On the Copy Complexity of Width 3 Horn Constraint Systems

Abstract

In this paper, we analyze the copy complexity of unsatisfiable width 3 Horn constraint systems, under the ADD refutation system. Recall that a linear constraint of the form $\sum _{i=1}^{n} a_{i}\cdot x_{i} \ge b$, is said to be a Horn constraint if all the $a_{i} \in \{0,1,-1\}$ and at most one of the $a_{i}$s is positive. A conjunction of such constraints is called a Horn constraint system (HCS). An HCS is said to have width 3, if there are at most 3 variables with non-zero coefficients per constraint. Horn constraints arise in a number of domains including but not limited to program verification, power systems, econometrics, and operations research. The ADD refutation system is both sound and complete. Additionally, it is the simplest and most natural refutation system for refuting the feasibility of a system of linear constraints. The copy complexity of an infeasible linear constraint system (not necessarily Horn) in a refutation system is the minimum number of times each constraint needs to be replicated, in order to obtain a read-once refutation. In this paper, we analyze width 3 HCSs from the perspective of copy complexity.

K. Subramani—This research was supported in part by the Air-Force Office of Scientific Research through Grant FA9550-19-1-0177 and in part by the Air-Force Research Laboratory, Rome through Contract FA8750-17-S-7007.