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Nominal rewriting
 Information and Computation
"... Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting ma ..."
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Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the metalanguage (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced firstorder character, since substitution of terms for variables is not captureavoiding. We show how good properties of firstorder rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem
A formal calculus for informal equality with binding
 In WoLLIC’07: 14th Workshop on Logic, Language, Information and Computation, volume 4576 of LNCS
, 2007
"... Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along w ..."
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Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as firstorder logic or the lambdacalculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1
Oneandahalfth order terms: CurryHoward and incomplete derivations
"... Abstract. The CurryHoward correspondence connects Natural Deduction derivation with the lambdacalculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. ..."
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Cited by 5 (4 self)
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Abstract. The CurryHoward correspondence connects Natural Deduction derivation with the lambdacalculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. At intermediate stages we may have an ‘incomplete derivation’, with ‘holes’. This is natural in informal practice; the challenge is to formalise it. To this end we use a oneandahalfth order technique based on nominal terms, with two levels of variable. Predicates are types, derivations are terms — and the two levels of variable are respectively the assumptions and the ‘holes ’ of an incomplete derivation. 1
The lambdacalculus is nominal algebraic
"... The λcalculus is fundamental in the study of logic and computation. Partly this is because it is a tool to study functions and functions are an important object of study in this field. Partly this is because the λcalculus seems to be, for homo sapiens, an ergonomic formal syntax. ..."
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The λcalculus is fundamental in the study of logic and computation. Partly this is because it is a tool to study functions and functions are an important object of study in this field. Partly this is because the λcalculus seems to be, for homo sapiens, an ergonomic formal syntax.
A nominal axiomatisation of the lambdacalculus
"... The lambdacalculus is a fundamental syntax in computer science. It resists an algebraic treatment because of captureavoidance sideconditions. Nominal algebra is a logic of equality designed with formalisation of specifications involving binding in mind. In this paper we axiomatise the lambdacalc ..."
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The lambdacalculus is a fundamental syntax in computer science. It resists an algebraic treatment because of captureavoidance sideconditions. Nominal algebra is a logic of equality designed with formalisation of specifications involving binding in mind. In this paper we axiomatise the lambdacalculus using nominal algebra, demonstrate how proofs with these axioms reflect the informal arguments on syntax, and we prove the axioms sound and complete. This makes a formal connection between a ‘nominal’ approach to variables, and the more traditional view of variables as a syntactic convenience for describing functions.
WFLP 2007 Twoandahalfth order lambdacalculus
"... Twoandahalfth order lambdacalculus is designed to provide a mathematical model of capturing substitution, which is a feature of the informal metalevel. There are two levels of variable; atoms representing objectlevel variables, and unknowns representing metavariables. Lambdaabstraction and ..."
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Twoandahalfth order lambdacalculus is designed to provide a mathematical model of capturing substitution, which is a feature of the informal metalevel. There are two levels of variable; atoms representing objectlevel variables, and unknowns representing metavariables. Lambdaabstraction and betareduction exist for both atoms and unknowns. This extends the twolevels of variable in nominal terms with a functional meaning.