This chapter contains the second part of our presentation of set notation. It is devoted to the study of the construction of mathematical objects: that is, finite sets, natural numbers, finite sequences, and finite trees. All such constructions are realized by using the same inductive method based on the fixpoint theorem of Knaster and Tarski. We shall also describe the way recursive functions on these objects can be formally defined.

At the beginning of the chapter generalized intersection and union are presented, then we introduce the framework that we shall use in order to construct our objects. Then the construction of each of the mentioned objects follows in separate sections. Finally, at the end of the chapter, we present the concept of well-foundedness, which unifies completely each of the previous constructs.

Hurried readers, who are not interested in the details of these formal constructions, can skip them. They can go to Appendix A, where the notations are all listed.

Generalized Intersection and Union

Before presenting the framework that will allow us to construct mathematical objects in a systematic manner (section 3.2), we need first to introduce two important set theoretic concepts called generalized intersection and generalized union.

As usual, we first introduce our concepts syntactically, then we give an informal explanation and some examples, and finally we state and prove their basic properties. We now present the syntax of the new constructs.

The main objective of this chapter, and of the next one, is to make precise part of the notation that we intend to use in order to specify and design software systems. As a matter of fact, this notation is nothing but the one used, more or less formally, by every scientist: that of set theory.

Clearly, in order to perform our subsequent formal developments, we could have stayed within some extensions of the logic introduced in the previous chapter. For the following three reasons it is my view that set theory is a better choice.

First, when using set theory, one remains within first order logic while it is clearly possible to manipulate objects of “high order” such as sets and relations of any depth (that is, sets and relations built themselves on sets and relations, and so on). In any case, such constructs are objects (not predicates) and are thus licit candidates for first order quantifications.

Second, and more importantly, the use of set theory allows one, very often, to eliminate quantifiers. In fact, most of the set-theoretic concepts are introduced precisely because they hide some quantifications. Typical examples are: set inclusion, relational composition, and generalized intersection or union of sets of sets.

Third, set theory has the (apparently innocent but, in fact, very important) property of handling negation in a very controlled way: it is simply because the complement of a set is still a set. That is, negation does not extend beyond certain limits.

This book is much more than a new programming manual. It introduces a method in which the program design is included in the global process that goes from understanding the problem to the validation of its solution.

The mathematical basis of the method provides the exactness while the proposed notation eliminates the ambiguities of the vernacular language. At the same time, the process is simple enough for an industrial use “Industrial” is in fact the key word.

The general aim of formal methods is to provide correctness of the problem specification. Here we can see how the solution can be found, step by step, by a continuously monitored process. The mathematical verification of each step is so closely bound to the refinement activity that it is no longer possible to separate the design choices from the checking process. Imagination is helped by exactness!

But how about the efficiency? Isn't the design too long? Are the design people able to do this work? Are the machines powerful enough to implement the method? The answers are easy to give. Let me tell you.

My company has been involved, since the sixties, in the realisation of train control systems, which must meet stringent safety requirements. As soon as we began to use programmed logic (end of the seventies) we had to solve the problem of software correctness. Together with other methods, we chose to use the program proving method proposed by C.A.R. Hoare. In 1986, J.-R. Abrial introduced us to the B method. We decided to learn it and to use it. The tools did not exist at the time.

Abstract

We discuss a number of open problems and conjectures in the theory and application of finite fields. We also provide a brief discussion of the status as well as references related to each problem.

Introduction

In this paper we try to summarise some interesting and/or important questions in the theory and application of finite fields. These questions obviously reflect our personal tastes but we have indeed tried to consider questions of general interest. We hope that these questions and even more, the methods developed for their solutions, will be of interest to other researchers. The reader may wonder why some questions we call ‘Problems’ and some we call ‘Conjectures’. Roughly speaking, we use the term Conjecture if we (and very often others) believe the statement to be true while the term Problem is used to indicate all other statements. In general we feel that our conjectures may be more difficult to resolve than our problems.

The standard notions of the theory of finite fields which we use can be found in the finite field bible by Lidl and Niederreiter.

Combinatorics

There are numerous open problems in combinatorics which are related to finite fields. In this section we briefly describe several of these. We begin with several questions related to latin squares. A latin square of order n is an n × n array based upon n distinct symbols with the property that each row and each column contains each of the n symbols exactly once.

This volume represents the refereed proceedings of the Third International Conference on Finite Fields and Applications held at the University of Glasgow, Scotland, 11–14 July, 1995, where it was hosted by the Department of Mathematics. The conference, often referred to as Fq3, was the successor of two other international conferences concerning finite fields held at the University of Nevada, Las Vegas, USA, in August 1991 and in August 1993. The Organising Committee comprised Steve Cohen, Stuart Hoggar, Bob Odoni (all of the University of Glasgow), James Hirschfeld (University of Sussex), Gary Mullen (Pennsylvania State University), Harald Niederreiter (Austrian Academy of Sciences) and Peter Shiue (University of Nevada, Las Vegas).

Finite fields with their tight structure are intrinsically fascinating; further, their study is now recognised to be extremely useful in diverse areas of pure and applicable mathematics, including aspects of number theory, algebra, analysis and algebraic geometry and, at the same time, manifold aspects of information theory, computer science and engineering. For example, coding theory is enriched by deep ideas, crucially involving finite fields, on exponential sums, function fields and linear algebra, and in turn, has stimulated further questions for finite field research. Indeed, what is particularly exciting in current activity is the interplay between various areas, within pure mathematics itself and within those having definite applications. A further sign of this vitality is the emergence in 1995 of the journal Finite Fields and Their Applications published by Academic Press.