Nominal rewriting

Abstract. Gabbay and Pitts proved that lambda-terms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alpha-equivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, Lévy-Longo and Berarducci trees).

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“…It will also be interesting to explore whether nominal rewriting [10] has applications in the infinitary λ-calculus.…”

Section: Resultsmentioning

confidence: 99%

An Alpha-Corecursion Principle for the Infinitary Lambda Calculus

Kurz

Petrişan

Severi

et al. 2012

Coalgebraic Methods in Computer Science

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“…It reduces parentheses to give substitution a higher priority than permutation and abstraction, so we do. The following commutation is easy to prove [9,6]:…”

Section: A Derivation Systemmentioning

confidence: 99%

“…There is no problem in principle with extending the system with a sort or type system if convenient, perhaps along the lines of other work [6,7].…”

Section: Syntaxmentioning

confidence: 99%

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A Formal Calculus for Informal Equality with Binding

Gabbay¹,

Mathijssen

2007

Logic, Language, Information and Computation

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Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing capture-avoidance 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 first-order logic or the lambda-calculus 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.

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“…A rewrite rule ∇ l → r may trigger a rewrite in a term t when (an instance of) l is provably equal in CORE to some subterm of t, and the corresponding instance of ∇ is derivable using the ambient context of freshness assumptions ∆ (so ∇ is freshness conditions on the rewrite rule) [8,9,10]. Rewrite rules are given in Fig.…”

Section: Nominal Algebramentioning

confidence: 99%

Capture-avoiding substitution as a nominal algebra

Gabbay

Mathijssen

2008

Form. Asp. Comput.

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Abstract. Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and beta-reduction in the lambda-calculus. So is substitution something we define on syntax on a case-by-case basis, or can we turn the idea of 'substitution' into a mathematical object? We exploit the new framework of Nominal Algebra to axiomatise substitution. We prove our axioms sound and complete with respect to a canonical model; this turns out to be quite hard, involving subtle use of results of rewriting and algebra.

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Nominal rewriting

Cited by 94 publications

References 22 publications

An Alpha-Corecursion Principle for the Infinitary Lambda Calculus

An Alpha-Corecursion Principle for the Infinitary Lambda Calculus

A Formal Calculus for Informal Equality with Binding

Capture-avoiding substitution as a nominal algebra

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