Herbrand Sequent Extraction

Hetzl, Stefan; Leitsch, Alexander; Weller, Daniel S.; Paleo, Bruno Woltzenlogel

doi:10.1007/978-3-540-85110-3_38

Cited by 25 publications

(22 citation statements)

References 11 publications

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“…But if we consider proofs of prenex end-sequents we can extract Herbrand sequents directly, instead of extracting expansion proofs and computing their deep functions. The method for Herbrand sequent extraction in the prenex case is based on collecting instances, and is described in [6,20].…”

Section: Proposition 1 the Transformation Et Is Sound: If ϕ Is A Proomentioning

confidence: 99%

“…proof, a proof of (one direction of) the equivalence of different definitions of the concept of a lattice. For background information about this proof and an informal version, we refer to Section 5 of [20]. Another interesting proof for analyzing and comparing the implemented methods is tape .…”

Section: Implementation and Experiments In Gaptmentioning

confidence: 99%

“…There are efficient algorithms for extracting Herbrand sequents from cut-free proofs, see e.g. [20]. Though every formula (and any sequent) can be transformed to prenex form such a transformation is unnatural and can have a disastrous impact on proof complexity (see [3]).…”

Section: Introductionmentioning

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: Proposition 1 the Transformation Et Is Sound: If ϕ Is A Proomentioning

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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“…In the case of sequent calculus proofs, Herbrand sequent extraction [35,62] was one of the earliest proposed techniques. It consists of obtaining a propositionally valid sequent from a first-order proof with the same propositional structure of the proof's end-sequent but with all needed instantiations for the quantified variables.…”

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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“…It plays the key role in Luckhardt's (manual) analysis [5] of Roth's theorem where it has been used to obtain polynomial bounds (which were obtained independently and by purely mathematical as opposed to logical methods by Bombieri and van der Poorten in [6]). The extraction and analysis of Herbrandsequents as described by Hetzl et al [7] has also been used in the computerassisted analysis of Fürstenberg's topological proof of the infinity of primes by Baaz et al [8] which yielded Euclid's original argument via cut-elimination. Herbrand's theorem and methods based on it have furthermore also been used in a number of smaller case studies such as [9] by Baaz et al or [10] by Urban.…”

Section: Introductionmentioning

confidence: 99%

Understanding Resolution Proofs through Herbrand’s Theorem

Hetzl

Libal

Riener

et al. 2013

Lecture Notes in Computer Science

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Abstract. Computer-generated proofs are usually difficult to grasp for a human reader. In this paper we present an approach to understanding resolution proofs through Herbrand's theorem and the implementation of a tool based on that approach. The information we take as primitive is which instances have been chosen for which quantifiers, in other words: an expansion tree. After computing an expansion tree from a resolution refutation, the user is presented this information in a graphical user interface that allows flexible folding and unfolding of parts of the proof. This interface provides a convenient way to focus on the relevant parts of a computer-generated proof. In this paper, we describe the prooftheoretic transformations, the implementation and demonstrate its usefulness on several examples.

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Herbrand Sequent Extraction

Cited by 25 publications

References 11 publications

Extraction of Expansion Trees

Extraction of Expansion Trees

Greedy pebbling for proof space compression

Understanding Resolution Proofs through Herbrand’s Theorem

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