Bruno Buchberger

Johannes Kepler University of Linz | JKU · Research Institute of Symbolic Computation (RISC)

Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly connected. We introduce a new project SC 2 to build a...

Mathematics - The Art of Thinking Manaemgent - The Art of Acting Meditation - The Art of Not Thinking and Not Acting. In this book Buchberger answers questions about mathematics, management and medition in the form of an interview. For hasty readers, each question first receives a short answer, which sometimes is provocative. Then the author goes...

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly...

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly...

We give a report on the EuKIM project, which was recently submitted to the EU Horizon 2020 program, INFRAIA-02-2017 (Integrating Activities for Starting Communities) topic, by a consortium of twelve European research groups. The project aims at building up a “Global Digital Math Library” (knowledge base) integrating and extending current efforts wo...

In the frame of the work of the Working Group “Global Digital Mathematical Library”, Jim Pitman proposed Aart Stam’s collection of combinatorial identities as a benchmark for “digitizing” mathematical knowledge. This collection seems to be a challenge for “digitization” because of its size (1300 pages in a .pdf file) and because of the fact that, f...

2016 SC-Square Project Poster

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...

We develop a new theory for treating boundary problems for linear ordinary differential equations whose fundamental system may have a singularity at one of the two endpoints of the given interval. Our treatment follows an algebraic approach, with (partial) implementation in the Theorema software system (which is based on Mathematica). We study an a...

In this talk we show how the theory of Groebner bases can be represented in the computer system Theorema, a system initiated by Bruno Buchberger in the mid-nineties. The main purpose of Theorema is to serve mathematical theory exploration and, in particular, automated reasoning. However, it is also an essential aspect of the Theorema philosophy tha...

In this talk we present the formalization and formal verification of the complexity analysis of Buchberger’s algorithm in the bivariate case in the computer system Theorema as a case study for using the system in mathematical theory exploration. We describe how Buchberger’s original complexity proof for Groebner bases can be carried out within the...

In this talk we argue that mathematics is essentially software. In fact, from the beginning of mathematics, it was the goal of mathematics to automate problem solving. By systematic and deep thinking, for problems whose solution was difficult in each individual instance, systematic procedures were found that allow to solve each instance without fur...

We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic repre...

In this paper, we summarize our recent work on establishing, for the first time, an algorithm for the symbolic solution of linear boundary problems. We put our work in the frame of Wen-Tsun Wu’s approach to algorithmic problem solving in analysis, geometry, and logic by mapping the significant aspects of the underlying domains into algebra. We brie...

Boundary value problems are of utmost importance for science and engineering. In fact, most differential equations come along with boundary conditions of some sort. It is therefore surprising that such problems—even in the linear case—have gained little attention in Symbolic Computation. Consequently, their coverage in computer algebra systems is r...

In our symbolic approach to boundary problems for linear ordinary differential equations we use the algebra of integro-differential operators as an algebraic analogue of differential, integral and boundary operators (Section 2). They allow to express the problem statement (differential equation and boundary conditions) as well as the solution opera...

In general, a function \(S\) is called a canonical simplifier for an equivalence relation \(\sim\) if \(S\) maps all elements of an equivalence class to the same representative of that class, i.e., whenever \(f\sim g\ ,\) then \(S(f)=S(g)\ .\) In our case, the equivalence relation is the congruence modulo a fixed basis \(B\ ,\) sometimes denoted by...

The development of computer technology has brought forth a renaissance of algorithmic mathematics which gave rise to the creation of new disciplines like Computational Mathematics. Symbolic Computation, which constitutes one of its major branches, is the main research focus of the Research Institute for Symbolic Computation (RISC). In the first Sec...

We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Green’s operators, composing boundary problems and integro-differential operators, and factoring boundary problems. Based on our factorization...

PCS (Proving-Computing-Solving) introduced by B. Buchberger in 2001 and S-Decomposition, introduced by T. Jebelean in 2001, are strategies for handling proof problems by combining logic inference steps (e.g., modus ponens, Skolemization, instantiation) with rewriting steps (application of definitions) and solving procedures based on algebraic techn...

Archives are implemented as an extension of Theorema for repre-senting mathematical repositories in a natural way. An archive can be con-ceived as one large formula in a language consisting of higher-order predicate logic together with a few constructs for structuring knowledge: attaching la-bels to subhierarchies, disambiguating symbols by the use...

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...

This book is a synopsis of basic and applied research done at the various research institutions of the Softwarepark Hagenberg in Austria. Starting with 15 coworkers in my Research Institute for Symbolic Computation (RISC), I initiated the Softwarepark Hagenberg in 1987 on request of the Upper Austrian Government with the objective of creating a sci...

Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing...

This report reviews the literature relevant for the research project "MathAgents: Mathematical Journals as Reasoning Agents" proposed by Bruno Buchberger as a technology transfer project based on the results of the SFB Project "Scientific Computing", in particular the project SFB 1302, "Theorema". The project aims at computer−supporting the referee...

We outline a prototype implementation of the algorithms for integro-differential operators/polynomials in [12]. Our approach based on a generic implementation of noncommutative monoid rings with reduction, programmed in the functors language of the THEOREMA system. The integro-differential operators—realized by a suitable quotient of noncommutative...

This booklet contains the invited talk by Kiyoshi Asai and the extended abstracts of the short communications session of the Third International Conference on

As in any other scientific field, the quality control (by peer reviewing), archiving and distribution of knowledge in mathematics is organized by journals. Currently, mathematical journals represent huge chunks of knowledge that sit passively on a shelf and wait for a human user to read or query them. In contrast, what we want to achieve in this pr...

We describe a design for a system for mathematical theory exploration that can be extended by implementing new reasoners using the logical input language of the system. Such new reasoners can be applied like the built-in reasoners, and it is possible to reason about them, e.g. proving their soundness, within the system. This is achieved in a practi...

In this paper we identify the organizational problems of mathematical knowledge management and describe tools that address some of these problems, namely the management of composite, hierarchical labels for formalized knowledge and instances of the problem of mathematical knowledge retrieval.

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...

Mathematics is characterized by its method of gaining knowledge, namely reasoning. The automation of reasoning has seen significant advances over the past decades and, thus, the expectation was that these advances would also have significant impact on the practice of doing mathematics. However, so far, this impact is small. We think that the reason...

In an oversimplified and abstract view, given a logic (syntax, semantics, inference system) L and a knowledge base K of formulae in L. “computer” for K enables the user to provide an expression (term, formula, program) T with a free variable x and a value υ (from an appropriate domain) and “evaluates” T x←υ (T with υ substituted for x) w.r.t. K in...

Let I be a zero-dimensional ideal in a polynomial ring F[s]:=F[s"1,...,s"n] over an arbitrary field F. We show how to compute an F-basis of the inverse system I^@? of I. We describe the F[s]-module I^@? by generators and relations and characterise the ...

This is the English translation (by Michael P. Abramson) of the PhD thesis of Bruno Buchberger, in which he introduced the algorithmic theory of Gröbner bases. Some comments by Buchberger on the translation and the thesis are given in an additional short paper in this issue of the Journal of Symbolic Computation.

Construction of geometrical objects by origami, the Japanese traditional art of paper folding, is enjoyable and intriguing. It attracted the minds of artists, mathematicians and computer scientists for many centuries. Origami will become a more rigorous, effective and enjoyable art if the origami constructions can be visualized on the computer and...

We present an algorithm for constructing a basis of the ideal of all polynomials, which vanish at a preassigned set of points {y1,...,ym} Kn, K a field. The algorithm yields also Newton-type polynomials for pointwise interpolation. These polynomials admit an immediate construction of interpolating polynomials and allow to shorten the algorithm, if...

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

This file may be copied and stored in data bases under the following conditions:- The file is kept unchanged (including this copyright note).- A message is sent to Bruno.Buchberger@jku.at.- If you use material contained in this file, cite it appropriately referring to the above talk and workshop.

The common goal of self-validating methods and computer algebra methods is to solve mathematical problems with complete rigor and with the aid of computers. The seminar focused on several aspects of such methods for computer-assisted proofs. @InProceedings{buchberger_et_al:DSP:2006:454, author = {Bruno Buchberger and Christian Jansson and Shin'ichi...

From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of th...

Since approximately 1960, symbolic computation added algebraicalgorithms (polynomial algorithms, simplification algorithms forexpressions, algorithms for integration, algorithms for theanalysis of algebraic structures like groups etc.) to numerics andprovided both numerical and algebraic algorithms in the frame ofpowerful integrated mathematical so...

This report contains a detailed case study in algorithm synthesis, the synthesis of sorting algorithms for tuples. We apply the lazy thinking method for algorithm synthesis proposed by the second author in several cycles of exploration which will illustrate both the details of the application of the lazy thinking method as well as some aspects conn...

In this paper we present an improvement of the lazy thinking method for program synthesis. We apply the lazy thinking method to a certain general class of problems (explicit problems) using a certain general algorithm scheme (divide-and-conquer) thus obtaining specifications for the subalgorithms of the scheme. This can be used in particular case s...

Origami (paper folding) has a long tradition in Japan’s culture and education. We are developing a computational origami system, based on symbolic computation system Mathematica, for performing and reasoning about origami on the computer. This system is based on the implementation of the six fundamental origami folding steps (origami axioms) formul...

We present a personal view and strategy for algorithm-supported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottom-up mathematical invention, the algorithmic generation of conjectures from failing proofs for...

Recently, as part of a general formal (i.e. logic based) method-ology for mathematical knowledge management we also introduced a method for the automated synthesis of correct algorithms, which we called the lazy thinking method. For a given concrete problem specifica-tion (in predicate logic), the method tries out various algorithm schemes and deri...

We extend first-order logic with sequence variables and sequence functions. We describe syntax, semantics and inference system for the extension, define an inductive theory with sequence variables and formulate induction rules. The calculus forms a basis for the top-down systematic theory exploration paradigm.

In this paper we identify the organizational problems of Mathematical Knowl-edge Management and describe tools that address one of these problems, namely, the additional annotation of formalized knowledge. We describe, then, how the tools are realized in the frame of the Theorema system.

Origami paper folding has a long tradition in Japan’s culture and education. The second author has recently developed a software system, based on functional logic programming and web-technology, for simulating origami paper folding on the computer (the "origami computing problem" or the "forward origami problem"). This system is based on the implem...

Acknowledgement: Sponsored by FWF (Österreichischer Fonds zur Förderung der Wissenschaftlichen Forschung; Austrian Science Foundation), project SFB 1302 ("Theorema") of the SFB 013 ("Numerical and Symbolic Scientific Computing").

Recently, we proposed a systematic method for top-down synthesis and verification of lemmata and algorithms called “lazy thinking method” as a part of systematic mathematical theory exploration (mathematical knowledge management). The lazy thinking method is characterized: •by using a library of theorem and algorithm schemes•and by using the inform...

We describe an extension of first-order logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, Löwenheim-Skolem, and Model Existence theorems remain valid. The obtained logic can be encoded as a special order-sorted first-order theory. We also defin...

Construction of geometrical objects by origami, the Japanese traditional art of paper folding, is enjoyable and intriguing. It attracted the minds of artists, mathematicians and computer scientists for many centuries. Origami will become a more rigorous, effective and enjoyable art if the origami constructions can be visualized on the computer and...

In this paper we identify the organizational problems of Mathematical Knowledge Management and describe tools that address one of these problems, namely, the management of composite, hierar-chical labels for formalized knowledge. We describe how the tools are realized in the frame of the Theorema system. Acknowledgments This work was partially supp...

We discuss the question of whether the central result of algorithmic Gröbner bases theory, namely the notion of S-polynomials together with the algorithm for constructing Grobner bases using S-polynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) algorithm synthesis method. We present the "lazy thinking" method...

We present a personal view and strategy for algorithm-supported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottom-up mathematical invention, the algorithmic generation of conjectures from failing proofs for...

We describe a tool, "Mathematical Knowledge Editor", which will fill an important gap in current technology for Mathematical Knowledge Management: the translation of mathematical knowledge contained in computer-processable files like, for example, LaTeX files into fully formal mathematical knowledge in the frame of a logic language like, for exampl...

This paper has two objectives: - provide a major case study for the use of categories and functors in Theorema, introduced in [Buchberger 1996] - provide a formulation of the main algorithms in Groebner bases theory in a generic way based on (a new variant of) the notion of reduction ring introduced in [Buchberger 1984] All the Theorema functor pro...

Algorithm retrieval is a special case of mathematical knowledge retrieval, which is one of the fundamental problems of mathematical knowledge management. In this paper, we distinguish between various versions of the problem of algorithm retrieval focusing on the version which can only be appropriately formulated in the frame of formal logic. This i...

We present a new method for solving regular boundary value problems for linear ordinary differential equations. Unlike existing methods that reduce everything to the functional level via the Green's function, our approach works on the level of operators throughout. We proceed by representing the operators needed as noncommutative polynomials using...

“Computational mathematics” (algorithmic mathematics) is the part of mathematics that strives at the solution of mathematical problems by algorithms. In a superficial view, some people might believe that computational mathematics is the easy part of mathematics in which trivial mathematics is made useful by repeating trivial steps sufficiently many...

A new approach for symbolically solving linear boundary value problems is presented. Rather than using general-purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable B...

In this paper, we study algorithm invention and verification as aspecific variant of systematic theory exploration and propose the "lazy thinking paradigm" for inventing and verifying algorithms automatically; i.e., for a given predicate logic specification of the problem in terms of a set of operations (functions and predicates), the method produc...

Origami (paper folding) has a long tradition in Japan's culture and education. We are developing a computational origami system, based on symbolic computation system Mathematica, for performing and reasoning about origami on the computer. This system is based on the implementation of the six fundamental origami folding steps (origami axioms) formul...