## Nominal Logic: A First Order Theory of Names and Binding (2001)

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Venue: | Information and Computation |

Citations: | 163 - 15 self |

### BibTeX

@ARTICLE{Pitts01nominallogic:,

author = {Andrew M. Pitts},

title = {Nominal Logic: A First Order Theory of Names and Binding},

journal = {Information and Computation},

year = {2001},

volume = {186},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

This paper formalises within first-order 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 first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...

### Citations

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

207 | A new approach to abstract syntax with variable binding. Formal Aspects of Computing 13
- GABBAY, PITTS
- 2002
(Show Context)
Citation Context ...pping and for freshness of names, from which a notion of binding can be derived. Its axioms express key properties of these primitives, which are satised by the FM-sets model of syntax introduced in [=-=11]-=-. Nominal Logic serves as a vehicle for making two general points. Firstly, nameswapping has much nicer logical properties than more general forms of renaming while at the same time providing a sucien... |

164 |
Isabelle: A generic theorem prover, volume 828
- Paulson
- 1994
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Citation Context ...9]. On the other hand, the Theory of Contexts is able to take advantage of existing machineassisted infrastructure (namely Coq [6]) quite easily, whereas Gabbay had to work hard to adapt the Isabelle =-=[31-=-] set theory package to produce his Isabelle/FM-sets package: see [13, Chapter III] and [15]. The notion of abstraction that is denable within Nominal Logic (see Section 7) captures something of thesr... |

154 |
de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem
- G
- 1972
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Citation Context ... by informal practice become explicit, but rather that many of these details have to be revisited on a case-by-case basis for each object language. The use of parse trees containing de Bruijn indices =-=[5]-=- is more elegant; but this has its own complications and also side-steps the issue of formalising informal practice to do with named bound variables. Abstractions as functions from terms to terms. The... |

154 |
The Lambda Calculus: its Syntax and Semantics. NorthHolland, revised edition
- Barendregt
- 1984
(Show Context)
Citation Context ...e of the possible `capture ' of b by binders b:( ) occurring in t. (For example b:a c:a holds, but fb=ag(b:a) = b:b c:b = fb=ag(c:a).) In the development of the theory of lambda calculus [1], this inconvenient fact immediately leads to the formulation of more complicated, `capture-avoiding' notions of renaming and substitution. However, it is possible to go in the other direction and rep... |

145 | A new approach to abstract syntax involving binders
- Gabbay, Pitts
- 1999
(Show Context)
Citation Context ...ill be proved by structural induction, so we are using structural recursion/induction anyway, but somehow not in the direct way we would like. We can do better than this. Indeed, the work reported in =-=[10, 11, 24-=-] does so, by providing a mathematical notion of `suciently fresh name' that remains very close to the informal practice described above while enabling -equivalence classes of parse trees to gain usef... |

142 |
Abstract syntax and variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ... ofsrst-order logic programming extended with Nominal Logic's primitives of swapping and freshness of atoms (cf. Hamana's logic programming language [21] based on the presheaf semantics of binding in =-=[11]-=-). However, if one wants a single, expressive foundational theory in which to de25 velop the mathematics of syntax in the style of this paper, one can use FM-set theory (and its automated support with... |

136 | A spatial logic for concurrency (part I
- Caires, Cardelli
- 2001
(Show Context)
Citation Context ...n inductive arguments. We gave some illustrations of this in Section 2. A further example is provided by the work of Caires and Cardelli on modal logic for the spatial structure of concurrent systems =-=[3, 4-=-]; this and the related work [5] make use of the freshness quantier of Section 6. See also [22] for the use of permutative renaming to treat naming aspects of process calculi. APPENDIX A: SYNTAX AND A... |

123 | Primitive recursion for higherorder abstract syntax
- Despeyroux, Pfenning, et al.
- 1997
(Show Context)
Citation Context ...lculus. It does not lend itself to principles of structural recursion and induction for the encoded object-language that are particularly straightforward, but such principles have been developed: see =-=[6, 25]-=-. Abstractions as functions from names to terms. The Theory of Contexts [15] reconciles the elegance of higher-order abstract syntax with the desire to deal with names at the object-level and have rel... |

90 | Reasoning with higher-order abstract syntax in a logical framework
- McDowell, Miller
(Show Context)
Citation Context ... McDowell and Miller have developed a way of using Hallnas' notion of partial inductive denitions [20] to enable inductive reasoning about HOAS specications in an intuitionistic higher-order logic [27]; and Despeyroux, Pfenning and Schurmann have developed a modal typed -calculus that allows primitive recursive functions on HOAS-encoded object-language syntax without destroying the adequacy of th... |

70 |
The Axiom of Choice
- Jech
- 1973
(Show Context)
Citation Context ...-order logic that entails that all sorts are non-empty. Possibly empty sorts, like the empty set, have their uses!sAndrew Pitts fails to hold in the Fraenkel-Mostowski permutation model of set theory =-=[17]-=-. However, there is no problem with principles of unique choice. For example, if a Nominal Logic theory has a model in Nom satisfying the sentence (8~x : ~ S)(9!x 0 : S 0 ) '(~x; x 0 ) (53) then the t... |

55 | A spatial logic for concurrency (part II
- Caires, Cardelli
- 2002
(Show Context)
Citation Context ... N a : A)';s( N -right) ; a # ~x; ' `s; ( N a : A)' `s( N -left) where fv (;s) ~x and fv(') a; ~x. The proof theoretical properties of these formulations have yet to be explored. (However, see [4] for a sequent calculus admitting cut-elimination for a modal process logic involving the N -quantier, in which freshness predicates like a # x appear as side-conditions rather than as formulas in se... |

52 | Five axioms of alpha-conversion
- Gordon, Melham
- 1997
(Show Context)
Citation Context ...s our notion of freshness is a purely logical property, independent of any particular object syntax. (The same remark applies to the axiomatization of -conversion of -terms in higher order logic in [1=-=2]-=-; and to the extension ofsrst-order logic with binders studied in [7].) Furthermore, the use of total functions on names to model abstraction means that the Theory of Contexts is incompatible with 240... |

46 | Logical properties of name restriction
- Cardelli, Gordon
- 2001
(Show Context)
Citation Context ...f this in Example 2.1. A further example is provided by the work of Caires and Cardelli on modal logic for the spatial structure of Nominal Logic 241 concurrent systems [2]; this and the related work =-=[3-=-] make use of the freshness quantier of Sect. 5. See also [14] for the use of permutative renaming to treat naming aspects of process calculi. Acknowledgements. The work described here draws upon join... |

41 |
First Order Categorical Logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...ay, as subobjects of [[S 1 ]] [[S n ]]. Indeed, Nom has sucient properties to soundly interpret classicalsrst-order logic with equality 2 using the usual techniques of categorical logic|see [26], or [33, Section 5] for a brief overview. In fact, readers unfamiliar with such techniques need not become so just to understand the interpretation ofsrst-order logic in the category of nominal sets,... |

35 |
A spatial logic for concurrency
- Cardelli, Caires
- 2001
(Show Context)
Citation Context ... We gave a small illustration of this in Example 2.1. A further example is provided by the work of Caires and Cardelli on modal logic for the spatial structure of Nominal Logic 241 concurrent systems =-=[2-=-]; this and the related work [3] make use of the freshness quantier of Sect. 5. See also [14] for the use of permutative renaming to treat naming aspects of process calculi. Acknowledgements. The work... |

31 | A compact representation of proofs - Miller - 1987 |

30 | HOL-λσ an intentional first-order expression of higher-order logic - Dowek, Hardin, et al. |

27 |
Partial inductive definitions
- Hallnäs
- 1991
(Show Context)
Citation Context ... for the encoded object-language, but nevertheless such principles have been developed. For example, McDowell and Miller have developed a way of using Hallnäs’ notion of partial inductive definitions =-=[20]-=- to enable inductive reasoning about HOAS specifications in an intuitionistic higher-order logic [27]; and Despeyroux, Pfenning and Schürmann have developed a modal typed λ-calculus that allows primit... |

21 |
An axiomatic approach to metareasoning on systems in higher-order abstract syntax
- Honsell, Miculan, et al.
- 2001
(Show Context)
Citation Context ...tion for the encoded object-language that are particularly straightforward, but such principles have been developed: see [6, 25]. Abstractions as functions from names to terms. The Theory of Contexts =-=[15]-=- reconciles the elegance of higher-order abstract syntax with the desire to deal with names at the object-level and have relatively simple forms of structural recursion/induction. It does so by axioma... |

19 | Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebeite - Beeson - 1985 |

19 | Elementary structures in process theory (1): Sets with renaming
- Honda
(Show Context)
Citation Context ...ork of Caires and Cardelli on modal logic for the spatial structure of Nominal Logic 241 concurrent systems [2]; this and the related work [3] make use of the freshness quantier of Sect. 5. See also [=-=14]-=- for the use of permutative renaming to treat naming aspects of process calculi. Acknowledgements. The work described here draws upon joint work with Gabbay described in [10, 24, 11]. I also gratefull... |

19 | An axiomatic approach to metareasoning on nominal algebras in hoas
- Honsell, Miculan, et al.
(Show Context)
Citation Context ...ce fails to hold in the Fraenkel-Mostowski permutation model of set theory [25]. However, there is no problem with principles of unique choice (in contrast to the situation for the Theory of Contexts =-=[23]-=-, a close cousin of Nominal Logic). For example, if a Nominal Logic theory has a model in Nom satisfying the sentence (8~x : ~ S)(9!x 0 : S 0 ) '(~x; x 0 ) (68) then the theory extended by a function ... |

17 |
A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
- Gabbay
- 2000
(Show Context)
Citation Context ... ↦→ [f ]M (d, n) is injective. ⊓⊔ Even though there is this incompleteness, it appears that the axioms of Nominal Logic are sufficient for a useful theory of names and name-binding along the lines of =-=[11, 9]-=-. The following sections give some evidence for this claim. 5 The Freshness Quantifier We begin by proving within Nominal Logic the characteristic ‘some/any’ property of fresh atoms (cf. [11, Proposit... |

12 |
Some type theory and lambda calculus formalised
- McKinna, Pollack
- 1999
(Show Context)
Citation Context ... a; a 00 and d does not occur in t or t 00 , we can apply rule (4) to this to deduce a:t a 000 :t 000 , i.e. a:t t 00 , as required. Example 2 (Weakening in type theory). McKinna and Pollack [28] note that in the nave approach to named bound variables referred to in the Introduction, there is a diculty with proving the weakening property of type systems by rule 5 induction. For example, con... |

12 | Developing (meta)theory of lambda-calculus in the theory of contexts
- Miculan
- 2001
(Show Context)
Citation Context ... names to model abstraction means that the Theory of Contexts is incompatible with the Axiom of Unique Choice (cf. Section 8), forcing the theory to have a relational rather than functional feel: see =-=[29]-=-. On the other hand, the Theory of Contexts is able to take advantage of existing machineassisted infrastructure (namely Coq [6]) quite easily, whereas Gabbay had to work hard to adapt the Isabelle [3... |

11 |
FM-HOL, a higher-order theory of names
- Gabbay
- 2002
(Show Context)
Citation Context ...ressive foundational theory in which to de25 velop the mathematics of syntax in the style of this paper, one can use FM-set theory (and its automated support within Isabelle); or, as Gabbay argues in =-=[14]-=-, a version of higher-order logic incorporating atoms, swapping and freshness. Finally, even if one does not care about the details of Nominal Logic, I think that two simple, but important ideas under... |

9 |
The axiom of choice
- Freyd
- 1980
(Show Context)
Citation Context ... f(a), i.e. a 0 = f(a) ) a 0 = a. Taking a 0 to be f(a), we get f(a) = a. ut This phenomenon is a re ection of the fact that the category Nom of nominal sets fails to satisfy the Axiom of Choice (see =-=[8]-=- for a categorical treatment of choice), which in turn re ects the fact that, by design, the Axiom of Choice 2 The reader can deduce at this point that the author, being of a category-theoretic bent, ... |

6 | A logic programming language based on binding algebras
- Hamana
(Show Context)
Citation Context ...ogic of freshness: work is in progress on a version ofsrst-order logic programming extended with Nominal Logic's primitives of swapping and freshness of atoms (cf. Hamana's logic programming language =-=[21]-=- based on the presheaf semantics of binding in [11]). However, if one wants a single, expressive foundational theory in which to de25 velop the mathematics of syntax in the style of this paper, one ca... |

3 |
A Theory of Inductive De with -Equivalence: Semantics, Implementation, Programming Language
- Gabbay
- 2000
(Show Context)
Citation Context ... 7! [[f ]] M (d; n) is injective. ut Even though there is this incompleteness, it appears that the axioms of Nominal Logic are sucient for a useful theory of names and name-binding along the lines of =-=[11, 9-=-]. The following sections give some evidence for this claim. 5 The Freshness Quantier We begin by proving within Nominal Logic the characteristic `some/any' property of fresh atoms (cf. [11, Propositi... |

3 | HOL-lambda-sigma: an intentional expression of higher-order logic - Dowek, Hardin, et al. - 2001 |

3 |
Partial inductive de
- Hallnas
- 1991
(Show Context)
Citation Context ... for the encoded object-language, but nevertheless such principles have been developed. For example, McDowell and Miller have developed a way of using Hallnas' notion of partial inductive denitions [20] to enable inductive reasoning about HOAS specications in an intuitionistic higher-order logic [27]; and Despeyroux, Pfenning and Schurmann have developed a modal typed -calculus that allows primit... |

2 | Endliche Gruppen I, volume 134 of Grundlehren Math - Huppert - 1967 |

1 |
A completeness theorem for an extension of logic with binders
- Dowek, Hardin, et al.
- 2001
(Show Context)
Citation Context ... any particular object syntax. (The same remark applies to the axiomatization of -conversion of -terms in higher order logic in [12]; and to the extension ofsrst-order logic with binders studied in [7=-=]-=-.) Furthermore, the use of total functions on names to model abstraction means that the Theory of Contexts is incompatible with 240 Andrew Pitts the Axiom of Unique Choice (cf. Sect. 7), forcing the t... |

1 |
A completeness theorem for an extension of first-order logic with binders
- Dowek, Hardin, et al.
- 2001
(Show Context)
Citation Context ...ny particular object syntax. (The same remark applies to the axiomatization of α-conversion of λ-terms in higher order logic in [12]; and to the extension of first-order logic with binders studied in =-=[7]-=-.) Furthermore, the use of total functions on names to model abstraction means that the Theory of Contexts is incompatible withs240 Andrew Pitts the Axiom of Unique Choice (cf. Sect. 7), forcing the t... |