This book, obviously, is about multidimensional declarative programming. Many different languages and paradigms may be claimed to result in multidimensional declarative programming. We are most concerned with a programming paradigm where the existence of different dimensions (even temporary dimensions) enables algorithms to be expressed naturally. The latest version of the language Lucid uses dimensions in this way and we will use Lucid throughout most of this book. We will emphasize that Lucid is a language that embodies a particular style or paradigm of programming that we call intensional programming. Toward the end of the book there will be a chapter that indicates how other language features and capabilities could fit into this same paradigm. Therefore this book is about multidimensional, declarative, intensional programming, with examples given mostly in the latest version of Lucid. Lucid has been around since the mid-seventies [7]. Nevertheless, with its new poly dimensionality capabilities it provides a fresh approach to programming and, we hope, introduces declarative programming to a whole new audience. In addition, we often find it is useful to think of Lucid programs as manipulating geometrical entities, even though programs are actually implemented very simply i.e., without any data structures corresponding to those geometrical entities. Thinking of programs as manipulating geometrical entities adds a visual aspect to programs, while the fact that there are no actual structures in the implementation emphasizes that Lucid programs can be thought of in several different ways. We tend to think of programs, and variables in programs, at a very high level when writing programs and ignore the implementation details. Yet for a very high level language, those implementation details are surprisingly simple. Programs are written while thinking of large, multidimensional entities, and yet the implementation deals only with small fragments. In fact, we might modify an environmental movement slogan and say “program globally and implement locally.” By the end of the book the appropriateness and meaning of this slogan should be apparent. This introductory chapter begins by illustrating some of the novelty of the new Lucid with a very simple program for transposing a matrix.
