Atomic Cut Introduction by Resolution: Proof Structuring and Compression

Paleo, Bruno Woltzenlogel

doi:10.1007/978-3-642-17511-4_26

Cited by 13 publications

(6 citation statements)

References 16 publications

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“…As every cut inference involves the derivation of a lemma (in its left premise) and its use (in the right premise), introducing cuts requires solving the difficult task of synthesizing lemmas. 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%

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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Section: Related Workmentioning

confidence: 99%

Greedy pebbling for proof space compression

Fellner

Paleo

2017

Int J Softw Tools Technol Transfer

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“…Work on cut-introduction can be found at a number of different places in the literature. Closest to our work are other approaches which aim to abbreviate or structure a given input proof: [35] is an algorithm for the introduction of atomic cuts that is capable of exponential proof compression. The method [15] for propositional logic is shown to never increase the size of proofs more than polynomially.…”

Section: Introductionmentioning

confidence: 99%

On the Generation of Quantified Lemmas

et al. 2018

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In this paper we present an algorithmic method of lemma introduction. Given a proof in predicate logic with equality the algorithm is capable of introducing several universal lemmas. The method is based on an inversion of Gentzen's cut-elimination method for sequent calculus. The first step consists of the computation of a compact representation (a so-called decomposition) of Herbrand instances in a cut-free proof. Given a decomposition the problem of computing the corresponding lemmas is reduced to the solution of a secondorder unification problem (the solution conditions). It is shown that that there is always a solution of the solution conditions, the canonical solution. This solution yields a sequence of lemmas and, finally, a proof based on these lemmas. Various techniques are developed to simplify the canonical solution resulting in a reduction of proof complexity. Moreover, the paper contains a comprehensive empirical evaluation of the implemented method and gives an application to a mathematical proof.

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“…Closest to our work are other approaches which aim to abbreviate or structure a given input proof. In [19] an algorithm for the introduction of atomic cuts that is capable of exponential proof compression is presented. The method [5] for propositional logic is shown to never increase the size of proofs more than polynomially.…”

Section: Introductionmentioning

confidence: 99%

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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Atomic Cut Introduction by Resolution: Proof Structuring and Compression

Cited by 13 publications

References 16 publications

Greedy pebbling for proof space compression

Greedy pebbling for proof space compression

On the Generation of Quantified Lemmas

Introducing Quantified Cuts in Logic with Equality

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