## A new approach to abstract syntax with variable binding (2002)

### Cached

### Download Links

- [www.cl.cam.ac.uk]
- [www.cl.cam.ac.uk]
- [www.cl.cam.ac.uk]
- [www.cl.cam.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | Formal Aspects of Computing |

Citations: | 206 - 44 self |

### BibTeX

@ARTICLE{Gabbay02anew,

author = {Murdoch J. Gabbay and Andrew M. Pitts},

title = {A new approach to abstract syntax with variable binding},

journal = {Formal Aspects of Computing},

year = {2002},

volume = {13},

pages = {341--363}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘name-abstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction 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 syntax-manipulating 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.

### Citations

847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ... renaming and substitution to the meta-level where their properties are established once and for all. This is the ‘higher order abstract syntax’ (HOAS) approach [PfE88] — an idea going back to Church =-=[Chu40]-=- and Martin-Löf [Mar84] that 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 loses the ability to defin... |

809 | Mobile Ambients - Cardelli, Gordon - 1998 |

785 | A calculus for cryptographic protocols: The spi calculus - Abadi, Gordon - 1997 |

500 | T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge - Melham - 1993 |

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...pos, and its associated FM-set theory, have any advantage over these other, related categories? It is well known that toposes correspond to theories in extensional, higher-order, intuitionistic logic =-=[LaS86]-=-. Unlike presheaf toposes in general, the Schanuel topos models classical rather than intuitionistic higher-order logic; furthermore, its higher-order structure (function and power objects) is rather ... |

342 |
Intuitionistic type theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...ion to the meta-level where their properties are established once and for all. This is the ‘higher order abstract syntax’ (HOAS) approach [PfE88] — an idea going back to Church [Chu40] and Martin-Löf =-=[Mar84]-=- that 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 loses the ability to define functions on syntax b... |

303 | N.: Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation.; Indag
- Bruijn
- 1972
(Show Context)
Citation Context ...mportance: the ability to manipulate names of bound variables explicitly in computation and proof. 2 The same criticism applies to approaches to variable binding based upon de Bruijn’s nameless terms =-=[dB72]-=- or categorical combinators [Cur93]: these are good for machine implementations, but not, we would argue, for representations intended for machine-assisted human reasoning. Instead of the HOAS approac... |

285 | Interpre'tation fonctionelle et e'limination des coupures duns l'arithme'tique d'ordre supe'rieure - Girard - 1972 |

267 | Semantics of Programming Languages: Structures and Techniques. Foundations of Computing - Gunter - 1992 |

185 |
Initial Algebra Semantics and Continuous Algebras
- Goguen, Thatcher, et al.
- 1977
(Show Context)
Citation Context ...ice of specifying and reasoning about syntactical structures that do not involve variablebinding constructs is well understood. The theory involves such indispensable concepts as algebraic data types =-=[GTWW77]-=-, inductively defined sets, definition by structural recursion and proof by structural induction [Bur69]; the practice can be seen in several general-purpose systems for machine-assisted proof (such a... |

165 |
An Abstract View of Programming Languages
- Moggi
- 1989
(Show Context)
Citation Context ...ams, of call-by-value higher order functions and dynamically created names (see also [JeR99]). In [Sta96a], Stark studies a model of the ν-calculus based on one of Moggi’s ‘dynamic allocation’ monads =-=[Mog89]-=- 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... |

145 | A new approach to abstract syntax involving binders
- Gabbay, Pitts
- 1999
(Show Context)
Citation Context ...omputer Laboratory, J. J. Thompson Avenue, Cambridge CB3 0FD, UK. e-mail: Andrew.Pitts@cl.cam.ac.uk 1 This paper is dedicated to Rod Burstall. It is a revised and expanded version of Gabbay and Pitts =-=[GaP99]-=-.s342 M. J. Gabbay and A. M. Pitts the particular use of such a metatheory of variable binding that most concerns us, namely machine-assisted proofs about the structural operational semantics [Plo81] ... |

142 |
Abstract syntax and variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ... categories. Thus if one is looking for a single, general-purpose setting for modelling variable-binding syntax, the logic of the Schanuel topos is both a bit more powerful and familiar. One can view =-=[FPT99]-=- as establishing, amongst other things, a very nice categorical algebra for the de Buijn, ‘nameless’ style of treating variable-binding and substitution. By contrast, here we have presented a useful l... |

123 | Primitive recursion for higherorder abstract syntax - Despeyroux, Pfenning, et al. - 1997 |

96 | Semantical analysis of higher-order abstract syntax - Hofmann - 1999 |

90 | Reasoning with higher-order abstract syntax in a logical framework - McDowell, Miller |

89 | Proving properties of programs by structural induction
- Burstall
- 1969
(Show Context)
Citation Context ...ts is well understood. The theory involves such indispensable concepts as algebraic data types [GTWW77], inductively defined sets, definition by structural recursion and proof by structural induction =-=[Bur69]-=-; the practice can be seen in several general-purpose systems for machine-assisted proof (such as [GoM93, PaM93]). This algebraic, ‘no binders’ machinery is often applied to syntax that does involve b... |

70 | The Axiom of Choice - Jech - 1973 |

60 | Sheaves in Geometry and Logic: A First Introduction to Topos Theory - MacLane, Moerdijk - 1992 |

52 | Five axioms of alpha-conversion
- Gordon, Melham
- 1997
(Show Context)
Citation Context ...h remains close to informal practice in its forms of expression. This, and the focus on α-conversion rather than substitution as primitive, makes our work close in spirit to that of Gordon and Melham =-=[GoM96]-=-, who axiomatise a type of untyped λ-terms modulo α-conversion within Church’s higherorder logic. However, we take a more foundational approach, in that the necessary properties of α-conversion become... |

46 | Logical properties of name restriction
- Cardelli, Gordon
- 2001
(Show Context)
Citation Context ...alculus processes in [MoP00]. Cardelli and Gordon have taken up some of the ideas presented here and used permutative renaming and theN -quantifier as part of a modal logic for their ambient calculus =-=[CaG00a]-=-. This paper is a revised and expanded version of [GaP99]. Since that paper appeared, the theory we have described here has been developed and applied in two somewhat different directions. The first a... |

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

26 |
Structured coalgebras and minimal HD-automata for the π-calculus
- Montanari, Pistore
(Show Context)
Citation Context ...vered independently by Honda in his work on a general framework for processes [Hon00]. It also features in work on finite state transition systems for checking bisimilarity of π-calculus processes in =-=[MoP00]-=-. Cardelli and Gordon have taken up some of the ideas presented here and used permutative renaming and theN -quantifier as part of a modal logic for their ambient calculus [CaG00a]. This paper is a re... |

23 | Towards a theory of bisimulation for local names
- Jeffrey, Rathke
- 1999
(Show Context)
Citation Context ...he second author and Stark [PiS93a, Sta95] to explore the properties, with respect to semantic equivalence of programs, of call-by-value higher order functions and dynamically created names (see also =-=[JeR99]-=-). In [Sta96a], Stark studies a model of the ν-calculus based on one of Moggi’s ‘dynamic allocation’ monads [Mog89] in the presheaf category Set I , where I is the category of finite ordinals and inje... |

22 |
Sheaf models for set theory
- Fourman
- 1980
(Show Context)
Citation Context ... further, we introduce a suitable theory of atoms and FM-sets within classical first-order logic with equality. This theory is based upon ZFA — Zermelo-Fraenkel set theory with Atoms (see for example =-=[Fou80]-=-, or [JoM95, §5]). This has a signature containing not only a binary relation symbol (10)s348 M. J. Gabbay and A. M. Pitts ‘∈’ for membership, but also a constant A for the set of atoms, A. The axioms... |

19 | Elementary structures in process theory (1): Sets with renaming
- Honda
(Show Context)
Citation Context ...ns’ [HMS98, Des00] are not entirely satisfactory. The formalism of permutation actions and finite support has been rediscovered independently by Honda in his work on a general framework for processes =-=[Hon00]-=-. It also features in work on finite state transition systems for checking bisimilarity of π-calculus processes in [MoP00]. Cardelli and Gordon have taken up some of the ideas presented here and used ... |

17 |
A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
- Gabbay
- 2000
(Show Context)
Citation Context ...Isabelle generic theorem prover [Pau94]. This required a good deal of ‘proof engineering’ to reuse the existing Isabelle ZF theory; an account may be found in Chapter III of the first author’s thesis =-=[Gab00]-=- (Chapter II of which contains a development of FM-set theory more detailed than the one given here). Secondly, we have begun to design an ML-style metalanguage, FreshML, for programming with recursiv... |

12 | Design considerations for a functional programming language. In The software revolution: state-of-the art conference, Pergamon - Burstall - 1977 |

8 |
Algebraic set theory. Number 220
- Joyal, Moerdijk
- 1995
(Show Context)
Citation Context ...e Schanuel topos relates to the cumulative hierarchy of FM-sets defined by equation (11) much as the usual von Neumann cumulative hierarchy, defined by equation (9), relates to the topos of sets (see =-=[JoM95]-=- for more on the category theory of universes of sets); on the other hand, the Schanuel topos is a sheaf subtopos of the presheaf category Set I , with the inclusion sending the FM-set of atoms A to t... |

6 | Should your speci language be typed - Lamport, Paulson - 1999 |

5 | A higher-order specification of the pi-calculus - Despeyroux - 2000 |

5 | Quotients of decidable objects in a topos - Johnstone - 1983 |

4 |
calculus in (co)inductive type theory. Technical report, Dipartimento di Matematica e Informatica, Universita degli Studi di Udine
- Honsell, Miculan, et al.
- 1998
(Show Context)
Citation Context ... permutation model of set theory to abstract syntax with variable binding is rather far from this 2 Of course, one can introduce a type of `names' in a HOAS-style signature, as is done for example in =-=[HMS98]-=-; but as the authors of that work say (p. 26): `The main drawback of HOAS is the diculty of dealing with metatheoretic issues concerning names. . . . As a consequence, some metatheoretic properties in... |

3 |
Scagnetto, I.: π-Calculus in (co)inductive type theory. Technical report, Dipartimento di Matematica e Informatica, Università degli Studi di Udine
- Honsell, Miculan, et al.
- 1998
(Show Context)
Citation Context ... permutation model of set theory to abstract syntax with variable binding is rather far from this 2 Of course, one can introduce a type of ‘names’ in a HOAS-style signature, as is done for example in =-=[HMS98]-=-; 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... |

3 | A higher-order speci of the pi-calculus - Despeyroux - 2000 |

3 | A fully abstract model for the -calculus (extended abstract - Fiore, Moggi, et al. - 1996 |

3 |
A Theory of Inductive De with -Equivalence: Semantics, Implementation, Programming Language
- Gabbay
- 2000
(Show Context)
Citation Context ...e Isabelle generic theorem prover [Pau94]. This required a good deal of `proof engineering' to reuse the existing Isabelle ZF theory; an account may be found in Chapter III of thesrst author's thesis =-=[Gab00]-=- (Chapter II of which contains a development of FM-set theory more detailed than the one given here). Secondly, we have begun to design an ML-style metalanguage, FreshML, for programming with recursiv... |

2 | A fully abstract model for the π-calculus (extended abstract - Fiore, Moggi, et al. - 1996 |

1 |
P.: The Lambda Calculus: Its Syntax and Semantics (rev. edn
- Barendregt
- 1984
(Show Context)
Citation Context ...onclude by mentioning some ways in which the theory presented here is being developed and applied. 2. Permutative Renaming As a motivating example, consider the terms M of the untyped lambda calculus =-=[Bar84]-=-: M ::= a | M M | λa.M (1) where a ranges over some countably infinite set A of names of variables. Definition 2.1. The countably infinite set A will be fixed throughout this paper. We call its elemen... |