arXiv subjects

Alexander Leitsch

Publications and source records attributed to Alexander Leitsch.

12 recordsLinked to original sources

arXiv2026

Schemata, Cyclic Proofs and Herbrand Systems

Inductive proofs can be represented by proof schemata, a formalism that represents infinite sequences of proofs by recursive definitions. Since proof schemata avoid the explicit application of induction rules, they admit novel applications, one of which is the realization of Herbrand's theorem in the presence of induction. In this paper, we develop a new type of proof schema based on point transition systems. For skolemized proof schemata without quantified cuts, so-called Herbrand systems, that is, schemata of Herbrand instances of quantified formulas, can be computed. Herbrand systems also allow the representation of schemata of Herbrand sequents, thereby realizing Herbrand's theorem for proof schemata. We compare proof schemata with cyclic proofs and define a transformation from a large class of cyclic proofs to proof schemata. Finally, we show that proof schemata based on point transition systems are capable of proving the 2-Hydra statement, a well-known example that is provable by the cyclic proof system CLKID\omega but not in LKID.

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arXiv2025

Towards an Analysis of Proofs in Arithmetic

Inductive proofs can be represented as proof schemata, i.e. as parameterized sequences of proofs defined in a primitive recursive way. Applications of proof schemata can be found in the area of automated proof analysis where the schemata admit (schematic) cut-elimination and the construction of Herbrand systems. This work focuses on the expressivity of proof schemata. We show that proof schemata can simulate primitive recursive arithmetic. The translation of proofs in arithmetic to proof schemata can be considered as a crucial step in the analysis of inductive proofs.

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arXiv2024

Herbrand's Theorem in Refutation Schemata

An inductive proof can be represented as a proof schema, i.e. as a parameterized sequence of proofs defined in a primitive recursive way. A corresponding cut-elimination method, called schematic CERES, can be used to analyze these proofs, and to extract their (schematic) Herbrand sequents, even though Herbrand's theorem in general does not hold for proofs with induction inferences. This work focuses on the most crucial part of the schematic cut-elimination method, which is to construct a refutation of a schematic formula that represents the cut-structure of the original proof schema. We develop a new framework for schematic substitutions and define a unification algorithm for resolution schemata. Moreover, we show that this new formalism allows the extraction of a structure from the refutation schema, called a Herbrand schema, which represents its Herbrand sequent.

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arXiv2019

Schematic Refutations of Formula Schemata

Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

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arXiv2016

The problem of Pi_2-cut-introduction

We describe an algorithmic method of proof compression based on the introduction of Pi_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 Pi_1-cuts. The Herbrand instances of a cut-free proof pi 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 Pi_2-cut formula A and a proof phi 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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arXiv2016

Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]

In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the formal language provided in [8]. Without this formalization a Herbrand system cannot be algorithmically extracted. In this work, we provide a restriction of the proof found in [5], the ECA-schema (Eventually Constant Assertion), or ordered infinitary pigeonhole principle, whose analysis can be completely carried out in the framework of [8], this is the first time the framework is used for proof analysis. From the refutation of the clause set and a substitution schema we construct a Herbrand system.

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arXiv2015

Analysis of Clause set Schema Aided by Automated Theorem Proving: A Case Study [Extended Paper]

The schematic CERES method [8] is a recently developed method of cut elimination for proof schemata, that is a sequence of proofs with a recursive construction. Proof schemata can be thought of as a way to circumvent adding an induction rule to the LK-calculus. In this work, we formalize a schematic version of the infinitary pigeonhole principle, which we call the Non-injectivity Assertion schema (NiA-schema), in the LKS-calculus [8], and analyse the clause set schema extracted from the NiA-schema using some of the structure provided by the schematic CERES method. To the best of our knowledge, this is the first appli- cation of the constructs built for proof analysis of proof schemata to a mathematical argument since its publication. We discuss the role of Automated Theorem Proving (ATP) in schematic proof analysis, as well as the shortcomings of the schematic CERES method concerning the formalization of the NiA-schema, namely, the expressive power of the schematic resolution calculus. We conclude with a discussion concerning the usage of ATP in schematic proof analysis.

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arXiv2014

Introducing Quantified Cuts in Logic with Equality

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 $\forall x.A$ (for quantifier-free $A$) to a method generating lemmas of the form $\forall x_1\ldots\forall x_n.A$. 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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arXiv2014

Algorithmic Introduction of Quantified Cuts

We describe a method for inverting Gentzen's cut-elimination in classical first-order logic. Our algorithm is based on first computign a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize such a compression. Finally, a proof using these cut-formulas is constructed. This method allows an exponential compression of proof length. It can be applied to the output of automated theorem provers, which typically produce analytic proofs. An implementation is available on the web and described in this paper.

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arXiv2013

PROOFTOOL: a GUI for the GAPT Framework

This paper introduces PROOFTOOL, the graphical user interface for the General Architecture for Proof Theory (GAPT) framework. Its features are described with a focus not only on the visualization but also on the analysis and transformation of proofs and related tree-like structures, and its implementation is explained. Finally, PROOFTOOL is compared with three other graphical interfaces for proofs.

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arXiv2013

CERES for First-Order Schemata

The cut-elimination method CERES (for first- and higher-order classical logic) is based on the notion of a characteristic clause set, which is extracted from an LK-proof and is always unsatisfiable. A resolution refutation of this clause set can be used as a skeleton for a proof with atomic cuts only (atomic cut normal form). This is achieved by replacing clauses from the resolution refutation by the corresponding projections of the original proof. We present a generalization of CERES (called CERESs) to first-order proof schemata and define a schematic version of the sequent calculus called LKS, and a notion of proof schema based on primitive recursive definitions. A method is developed to extract schematic characteristic clause sets and schematic projections from these proof schemata. We also define a schematic resolution calculus for refutation of schemata of clause sets, which can be applied to refute the schematic characteristic clause sets. Finally the projection schemata and resolution schemata are plugged together and a schematic representation of the atomic cut normal forms is obtained. A major benefit of CERESs is the extension of cut-elimination to inductively defined proofs: we compare CERESs with standard calculi using induction rules and demonstrate that CERESs is capable of performing cut-elimination where traditional methods fail. The algorithmic handling of CERESs is supported by a recent extension of the CERES system.

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arXiv2008

How to acknowledge hypercomputation?

We discuss the question of how to operationally validate whether or not a "hypercomputer" performs better than the known discrete computational models.

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