In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and crypto-systems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...

The mobile robot planning domain is dynamic, with goals becoming active asynchronously. In order to successfully operate in this environment, a robot must be able to interrupt and reformulate its plans of action on-the-fly. This paper investigates a method for incorporating the accomplishment of a new goal into a partially executed plan. A decision theoretic approach using net present value as the decision criterion serves as the basis for doing dynamic goal ordering. The appropriateness of net present value over benefit-cost ratio is argued. The approach has been implemented on a robot operating in an office setting. Examples from this domain are used to show the advantages of the approach with respect to fixed priority and heuristic based approaches. 1

. Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In this paper we try to get the best of both worlds by careful exploitation of a link between a theorem prover and a computer algebra system. 1 Motivation In the HOL theorem prover[5], a theory of real numbers has been developed, using a rigorous definition in terms of Dedekind cuts [8]. It is therefore possible to apply HOL to areas traditionally within the purview of Computer Algebra Systems (CASs). This offers two main benefits. Firstly, theorem provers are designed to manipulate proofs and theorems in a coherent and structured way, with all concepts clearly defined. By contrast, most CASs have no concept of `logic' as such -- they usually take an algebraic expression and return another pur...

Abstract. Any tree can be represented in a max/ma//y compact form as a directed acyclic graph where common subtrees are factored and shared, being represented only once. Such a compaction can be effected in linear time. It is used to save storage in implementations of functional programming languages, as well as in symbolic manipulation and computer a/gebra systems. In compiling, the compaction problem is known as the "common subexpression problem " and it plays a central r61e in register allocation, code generation and optimisation. We establish here that, under a variety of probabilistic models, a tree of size n has a compacted form of expected size asymptotically n C.--ogWi- ' where the constant C is explicitly related to the type of trees to be compacted and to the statistical model reflecting tree usage. In particular the savings in storage approach 100 % on average for large structures, which overperforms the commonly used form of sharing that is restmcted to leaves (atoms). Introduction. A tree can be compacted by representing occurrences of repeated subtrees only once. In that case, several pointers will point to the representation of any common subtree,

, the knowledge base MBase [3], a presentation planner (see x2), a user model, a pedagogical module, and deduction and The project which led to the results presented in this paper was funded by the German Bundesministerium fur Bildung und Forschung, contract 01AK008/2. The authors are responsible for the content of this publication. 2 computation service systems such as the proof planner of\Omega mega [6] and the Computer Algebra System (CAS) Maple [2]. The components can communicate over the Internet by a standardized xml-rpc protocol[7]. In Figure 1, the arrows represent the data flow between the components. For instance, requests of the user and (in the other direction) html-pages are communicated via a web-server. The presentation planner requests and processes information from MBase, from the user model, and from the pedagogical module in order to gen

CONTENTS CHAPTER PAGE 1 INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2 EFFICIENT NUMERICAL SIMULATION OF ELECTRON STATES IN QUANTUM WIRES : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 The equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.3 Reformulation as a fixed point problem : : : : : : : : : : : : : : : : : : : 13 2.4 The Eigenvalue Problem for Schrodinger's Equation : : : : : : : : : : : : 16 2.5 The Solution of the Nonlinear Poisson Equation : : : : : : : : : : : : : : 18 2.6 Stabilization and Acceleration : : : : : : : : : : : : : : : : : : : : : : : : 21 2.6.1 Stabilization by Adaptive Underrelaxation : : : : : : : : : : : : : 22 2.6.2 Acceleration by New

We present a numerical approach to the simulation of dielectric waveguides that is free of spurious modes and is based on the solution of an eigenvalue problem for the two transverse components of the magnetic field. We introduce a new discretization which has several computational advantages. In particular, by careful design of the discretization procedure we obtain systems of equations for the two components which are equivalent in the sense that a rotation over 90 ffi corresponds to a suitable permutation of indices. The eigenvalue problem is solved iteratively by using an adapted version of the Chebyshev-- Arnoldi algorithm. This approach takes full advantage of the sparsity of the matrix and circumvents the large memory requirements and the large computational complexity associated with dense methods. This allows us to employ meshes that are sufficiently fine to resolve higher modes without large discretization errors. 1 Introduction Dielectric channel waveguides are widely u...

. The `Interactive Textbook' project provides an architecture, basic knowledge representations, and techniques for second-generation interactive mathematics documents (textbooks, courses, tutorials) exhibiting, among others, the following features: separation of knowledge representation and functionalities, knowledge representation in an XML format that extends OpenMath, and adaptivity. The `Interactive Textbook ' includes an interactive proof planning system and computer algebra systems for exploratory learning purposes and uses knowledge acquired for automated proof planning to support active learning of mathematics. A prototypical interactive textbook is being implemented that will realize the dierent features explained and motivated in this paper. 1 Introduction Our previous work resulted in systems such as MathWeb [9] and the prototypical proof assistant system mega [1] including knowledge-based proof planning [12] and in ideas for a mathematical knowledge base [8]. ...

We report on new codes for sparse multivariate polynomial multiplication and division over the integers that we have integrated into Maple 14’s expand and divide commands. We describe our polynomial data structure and compare it with the one used by Maple. We then present some benchmarks comparing our software with the Magma, Maple, Singular, Trip and Pari computer algebra systems and also illustrating how multivariate polynomial factorization benefits from our improvements to polynomial multiplication and division. This work was supported by the MITACS NCE of Canada and Maplesoft. 1

Intelligent tutoring systems provide a promising application area for techniques from many subfields of Artificial Intelligence (AI), including knowledge representation, user modelling, rule-based systems, automated diagnosis, automated reasoning, adaptive hypermedia, natural language processing