Bücher der Reihe Progress in Theoretical Computer Science

This monograph studies the logical aspects of domains as used in de- notational semantics of programming languages. Frameworks of domain logics are introduced; these serve as foundations for systematic derivations of proof systems from denotational semantics of programming languages. Any proof system so derived is guaranteed to agree with denotational se- mantics in the sense that the denotation of any program coincides with the set of assertions true of it. The study focuses on two categories for dena- tational semantics: SFP domains, and the less standard, but important, category of stable domains. The intended readership of this monograph includes researchers and graduate students interested in the relation between semantics of program- ming languages and formal means of reasoning about programs. A basic knowledge of denotational semantics, mathematical logic, general topology, and category theory is helpful for a full understanding of the material. Part I SFP Domains Chapter 1 Introduction This chapter provides a brief exposition to domain theory, denotational se- mantics, program logics, and proof systems. It discusses the importance of ideas and results on logic and topology to the understanding of the relation between denotational semantics and program logics. It also describes the motivation for the work presented by this monograph, and how that work fits into a more general program. Finally, it gives a short summary of the results of each chapter. 1. 1 Domain Theory Programming languages are languages with which to perform computa- tion.

Types can be consid ered as weak specifications of programs and checking that a program is of a certain type provides a verification that a program satisfies such a weak speci fication.

Equations occur in many computer applications, such as symbolic compu- tation, functional programming, abstract data type specifications, program verification, program synthesis, and automated theorem proving. Rewrite systems are directed equations used to compute by replacing subterms in a given formula by equal terms until a simplest form possible, called a normal form, is obtained. The theory of rewriting is concerned with the compu- tation of normal forms. We shall study the use of rewrite techniques for reasoning about equations. Reasoning about equations may, for instance, involve deciding whether an equation is a logical consequence of a given set of equational axioms. Convergent rewrite systems are those for which the rewriting process de- fines unique normal forms. They can be thought of as non-deterministic functional programs and provide reasonably efficient decision procedures for the underlying equational theories. The Knuth-Bendix completion method provides a means of testing for convergence and can often be used to con- struct convergent rewrite systems from non-convergent ones. We develop a proof-theoretic framework for studying completion and related rewrite- based proof procedures. We shall view theorem provers as proof transformation procedures, so as to express their essential properties as proof normalization theorems.

by Luea Cardelli Ever since Strachey's work in the 1960's, polymorphism has been classified into the parametric and overloading varieties. Parametric polymorphism has been the subject of extensive study for over two decades. Overloading, on the other hand, has often been considered too ad hoc to deserve much attention even though it has been, in some form, an ingredient of virtually every programming lan- guage (much more so than parametric polymorphism). With the introduction of object-oriented languages, and in particular with multiple-dispatch object-oriented languages, overloading has become less of a programming convenience and more of a fundamental feature in need of proper explanation. This book provides a compelling framework for the study of run-time over- loading and of its interactions with subtyping and with parametric polymorphism. The book also describes applications to object-oriented programming. This new framework is motivated by the relatively recent spread of programming languages that are entirely based on run-time overloading; this fact probably explains why this subject was not investigated earlier. Once properly understood, overloading reveals itself relevant also to the study of older and more conventional (single- dispatch) object-oriented languages, clarifying delicate issues of covariance and contravariance of method types, and of run-time type analysis. In the final chapters, a synthesis is made between parametric and overloading polymorphism.

To construct a compiler for a modern higher-level programming languagel one needs to structure the translation to a machine-like intermediate language in a way that reflects the semantics of the language. little is said about such struc- turing in compiler texts that are intended to cover a wide variety of program- ming languages. More is said in the Iiterature on semantics-directed compiler construction [1] but here too the viewpoint is very general (though limited to 1 languages with a finite number of syntactic types). On the other handl there is a considerable body of work using the continuation-passing transformation to structure compilers for the specific case of call-by-value languages such as SCHEME and ML [21 3]. ln this paperl we will describe a method of structuring the translation of ALGOL-like languages that is based on the functor-category semantics devel- oped by Reynolds [4] and Oles [51 6]. An alternative approach using category theory to structure compilers is the early work of F. L. Morris [7]1 which anticipates our treatment of boolean expressionsl but does not deal with procedures. 2 Types and Syntax An ALGOL-like language is a typed lambda calculus with an unusual repertoire of primitive types. Throughout most of this paper we assume that the primi- tive types are comm(and) int(eger)exp(ression) int(eger)acc(eptor) int(eger)var(iable) I and that the set 8 of types is the least set containing these primitive types and closed under the binary operation -.

Our Subjects and Objectives. This book is about algebraic and symbolic computation and numerical computing (with matrices and polynomials). It greatly extends the study of these topics presented in the celebrated books of the seventies, [AHU] and [BM] (these topics have been under-represented in [CLR], which is a highly successful extension and updating of [AHU] otherwise). Compared to [AHU] and [BM] our volume adds extensive material on parallel com- putations with general matrices and polynomials, on the bit-complexity of arithmetic computations (including some recent techniques of data compres- sion and the study of numerical approximation properties of polynomial and matrix algorithms), and on computations with Toeplitz matrices and other dense structured matrices. The latter subject should attract people working in numerous areas of application (in particular, coding, signal processing, control, algebraic computing and partial differential equations). The au- thors' teaching experience at the Graduate Center of the City University of New York and at the University of Pisa suggests that the book may serve as a text for advanced graduate students in mathematics and computer science who have some knowledge of algorithm design and wish to enter the exciting area of algebraic and numerical computing. The potential readership may also include algorithm and software designers and researchers specializing in the design and analysis of algorithms, computational complexity, alge- braic and symbolic computing, and numerical computation.

In 1959 John Backus presented a paper on a proposed international algebraic language which evolved into ALGOL 60. This set of two volumes aims to review the attempts over recent years to use programming languages based on ALGOL 60, using Backus' original document as an introduction.