On Unit Read-Once Resolutions and Copy Complexity

Abstract

In this paper, we discuss the copy complexity of unit resolution with respect to Horn formulas. A Horn formula is a boolean formula in conjunctive normal form (CNF) with at most one positive literal per clause. Horn formulas find applications in a number of domains such as program verification and logic programming. Resolution as a proof system for boolean formulas is both sound and complete. However, resolution is considered an inefficient proof system when compared to other stronger proof systems for boolean formulas. Despite this inefficiency, the simple nature of resolution makes it an integral part of several theorem provers. Unit resolution is a restricted form of resolution in which each resolution step needs to use a clause with only one literal (unit literal clause). While not complete for general CNF formulas, unit resolution is complete for Horn formulas. A read-once resolution (ROR) refutation is a refutation in which each clause (input or derived) may be used at most once in the derivation of a refutation. As with unit resolution, ROR refutation is incomplete in general and complete for Horn clauses. This paper focuses on a combination of unit resolution and read-once resolution called Unit read-once resolution (UROR). UROR is incomplete for Horn clauses. In this paper, we study the copy complexity problem in Horn formulas under UROR. Briefly, the copy complexity of a formula under UROR is the smallest number k such that replicating each clause k times guarantees the existence of a UROR refutation. This paper focuses on two problems related to the copy complexity of unit resolution. We first relate the copy complexity of unit resolution for Horn formulas to the copy complexity of the addition rule in the corresponding Horn constraint system. We also examine a form of copy complexity where we permit replication of derived clauses, in addition to the input clauses. Finally, we provide a polynomial time algorithm for the problem of checking if a 2-CNF formula has a UROR refutation.