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We show that is hard to find regular or even ordered (also known as Davis-Putnam) Resolution proofs, extending the breakthrough result for general Resolution from Atserias and Muller to these restricted forms. Namely, regular and ordered Resolution are automatable if and only if P = NP. Specifically, for a CNF formula F the problem of distinguishing between the existence of a polynomial-size ordered Resolution refutation of F and an at least exponential-size general Resolution proof being required to refute F is NP-complete.
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Regular resolution effectively simulates resolution
Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs in regular resolution while admitting polynomial-size proofs in resolution. Thus, with respect to the usual notion of simulation, regular resolution is separated from resolution. An alternative, and weaker, notion for comparing proof systems is that of an “effective simulation,” which allows the translation of the formula along with the proof when moving between proof systems. We prove that regular resolution is equivalent to resolution under effective simulations. As a corollary, we recover in a black-box fashion a recent result on the hardness of automating regular resolution.
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- Award ID(s):
- 2015445
- PAR ID:
- 10513095
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Information Processing Letters
- Volume:
- 186
- Issue:
- C
- ISSN:
- 0020-0190
- Page Range / eLocation ID:
- 106489
- Subject(s) / Keyword(s):
- resolution regular resolution effective simulation automatability proof complexity
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Tauman Kalai, Yael (Ed.)We study the complexity of proof systems augmenting resolution with inference rules that allow, given a formula Γ in conjunctive normal form, deriving clauses that are not necessarily logically implied by Γ but whose addition to Γ preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in Γ, the systems we consider become equivalent to extended resolution. We are concerned with the versions of these systems without new variables. They are called BC⁻, RAT⁻, SBC⁻, and GER⁻, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold --- without loss of generality --- which is commonly used informally to simplify or shorten proofs. Except for SBC⁻, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. By taking advantage of this fact, we construct formulas that separate RAT⁻ from GER⁻ and vice versa. With the same strategy, we also separate SBC⁻ from RAT⁻. Additionally, we give polynomial-size SBC⁻ proofs of the pigeonhole principle, which separates SBC⁻ from GER⁻ by a previously known lower bound. These results also separate the three systems from BC⁻ since they all simulate it. We thus give an almost complete picture of their relative strengths.more » « less
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Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that—from the viewpoint of proof complexity—these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems.more » « less
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We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula F, it is NP-hard to find a refutation of F in the Nullstellensatz, Polynomial Calculus, or Sherali--Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Muller (FOCS 2019) that established an analogous result for Resolution.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ipl.2024.106489
