On the satisfiability of random regular signed SAT formulas

TL;DR

This paper investigates the satisfiability of random regular signed SAT formulas, revealing how the probability of satisfiability depends on variable set size, clause-to-variable ratio, and identifying a phase transition for the case k=2.

Contribution

It answers open questions about the behavior of random regular signed SAT formulas, including phase transition points and bounds on satisfiability probability.

Findings

Probability increases with |V|

Existence of a critical clause-to-variable ratio

Phase transition at m/n=2 for k=2 and V=[0,1]

Abstract

Regular signed SAT is a variant of the well-known satisfiability problem in which the variables can take values in a fixed set V \subset [0,1], and the `literals' have the form "x \le a" or "x \ge a". We answer some open question regarding random regular signed k-SAT formulas: the probability that a random formula is satisfiable increases with |V|; there is a constant upper bound on the ratio m/n of clauses m over variables n, beyond which a random formula is asypmtotically almost never satisfied; for k=2 and V=[0,1], there is a phase transition at m/n=2.