## A New Approach to Abstract Syntax Involving Binders (1999)

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Venue: | In 14th Annual Symposium on Logic in Computer Science |

Citations: | 145 - 14 self |

### BibTeX

@INPROCEEDINGS{Gabbay99anew,

author = {Murdoch Gabbay and Andrew Pitts},

title = {A New Approach to Abstract Syntax Involving Binders},

booktitle = {In 14th Annual Symposium on Logic in Computer Science},

year = {1999},

pages = {214--224},

publisher = {IEEE Computer Society Press}

}

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### Abstract

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 Fraenkel-Mostowski permutation model of set theory with atoms (FM-sets) can serve as the semantic basis of meta-logics for specifying and reasoning about formal systems involving name binding, ff-conversion, capture avoiding substitution, and so on. We show that in FM-set theory one can express statements quantifying over `fresh' names and we use this to give a novel set-theoretic interpretation of name abstraction. Inductively defined FM-sets involving this name-abstraction set former (together with cartesian product and disjoint union) can correctly encode object-level syntax modulo ff-conversion. 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...

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Citation Context ... Gen([T ]Term) j Spec(Term \Theta Type): Equivariant SOS We believe that FM-set theory will be a useful setting for developing programming language semantics based on structural operational semantics =-=[34]-=-. Syntax-directed, rule-based inductive definitions of relations quite often contain side-conditions to do with freshness of variables, and the hope is that these can be assimilated and manipulated co... |

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Citation Context ... 1 ; : : : ; xn : X #a n if \Gamma is the type environment x 1 : X 1 ; : : : ; xn : Xn . Thus, rather in the spirit (though not the letter) of [25], one goal we are aiming for is an SML-like language =-=[27]-=- for meta-programming that combines user-declared data types with primitives for atomic types A (i.e. sorts of atoms), abstraction types [A]X , and apartness X #a . A practically important part of suc... |

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Citation Context ...n extract introduction and elimination rules for the N -quantifier from the above lemma, provided one uses sequents tagged with sets of possibly-free variables (a common practice in categorical logic =-=[20]-=-)---e.g. sequents of the form \Gamma ` ~x OE, where \Gamma is a finite set of formulas, OE a formula, and ~x a finite set of variables containing those occurring freely in \Gamma and OE. Then we can d... |

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Citation Context ...nd substitution to the meta-level where their properties are established once and for all. This is the `higher order abstract syntax' (HOAS) approach---an idea going back to Church [2] and MartinL of =-=[23]-=- which has found its way into many of the current logical frameworks and proof assistants. Its big drawback, in its original form at least, is that one looses the ability to define functions on syntax... |

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Citation Context ...and call it the A -abstraction determined by a 2 A and x 2 VFM (A ). (We will see below that [a]x 2 VFM (A ).) This is a form of `abstraction as information hiding' (like that for abstract data types =-=[28]-=-), since [a]x turns out to behave like a pair (a; x) in which the identity of a is hidden. However, and quite remarkably, A -abstractions also embody a notion of `abstraction as function' (analogous t... |

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Citation Context ... specifying and reasoning about syntactical structures that do not involve binding constructs is well understood. The theory involves such indispensable concepts as user-declared algebraic data types =-=[10]-=- (inductively defined sets), structural recursion over such data, and proof by structural induction [1]; the practice can be seen in several general-purpose systems for machine-assisted proof (such as... |

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Citation Context ... and dynamically created names introduced by the second author and Stark [32, 36] (see also [17]). In [37], Stark studies a model of the -calculus based on one of Moggi's `dynamic allocation ' monads =-=[29]-=- in the presheaf category Set I , where I is the category of finite ordinals and injective functions between them. Crucial ingredients of the dynamic allocation monad used there are the `object of nam... |

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Citation Context ...(inductively defined sets), structural recursion over such data, and proof by structural induction [1]; the practice can be seen in several general-purpose systems for machine-assisted proof (such as =-=[12, 30]-=-). This algebraic, `no binders' machinery is often applied to syntax that does involve binders; but in that case it yields overly-concrete representations in which large numbers of essentially routine... |

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Citation Context ...sely such problems with names which we claim our approach overcomes in a simple way. Similar criticisms apply to approaches to binding based upon de Bruijn's nameless terms or categorical combinators =-=[4, 3, 7]-=-: these are good for machine implementations, but not, we would argue, for representations intended for machine-assisted humansreasoning. Instead of the HOAS approach of moving both ffconversion and s... |

142 |
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Citation Context ...sely such problems with names which we claim our approach overcomes in a simple way. Similar criticisms apply to approaches to binding based upon de Bruijn's nameless terms or categorical combinators =-=[4, 3, 7]-=-: these are good for machine implementations, but not, we would argue, for representations intended for machine-assisted humansreasoning. Instead of the HOAS approach of moving both ffconversion and s... |

134 | G.: Towards a mathematical operational semantics
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Citation Context ...ng in the style of Miller and Nadathur [26] involving this quantifier may be possible.) It is also possible that the approach presented here will help extend the `functorial' operational semantics of =-=[40]-=- to encompass languages involving binders. FM type theory An extensional set theory is by no means the only setting in which to consider the notions of permutation action and finite support (although ... |

123 | Primitive recursion for higherorder abstract syntax
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Citation Context ... recursion and to prove properties by structural induction---absolutely essential tools for our intended applications to operational semantics. There are recent proposals to overcome this shortcoming =-=[24, 5]-=-. They result in systems which are technically very interesting but force the designer of algorithms and proofs `modulo-ff-conversion' to use forms of expression in our view rather far from familiar i... |

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Citation Context ...elation to presheaf models One origin of the work presented here lies in the `- calculus', a calculus of higher order functions and dynamically created names introduced by the second author and Stark =-=[32, 36]-=- (see also [17]). In [37], Stark studies a model of the -calculus based on one of Moggi's `dynamic allocation ' monads [29] in the presheaf category Set I , where I is the category of finite ordinals ... |

104 |
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Citation Context ...(inductively defined sets), structural recursion over such data, and proof by structural induction [1]; the practice can be seen in several general-purpose systems for machine-assisted proof (such as =-=[12, 30]-=-). This algebraic, `no binders' machinery is often applied to syntax that does involve binders; but in that case it yields overly-concrete representations in which large numbers of essentially routine... |

96 |
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Citation Context ...signatures with binders. We give examples of this in Section 5. Section 6 sketches the relationship of our approach to recent work on modelling variable-binding abstract syntax in presheaf categories =-=[7, 14]-=-. Finally, Section 7 mentions some of the many things that remain to be done to develop our set-theoretic modelling of abstract syntax involving binders. One of our main motivations is to produce a me... |

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Citation Context ... recursion and to prove properties by structural induction---absolutely essential tools for our intended applications to operational semantics. There are recent proposals to overcome this shortcoming =-=[24, 5]-=-. They result in systems which are technically very interesting but force the designer of algorithms and proofs `modulo-ff-conversion' to use forms of expression in our view rather far from familiar i... |

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Citation Context ...derstood. The theory involves such indispensable concepts as user-declared algebraic data types [10] (inductively defined sets), structural recursion over such data, and proof by structural induction =-=[1]-=-; the practice can be seen in several general-purpose systems for machine-assisted proof (such as [12, 30]). This algebraic, `no binders' machinery is often applied to syntax that does involve binders... |

77 | Elf: A language for logic definition and verified meta-programming
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Citation Context ...upport (although we believe it is the simplest place to start). We have begun to investigate what our approach might look like in the setting of constructive type theory [23, 30] and meta-programming =-=[31]-=-. For one thing, it is evident that Definition 4.2 generalises to a dependently typed version of abstraction [a 2 A ]X(a) def = ff 2 Abs(A ) j N a 2 A : f(a) 2 X(a)g which bears the same relationship ... |

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Citation Context ...elation to presheaf models One origin of the work presented here lies in the `- calculus', a calculus of higher order functions and dynamically created names introduced by the second author and Stark =-=[32, 36]-=- (see also [17]). In [37], Stark studies a model of the -calculus based on one of Moggi's `dynamic allocation ' monads [29] in the presheaf category Set I , where I is the category of finite ordinals ... |

52 | Five axioms of alpha-conversion
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Citation Context ...ive power of classical set theory which remains close to informal practice in its forms of expression. This, and the focus on ffconversion, makes our work close in spirit to that of Gordon and Melham =-=[11]-=-, who axiomatise a type of untyped -terms modulo ff-conversion within Church's higher order logic. However, we take a more foundational approach, in that the necessary properties of renaming become pa... |

49 | A logic programming approach to manipulating formulas and programs
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Citation Context ... of variables, and the hope is that these can be assimilated and manipulated conveniently via the N -quantifier we have introduced here. (Indeed, logic programming in the style of Miller and Nadathur =-=[26]-=- involving this quantifier may be possible.) It is also possible that the approach presented here will help extend the `functorial' operational semantics of [40] to encompass languages involving binde... |

45 | Should your specification language be typed - Lamport, Paulson - 1999 |

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Citation Context ...e origin of the work presented here lies in the `- calculus', a calculus of higher order functions and dynamically created names introduced by the second author and Stark [32, 36] (see also [17]). In =-=[37]-=-, Stark studies a model of the -calculus based on one of Moggi's `dynamic allocation ' monads [29] in the presheaf category Set I , where I is the category of finite ordinals and injective functions b... |

38 |
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Citation Context ...s morphisms are the S A -equivariant functions. Thus on the one hand, the Schanuel topos relates to the FM-universe VFM (A ) much as the usual cumulative hierarchy V relates to the topos of sets (see =-=[19]-=- for more on the category theory of universes of sets); on the other hand, the Schanuel topos is a sheaf subtopos of the presheaf categorysSet I , with the inclusion sending the FM-set of atoms A to t... |

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Citation Context ...#a \Gamma ` (new a in t) : X where \Gamma #a = x 1 : X #a 1 ; : : : ; xn : X #a n if \Gamma is the type environment x 1 : X 1 ; : : : ; xn : Xn . Thus, rather in the spirit (though not the letter) of =-=[25]-=-, one goal we are aiming for is an SML-like language [27] for meta-programming that combines user-declared data types with primitives for atomic types A (i.e. sorts of atoms), abstraction types [A]X ,... |

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Citation Context ... models One origin of the work presented here lies in the `- calculus', a calculus of higher order functions and dynamically created names introduced by the second author and Stark [32, 36] (see also =-=[17]-=-). In [37], Stark studies a model of the -calculus based on one of Moggi's `dynamic allocation ' monads [29] in the presheaf category Set I , where I is the category of finite ordinals and injective f... |

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Citation Context ...upp(x) (leaving implicit which S A -set X is being referred to). We say that an atom a is apart from x, and write a # x, if a = 2 supp(x). Recall the usual von Neumann cumulative hierarchy of sets, V =-=[35]-=-. We can build the notions of `permutation action ' and `finite support property' into a set-theoretic universe by replacing V with the Fraenkel-Mostowski universe, VFM (A ), which by definition is th... |

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Citation Context ...t of names’, given by the inclusion functor I ↩→ Set, and the shift functor δ : Set I → Set I , given by δX(n) =X(n +1). These ingredients also occur in the work on modelling πcalculus names in Set I =-=[6, 38]-=- and the recent work on modelling variable-binding abstract syntax [7, 14], where other presheaf categories besides Set I are considered. Now, a somewhat overlooked model of the ν-calculus, mentioned ... |

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Citation Context ...mes', given by the inclusion functor I ,! Set, and the shift functorsffi : Set I ! Set I , given by ffiX(n) = X(n + 1). These ingredients also occur in the work on modelling - calculus names in Set I =-=[6, 38]-=- and the recent work on modelling variable-binding abstract syntax [7, 14], where other presheaf categories besides Set I are considered. Now, a somewhat overlooked model of the -calculus, mentioned i... |

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Citation Context ...tical importance: the ability to manipulate names of bound variables explicitlysin computation and proof. Of course, one can introduce a type of `names' in a HOAS signature, as for example is done in =-=[15]; but-=- as the authors of that work say [p 26] "The main drawback of HOAS is the difficulty of dealing with metatheoretic issues concerning names : : : . As a consequence, some metatheoretic properties ... |

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Citation Context ...mes', given by the inclusion functor I ,! Set, and the shift functorsffi : Set I ! Set I , given by ffiX(n) = X(n + 1). These ingredients also occur in the work on modelling - calculus names in Set I =-=[6, 38]-=- and the recent work on modelling variable-binding abstract syntax [7, 14], where other presheaf categories besides Set I are considered. Now, a somewhat overlooked model of the -calculus, mentioned i... |

3 |
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Citation Context ...tical importance: the ability to manipulate names of bound variables explicitly in computation and proof. Of course, one can introduce a type of ‘names’ in a HOAS signature, as for example is done in =-=[15]-=-; but as the authors of that work say [p 26] “The main drawback of HOAS is the difficulty of dealing with metatheoretic issues concerning names ... . As a consequence, some metatheoretic properties in... |

2 |
A fully abstract model for the π-calculus (extended abstract
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Citation Context ...t of names’, given by the inclusion functor I ↩→ Set, and the shift functor δ : Set I → Set I , given by δX(n) =X(n +1). These ingredients also occur in the work on modelling πcalculus names in Set I =-=[6, 38]-=- and the recent work on modelling variable-binding abstract syntax [7, 14], where other presheaf categories besides Set I are considered. Now, a somewhat overlooked model of the ν-calculus, mentioned ... |