Research resume and selected publications

A. The Baker-Beynon duality

The following papers relate to a categorical duality between finitely generated lattice ordered Abelian groups / vector lattices and the polyhedral category [2,3,4]. They describe research that stems from my doctoral studies, carried out under the supervision of Philip Higgins at KCL. A key theorem (for which the 'if' implication was observed by Kirby Baker of UCLA) shows that a map is piecewise linear if and only if it can be expressed by applying pointwise max and min to a finite set of linear functionals [1]. It establishes what is now known as the Baker-Beynon duality (see Encyclopaedia of Mathematics, by Michiel Hazewinkel, Springer, 2001, p75). This duality is currently being exploited in mathematical research by Andrew Glass (Cambridge) and Daniele Mundici (Florence).

1. Combinatorial aspects of piecewise-linear maps
Journal London Math Soc (2) 7, 1974, 719-727
2. Duality theorems for finitely-generated vector lattices
Proc London Math Soc (3) 31, 1975, 114-128
3. Applications of duality in the theory of finitely-generated lattice-ordered Abelian groups
Can J Math 29 (2) 1977, 243-254
4. Vector lattices freely generated by distributive lattices
Math Proc Camb Phil Soc 81, 1977, 193-200

B. The computation of group-theoretic invariants

The following papers relate to the computation of two different classes of group-theoretic invariants using algorithms devised by George Lustzig [1] and Nicolas Spaltenstein [2] at the Mathematics Institute at Warwick. The computation of the Green functions of finite Chevalley groups [2,3] was the most technically demanding programming task I have ever carried out. The entire project involved six months of intensive work: a number of auxiliary programs in C and APL were written to generate the input data (comprising some 36,000 integers) for one of the three main C programs.

1. (with George Lustzig) Some numerical results on the characters of exceptional Weyl groups
Math Proc Camb Phil Soc 84, 1978, 417-426
2. (with Nicolas Spaltenstein) Green functions of finite Chevalley groups of type En (n=6,7,8)
Journal of Algebra, Vol 88 (2), June 1984, 584-614
3. (with Nicolas Spaltenstein) The computation of Green functions of finite Chevalley groups of type En (n=6,7,8) (16pp+157pp of tables)
Computer Centre Report 23, University of Warwick 1982

C. Miscellaneous contributions to Theoretical Computer Science

The early years of my research career in Computer Science at the University of Warwick were devoted to studying mathematical aspects of computer science alongside research groups led by Mike Paterson and David Park. A general theme in this research is that of applying algebraic theories to computing [1,2], with particular reference to how the theory of distributive lattices relates to the study of boolean circuits. My main contribution in this area has been to develop a theory of computational equivalence and replaceability [3] that can be used to characterise those monotone boolean functions which can be computed by planar monotone circuits [4]. I have also devoted several years of effort to seeking algorithmic implications of the characterisation of monotone boolean functions as combinatorially piecewise linear maps (see [A1]) that I discovered in my doctoral research [5].

1. On the structure of free finite state machines
Theoretical Computer Science 11, 1980, 167-180
2. A formal account of some elementary continued fraction algorithms
Journal of Algorithms 4, 1983, 221-240
3. Replaceability and computational equivalence for monotone boolean functions
Acta Informatica 22, 1985, 433-449
4. (with John Buckle) On the planar monotone computation of Boolean functions
Theoretical Computer Science 53, 1987, 267-279
5. Boolean Function Complexity: a Lattice-Theoretic Perspective
Boolean Function Complexity, ed. M. S. Paterson, LMS Lecture Notes Series 169, CUP 1992, 35-56

D. Empirical Modelling principles

Empirical Modelling (EM) is an alternative way of thinking about computing that has its origins in the problematic conceptual relationship between '=' as it is used in procedural programming and '=' as it is used in mathematics or declarative programming. Where '=' is used in procedural and declarative paradigms for specifying 'action' and 'static relationships' respectively, EM exploits '=' in a way similar to that in which it is used in a spreadsheet, to record dynamic relationships between values ('dependencies') as they are encountered through observation and interaction. The following papers document how the concept of EM has emerged with reference to:

• personal interaction and sense-making activity [1,3,7,11];
• the comprehension of 'commonsense concurrency' through the projection of personal models of interaction on to other agents [2,5,6];
• the role that observation plays in system and software development [9,10].

1. Definitive notations for interaction
Proc hci'85, ed Johnson & Cook, Cambridge University Press, 1985, 23-34
2. The LSD Notation for Communicating Systems
CS-RR-087, University of Warwick, 1986
3. Definitive principles for interactive graphics
NATO ASI Series F, Vol 40, Springer-Verlag 1988, 1083-1097
4. (with Steve Russ) The Development and Use of Variables in Mathematics and Computer Science
The Mathematical Revolution Inspired by Computing, IMA Conf Series 30, 1991, 285-95
5. Agent-oriented Modelling and the Explanation of Behaviour
Proc. International W/S "Shape Modeling: Parallelism, Interactivity and Applications", Department of Computer Software, TR 94-1-040, Univ. Aizu, Japan, Sept. 1994, 54-63
6. (with Steve Russ and Paul Ness) Worlds Before and Beyond Words
Proc. Virtual Futures '95, Warwick University, 1995
7. Empirical Modelling for Educational Technology
Proc Cognitive Technology '97, IEEE, 54-68, 1997
8. Empirical Modelling and the Foundations of Artificial Intelligence
Lecture Notes in AI 1562, Springer, 322-364, 1999
9. (with Jaratsri Rungrattanaubol and Jane Sinclair) Formal Specification from an Observation-oriented Perspective
Journal of Universal Computer Science, Vol. 6(4), 2000, 407-421
10. Liberating the Computer Arts
Proc DALI'2001, First International Conference on Digital and Academic Liberty of Information, University of Aizu, Japan, March 2001 (invited paper, 25pp)
11. (co-authored by the EM research group) The Computer as Instrument
Proc. Cognitive Technology: Instruments of Mind, University of Warwick, August 2001, LNAI 2116, Springer-Verlag, 476-489
12. Radical Empiricism, Empirical Modelling and the Nature of Knowing
In (ed. Itiel E Dror) Cognitive Technologies and the Pragmatics of Cognition: Special Issue of Pragmatics and Cognition, 13:3, December 2005, 615-646.

E: Empirical Modelling applications

The development of EM principles has been complemented by a wide-ranging programme of collaborative research. Themes represented in this research include: computer-aided geometric and conceptual design [1,6], parallel architectures [2], concurrent systems modelling and simulation [3,7], visualisation [4], concurrent engineering [5], business process re-engineering [8,9], educational technology [10,11] and humanities computing [12,13].

1. (with Alan Cartwright) A definitive programming approach to the implementation of CAD software
Intelligent CAD Systems II: Implementation Issues, Springer-Verlag 1989, 126-45
2. (with Mike Slade and Edward Yung) Parallel computation in definitive models
CONPAR'88, British Computer Society WS Series CUP 1989, 359-367
3. (with Mark Norris, Rodney Orr and Mike Slade) Definitive specification of concurrent systems
Proc UKIT'90, IEE Conference Publications 316, 1990, 52-57
4. (with Simon Yung) Definitive Interfaces as a Visualisation Mechanism
Proc Graphics Interface '90, Canadian Information Processing Soc., 1990, 285-292
5. (with Valery Adzhiev, Alan Cartwright and Simon Yung) A Computational Model for Multi-agent Interaction in Concurrent Engineering
Proc. CEEDA'94, Bournemouth Univ., 1994, 227-232
6. (with Valery Adzhiev, Alan Cartwright and Simon Yung) A New Computer-Based Tool for Conceptual Design
Proc. Workshop Computer Tools for Conceptual Design, Univ. of Lancaster, 1994, 171-188
7. (with Patrick Sun) Computer-mediated Communication: a Distributed Empirical Modelling Perspective
Proceedings of Cognitive Technology 99, San Francisco, 1999 http://www.added.com.au/cogtech/CT99/
8. (with Soha Maad) Integrated Environments for Virtual Collaboration: an Empirical Modelling Perspective
Proceedings of the Fifth World Conference On Integrated Design & Process Technology, Texas 2000
9. (with Kiran Fernandes, Vinesh Raja, John Keast, Pui Shan Chan and Mike Joy) Business and IT Perspectives on AMORE: a Methodology for Object-Orientation in Re-engineering Enterprises
Systems Engineering for Business Process Change: New Directions, (ed P. Henderson) Springer-Verlag 2002, 274-297
10. (with Abhir Bhalerao, Chris Roe and Ashley Ward) A computer-based environment for the study of relational query languages
Proc. LTSN Workshop on Teaching. Learning and Assessment in Databases, Coventry, UK, July 2003, 104-108
11. (with Chris Roe) Enriching computer support for constructionism
In (ed. Eshaa Alkhalifa) Cognitively Informed Systems: Utilizing Practical Approaches to Enrich Information Presentation and Transfer, Idea Group Publishing, 2006, 209-233.
12. (with Steve Russ and Willard McCarty) Human Computing: Modelling with Meaning
Literary and Linguistic Computing 21(2), 2006, 141-157.
13. Mathematics and Music - Models and Morals
In Conference Proceedings, Bridges London: Mathematical Connections in Art, Music, and Science (eds. Sarhangi and Sharp), Tarquin Books, 2006, 437-444.

F. Practical exercises in Empirical Modelling

The practical development of EM depended crucially upon the implementation of both definitive notations and mechanisms for effecting agency. Two different types of implementation were devised around 1987: Edward Yung's EDEN, based on an extension of conventional procedural languages through the addition of dependency and triggered action, and Mike Slade's am, based on an abstract computational model known as the abstract definitive machine (see [E2]). To date, most of the model-building in EM has been carried out by students at Warwick working under the supervision of myself and Steve Russ, using an extension of the EDEN interpreter that has been developed initially by brothers Edward and Simon Yung and latterly by Ashley Ward. The current web archive of EM models, developed by Ward with the help of Chris Roe, includes about 150 such models. The principal models to which I have made a significant contribution in conception and/or construction are:

1. A telephone simulation (devised by Slade from an LSD account developed by Beynon at BTRL in 1986 [D2])
2. The 'jugs' educational program for schools (jugsBeynon1988)
3. Planar configurations of lines (linesBeynon1991)
4. Variations on noughts-and-crosses (oxoJoy1994, oxoGardner1999)
5. A digital watch, after David Harel (digitalwatchCartwright1995)
6. Sorting algorithms (bubblesortBeynon1998, heapsortBeynon1998, heapsortextendRun-bol2001)
7. Historic railway accident scenarios (claytontunnelSun1999)
8. Planar monotone boolean functions (planarcomputeRun-bol2001)
9. A project oral timetabling environment (projecttimetableKeen2000)
10. An interactive environment for studying relational query languages (sqleddiBeynon2001, sqleddiWard2003)
11. Representations of the free distributive lattice on 4 generators (fdl4Beynon2002)
12. Travails in a lift (liftBeynon2003)
13. A model of Schubert's setting of Goethe's ballad Erlkoenig (see poster in kaleidoscopeBeynon2005)

G. Postgraduate teaching and supervision

Empirical Modelling research has supplied the material for masters level modules that have been taught at Warwick by myself with input from Steve Russ and several research students from 1992 to 1999 and since 2002.

Many of the themes represented in my research have been amplified in postgraduate theses developed under my supervision: