# Russell Bradford's research while affiliated with University of Bath and other places

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## Publications (50)

seeks to introduce the ISSAC community to the DEWCAD project, which is based at Coventry University and the University of Bath, in the United Kingdom. The project seeks to push back the Doubly Exponential Wall of Cylindrical Algebraic Decomposition, through the integration of SAT/SMT technology, the extension of Lazard projection theory, and the de...

seeks to introduce the ISSAC community to the DEWCAD project, which is based at Coventry University and the University of Bath, in the United Kingdom. The project seeks to push back the Doubly Exponential Wall of Cylindrical Algebraic Decomposition, through the integration of SAT/SMT technology, the extension of Lazard projection theory, and the de...

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic...

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community, as a tool to identify satisfying solutions of problems in nonlinea...

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear...

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic...

We consider problems originating in economics that may be solved automatically using mathematical software. We present and make freely available a new benchmark set of such problems. The problems have been shown to fall within the framework of non-linear real arithmetic, and so are in theory soluble via Quantifier Elimination (QE) technology as usu...

We consider the use of Quantifier Elimination (QE) technology for automated reasoning in economics. QE dates back to Tarski's work in the 1940s with software to perform it dating to the 1970s. There is a great body of work considering its application in science and engineering but we show here how it can also find application in the social sciences...

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combination...

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combination...

When building a cylindrical algebraic decomposition (CAD) savings can be made
in the presence of an equational constraint (EC): an equation logically implied
by a formula.
The present paper is concerned with how to use multiple ECs, propagating
those in the input throughout the projection set. We improve on the approach of
McCallum in ISSAC 2001 by...

We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
Th...

Accurate and comprehensible knowledge about the position of branch cuts is
essential for correctly working with multi-valued functions, such as the square
root and logarithm. We discuss the new tools in Maple 17 for calculating and
visualising the branch cuts of such functions, and others built up from them.
The cuts are described in an intuitive a...

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems
in real algebraic geometry and beyond. In recent years a new approach has been
developed, where regular chains technology is used to first build a
decomposition in complex space. We consider the latest variant of this which
builds the complex decomposition incrementally by...

Cylindrical algebraic decomposition (CAD) is an important tool, both for
quantifier elimination over the reals and a range of other applications.
Traditionally, a CAD is built through a process of projection and lifting to
move the problem within Euclidean spaces of changing dimension. Recently, an
alternative approach which first decomposes comple...

Cylindrical algebraic decompositions (CADs) are a key tool for solving
problems in real algebraic geometry and beyond. We recently presented a new CAD
algorithm combining two advances: truth-table invariance, making the CAD
invariant with respect to the truth of logical formulae rather than the signs
of polynomials; and CAD construction by regular...

Cylindrical Algebraic Decompositions (CADs) have been studied since their
creation as a tool for working with semi-algebraic sets and eliminating
quantifiers of the reals. In this paper we are concerned with Cylindrical
Algebraic Sub-Decompositions (sub-CADs), defined as subsets of CADs sufficient
to describe the solutions for given formulae. We di...

A new algorithm to compute cylindrical algebraic decompositions (CADs) is
presented, building on two recent advances. Firstly, the output is truth table
invariant (a TTICAD) meaning given formulae have constant truth value on each
cell of the decomposition. When investigated in ISSAC '13, TTICAD theory was
shown to provide lower cell counts and a d...

It has long been known that cylindrical algebraic decompositions (CADs) can
in theory be used for robot motion planning. However, in practice even the
simplest examples can be too complicated to tackle. We consider in detail a
"Piano Mover's Problem" which considers moving an infinitesimally thin piano
(or ladder) through a right-angled corridor.
P...

A little over a decade ago, Goto and van de Geijn wrote about the importance of the treatment of the translation lookaside buffer (TLB) on the performance of matrix multiplication. Crucially, they did not say how important, nor did they provide results that would allow the reader to make his own judgement. In this paper, we revisit their work and l...

This article makes the key observation that when using cylindrical algebraic
decomposition (CAD) to solve a problem with respect to a set of polynomials, it
is not always the signs of those polynomials that are of paramount importance
but rather the truth values of certain quantifier free formulae involving them.
This motivates our definition of a...

We assume some standard choices for the branch cuts of a group of functions
and consider the problem of then calculating the branch cuts of expressions
involving those functions. Typical examples include the addition formulae for
inverse trigonometric functions. Understanding these cuts is essential for
working with the single-valued counterparts,...

Cylindrical algebraic decomposition (CAD) is an important tool for the study
of real algebraic geometry with many applications both within mathematics and
elsewhere. It is known to have doubly exponential complexity in the number of
variables in the worst case, but the actual computation time can vary greatly.
It is possible to offer different form...

Cylindrical Algebraic Decomposition (CAD, first introduced in [Col75]) of Euclidean space has become an important tool in mathematics and allows for practical quantifier elimination (QE) over the reals. Much research has gone into improving the projection operator (e.g. [McC85]), the use of partial CADs (e.g. [CH91]), or into alternative algorithms...

In considering the reliability of numerical programs, it is normal to "limit
our study to the semantics dealing with numerical precision" (Martel, 2005). On
the other hand, there is a great deal of work on the reliability of programs
that essentially ignores the numerics. The thesis of this paper is that there
is a class of problems that fall betwe...

Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought
of as two, rather different, methods of looking at systems of equations and, in
the case of Cylindrical Algebraic Decomposition, inequalities. However, even
for a mixed system of equalities and inequalities, it is possible to apply
Gr\"obner bases to the (conjoined) equal...

The word `Simplification' is often used in Computer Algebra. However, the task is not simple even with the aid of modern Computer Algebra Systems (CASs). Not only ones face questions such as `is expression $f$ simpler than expression $g$?', but many well-known formulae are not true on some points, lines or regions due to the fact that the elementar...

Recent years have witnessed an exceptional research interest in cryptographic hash functions , especially after the popular attacks against MD5 and SHA-1 in 2005. In 2007, the U.S. National Institute of Standards and Technology (NIST) has also significantly boosted this interest by announcing a public competition to select the next hash function st...

How do we recognize when an answer is \right"? This is a question that has bedevilled the use of computer systems in mathemat- ics (as opposed to arithmetic) ever since their introduction. A computer system can certainly say that some answers are denitely wrong, in the sense that they are provably not an answer to the question posed. How- ever, an...

One of the problems with manipulating function identities in computer algebra systems is that they often involve functions
which are multivalued, whilst most users tend to work with single-valued functions. The problem is that many well-known identities
may no longer be true everywhere in the complex plane when working with their single-valued coun...

Given an elementary function with algebraic branch cuts, we show how to decide which sheet of the associated Riemann surface we are on at any given point. We do this by establishing a correspondence between the Cylindrical Algebraic Decomposition (CAD) of the complex plane defined by the branch cuts and a finite subset of sheets of the Riemann surf...

Simplification has been long recognised to be a fundamental problem within computer algebra. One of the main concerns of this paper is to investigate the feasibility of the simplification system proposed by the authors in ISSAC 2004. This will be based on a set of examples which one would might realistically meet in practice. Cylindrical Algebraic...

Simplification has been long recognised to be a fundamental problem within computer algebra [17]. However, even for the class of elementary functions, it has not been resolved in a satisfactory way.Algorithms were presented in [4, 2] to solve this problem, and it was seen that both methods had their own strengths and weaknesses. Also, not all funct...

In [5], we introduced an algorithm for deciding whether a proposed simplification of elementary functions was correct in the presence of branch cuts. This algorithm used multivalued function simplification followed by verification that the branches were consistent.In [14] an algorithm was presented for zero-testing functions defined by ordinary dif...

We present an algorithm for simplifying a large class of elementary functions in the presence of branch cuts. This algorithm works by:(a) verifying that the proposed simplification is correct as a simplification of multi-valued functions;(b) decomposing C (or Cn in the case of multivariate simplifications) according to the branch cuts of the releva...

There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make "howlers" or not to simplify enough. In this paper we outline the "unwinding number" approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have n...

We present our experience withEuLisp as a teaching language, focussing on the level of the language which was specifically designed for this purpose (level-0).EuLisp has been used in undergraduate and postgraduate teaching since 1990, in lectures and laboratories, where in many cases it has replaced Scheme or Common Lisp. It has been used extensive...

this document is to define the programming language EULISP. EULISP is a dialect of Lisp and as such owes much to the great body of work that has been done on language design in the name of Lisp over the last thirty years. EULISP is the outcome of efforts on the part of many people in countries of the European Community since 1986. The guiding princ...

Many current algebra systems have a lax attitude to the simplification of expressions involving functions like log and , leading to the ability to prove equalities like –1=1 in such systems. In fact, only a little elementary arithmetic is needed to devise what the correct simplifications should be. We detail some of these simplification rules, and...

We describe the approach to the concurrent execution of object-oriented programs that is being research at the University of Bath. The aim of this project is the concurrent execution of both new Lisp programs and existing Lisp applications. We are pursuing this goal by addressing the problems of concurrent execution at several levels: static analys...

We have experimented with a little elementary parallelism applied to Buchberger's algorithm. This is in contrast to [Ponder] and [Vidal], as gains can be achieved by using the method even on a single processor.

The defect of an algebraic number field (or, more correctly, of a presentation of the field) is the largest rational integer that divides the denominator of any algebraic integer in the field when written using that presentation. Knowing the defect, or estimating it accurately is extremely valuable in many algorithms, the factorization of polynomia...

We present two efficient tests that determine if a given polynomial is cyclotomic, or is a product of cyclotomics. The first method uses the fact that all the roots of a cyclotomic polynomial are roots of unity, and the second the fact that the degree of a cyclotomic polynomial is a value of (n), for some n. We can also find the cyclotomic factors...

The problem of factorising polynomials: that is to say, given a polynomial with integer coefficients, to find the irreducible polynomials that divide it, is one with a long history. While the last word has not been said on the subject, we can say that the past 15 years have seen major break-throughs, and many computer algebra systems now include ef...

We describe the methods available in the current REDUCE system for introducing new mathematical domains, and illustrate these by discussing several new domains that significantly increase the power of the overall system.

This paper describes a package implemented in REDUCE 3.2 for the manipulation of algebraic numbers. The package regards algebraic numbers as elements of abstract extensions of the rational numbers, not as particular real or complex numbers. We describe in this paper the various design choices that were made, and the current state of the package, as...

We are concerned with factoring polynomials over algebraic extensions of the rationals, and have implemented a variant of Trager's [1976] algorithm, which reduces this problem to that of factoring polynomials over the integers, for which we use Wang's [1978] algorithm (as implemented in REDUCE [Hearn 82] by Moore and Norman [1981]). However, Trager...

## Citations

... (38) and (39), respectively. Naturally, the above quantified formula (41) is simply an appropriate slight modification of the related quantified formula (37) having been used in the previous subsection for only two high-reliability components. Now our task is to derive the related QFF (quantifier-free formula) again by using the method of quantifier elimination and its powerful implementation in Mathematica [34]. ...

... The original idea was introduced by George Collins [41] as efficient computational method for quantifier elimination. A modern description with applicable algorithms can be found in [42]. ...

... More recent work focuses on understanding and finding the multistationarity region, but here progress is scarce and often restricted to special systems, e.g. [8][9][10][11][12][13][14]. ...

... Besides CRNT and linear/nonlinear programming, there are a number of alternative approaches for gaining insights into an ODE system's behavior. For example, algebraic methods utilizing the Gröbner basis approach [13,16], cylindrical algebraic decomposition [17] and sign pattern analysis of the Jacobian [18] have been developed in concert with CRNT. The goal of these methods is to obtain analytical solutions to the system of ODEs at the steady state and then further analyze the solutions to detect bistability. ...

... Robust implementations are freely available [9,19] and well supported. They have been successfully applied to problems in chemical reaction network theory during the past decade [7,8,[21][22][23]33,67,71,72,78]. Such a generalization is not quite straightforward. ...

... Theoretically, these two problems can be addressed using CAD [6,13,32]. However, this method is impractical as it is double exponential in the total number of variables and parameters, and depends also on the number and degree of the polynomials [18]. ...

... There is a large set derived from university mathematics entrance exams [73], which but may be publicly available in the future. An obvious extension of Case Study A would be to test additional heuristics, such as the greedy sotd heuristic [41], one based on the number of full dimensional cells [99], or those developed for CAD by Regular Chains [46]. For Case Study B there are now further CAD optimisations for problems with multiple equalities under development [39,44,47] which may affect the role of GB preconditioning from CAD. ...

... • The variable ordering heuristics for Case Study A were calculated using tools in the authors ProjectionCAD library [49]. • The GB calculations for Case Study B used the implementation that ships with Maple: a meta algorithm calling multiple GB implementations. ...

... In our example using ≺ ≺ requires 27 cells but ≺ ≺ requires 115. Human-designed heuristics [14], [3], [2], [16] usually make the choice in implementations. In 2014, we trained a Support Vector Machine (SVM) to choose which of these heuristic to follow [24]. ...

... There is a collection [WBD12], not quite as old as [BM96] (2014 was the last update), but still completely static. The DEWCAD project [BDE + 21] might update this, but there are still issues of long-term conservation. ...