Wolfgang Windsteiger

Johannes Kepler University Linz | JKU · Institute of Symbolic Computation

We report on using logic software in a novel course-format for an undergraduate logic course for students in computer science or artificial intelligence. Although being designed as the students’ basic introduction to the field of logic, the course features a novel structure and it adds some modern content, such as SAT and SMT solving, to the tradit...

We report on several scenarios of using automated theorem proving software in university education. In particular, we focus on using the Theorema system in a software-enhanced logic-course for students in computer science or artificial intelligence. The purpose of using logic-software in our teaching is not to teach students the proper use of a par...

In earlier work presented at CICM, four theorem provers (Isabelle, Mizar, Hets/CASL/TPTP, and Theorema) were compared based on a case study in theoretical economics, the formalization of the landmark Theorem of Vickrey in auction theory. At the time of this comparison the Theorema system was in a state of transition: The original Theorema system (T...

The Theorema project aims at the development of a computer assistant for the working mathematician. Support should be given throughout all phases of mathematical activity, from introducing new mathematical concepts by definitions or axioms, through first (computational) experiments, the formulation of theorems, their justification by an exact proof...

Theorema 2.0 stands for a re-design including a complete re-implementation of the Theorema system, which was originally designed, developed, and implemented by Bruno Buchberger and his Theorema group at RISC. In this talk, we want to present the current status of the new implementation, in particular the new user interface of the system.

Theorema 2.0 stands for a re-design including a complete re-implementation of the Theorema system, which was originally designed, developed, and implemented by Bruno Buchberger and his Theorema group at RISC. In this paper, we present the first prototype of a graphical user interface (GUI) for the new system. It heavily relies on powerful interacti...

Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machine-checked proofs. We investig...

Auctions allocate trillions of dollars in goods and services every year. Auction design can have significant consequences, but its practice outstrips theory. We seek to advance auction theory with help from mechanised reasoning. To that end we are developing a toolbox of formalised representations of key facts of auction theory, which will allow au...

Theorema 2.0 stands for a redesign including a complete reimplementation of the Theorema system, which was originally designed, developed, and implemented by Bruno Buchberger and his Theorema group at RISC. In this paper, we present the first prototype of a graphical user interface (GUI) for the new system. It heavily relies on powerful interactive...

Funktionen sind ein zentrales Hilfsmittel der Mathematik und erlauben, die Abhängigkeit gewisser Größen von anderen zu beschreiben, etwa die Abhängigkeit des Preises eines Produkts von Angebot und Nachfrage. Da durch sie Objekte einer Menge auf Objekte einer anderen Menge abgebildet werden, wird auch oft von Abbildungen gesprochen. Funktionen könne...

Die in diesem Kapitel vorgestellten Algorithmen behandeln m × n-Matrizen, worunter wir uns zunächst „Rechtecksschemata“ bestehend aus m Zeilen und n Spalten vorstellen wollen. Unser Hauptaugenmerk liegt auf dem Lösen reeller linearer Gleichungssysteme der Form $$ \begin{array}{*{20}c} {A_{11} x_1 } & { + \ldots + } & {A_{1n} x_n } & = & {b_1 } \\ \...

Gegenstand dieses Kapitels sind Gleichungssysteme der Gestalt $$ \begin{array}{*{20}c} {F_1 (x_1 , \ldots ,x_n ) = 0,} \\ \vdots \\ {F_m (x_1 , \ldots ,x_n ) = 0,} \\ \end{array} $$ bei denen zu m gegebenen reell-wertigen Funktionen F j : ℝn → ℝ nach einer Lösung x̄1,...,x̄n gesucht wird. Unter Verwendung der vektorwertigen Funktion F: ℝn → ℝm mit...

Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can—at least in principle—also be carried through in formal systems such as Theorema. In this paper we describe experiments c...

Zahlen sind die Grundbausteine der Mathematik. Wir besprechen daher in diesem Kapitel verschiedene Zahlenmengen und ihre Darstellung am Computer. Für eine Menge A und n∈ℕ ist An : = A ×¼×A: = { (a1 , ¼,an ) | ai Î A f[(u)\ddot]r 1 \leqslant i \leqslant n} , A^n : = A \times \cdots \times A: = \{ (a_1 , \ldots ,a_n ) | a_i \in {\rm A} f\ddot ur 1...

In diesem Kapitel schaffen wir die Voraussetzungen für einen algorithmisch orientierten Zugang zur Lösung mathematischer Fragestellungen. Dazu überlegen wir uns, wie mathematische Probleme spezifiziert und welche Aussagen über deren Lösung getroffen werden können. Anhand einführender Beispiele wenden wir uns Algorithmen zur Problemlösung zu und dis...

In diesem Kapitel beschäftigen wir uns mit den Grundlagen zu univariaten Polynomen. Diese stellt man sich häufig als Summe von Potenzen einer Variable vor, auf der dann auch die Termdarstellung der Polynomfunktionen beruht. So besteht etwa ein Naheverhältnis zwischen dem Polynom 7−x+2x3 und der Funktion x→7−x+2x3. Wir erklären, wie man mit Polynome...

This book is a synopsis of the basic and applied research carried out at the various research institutions of Softwarepark Hagenberg in Austria. Started in 1987, following a decision of the government of Upper Austria to create a scientific, technological, and economic impulse for the region and for the international community, Softwarepark Hagenbe...

Observing is the process of obtaining new knowledge, expressed in language, by bringing the senses in contact with reality. Reasoning, in contrast, is the process of obtaining new knowledge from given knowledge, by applying certain general transformation rules that depend only on the form of the knowledge and can be done exclusively in the brain wi...

Ein Vektorraum (über einem Körper K) ist eine mathematische Struktur, deren Objekte — die Vektoren — addiert und mit Elementen aus K multipliziert werden können. Die aus der Schule bekannten Vektoren mit zwei oder drei Koordinaten liefern geometrisch auschauliche Beispiele für Vektorräume (über ℝ), die Elemente eines Vektorraumes können aber auch v...

The Mutilated Checkerboard Problem has some tradition as a bench-mark problem for automated theorem proving systems. Informally speaking, it states that an 8 by 8 checkerboard with the two opposite corners removed cannot be covered by dominoes. Various solutions using different approaches have been presented since its original statement by John McC...

We present an environment for learning and teaching mathematics that aims at inspiring the creative potential of students by enabling the learners to perform various kinds of interactive experiments during their learning process. Computer interactions are both of visual and purely formal mathematical nature, where the computer-algebra system Mathem...

CreaComp provides an electronic environment for learning and teaching math- ematics that aims at inspiring the creative potential of students. During their learning process, students are encouraged to engage themselves in various kinds of interactive experiments, both of visual and purely formal mathematical na- ture. The computer-algebra system Ma...

Theorema is a project that aims at supporting the entire process of mathematical theory exploration within one coherent logic and software system. This survey paper illustrates the style of Theorema-supported mathematical theory exploration by a case study (the automated synthesis of an algorithm for the construction of Gröbner Bases) and gives an...

This paper presents some fundamental aspects of the design and the im- plementation of an automated prover for Zermelo-Fraenkel set theory within the well-known Theorema system. The method applies the \Prove- Compute-Solve"-paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory.

Statement Theorem[”gsqrt[2] irrational”, ¬ rat[\(\sqrt{2}\)]]

Analytica V is a theorem proving system that is built on top of the symbolic compu-tation system Mathematica. It was originally designed by E. Clarke and X. Zhao in the early 1990's. We describe here a redesign of the system that extends its abilities to reasoning about some aspects of number theory.

Short descriptions of computer algebra systems are presented in three sections: major systems, special purpose systems, and packages. However, the separation between special purpose systems and packages is not to be taken too literally. An older survey is the paper by Calmet and van Hulzen in [Buchberger et al. 1982]. There is now an excellent new...

In this paper, we present the Theorema Set Theory Prover. This prover is designed for proving statements involving notions from set theory using natural deduction inference rules for set theory. Moreover, it applies the PCS paradigm (Proving-Computing-Solving) for generating natural proofs that has already been used in other provers in the Theorem...

The THEOREMA project aims at supporting, within one consistent logic and one coherent software system, the entire mathematical exploration cycle including the phase of proving. In this paper we report on some of the new features of THEOREMA that have been designed and implemented since the first expository version of THEOREMA in 1997. These feature...

The world of mathematical domains is structured hierarchically. There are elementary domains and there are well-known techniques how to build up new domains from existing ones. Which of the domains to view as the actual basis of the hierarchy is the freedom of the mathematician who wants to work with these domains and it depends of course on the in...

. The Theorema project aims at integrating computation and deduction in a system that can be used by the working scientist for building and checking mathematical models, including the design and verification of new algorithms. Currently, the system uses the rewrite engine of the computer algebra system Mathematica for building and combining a numbe...

Interactive software systems that are designed to offer proving and computing facilities at the same time face the problem of evaluation of formulae: In the situation of computing, a formula given to the system should be evaluated whereas in the situation of proving the formula should be kept unevaluated. Also, in the Theorema project we use the sa...

This document should serve as a User's Guide to the library GR OBNER and its accompanying I/O library GR OBNER-IO for those users who do not intend to change any part of the library but rather want to use it as a "Grobner--basis--calculator". Therefore, no deep insight into the structure of the programs, software-technological issues, and data stru...

Almost every Computer Algebra System contains some implementation of the Gröbner bases algorithm. The present implementation has the following specific features: - The source code is distributed and publically available free of charge. - The library is written in C. - A simple but efficient mechanism of polymorphism is implemented that enables the...

This paper shows one possibility how to achieve some aspects of object-oriented programming in C. Since this should be used in Algebraic Computation, the most important feature one desires to have, is the possibility to use the same names for algorithms that perform the same actions. For example the multiplication algorithm should always be called

The main aim of our work with PCL, a kind of "object-oriented" expansion of COMMON-LISP, is to test whether it is suitable for the implementation of our Grobner bases package. Criteria for this are speed and the facility of generic programming. For this purpose we implemented the domain of Rational Functions . For arithmetic in this field it is nec...

Introduction We present an environment for learning and teaching mathematics that aims at inspiring the creative potential of students by enabling the learners to perform various kinds of interactive computer experiments during their learning process. Computer interactions are both of visual and purely formal mathematical nature, where the computer...

Almost every Computer Algebra System contains some implementation of the Gröbner bases algorithm. The present implementation has the following specific features: - The source code is distributed and publically available free of charge. - The library is written in C. - A simple but efficient mechanism of polymorphism is implemented that enables the...

We present a case study using the Theorema system to explore an algorithm for polynomial interpolation. The emphasis of the case study lies on formulating mathematical knowledge in one language that appears in its syntax close to common mathematical language but is precise enough to formulate all details necessary for proving. Moreover, the languag...

Kurzfassung Transformation beschreibt im Wesentlichen die algorithmische Rückführung eines Prob-lems auf ein einfacheres Problem durch sogenannte Transformationsfunktionen. Ist für das einfachere Problem ein Lösungsalgorithmus bekannt, so bekommen wir dadurch automatisch auch einen Lösungsalgorithmus für das ursprüngliche Problem. Wir bes-chreiben...