Algorithmic introduction of quantified cuts

Hetzl, Stefan; Leitsch, Alexander; Reis, Giselle; Weller, Daniel S.

doi:10.1016/j.tcs.2014.05.018

Cited by 30 publications

(31 citation statements)

References 33 publications

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“…For example, motivated by the aim to structure and compress automatically generated proofs, an algorithm for Π 1 cut-introduction based on proof grammars has been developed in [7]. This method has been implemented and empirically evaluated with good results in [8].…”

Section: Discussionmentioning

confidence: 99%

Herbrand's theorem revisited

Afshari

Hetzl

Leigh

2016

Proc Appl Math and Mech

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We present recent results on the deepening connection between proof theory and formal language theory. To each first-order proof with prenex cuts of complexity at most Πn/Σn, we associate a typed (non-deterministic) tree grammar of order n (equivalently, an order n recursion scheme) that abstracts the computation of Herbrand sets obtained through Gentzen-style cut elimination. Apart from offering a means to compute Herbrand expansions directly from proofs with cuts, these grammars provide a structural counterpart to Herbrand's theorem that opens the door to tackling a number of questions in proof theory such as proof equivalence, proof compression and proof complexity.

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Section: Discussionmentioning

confidence: 99%

Herbrand's theorem revisited

Afshari

Hetzl

Leigh

2016

Proc Appl Math and Mech

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“…We define the notion of an extended Herbrand sequent as in [16]; for simplicity we do not consider blocks of quantifiers in the cuts, but only formulas of the form ∀x∃y.A where A is quantifier-free, V (A) ⊆ {x, y}, and V (A) denotes the set of variables in A. As in the case of ∀-cuts, extended Herbrand sequents represent proofs with cuts by encoding the cuts by implication formulas.…”

Section: Proof-theoretic Infrastructurementioning

confidence: 99%

The problem of Π2-cut-introduction

Leitsch

Lettmann

2018

Theoretical Computer Science

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We describe an algorithmic method of proof compression based on the introduction of Π 2 -cuts into a cut-free LK-proof. The current approach is based on an inversion of Gentzen's cut-elimination method and extends former methods for introducing Π 1 -cuts. The Herbrand instances of a cut-free proof π of a sequent S are described by a grammar G which encodes substitutions defined in the elimination of quantified cuts. We present an algorithm which, given a grammar G, constructs a Π 2 -cut formula A and a proof π ′ of S with one cut on A. It is shown that, by this algorithm, we can achieve an exponential proof compression.

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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%

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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Algorithmic introduction of quantified cuts

Cited by 30 publications

References 33 publications

Herbrand's theorem revisited

Herbrand's theorem revisited

The problem of Π2-cut-introduction

Greedy pebbling for proof space compression

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