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The theory of calculi with explicit substitutions revisited
 CSL 2007
, 2007
"... Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we fi ..."
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Cited by 10 (1 self)
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Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambdacalculus which enjoys fundamental properties such as simulation of onestep betareduction, confluence on metaterms, preservation of betastrong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proofnets.
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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Cited by 6 (3 self)
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We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
Patterns as first class citizens
, 2008
"... Abstract. The pure pattern calculus generalises the pure lambdacalculus by basing computation on patternmatching instead of betareduction. The simplicity and power of the calculus derive from allowing any term to be a pattern. As well as supporting a uniform approach to functions, it supports a u ..."
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Abstract. The pure pattern calculus generalises the pure lambdacalculus by basing computation on patternmatching instead of betareduction. The simplicity and power of the calculus derive from allowing any term to be a pattern. As well as supporting a uniform approach to functions, it supports a uniform approach to data structures which underpins two new forms of polymorphism. Path polymorphism supports searches or queries along all paths through an arbitrary data structure. Pattern polymorphism supports the dynamic creation and evaluation of patterns, so that queries can be customised in reaction to new information about the structures to be encountered. In combination, these features provide a natural account of tasks such as programming with XML paths. As the variables used in matching can now be eliminated by reduction it is necessary to separate them from the binding variables used to control scope. Then standard techniques suffice to ensure that reduction progresses and to establish confluence of reduction. 1
Basic Pattern Matching Calculi: Syntax, Reduction, Confluence, and Normalisation
, 2003
"... The pattern matching calculus is a refinement of λcalculus that integrates mechanisms appropriate for finegrained modelling of nonstrict pattern matching. In comparison with the functional rewriting strategy that is usually employed to define the operational semantics of patternmatching in nons ..."
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The pattern matching calculus is a refinement of λcalculus that integrates mechanisms appropriate for finegrained modelling of nonstrict pattern matching. In comparison with the functional rewriting strategy that is usually employed to define the operational semantics of patternmatching in nonstrict functional programming languages like Haskell or Clean, the pattern matching calculus allows simpler and more local definitions to achieve the same effects. The main device of the calculus is to further emphasise the clear distinction between matching failure and undefinedness already discussed in the literature by embedding into expressions the separate syntactic category of matchings. This separation is also important to properly restrain the possible effects of the nonmonotonicity that a naïve treatment of matching alternatives would exhibit. The language arising from that distinction turns out to naturally encompass the pattern guards of Peyton Jones and Erwig and conventional Boolean guards as special cases of the intermediate stages of matching reduction. By allowing a confluent reduction system and a normalising strategy, the pattern matching calculus
This work is licensed under the Creative Commons Attribution License. On the Implementation of Dynamic Patterns
"... The evaluation mechanism of pattern matching with dynamic patterns is modelled in the Pure Pattern Calculus by one single metarule. This contribution presents a refinement which narrows the gap between the abstract calculus and its implementation. A calculus is designed to allow reasoning on matchi ..."
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The evaluation mechanism of pattern matching with dynamic patterns is modelled in the Pure Pattern Calculus by one single metarule. This contribution presents a refinement which narrows the gap between the abstract calculus and its implementation. A calculus is designed to allow reasoning on matching algorithms. The new calculus is proved to be confluent, and to simulate the original Pure Pattern Calculus. A family of new, matchingdriven, reduction strategies is proposed.