Using a Language of Functions and Relations

This paper describes the T-Ruby language for specifying VLSI circuits in terms of relational abstractions of their behaviour. The language is based on a typed lambda calculus with a type system closely related to that of Edinburgh LF. Terms in the language may have simple, non-parameterised types or types which are constructed from other types, possibly together with terms. The dependent types which can be constructed in this way are useful for describing param-eterised regular structures which commonly appear in VLSI circuits. The language forms the basis for a tool which permits term rewriting in a user-directed transformational style according to a set of proved term equivalences. Terms which describe causal relations may then be translated into a VLSI speciication in a standard VLSI description language. The rewriting process is illustrated by the derivation of a relation describing a circuit for 2-dimensional convolution. 1 Introduction Ruby 3] is a language intended for specifying VLSI circuits in terms of relational abstractions of their behaviour. A circuit is described by a binary relation, and simple relations can be composed into more complex ones by the use of a variety of combining forms which are higher-order functions. The basic relations and their combining forms generate an algebra, which within a given interpretation deenes sets of equivalences between relational expressions. These equivalences , which may take the form of equalities or conditional equalities, are essential for the design process, which typically involves transformation from a \speciication" in the form of one Ruby expression to an \implementation" in the form of another Ruby expression, in a calculation-oriented style 4, 7, 11]. This paper describes some aspects of a computer-based system, known as T-Ruby 9], for supporting such calculations. T-Ruby is based on a formalisation of Ruby as a language of functions and relations, with a conventional set of elementary types (integers, Booleans, bits, characters ,...), and a set of type constructors for forming constructed types, such as the types of pairs, lists and binary