, that is, the guarantee that it has a good behaviour under unit propagation. Several encodings with propagation strength guarantees rely upon prior compilation of the constraints into DNNF (decomposable negation normal form), BDD (binary decision diagram), or some other sub-variants. However it has been shown that there exist PB-constraints whose ordered BDD (OBDD) representations, and thus the inferred CNF encodings, all have exponential size. Since DNNFs are more succinct than OBDDs, preferring encodings via DNNF to avoid size explosion seems a legitimate choice. Yet in this paper, we prove the existence of PB-constraints whose DNNFs all require exponential size.