Introduction

In this Chapter we describe the relationship between HRDM and the logic ILs and its model theory. This relationship is first presented formally, and is then followed by an informal discussion that emphasizes insights that it can provide into the way that a database models the real world, and into the nature of entities and relationships, of key and non-key attributes, of queries and data constraints, and of the interaction of time with all of these concepts. The formalism is presented in the interests of completeness and rigor, but it is easy to get lost in some of the notation; the informal discussion provides a better overview both of how the temporal dimension is incorporated into, and how it affects, the traditional relational model.

In Chapter 2 we described the syntax and semantics of the language ILs. To be more precise we should rather say the family of ILs languages, any particular language in this family being determined by the set C of non-logical constants. The historical relational database concept presented in Chapter 3 will now be related to the discussion of the intensional logic as follows. First we show that a particular HRDB scheme defines a particular logic in the family of ILs languages that provides a formal expression of the historical database semantics and that serves as the target language for translations from our English Query Language which will be described in the latter half of this work.

Summary

The Historical Relational Database Model

Under the general assumption that in any scientific enterprise the development of a formal logical theory is a fruitful endeavor, we have attempted in this work to develop such formal theories for the incorporation of a temporal component into a database model, and for the specification of an English query language facility for such a database. Because of the power of the logical formalism of IL developed by Richard Montague [Mon74] we have been able to express both of these theories in terms of the same logical apparatus. Thus two areas which may have seemed only marginally related at the outset were seen to have a closer connection.

Beginning first with formalizing a historical database model, we presented the model HRDM. This model turned out to be a simple extension to an entity-relationship semantics imposed upon the relational database model. The temporal component was included in a very simple but fundamental way as a basic component of the model, and its significance from both the informal, intuitive view of three-dimensional relations extending in both directions of time, and from the formal, model-theoretic view, were discussed. In order to present as general a model of time as possible, we put only the barest of constraints upon the temporal dimension – linear order and density.

The HRDM model was seen as providing a number of insights into the nature of a database and how it functions as a model of the real world.

This book is a first step in a new direction: to modify existing theory from a constructive point of view and to stimulate the readers to make their own computational experiments. We are thoroughly convinced that their observations will help to build a new basis from which to venture into new theory on algebraic numbers. History shows that in the long run, number theory always followed the cyclic movement from theory to construction to experiment to conjecture to theory.

Consequently, this book is addressed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to (constructive) algebraic number theory and is therefore especially suited as a textbook for a course on that subject. On the other hand, many parts go far beyond an introduction and make the user familiar with recent research in the field. For experimental number theoreticians we developed new methods and obtained new results (e.g., in the tables at the end of the book) of great importance for them. Both computer scientists interested in higher arithmetic and in the basic makeup of digital computers, and amateurs and teachers liking algebraic number theory will find the book of value.

Many parts of the book have been tested in courses independently by both authors. However, the outcome is not presented in the form of lectures, but, rather, in the form of developed methods and problems to be solved. Algorithms occur frequently throughout the presentation.

Introduction

Since the first printing of this book in 1989 algorithmic algebraic number theory has attracted rapidly increasing interest. This is documented, for example, by a regular meeting, ANTS (algebraic number theory symposium), every two years whose proceedings give a good survey about ongoing research. Also there are several computer algebra packages concentrating on number theoretical computations. At present the most prominent ones, which are available for free, are KANT, PARI and SIMATH. KANT comes with a data base for algebraic number fields, already containing more than a million fields of small degree. KANT is developed by the research group of the author at Berlin and will be presented in some detail in this chapter. We note that almost all of KANT and PARI is also contained in the MAGMA system.

In the sequel we shortly discuss the improvements which were obtained for the computation of the important invariants of algebraic number fields. On the other hand, in computational algebraic number theory the interest has gradually turned from absolute extensions to relative extensions of number fields and we will sketch the important developments in that area. If subfields exist, the information about the invariants of those subfields can be lifted and used in the field under consideration. This relative point of view permits computations in fields of much larger degrees and has important applications, for example to class field computations.

A large proof should be organized as a collection of theories. An LCF theory has a signature: its type constructors, constants, infixes, predicates. It may have parent theories, inheriting all their symbols and axioms. This rich environment may be extended by new axioms. Theorems may be proved and recorded in the theory.

Existing theories may become the parents of a new theory if their signatures are disjoint. Names of types, constants, infixes, and predicates cannot be hidden or renamed to avoid clashes. Each theory has separate name spaces for its axioms and theorems. An axiom is designated by the pair (theory name, axiom name), a theorem by (theory name, theorem name).

Theories do not have a tree structure: sharing always occurs. In Figure 6.1, the theory T has parents T1 and T2. They both have T4 as a parent. Both the theories T3 and T4 have T5 as a parent; both T4 and T5 have PPλ as a parent. Symbols declared in a common ancestor are shared. If the symbol + is declared in T4 then it is visible in both T1 and T2 and does not cause a clash between the two theories. But if it were declared in both T2 and T3 then T1 and T2 would clash during the construction of T.

Every theory is ultimately descended from PPλ. A theory is the child of its parents.

When representing a logic on a computer, how should we treat inference rules? A rule, given appropriate premises, delivers a conclusion, so LCF represents an inference rule by a function from theorems to theorems. Theorems of PPλ are represented by the ML data type thm. Axioms and rules of PPλ are predefined identifiers. Each axiom is a theorem; applying rules to axioms computes new theorems.

This chapter lists the axioms, inference rules, and predefined theorems of PPλ. The examples include formal proofs from previous chapters, performed in LCF.

The representation of inference rules

A rule may hold only if its premises have a certain form; otherwise the corresponding ML function fails, raising exception rule of type string × thm list. The exception includes an error message and the offending premises.

In certain inference rules, the premises do not contain enough information to completely determine the conclusion. The corresponding function takes additional arguments, giving parts of the conclusion or even a formula stating the conclusion itself.

To implement quantifier rules, the abstract syntax functions of Chapter 5 perform substitution and enforce the provisos on eigenvariables. If a proviso is violated, the rule fails.

In the sequent Γ ├ A, the assumptions Γ are usually regarded as a set. ML represents a set of PPλ formulae as a list without repetitions. The LCF user need not be concerned with the order of the assumptions in the assumption list.