Computing Short SAT Implicants via Ising/QUBO Encodings

2026-05-07

AI Summary

This work addresses the limitation of traditional Ising/QUBO models, which only support total assignments and thus cannot directly encode short conjunctions over subsets of variables—i.e., partial assignments—central to SAT solving. The paper introduces, for the first time, a notion of “don’t-care” semantics and proposes an Ising/QUBO encoding framework that accommodates partial assignments. This is achieved by representing unassigned variables via bipolar encoding, incorporating quadratic penalty functions combined with objective fine-tuning to contract and project conjunctions, and employing a polarity-freezing strategy to guide the optimization process. Experimental results on random 3-SAT and non-CNF instances demonstrate that the approach can satisfy formulas while leaving approximately one-third of variables unassigned, and, with high probability, yields minimal or minimum-cardinality conjunctions through successive rounds of continuous freezing.

Abstract

Many reasoning tasks require short partial satisfying assignments (implicants), sometimes focusing on a set of important variables. SAT-to-Ising-QUBO formulations are implicitly designed so that ground states correspond to total assignments, since the Ising/QUBO model assigns a value to every spin and has no native representation of unassigned variables. We introduce an Ising/QUBO framework that incorporates "don't-care" semantics into the quadratic model via a dual-polarity representation, enabling the retrieval of short implicants. The encoding supports implicant shrinking and projection through minor objective modifications. We provide parameter regimes under which ground states correspond to short partial satisfying assignments, achieving minimality and, when the quadratic penalty function permits, minimum-cardinality. We empirically evaluate the encoding with simulated annealing on random 3-SAT enumeration benchmarks and non-CNF formulas, showing that it leaves about one-third of variables unassigned on random 3-SAT formulas while preserving satisfiability, and that consecutive polarity-freezing rounds achieve minimality (and minimum-cardinality) with high probability.

Problem

• SAT implicants
• Ising model
• QUBO
• partial satisfying assignments
• don't-care semantics

Innovation

• Ising/QUBO encoding
• short implicants
• don't-care semantics
• dual-polarity representation
• partial satisfying assignments