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Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward 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 ..."
Abstract

Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward 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 λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
Term Graphs and Principal Types for X
"... 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 ca ..."
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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 nonstandard notion of type assignment on term graph rewriting. We will show that this system has a principal type property.
An optimised term graph rewriting engine forX Measuring the cost of αconversion
"... This paper studies the calculus X, that has its foundation in Classical Logic; we present an implementation for X using term graph rewriting techniques, and discuss improvements thereof which result in an increasingly more efficient running of the reduction engine. We show that name capture can be d ..."
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This paper studies the calculus X, that has its foundation in Classical Logic; we present an implementation for X using term graph rewriting techniques, and discuss improvements thereof which result in an increasingly more efficient running of the reduction engine. We show that name capture can be dealt with ‘on the fly’, by realising the avoidance of capture through adding or modifying the rewrite rules. We study two different approaches to garbage collection, and compare the various implementations by presenting benchmarks. 1
Explicit Alpha Conversion and Garbage Collection in X
, 2006
"... This paper will present improvements on the term graph rewriting model of implementation for the (untyped) calculus X as presented in [3]. X is a new style calculus which embodies both substitution and context call, that has first been defined in [13] and was later extensively studied in [2]. ..."
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This paper will present improvements on the term graph rewriting model of implementation for the (untyped) calculus X as presented in [3]. X is a new style calculus which embodies both substitution and context call, that has first been defined in [13] and was later extensively studied in [2].