## Simple nominal type theory

Citations: | 8 - 1 self |

### BibTeX

@MISC{Cheney_simplenominal,

author = {James Cheney},

title = {Simple nominal type theory},

year = {}

}

### OpenURL

### Abstract

Abstract. Nominal logic is an extension of first-order logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standard type theories for propositional, first- or higher-order logic. This has proven difficult, largely because of complex interactions between nominal logic’s name-abstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory which enjoys properties such as type soundness and strong normalization, and which can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories. 1

### Citations

223 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
(Show Context)
Citation Context ...t notationally burdensome. We recapitulate some of the basic definitions concerning permutations, group actions and nominal sets needed for the semantics (introduced in prior work by Gabbay and Pitts =-=[6, 12, 15]-=-). We write Pfin(A) for the set of all finite sets of names and G = FSym(A) for the group of all (finite) permutations π on A; that is, invertible functions such that π(a) = a for all but finitely man... |

191 | The logic of bunched implications
- O’Hearn, Pym
- 1999
(Show Context)
Citation Context ...ce, α −◦ A, consisting of functions that may only be applied to a “fresh” name, similar to the “magic wand” connective/type constructor in the logic of bunched implications (BI) and its type theories =-=[11]-=-. Schöpp and Stark’s type theory takes the approach that name-abstractions have the introduction and elimination forms of both quotiented-pair and partial-function presentations, using BI-style bunche... |

174 | Nominal logic, a first order theory of names and binding
- Pitts
(Show Context)
Citation Context ... be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories. 1 Introduction Nominal logic =-=[12]-=- is a variant of first-order logic that axiomatizes name-binding and alpha-equivalence using permutative renamings, or swappings. So far, nominal logic has been studied primarily using classical model... |

148 |
Abstract syntax and variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...move) x : Λ → α list 3.3 Nominal recursion combinators Fig. 7. Examples of recursive definitions. One of the main advertised benefits of nominal logic (and related approaches such as binding algebras =-=[3]-=-) over other techniques has been the availability of principles for inductive reasoning and recursive definitions that extend well-known principles for induction and recursion for “first-order” langua... |

125 | Primitive Recursion for Higherorder Abstract syntax
- Schurmann, Despeyroux, et al.
(Show Context)
Citation Context ...s such as higher-order abstract syntax and de Bruijn representations have also been studied by Fiore, Plotkin and Turi [3] and Hofmann [7] using functor categories, Schürmann, Despeyroux and Pfenning =-=[19]-=- using a modal type system and by Washburn and Weirich [24] using parametricity. These approaches appear (subjectively) more difficult to use than recursion in SNTT; previous nominal techniques involv... |

118 | Observable properties of higher order functions that dynamically create local names, or: What’s new
- Pitts, Stark
- 1993
(Show Context)
Citation Context ...a fresh name a and use it within M. However, the behavior of νa:α.M as a proof term is problematic. Similar name-generation operators has been studied independently of name-binding by Pitts and Stark =-=[14]-=- and Odersky [10]. Pitts and Stark’s semantics for ν is, like FreshML’s let fresh, generative; conversely, Odersky’s approach is purely functional but it admits well-formed, “stuck” terms such as νa.a... |

99 | Semantical analysis of higher-order abstract syntax
- Hofmann
- 1999
(Show Context)
Citation Context ...ral authors [9, 21, 15]. Recursion principles for other techniques such as higher-order abstract syntax and de Bruijn representations have also been studied by Fiore, Plotkin and Turi [3] and Hofmann =-=[7]-=- using functor categories, Schürmann, Despeyroux and Pfenning [19] using a modal type system and by Washburn and Weirich [24] using parametricity. These approaches appear (subjectively) more difficult... |

94 | A metalanguage for programming with bound names modulo renaming
- Pitts, Gabbay
- 2000
(Show Context)
Citation Context ...osely related research in the body of the paper, and for the rest we refer to the papers [2, 15]. Pottier [16] has recently revisited the problem of inferring freshness information for “pure” FreshML =-=[13]-=-. This approach reduces freshness inference to set constraint solving and is aimed towards practical programming rather than deduction. Schürmann, Poswolsky, and Sarnat’s ∇-calculus is a core language... |

87 | Nominal Techniques in Isabelle/HOL
- Urban, Tasson
(Show Context)
Citation Context ...of theory [1, 5] to formalize explicit reasoning about equational and freshness properties. While this analysis lends itself to implementations within theorem proving systems based on classical logic =-=[23]-=-, nominal logic has resisted incorporation into constructive systems based on typed lambda-calculi. This is unfortunate, because nominal logic seems promising as a foundation for inductive reasoning a... |

65 | A proof theory for generic judgments
- Miller, Tiu
(Show Context)
Citation Context ... approaches compare in terms of expressiveness; it appears that SNTT’s convenience may come at the cost of expressiveness compared to the other nominal approaches [9, 21, 15]. Miller and Tiu’s F Oλ∆∇ =-=[8]-=- extends first-order logic over higher-order lambdaterms with a ∇-quantifier with similar (but not identical) properties to nominal logic’s -quantifier. F Oλ ∆∇ uses separate contexts for ordinary var... |

57 | Nominal unification - Urban, Pitts, et al. |

48 | Alpha-structural recursion and induction
- Pitts
(Show Context)
Citation Context ...al logic to internalize the freshness reasoning about name-abstractions that currently needs to be performed explicitly to justify recursion and induction for nominal abstract syntax (for example, in =-=[9, 15, 21]-=-). Although proof theories for intuitionistic nominal logic has been considered already in work by Gabbay and Cheney [5, 1, 4], those systems consider only provability, not proof terms. Although the p... |

37 | Boxes go bananas: Encoding higherorder abstract syntax with parametric polymorphism
- Washburn, Weirich
(Show Context)
Citation Context ...entations have also been studied by Fiore, Plotkin and Turi [3] and Hofmann [7] using functor categories, Schürmann, Despeyroux and Pfenning [19] using a modal type system and by Washburn and Weirich =-=[24]-=- using parametricity. These approaches appear (subjectively) more difficult to use than recursion in SNTT; previous nominal techniques involve checking freshness side-conditions and previous higherord... |

31 | A functional theory of local names
- Odersky
- 1994
(Show Context)
Citation Context ...d use it within M. However, the behavior of νa:α.M as a proof term is problematic. Similar name-generation operators has been studied independently of name-binding by Pitts and Stark [14] and Odersky =-=[10]-=-. Pitts and Stark’s semantics for ν is, like FreshML’s let fresh, generative; conversely, Odersky’s approach is purely functional but it admits well-formed, “stuck” terms such as νa.a that are not con... |

24 | Static name control for FreshML
- Pottier
- 2006
(Show Context)
Citation Context ...g informs and motivates this work; we cannot give a complete survey here. We have discussed closely related research in the body of the paper, and for the rest we refer to the papers [2, 15]. Pottier =-=[16]-=- has recently revisited the problem of inferring freshness information for “pure” FreshML [13]. This approach reduces freshness inference to set constraint solving and is aimed towards practical progr... |

22 | A sequent calculus for nominal logic
- Gabbay, Cheney
- 2004
(Show Context)
Citation Context ...ic that axiomatizes name-binding and alpha-equivalence using permutative renamings, or swappings. So far, nominal logic has been studied primarily using classical model theory [2, 12] or proof theory =-=[1, 5]-=- to formalize explicit reasoning about equational and freshness properties. While this analysis lends itself to implementations within theorem proving systems based on classical logic [23], nominal lo... |

21 | A simpler proof theory for nominal logic
- Cheney
- 2005
(Show Context)
Citation Context ...ic that axiomatizes name-binding and alpha-equivalence using permutative renamings, or swappings. So far, nominal logic has been studied primarily using classical model theory [2, 12] or proof theory =-=[1, 5]-=- to formalize explicit reasoning about equational and freshness properties. While this analysis lends itself to implementations within theorem proving systems based on classical logic [23], nominal lo... |

21 | A Recursion Combinator for Nominal Datatypes Implemented in Isabelle/HOL
- Urban, Berghofer
- 2006
(Show Context)
Citation Context ...al logic to internalize the freshness reasoning about name-abstractions that currently needs to be performed explicitly to justify recursion and induction for nominal abstract syntax (for example, in =-=[9, 15, 21]-=-). Although proof theories for intuitionistic nominal logic has been considered already in work by Gabbay and Cheney [5, 1, 4], those systems consider only provability, not proof terms. Although the p... |

16 | A logic for reasoning about generic judgments
- Tiu
(Show Context)
Citation Context ...teraction using lifting. Also, F Oλ ∆∇ has been studied only as an intuitionistic sequent calculus; proof terms or natural deduction systems for F Oλ ∆∇ have not been investigated. More recently, Tiu =-=[20]-=- introduced a logic LG ω that incorporates the idea of equivariance from nominal logic and drops the ∇-contexts. In both F Oλ ∆∇ and LG ω , definitions and induction over N can be used to perform 13si... |

15 | A dependent type theory with names and binding - Schöpp, Stark - 2004 |

14 | Recursive function definition for types with binders
- Norrish
- 2004
(Show Context)
Citation Context ...al logic to internalize the freshness reasoning about name-abstractions that currently needs to be performed explicitly to justify recursion and induction for nominal abstract syntax (for example, in =-=[9, 15, 21]-=-). Although proof theories for intuitionistic nominal logic has been considered already in work by Gabbay and Cheney [5, 1, 4], those systems consider only provability, not proof terms. Although the p... |

10 | Completeness and Herbrand theorems for nominal logic
- Cheney
(Show Context)
Citation Context ...ts which define the standard example of lambda-term syntax modulo alpha-equivalence using nominal terms: var : α → Λ, app : Λ × Λ → Λ, lam : 〈α〉Λ → Λ ∈ ΣΛ It has been shown in several places see e.g. =-=[2, 15]-=-) that it is possible to model this language using a least fixed point construction on nominal sets , since X ↦→ 〈〈A〉〉X is a continuous operator on nominal sets. The resulting nominal set Λ ∼ = A+Λ×Λ+... |

4 |
Fresh logic: proof-theory and semantics for FM and nominal techniques
- Gabbay
- 2007
(Show Context)
Citation Context ...fy recursion and induction for nominal abstract syntax (for example, in [9, 15, 21]). Although proof theories for intuitionistic nominal logic has been considered already in work by Gabbay and Cheney =-=[5, 1, 4]-=-, those systems consider only provability, not proof terms. Although the proof trees available in such systems could themselves be viewed as proof terms, doing so does not immediately yield a well-beh... |

4 |
Names and binding in type theory
- SCHÖPP
- 2006
(Show Context)
Citation Context ...d a well-behaved type theory. Moreover, both systems involve significant amounts of explicit reasoning about equality and freshness.sNominal type theory has also been investigated by Schöpp and Stark =-=[17, 18]-=-, who have developed a family of type theories based on categorical models of nominal logic. Their approach to nominal dependent type theory is very expressive, but motivated primarily by semantics. C... |