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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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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ipl.2024.106489
