Stefan Hetzl

A graph–theoretic approach to steganography

We suggest a graph-theoretic approach to steganography based on the idea of exchanging rather than overwriting pixels. We construct a graph from the cover data and the secret message. Pixels that need to be modified are represented as vertices and possible partners of an exchange are connected by edges. An embedding is constructed by solving the combinatorial problem of calculating a maximum cardinality matching. The secret message is then embedded by exchanging those samples given by the matched edges. This embedding preserves first-order statistics. Additionally, the visual changes can be minimized by introducing edge weights. We have implemented an algorithm based on this approach with support for several types of image and audio files and we have conducted computational studies to evaluate the performance of the algorithm.

CERES: An analysis of Fürstenberg's proof of the infinity of primes

The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzens famous cut-elimination theorem stating that synthetic proofs can be transformed into analytic ones. CERES (cut-elimination by resolution) is a cut-elimination method that has the advantage of considering the original proof in its full generality which allows the extraction of different analytic arguments from it. In this paper we will use an implementation of CERES to analyze Furstenbergs topological proof of the infinity of primes. We will show that Euclids original proof can be obtained as one of the analytic arguments from Furstenbergs proof. This constitutes a proof-of-concept example for a semi-automated analysis of realistic mathematical proofs providing new information about them.

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 .

Applying tree languages in proof theory

We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications.

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.

Cut-Elimination: Experiments with 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 constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts.

Proof transformation by 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 constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LK-proof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cut-elimination method. The system CERES already proved efficient in handling very large 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.

System Description: GAPT 2.0

GAPT General Architecture for Proof Theory is a proof theory framework containing data structures, algorithms, parsers and other components common in proof theory and automated deduction. In contrast to automated and interactive theorem provers whose focus is the construction of proofs, GAPT concentrates on the transformation and further processing of proofs. In this paper, we describe the current 2.0 release of GAPT.