Understanding Resolution Proofs through Herbrand’s Theorem

Hetzl, Stefan; Libal, Tomer; Riener, Martin; Rukhaia, Mikheil

doi:10.1007/978-3-642-40537-2_15

Cited by 11 publications

(17 citation statements)

References 20 publications

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“…Note that the expansion proof of a proof ϕ is a sequent of expansion trees, which are defined to be the expansion trees of all formulas in the end-sequent of ϕ. An algorithm for the extraction of expansion proofs from sequent calculus proofs is presented in [23] and modified algorithms (dealing with cuts and equality) are presented in [21,22]. There exist also algorithms for a transformation of resolution-trees into expansion-trees, see [24].…”

Section: Dp(e)mentioning

confidence: 99%

“…We will use an algorithm that is briefly described in [21]. In order to show how an expansion proof is extracted from a proof in LK = , we first need to define an operation on expansion trees.…”

Section: Dp(e)mentioning

confidence: 99%

“…Note that this method extracts an expansion proof from any LK-proof and not only from CERES normal forms. More information on expansion trees in Gapt can be found in [21,25]. The following demonstration shows how to obtain expansion proofs and Herbrand sequents (corresponding to the deep function of an expansion proof) from the CERES normal form p1 (defined in the demonstration above): scala> val exp = LKToExpansionProof( p1 ) scala> prooftool ( exp ) scala> val dp = exp.deep scala> prooftool ( dp )…”

Section: Implementation and Experiments In Gaptmentioning

confidence: 99%

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Extraction of Expansion Trees

Leitsch

Lolić

2018

J Autom Reasoning

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We define a new method for proof mining by CERES (cut-elimination by resolution) that is concerned with the extraction of expansion trees in first-order logic (see Miller in Stud Log 46(4):347-370, 1987) with equality. In the original CERES method expansion trees can be extracted from proofs in normal form (proofs without quantified cuts) as a postprocessing of cut-elimination. More precisely they are extracted from an ACNF, a proof with at most atomic cuts. We define a novel method avoiding proof normalization and show that expansion trees can be extracted from the resolution refutation and the corresponding proof projections. We prove that the new method asymptotically outperforms the standard method (which first computes the ACNF and then extracts an expansion tree). Finally we compare an implementation of the new method with the old one; it turns out that the new method is also more efficient in our experiments.

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Section: Dp(e)mentioning

confidence: 99%

Section: Dp(e)mentioning

confidence: 99%

Section: Implementation and Experiments In Gaptmentioning

confidence: 99%

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Extraction of Expansion Trees

Leitsch

Lolić

2018

J Autom Reasoning

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“…The first method [63] to address this problem introduced atomic cuts by using the resolution calculus, which is based on atomic cuts. A few years later, another method [34], based on discovering a grammar that could generate the Herbrand sequent of the proof to be compressed and then constructing a proof with cuts based on that grammar, was also proposed and implemented in GAPT [24,26,36,44,55].…”

Section: Related Workmentioning

confidence: 99%

“…All natural deduction, propositional and first-order resolution proof compression algorithms mentioned have been implemented by various people in Skeptik [12], whereas the sequent calculus algorithms have been implemented either in the CERes system [25] for cut-elimination by resolution, or in its successor GAPT [24,26,36,44,55] (the General Architecture for Proof Theory).…”

Section: Related Workmentioning

confidence: 99%

Greedy pebbling for proof space compression

Fellner

Paleo

2017

Int J Softw Tools Technol Transfer

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Automated reasoning tools for the verification and synthesis of software often produce proofs to allow independent certification of the correctness of the produced solutions. As proofs can be large, this paper considers the problem of compressing proofs with respect to their space, which is approximately proportional to the memory necessary to check them. Proof checking with a small amount of available memory is analogous to playing a pebbling game with a small number of pebbles. This paper exploits this analogy and describes novel algorithms for playing a pebbling game. The sequence of moves executed in the pebbling game then corresponds to an improved topological ordering of the nodes of the proof, leading to smaller memory consumption when the proof is checked. Because the number of possible pebbling strategies and topological orderings is too large, brute-force approaches to find optimal solutions are impractical, and hence, the new pebbling algorithms proposed here are based on heuristics for finding good, though not necessarily optimal, solutions. The algorithms are evaluated on the task of compressing the space of thousands of propositional resolution proofs generated by SAT-and SMT-solvers.

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Introducing Quantified Cuts in Logic with Equality

Hetzl

Leitsch

Reis

et al. 2014

Automated Reasoning

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Abstract. Cut-introduction is a technique for structuring and compressing formal proofs. In this paper we generalize our cut-introduction method for the introduction of quantified lemmas of the form ∀x.A (for quantifier-free A) to a method generating lemmas of the form ∀x1 . . . ∀xn.A. Moreover, we extend the original method to predicate logic with equality. The new method was implemented and applied to the TSTP proof database. It is shown that the extension of the method to handle equality and quantifier-blocks leads to a substantial improvement of the old algorithm.

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Understanding Resolution Proofs through Herbrand’s Theorem

Cited by 11 publications

References 20 publications

Extraction of Expansion Trees

Extraction of Expansion Trees

Greedy pebbling for proof space compression

Introducing Quantified Cuts in Logic with Equality

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