Ursula Martin

Ordered Rewriting and Confluence

One of the major problems in term rewriting theory is what to do with an equation which cannot be ordered into a rule. Many solutions have been proposed, including the use of special unification algorithms or of unfailing completion procedures.

Automating the Knuth Bendix ordering

Knuth and Bendix proposed a very versatile technique for ordering terms, based upon assigning weights to operators and then to terms by adding up the weights of the operators they contain. Our purpose in this paper is as follows. First we give some examples to indicate the flexibility of the method. Then we give a simple and practical algorithm, based on solving systems of linear inequalities, for determining whether or not a set of rules can be ordered by a Knuth Bendix ordering. We also describe how this algorithm may be incorporated in a completion procedure which thus considers all possible choices of weights which orient a given equation.

Automated theorem proving in support of computer algebra: symbolic definite integration as a case study

We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integration, and present a verifiable symbolic definite integral table look up: a system which matches a query comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of lemmas about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon FatemanA¢Â�Â�s web-based table by including parametric limits of integration and queries with side conditions.

Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS

We describe an interface between version 6 of the Maple computer algebra system with the PVS automated theorem prover. The interface is designed to allow Maple users access to the robust and checkable proof environment of PVS. We also extend this environment by the provision of a library of proof strategies for use in real analysis. We demonstrate examples using the interface and the real analysis library. These examples provide proofs which are both illustrative and applicable to genuine symbolic computation problems.

How to Choose Weights in the Knuth Bendix Ordering

Knuth and Bendix proposed a very versatile technique for ordering terms, based upon assigning weights to operators and then to terms by adding up the weights of the operators they contain. Our purpose in this paper is twofold. First we give some examples to indicate the flexibility of the method. Then we give a simple and practical algorithm, based on the simplex algorithm, for determining whether or not a set of rules can be ordered by a Knuth Bendix ordering.

A geometrical approach to multiset orderings

We survey different ways of ordering multisets, and give a classification of multiset orderings based on the notion of a cone in Rn. This enables us to derive new results about the dominance ordering and the standard multiset ordering.

Unification in Boolean rings

We show that two Boolean terms which are unifiable have a most general unifier, which can be described using the terms themselves and a single unifier. Techniques for finding a single unifier are given.

A Hoare logic for single-input single-output continuous-time control systems

This paper presents a Hoare-style logic for reasoning about the frequency response of control systems in the continuous-time domain. Two properties, the gain (amplitude) and phase shift, of a control system are considered. These properties are for a sinusoidal input of variable frequency. The logic operates over a simplified form of block diagram, including arbitrary transfer functions, feedback loops, and summation of signals. Reasoning is compositional, i.e. properties of a system can be deduced from properties of its subsystems. A prototype tool has been implemented in a mechanised theorem prover.

The order types of termination orderings on monadic terms, strings and multisets

Total well-founded orderings on monadic terms satisfying the replacement and full invariance properties are considered. It is shown that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega , omega /sup 2/, or omega /sup omega /. It is further shown that a familiar construction gives rise to continuum many such orderings of order type omega . A new family of such orderings of order type omega is constructed, and it is shown that there are only four such orderings of order type omega /sup omega /, the two familiar recursive path orderings and two closely related orderings. It is shown that any total well-founded ordering on N/sup n/ that is preserved under vector addition must have order type omega /sup lambda / for some 1<or=k<or=n; if k<n, there are continuum many such orderings, and if k=n, there are only n-factorial, namely the n-factorial lexicographic orderings.<<ETX>>

Almost all

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups with automorphism group a p-group. Yet the goal of this paper is to prove that almost all finite p-groups do have automorphism group a p-group when p is odd. The asymptotic sense in which the theorem holds involves bounding the Frattini length of the p-groups and letting the number of generators go to infinity. The proof of the theorem draws on a detailed analysis of the Frattini series of a free group and the combinatorics linking finite p-groups and representations of GL(n, Fp). The case of p = 2 remains open.