In this paper, we analyze the copy complexity of unsatisfiable 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). 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. We show that for an HCS with n variables and m constraints, the copy complexity is at most \(2^{n-1}\), in the ADD refutation system. Additionally, we analyze bounded-width HCSs from the perspective of copy complexity. Finally, we provide an empirical analysis of an integer programming formulation of the copy complexity problem in HCSs. (An extended abstract was published in FroCos 2021 [26].)

在本文中，我们分析了ADD反驳系统下不可满足的Horn约束系统的复制复杂度。回想一下，形式为\(\sum _{i=1}^{n} a_{i}\cdot x_{i} \ge b\) 的线性约束，如果所有\( a_{i} \in \{0,1,-1\}\)并且最多有一个\(a_{i}\) s 为正。此类约束的结合称为 Horn 约束系统 (HCS)。喇叭约束出现在许多领域，包括但不限于程序验证、电力系统、计量经济学和运筹学。ADD的反驳体系是健全的、完整的。此外，它是反驳线性约束系统可行性的最简单、最自然的反驳系统。反驳系统中不可行的线性约束系统（不一定是 Horn）的复制复杂度是每个约束需要复制的最小次数，以获得一次读取的反驳。我们证明，对于具有n 个变量和m 个约束的 HCS，在 ADD 反驳系统中，复制复杂度最多为\(2^{n-1}\) 。此外，我们从复制复杂性的角度分析有界宽度的 HCS。最后，我们对 HCS 中复制复杂性问题的整数规划公式进行了实证分析。（扩展摘要发表于 FroCos 2021 [26]。）