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21
A New Approach to Abstract Syntax Involving Binders
 In 14th Annual Symposium on Logic in Computer Science
, 1999
"... Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) ..."
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Cited by 145 (14 self)
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Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) can serve as the semantic basis of metalogics for specifying and reasoning about formal systems involving name binding, ffconversion, capture avoiding substitution, and so on. We show that in FMset theory one can express statements quantifying over `fresh' names and we use this to give a novel settheoretic interpretation of name abstraction. Inductively defined FMsets involving this nameabstraction set former (together with cartesian product and disjoint union) can correctly encode objectlevel syntax modulo ffconversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated n...
Primitive Recursion for HigherOrder Abstract Syntax
 Theoretical Computer Science
, 1997
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A Metalanguage for Programming with Bound Names Modulo Renaming
 Mathematics of Program Construction, volume 1837 of Lecture Notes in Computer Science
, 2000
"... This paper describes work in progress on the design of an MLstyle metalanguage FreshML for programming with recursively defined functions on userdefined, concrete data types whose constructors may involve variable binding. Up to operational equivalence, values of such FreshML data types can faithf ..."
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Cited by 88 (15 self)
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This paper describes work in progress on the design of an MLstyle metalanguage FreshML for programming with recursively defined functions on userdefined, concrete data types whose constructors may involve variable binding. Up to operational equivalence, values of such FreshML data types can faithfully encode terms modulo alphaconversion for a wide range of object languages in a straightforward fashion. The design of FreshML is `semantically driven', in that it arises from the model of variable binding in set theory with atoms given by the authors in [7]. The language has a type constructor for abstractions over names ( = atoms) and facilities for declaring locally fresh names. Moreover, recursive definitions can use a form of patternmatching on bound names in abstractions. The crucial point is that the FreshML type system ensures that these features can only be used in welltyped programs in ways that are insensitive to renaming of bound names.
Macros as multistage computations: Typesafe, generative, binding macros in MacroML
 in MacroML. In the International Conference on Functional Programming (ICFP ’01
, 2001
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MetaProgramming with Names and Necessity
, 2002
"... Metaprogramming is a discipline of writing programs in a certain programming language that generate, manipulate or execute programs written in another language. In a typed setting, metaprogramming languages usually contain a modal type constructor to distinguish the level of object programs (which ..."
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Cited by 34 (3 self)
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Metaprogramming is a discipline of writing programs in a certain programming language that generate, manipulate or execute programs written in another language. In a typed setting, metaprogramming languages usually contain a modal type constructor to distinguish the level of object programs (which are the manipulated data) from the meta programs (which perform the computations). In functional programming, modal types of object programs generally come in two flavors: open and closed, depending on whether the expressions they classify may contain any free variables or not. Closed object programs can be executed at runtime by the meta program, but the computations over them are more rigid, and typically produce less e#cient residual code. Open object programs provide better inlining and partial evaluation, but once constructed, expressions of open modal type cannot be evaluated.
Nominal logic programming
, 2006
"... Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especial ..."
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Cited by 23 (8 self)
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Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especially wellsuited for prototyping type systems, proof theories, operational semantics rules, and other formal systems in which bound names are present. In many cases, nominal logic programs are essentially literal translations of “paper” specifications. As such, nominal logic programming provides an executable specification language for prototyping, communicating, and experimenting with formal systems. We describe some typical nominal logic programs, and develop the modeltheoretic, prooftheoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via two examples.
The ∇calculus. Functional programming with higherorder encodings
 In Proceedings of the 7th International Conference on Typed Lambda Calculi and Applications
, 2005
"... Abstract. Higherorder encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate. In this paper we present the ∇calculus, a calculus for defining gener ..."
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Cited by 23 (3 self)
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Abstract. Higherorder encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate. In this paper we present the ∇calculus, a calculus for defining general recursive functions over higherorder encodings. To avoid problems commonly associated with using the same function space for representations and computations, we separate one from the other. The simplytyped λcalculus plays the role of the representationlevel. The computationlevel contains not only the usual computational primitives but also an embedding of the representationlevel. It distinguishes itself from similar systems by allowing recursion under representationlevel λbinders while permitting a natural style of programming which we believe scales to other logical frameworks. Sample programs include bracket abstraction, parallel reduction, and an evaluator for a simple language with firstclass continuations. 1
A Definitional Approach to Primitive Recursion over Higher Order Abstract Syntax
 In Proceedings of the 2003 workshop on Mechanized
, 2003
"... Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. ..."
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Cited by 22 (5 self)
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Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.
Abstract syntax and variable binding (extended abstract
 In Proc. 14 th LICS
, 1999
"... Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the ..."
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Cited by 21 (0 self)
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Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.
Focusing on binding and computation
 In IEEE Symposium on Logic in Computer Science
, 2008
"... Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the CurryHoward interpretation of a focused ..."
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Cited by 21 (6 self)
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Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the CurryHoward interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by patternmatching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation. 1