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A new approach to abstract syntax with variable binding
 Formal Aspects of Computing
, 2002
"... Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding op ..."
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Cited by 226 (48 self)
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Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding operations. Inductively defined FMsets involving the nameabstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. 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 notion of structural recursion for defining syntaxmanipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science. 1.
Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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Cited by 176 (15 self)
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
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 153 (15 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...
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 95 (17 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.
Monadic Presentations of Lambda Terms Using Generalized Inductive Types
 In Computer Science Logic
, 1999
"... . We present a denition of untyped terms using a heterogeneous datatype, i.e. an inductively dened operator. This operator can be extended to a Kleisli triple, which is a concise way to verify the substitution laws for calculus. We also observe that repetitions in the denition of the monad as wel ..."
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Cited by 81 (15 self)
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. We present a denition of untyped terms using a heterogeneous datatype, i.e. an inductively dened operator. This operator can be extended to a Kleisli triple, which is a concise way to verify the substitution laws for calculus. We also observe that repetitions in the denition of the monad as well as in the proofs can be avoided by using wellfounded recursion and induction instead of structural induction. We extend the construction to the simply typed calculus using dependent types, and show that this is an instance of a generalization of Kleisli triples. The proofs for the untyped case have been checked using the LEGO system. Keywords. Type Theory, inductive types, calculus, category theory. 1 Introduction The metatheory of substitution for calculi is interesting maybe because it seems intuitively obvious but becomes quite intricate if we take a closer look. [Hue92] states seven formal properties of substitution which are then used to prove a general substitution theor...
Nominal Unification
 Theoretical Computer Science
, 2003
"... We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the a ..."
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Cited by 58 (24 self)
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We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a `nominal' approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijnstyle representations) and yet get a version of this form of substitution that respects #equivalence and possesses good algorithmic properties. We achieve this by adapting an existing idea and introducing a key new idea. The existing idea is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The key new idea is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of the classical firstorder unification algorithm to this setting which retains the latter's pleasant properties: unification problems involving #equivalence and freshness are decidable; and solvable problems possess most general solutions.
Alphastructural recursion and induction
 Journal of the ACM
, 2006
"... The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically i ..."
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Cited by 48 (6 self)
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The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo αequivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higherorder logic of principles of αstructural recursion and induction for αequivalence classes from the ordinary versions of these principles for abstract syntax trees.
Macros as multistage computations: Typesafe, generative, binding macros in MacroML
 in MacroML. In the International Conference on Functional Programming (ICFP ’01
, 2001
"... ..."
AlphaProlog: A Logic Programming Language with Names, Binding, and AlphaEquivalence
, 2004
"... There are two wellknown approaches to programming with names, binding, and equivalence up to consistent renaming: representing names and bindings as concrete identifiers in a firstorder language (such as Prolog), or encoding names and bindings as variables and abstractions in a higherorder langua ..."
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Cited by 41 (9 self)
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There are two wellknown approaches to programming with names, binding, and equivalence up to consistent renaming: representing names and bindings as concrete identifiers in a firstorder language (such as Prolog), or encoding names and bindings as variables and abstractions in a higherorder language (such as LambdaProlog). However, both approaches have drawbacks: the former often involves stateful namegeneration and requires manual definitions for alphaequivalence and captureavoiding substitution, and the latter is semantically very complicated, so reasoning about programs written using either approach can be very di#cult. Gabbay and Pitts have developed a new approach to encoding abstract syntax with binding based on primitive operations of nameswapping and freshness. This paper presents AlphaProlog, a logic programming language that uses this approach, along with several illustrative example programs and an operational semantics.