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33
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 219 (28 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Captureavoiding Substitution as a Nominal Algebra
 Formal Aspects of Computing
, 2008
"... Abstract. Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and betareduction in the lambdacalculus. So is substitution something we define on syntax on a casebycase basis, or can we turn the idea of ‘substitution ’ into a mathematical objec ..."
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Cited by 15 (5 self)
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Abstract. Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and betareduction in the lambdacalculus. So is substitution something we define on syntax on a casebycase 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. 1
Nominal unification from a higherorder perspective
 In Proceedings of RTA’08
"... Abstract. Nominal Logic is an extension of firstorder logic with equality, namebinding, nameswapping, and freshness of names. Contrarily to higherorder logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows “variable capture”, ..."
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Cited by 12 (3 self)
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Abstract. Nominal Logic is an extension of firstorder logic with equality, namebinding, nameswapping, and freshness of names. Contrarily to higherorder logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows “variable capture”, breaking a fundamental principle of lambdacalculus. Despite this difference, nominal unification can be seen from a higherorder perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higherorder unification problems: higherorder patterns unification. This reduction proves that nominal unification can be decided in quadratic deterministic time. 1
AN EFFICIENT NOMINAL UNIFICATION ALGORITHM
"... Abstract. Nominal Unification is an extension of firstorder unification where terms can contain binders and unification is performed modulo αequivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearlyreduce nominal unification problems to ..."
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Cited by 7 (1 self)
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Abstract. Nominal Unification is an extension of firstorder unification where terms can contain binders and unification is performed modulo αequivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearlyreduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for firstorder unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently. 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
Initiality for Typed Syntax and Semantics
"... Abstract. We give an algebraic characterization of the syntax and semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purp ..."
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Abstract. We give an algebraic characterization of the syntax and semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purpose, we employ techniques developed in two previous works: in [2], we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In [1], we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we show that those techniques are modular enough to be combined: we thus characterize simply–typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations — that are semantically faithful by construction — between languages over possibly different sets of types. We specify a language by a 2–signature, that is, a signature on two levels: the syntactic level specifies the types and terms of the language, and associates a type to each term. The semantic level specifies, through inequations, reduction rules on the terms of the language. To any given
Modules over relative monads For Syntax and semantics
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2012
"... We give an algebraic characterization of the syntax and semantics of a class of functional programming languages. We introduce a notion of 2–signature: such a signature specifies not only the terms of a language, but also reduction rules on those terms. To any 2–signature S we associate a ..."
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Cited by 4 (2 self)
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We give an algebraic characterization of the syntax and semantics of a class of functional programming languages. We introduce a notion of 2–signature: such a signature specifies not only the terms of a language, but also reduction rules on those terms. To any 2–signature S we associate a
Twolevel Lambdacalculus
 Electron. Notes Theor. Comput. Sci
, 2009
"... Twolevel lambdacalculus is designed to provide a mathematical model of capturing substitution, also called instantiation. Instantiation is a feature of the ‘informal metalevel’; it appears pervasively in specifications of the syntax and semantics of formal languages. The twolevel lambdacalculus ..."
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Twolevel lambdacalculus is designed to provide a mathematical model of capturing substitution, also called instantiation. Instantiation is a feature of the ‘informal metalevel’; it appears pervasively in specifications of the syntax and semantics of formal languages. The twolevel lambdacalculus has two levels of variable. Lambdaabstraction and betareduction exist for both levels. A level 2 betareduct, triggering a substitution of a term for a level 2 variable, does not avoid capture for level 1 abstractions. This models metavariables and instantiation as appears at the informal metalevel. In this paper we lay down the syntax of the twolevel lambdacalculus; we develop theories of freshness, alphaequivalence, and betareduction; and we prove confluence. In doing this we give nominal terms unknowns — which are level 2 variables and appear in several previous papers — a functional meaning. In doing this we take a step towards longerterm goals of developing a foundation for theoremprovers which directly support reasoning in the style of nominal rewriting and nominal algebra, and towards a mathematics of functions which can bind names in their arguments.
An AlphaCorecursion Principle for the Infinitary Lambda Calculus
, 2012
"... Gabbay and Pitts proved that lambdaterms 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 lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecursion pri ..."
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Gabbay and Pitts proved that lambdaterms 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 lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alphaequivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, LévyLongo and Berarducci trees).