Jacques Calmet

Computer Algebra Systems

A survey is given of Computer algebra systems, with emphasis on design and implementation aspects, by presenting a review of the development of ideas and methods in a historical perspective, by us considered as instrumental for a better understanding of the rich diversity of now available facilities. We first indicate which classes of mathematical expressions can be stated and manipulated in different systems before we touch on different general aspects of usage, design and implementation, such as language design, encoding, dynamic storage allocation and a symbolic-numeric interface. Then we discuss polynomial and rational function systems, by describing ALTRAN and SAC-2. This is followed by a comparison of some of the features of MATHLAB-68, SYMBAL and FORMAC, which are pretended general purpose systems. Before considering giants (MACSYMA and SCRATCHPAD) and gnomes (muMATH-79), we give the main characteristics of TRIGMAN, CAMAL and REDUCE, systems we tend to consider as grown out special purpose facilities. Finally we mention some modern algebra systems (CAYLEY and CAMAC-79) in relation to recent proposals for a language for computational algebra. We conclude by stipulating the importance of documentation. Throughout this discussion related systems and facilities will be mentioned. Noticeable are ALKAHEST II, ALP AK, ANALITIK, ASHMEDAI, NETFORM, PM, SAC-1, SCHOONSCHIP, SHEEP, SMP, SYCOPHANTE and TAYLOR.

Specification and Integration of Theorem Provers and Computer Algebra Systems

Computer algebra systems (CASs) and automated theorem provers (ATPs) exhibit complementary abilities. CASs focus on efficiently solving domain-specific problems. ATPs are designed to allow for the formalization and solution of wide classes of problems within some logical framework. Integrating CASs and ATPs allows for the solution of problems of a higher complexity than those confronted by each class alone. However, most experiments conducted so far followed an ad-hoc approach, resulting in tailored solutions to specific problems. A structured and principled approach is necessary to allow for the sound integration of systems in a modular way. The Open Mechanized Reasoning Systems (OMRS) framework was introduced for the specification and implementation of mechanized reasoning systems, e.g. ATPs. The approach was recasted to the domain of computer algebra systems. In this paper, we introduce a generalization of OMRS, named OMSCS (Open Mechanized Symbolic Computation Systems). We show how OMSCS can be used to soundly express CASs, ATPs, and their integration, by formalizing a combination between the Isabelle prover and the Maple algebra system. We show how the integrated system solves a problem which could not be tackled by each single system alone.

Artificial Intelligence and Symbolic Mathematical Computation

This introductory paper summarizes the picture of the territory common to AI and SMC that has evolved from discussions following the presentation of papers given at the 1992 Karlsruhe conference. Its main objective is to highlight some patterns that can be used to guide both sketches of the state of the art in this territory and suggestions for future research activities.

Artificial Intelligence, Automated Reasoning, and Symbolic Computation

Many problems may be viewed as constraint satisfaction problems. Application domains range from construction scheduling to bioinformatics. Constraint satisfaction problems involve finding values for problem variables subject to restrictions on which combinations of values are allowed. For example, in scheduling professors to teach classes, we cannot schedule the same professor to teach two different classes at the same time. There are many powerful methods for solving constraint satisfaction problems (though in general, of course, they are NP-hard). However, before we can solve a problem, we must describe it, and we want to do so in an appropriate form for efficient processing. The Cork Constraint Computation Centre is applying artificial intelligence techniques to assist or automate this modelling process. In doing so, we address a classic dilemma, common to most any problem solving methodology. The problem domain experts may not be expert in the problem solving methodology and the experts in the problem solving methodology may not be domain experts. The author is supported by a Principal Investigator Award from Science Foundation Ireland. J. Calmet et al. (Eds.): AISC-Calculemus 2002, LNAI 2385, p. 1, 2002. c

KOMET - A System for the Integration of Heterogeneous Information Sources

We present KOMET, an architecture for the intelligent integration of heterogeneous information sources. It is based on the idea of a mediator, which is an independent software layer between an application and various knowledge sources which need to be accessed. We present an especially suitable logic-based language for encoding typical mediation tasks like conditional preference strategies, schema integration or data inconsistency resolution. Using annotated logic, KOMET is able to perform various common types of reasoning, such as probabilistic, fuzzy, paraconsistent and certain types of temporal and spatial reasoning. In combination with an extensible type system and the embedding of external knowledge sources as constraint domains, our mediation language offers a rich framework, which not only facilitates access to structured information, but as well supports unstructured and semi-structured information. A number of examples show the practical application of our approach.

A generic query-translation framework for a mediator architecture

A mediator is a domain-specific tool to support uniform access to multiple heterogeneous information sources and to abstract and combine data from different but related databases to gain new information. This middleware product is urgently needed for these frequently occurring tasks in a decision support environment. In order to provide a front end, a mediator usually defines a new language. If an application or a user submits a question to the mediator, it has to be decomposed into several queries to the underlying information sources. Since these sources can only be accessed using their own query language, a query translator is needed. This paper presents a new approach for implementing query translators. It supports conjunctive queries as well as negation. Care is taken to enable information sources of which processing capabilities do not allow conjunctive queries in general. Rapid implementation is guided by reusing previously prepared code. The specification of the translator is done declaratively and domain-independently.

Algorithmic Methods for Lie Pseudogroups

We present an algorithm to complete any given system of differential equations to an involutive system as needed e.g. for concrete applications of Lie pseudogroups. It is based on jet bundle formalism and formal theory. An implementation in the computer algebra system AXIOM is described.

Combining Theorem Proving and Symbolic Mathematical Computing

An intelligent mathematical environment must enable symbolic mathematical computation and sophisticated reasoning techniques on the underlying mathematical laws. This paper disscusses different possible levels of interaction between a symbolic calculator based on algebraic algorithms and a theorem prover. A high level of interaction requires a common knowledge representation of the mathematical knowledge of the two systems. We describe a model for such a knowledge base mainly consisting of type and algorithm schemata, algebraic algorithms and theorems.

Classification of Communication and Cooperation Mechanisms for Logical and Symbolic Computation Systems

The combination of logical and symbolic computation systems has recently emerged from prototype extensions of stand-alone systems to the study of environments allowing interaction among several systems. Communication and cooperation mechanisms of systems performing any kind of mathematical service enable one to study and solve new classes of problems and to perform efficient computation by distributed specialized packages. The classification of communication and cooperation methods for logical and symbolic computation systems given in this paper provides and surveys different methodologies for combining mathematical services and their characteristics, capabilities, requirements, and differences. The methods are illustrated by recent well-known examples. We separate the classification into communication and cooperation methods. The former includes all aspects of the physical connection, the flow of mathematical information, the communication language(s) and its encoding, encryption, and knowledge sharing. The latter concerns the semantic aspects of architectures for cooperative problem solving

From the OntoBayes Model to a Service Oriented Decision Support System

The aim of this paper is to propose a service oriented decision support system based on an ontology-driven uncertainty model (OntoBayes). OntoBayes consists of knowledge and decision model parts. The former is the integration of ontologies and Bayesian networks while the latter can describe different decision models. OntoBayes gives a solution to deal with uncertainty and structure complexity and provides decision models by decision making. In order to construct a decision support system, a service oriented framework and architecture was introduced finally.