## Alpha-structural recursion and induction (Extended Abstract) (2005)

Venue: | THEOREM PROVING IN HIGHER ORDER LOGICS, 18TH INTERNATIONAL CONFERENCE, TPHOLS 2005, OXFORD UK, AUGUST 2005, PROCEEDINGS, VOLUME 3603 OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Pitts05alpha-structuralrecursion,

author = {Andrew M. Pitts},

title = {Alpha-structural recursion and induction (Extended Abstract)},

booktitle = {THEOREM PROVING IN HIGHER ORDER LOGICS, 18TH INTERNATIONAL CONFERENCE, TPHOLS 2005, OXFORD UK, AUGUST 2005, PROCEEDINGS, VOLUME 3603 OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {2005},

pages = {17--34},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and name-binding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax trees modulo α-equivalence. At the heart of this formalisation 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 higher-order logic of principles of α-structural recursion and induction for α-equivalence classes from the ordinary versions of these principles for abstract syntax trees.

### Citations

1286 | A structural approach to operational semantics - Plotkin - 1981 |

847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...called nominal sets are introduced in [16] and I will use them here to express α-structural recursion and induction within “ordinary mathematics”, or more precisely, within Church’s higherorder logic =-=[3]-=-. (7)s20 Andrew Pitts 2 Nominal Syntax The usual principles of structural recursion and induction are parameterised by an algebraic signature that specifies the allowed constructors for forming ASTs o... |

434 | The π-calculus: a Theory of Mobile Processes - Sangiorgi, Walker - 2001 |

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...d on the fact that nominal sets form a model of higher-order logic (without choice functions—see Example 6). In the author’s opinion, the best way of explaining this model is to use topos theory (see =-=[13]-=-, for example). Call a function f ∈ X → Y between two nominal sets equivariant if it is supported by the empty set; in view of (12), this means that π · (f(x)) = f(π · x), for all π ∈ Perm and x ∈ X. ... |

302 |
Higher-order abstract syntax
- Pfenning, Elliot
- 1988
(Show Context)
Citation Context ...and Λ/=α for its quotient by α-equivalence =α, then capture-avoiding substitution of e � [t]α for x is a function ˆsx,e ∈ Λ/=α → Λ/=α. 1 This includes the metatheory of “higher-order abstract syntax” =-=[15]-=-, where the questions we are addressing are pushed up one meta-level to a single binding-form, λ-abstraction. (1)sAlpha-Structural Recursion and Induction 19 Every such function corresponds to a funct... |

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

206 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
- 2002
(Show Context)
Citation Context ...f constructions involving ASTs are independent of choice of bound names. A fully formal treatment has to prove such independence results and in this paper we examine ways, arising from the results of =-=[8, 16]-=-, to reduce the burden of such proofs. However, proving that pre-existing functions respect α-equivalence is only part of the story; in most cases a prior (or simultaneous) problem is to prove the exi... |

162 | Nominal Logic, A first order theory of names and binding
- Pitts
(Show Context)
Citation Context ...f constructions involving ASTs are independent of choice of bound names. A fully formal treatment has to prove such independence results and in this paper we examine ways, arising from the results of =-=[8, 16]-=-, to reduce the burden of such proofs. However, proving that pre-existing functions respect α-equivalence is only part of the story; in most cases a prior (or simultaneous) problem is to prove the exi... |

154 | The Lambda Calculus: its Syntax and Semantics. NorthHolland, revised edition - Barendregt - 1984 |

154 |
de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem
- G
- 1972
(Show Context)
Citation Context ...Ts. It is true that this level of abstraction, which identifies terms differing only in the names of bound entities, can be reconciled with an algebraic treatment of syntax by using de Bruijn indexes =-=[4]-=-. The well-known disadvantage of this device is that it necessitates a calculus of operations on de Bruijn indexes that does not have much to do with our intuitive view of the structure of syntax. As ... |

142 |
Abstract syntax and variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...pecify how bound occurrences of names in an AST are associated with a binding site further up the syntax tree. There are a number of such mechanisms in the literature of varying degrees of generality =-=[10, 17, 5, 12, 22]-=-. Here we will use the notion of nominal signature from [22]. It has the advantage of dealing with binding and α-equivalence independently of any considerations to do with variables, substitution and ... |

82 | Nominal techniques in Isabelle/HOL
- Urban
(Show Context)
Citation Context ...andard α-equivalence classes of abstract syntax trees. This is also the approach taken by Norrish [14], building on Gordon and Melham’s five axioms for α-equivalence [9]; and also by Urban and Tasson =-=[23]-=-. Norrish’s recursion principle [14, Fig. 1] has sideconditions requiring that the function being defined be well-behaved with respect to variable-permutations and with respect to fresh name generatio... |

52 | Five axioms of alpha-conversion
- Gordon, Melham
- 1997
(Show Context)
Citation Context ...refer directly to α-terms, i.e. standard α-equivalence classes of abstract syntax trees. This is also the approach taken by Norrish [14], building on Gordon and Melham’s five axioms for α-equivalence =-=[9]-=-; and also by Urban and Tasson [23]. Norrish’s recursion principle [14, Fig. 1] has sideconditions requiring that the function being defined be well-behaved with respect to variable-permutations and w... |

52 | Nominal Unification
- Urban, Pitts, et al.
- 2003
(Show Context)
Citation Context ...pecify how bound occurrences of names in an AST are associated with a binding site further up the syntax tree. There are a number of such mechanisms in the literature of varying degrees of generality =-=[10, 17, 5, 12, 22]-=-. Here we will use the notion of nominal signature from [22]. It has the advantage of dealing with binding and α-equivalence independently of any considerations to do with variables, substitution and ... |

27 |
Substitution revisited
- Stoughton
- 1988
(Show Context)
Citation Context ... (either by giving up structural properties and using a less natural recursion on the height of trees; or by using structural recursion to define a more general operation of simultaneous substitution =-=[21]-=-). An alternative approach, and one that works with the original simple specification, is to construct functions by giving rule-based inductive definitions of their graphs (with the rules encoding the... |

26 | On a monadic semantics for freshness - Shinwell, Pitts |

23 | Nominal Logic Programming
- Cheney
(Show Context)
Citation Context ...t X, each element x ∈ X possesses a smallest finite support, which we write as supp X(x) (or just supp(x), if X is clear from the context) and call the support of x in X. 5 Both Gabbay [7] and Cheney =-=[2]-=- develop more general notions of “small” supports. As Cheney’s work shows, such a generalisation is necessary for some techniques of classical model theory to be applied; but finite supports are suffi... |

19 | An axiomatic approach to metareasoning on nominal algebras in hoas
- Honsell, Miculan, et al.
(Show Context)
Citation Context ...pecify how bound occurrences of names in an AST are associated with a binding site further up the syntax tree. There are a number of such mechanisms in the literature of varying degrees of generality =-=[10, 17, 5, 12, 22]-=-. Here we will use the notion of nominal signature from [22]. It has the advantage of dealing with binding and α-equivalence independently of any considerations to do with variables, substitution and ... |

17 |
A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
- Gabbay
- 2000
(Show Context)
Citation Context ...App(Var a2, e)), as required for (24). 6 Assessment Mathematical Perspective. The results of this paper are directly inspired by my joint work with Gabbay on “FM-set” theory [8] and by his PhD thesis =-=[6]-=-; in particular those works contain structural recursion and induction principles for an inductively defined FM-set isomorphic to λ-terms modulo α-equivalence. Here I have taken an approach that is bo... |

15 | An illustrative theory of relations, in
- Plotkin
- 1990
(Show Context)
Citation Context |

13 | Recursive function definition for types with binders
- Norrish
- 2004
(Show Context)
Citation Context ...rphic to the set of α-terms, the recursion and induction principles refer directly to α-terms, i.e. standard α-equivalence classes of abstract syntax trees. This is also the approach taken by Norrish =-=[14]-=-, building on Gordon and Melham’s five axioms for α-equivalence [9]; and also by Urban and Tasson [23]. Norrish’s recursion principle [14, Fig. 1] has sideconditions requiring that the function being ... |

11 |
FM-HOL, a higher-order theory of names
- Gabbay
- 2002
(Show Context)
Citation Context ...in a nominal set X, each element x ∈ X possesses a smallest finite support, which we write as supp X(x) (or just supp(x), if X is clear from the context) and call the support of x in X. 5 Both Gabbay =-=[7]-=- and Cheney [2] develop more general notions of “small” supports. As Cheney’s work shows, such a generalisation is necessary for some techniques of classical model theory to be applied; but finite sup... |

9 |
Notational definition - a formal account
- Griffin
(Show Context)
Citation Context |