X is an untyped continuation-style formal language with a typed subset which provides a Curry-Howard isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λ-calculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.

Abstract. In this paper we study the calculus of circuits X, as first presented in [13] and studied in detail in [5]. We will present a number of new implementations of X using term graph rewriting techniques, which are improvements of the technique used in [6]. We will show that alpha conversion can be dealt with ‘on the fly’, and that explicit copying can be avoided, by presenting, discussing and comparing a number of solutions to these problems. We will define a notion of type assignment on circuits by labelling input and output connectors with types. This notion is then used to define a non-standard notion of type assignment on term graph rewriting. We will show that this system has a principal type property.