Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?

TL;DR

This paper investigates the complexity of the CardMinSat problem within the Krom fragment, revealing it remains hard (Θ₂-complete) despite structural restrictions, and explores implications for reasoning tasks in belief revision and abduction.

Contribution

It demonstrates that CardMinSat is Θ₂-complete for Krom formulas, contrasting with its ease in Horn formulas, and analyzes how Krom restrictions affect reasoning problem complexities.

Findings

CardMinSat remains Θ₂-complete for Krom formulas.

Restrictions to Krom formulas do not always reduce complexity.

Complexity varies across reasoning tasks under Krom restrictions.

Abstract

Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula $\phi$ and an atom $x$, whether $x$ is true in some cardinality-minimal model of $\phi$. This problem is easy for the Horn fragment, but, as we will show in this paper, remains $\Theta_2$-complete (and thus $\mathrm{NP}$-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems.