Karsten Homann

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.

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

Structures for Symbolic Mathematical Reasoning and Computation

Recent research towards integrating symbolic mathematical reasoning and computation has led to prototypes of interfaces and environments. This paper introduces computation theories and structures to represent mathematical objects and applications of algorithms occuring in algorithmic services. The composition of reasoning and computation theories and structures provide a formal framework for the specification of symbolic mathematical problem solving by cooperation of algorithms and theorems.

Towards the mathematics software bus

Abstract The Mathematics Software Bus is a software environment for combining heterogeneous systems performing any kind of mathematical computation. Such an environment will provide combinations of graphics, editing and computation tools through interfaces to already existing powerful software by flexible and powerful semantically integration. Communication and cooperation mechanisms for logical and symbolic computation systems enable to study and solve new classes of problems and to perform efficient computation in mathematics through cooperating specialized packages. We give an overview on the need for cooperation in solving mathematical problems and illustrate the advantages by several well-known examples. The needs and requirements for the Mathematics Software Bus and its architecture are demonstrated through some implementations of powerful interfaces between mathematical services.

Unified Domains and Abstract Computational Structures

This paper introduces a formalism to specify abstract computational structures (ACS) of mathematical domains of computation. This is a basic step of a project aiming at designing an environment for symbolic computing based upon knowledge representation and relying, when needed, on AI methods.