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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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Cited by 13 (2 self)
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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
A.: Nominal algebra
, 2006
"... Abstract. Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathem ..."
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Cited by 7 (2 self)
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Abstract. Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathematical usage and notation, where expressions such as λa.t or ∀a.φ are common, in which metavariables t and φ explicitly occur in the scope of a variable a. We describe the syntax and semantics of NA, and provide a sound and complete proof system for it. We also give some examples of axioms; other work has considered sets of axioms of particular interest in some detail. 1.
Strong normalization for System F by HOAS on top of FOAS
"... Abstract—We present a point of view concerning HOAS (HigherOrder Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (FirstOrder Abstract Syntax). As such, HOAS ..."
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Abstract—We present a point of view concerning HOAS (HigherOrder Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (FirstOrder Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higherorder view of a firstorder reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL.
Nominal Algebra and the HSP Theorem
"... Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. ..."
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Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitelysupported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to socalled freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a ‘nominal ’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atomsabstraction construction specific to nominalstyle semantics. Keywords: universal algebra, equational logic, nominal algebra, HSP or Birkhoff’s theorem, nominal sets, nominal terms 1
SecondOrder Equational Logic (Extended Abstract)
"... We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established. ..."
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We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.
GFOL: A TermGeneric Logic for Defining λCalculi
, 2006
"... Generic firstorder logic (GFOL) is a firstorder logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing th ..."
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Generic firstorder logic (GFOL) is a firstorder logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing that of FOL. An important fragment of GFOL, called HORN 2, possesses a much simpler Gentzen system, similar to traditional contextbased derivation systems of λcalculi. HORN 2 appears to be sufficient for defining virtually any λcalculi (including polymorphic and typerecursive ones) as theories inside the logic. GFOL endows its theories with a default loose semantics, complete for the specified calculi.