Daniel Weller

Towards algorithmic cut-introduction

We describe a method for abbreviating an analytic proof in classical first-order logic by the introduction of a lemma. Our algorithm is based on first computing a compressed representation of the terms present in the analytic proof and then a cut-formula that realizes such a compression. This method can be applied to the output of automated theorem provers, which typically produce analytic proofs.

Herbrand Sequent Extraction

Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzens sequent calculus LK for classical first-order logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES .

Algorithmic introduction of quantified cuts

Abstract We describe a method for inverting Gentzens cut-elimination in classical first-order logic. Our algorithm is based on first computing 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. Concerning asymptotic complexity, this method allows an exponential compression of quantifier complexity (the number of quantifier-inferences) of proofs.

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 ∀ x.A (for quantifier-free A) to a method generating lemmas of the form ∀ x 1 … ∀ 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.

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.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.

CERES in higher-order logic

We define a generalization CERES ! of the first-order cut-elimination method CERES to higher-order logic. At the core of CERES ! lies the computation of an (unsatisfiable) set of sequents CS(�) (the characteristic sequent set) from a proof � of a sequent S. A refutation of CS(�) in a higher-order resolution calculus can be used to transform cut-free parts of � (the proof projections) into a cutfree proof of S. An example illustrates the method and shows that CERES ! can produce meaningful cut-free proofs in mathematics that traditional cutelimination methods cannot reach.

Cut-Elimination and Proof Schemata

By Gentzens famous Hauptsatz the cut-elimination theorem every proof in sequent calculus for first-order logic with cuts can be transformed into a cut-free proof; cut-free proofs are analytic and consist entirely of syntactic material of the end-sequent the proven theorem. But in systems with induction rules, cut-elimination is either impossible or does not produce proofs with the subformula property. One way to overcome this problem is to formulate induction proofs as infinite sequences of proofs in a uniform way and to develop a method, which yields a uniform description of the corresponding cut-free proofs. We present such a formalism, as an alternative to systems with induction rules, and define a corresponding cut-elimination method based on the CERES-method for first-order logic. The basic tools of proof theory, such as sequent- and resolution calculi are enriched with inductive definitions and schemata of terms, formulas, proofs, etc. We define a class of inductive proofs which can be transformed into this formalism and subjected to schematic cut-elimination.

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. nWe 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.

System description: the proof transformation system CERES

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by extracting a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an ACNF, an LK-proof with only atomic cuts. n nThe system CERES, an implementation of the CERES-method has been used successfully in analyzing nontrivial mathematical proofs (see 4).In this paper we describe the main features of the CERES system with special emphasis on the extraction of Herbrand sequents and simplification methods on these sequents. We demonstrate the Herbrand sequent extraction and simplification by a mathematical example.

On the Generation of Quantified Lemmas

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 second-order 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.