## Synthetic domain theory and models of linear Abadi & Plotkin logic (2005)

Citations: | 5 - 4 self |

### BibTeX

@TECHREPORT{Møgelberg05syntheticdomain,

author = {Rasmus Ejlers Møgelberg and Lars Birkedal},

title = {Synthetic domain theory and models of linear Abadi & Plotkin logic},

institution = {},

year = {2005}

}

### OpenURL

### Abstract

Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPL-structure. This adds to the evidence that the notion of LAPL-structure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1

### Citations

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68 |
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(Show Context)
Citation Context ...initial algebras and final coalgebras in PILLY , which by the existence of fixed points gave solutions for general recursive domain equations, as in Freyd’s theory of algebraically compact categories =-=[9, 8, 10]-=-. This theory can be seen as an approach to axiomatic domain theory where the concept of linear and intuitionistic maps correspond to strict and non-strict continuous maps between domains, and where r... |

62 |
Remarks on algebraically compact categories
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(Show Context)
Citation Context ...initial algebras and final coalgebras in PILLY , which by the existence of fixed points gave solutions for general recursive domain equations, as in Freyd’s theory of algebraically compact categories =-=[9, 8, 10]-=-. This theory can be seen as an approach to axiomatic domain theory where the concept of linear and intuitionistic maps correspond to strict and non-strict continuous maps between domains, and where r... |

53 | Term assignment for intuitionistic linear logic (preliminary report
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(Show Context)
Citation Context ...l provide models as these embed into categories of classes as described in [27]. The rest of this section is devoted to showing that the category Dom⊥ has a linear category structure, in the sense of =-=[1]-=-, i.e. a symmetric monoidal closed category (SMCC) structure with a symmetric monoidal comonad ! and natural transformations e: !(−) → I and d: !(−) →!(−)⊗!(−) satisfying a number of axioms. Linear ca... |

53 | Types, Abstraction and Parametric Polymorphism, Part 2
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(Show Context)
Citation Context ...ose objects of the total category essentially are indexed relations, constructed from (5). This reflexive graph gives a relational interpretation of types of the model, essentially as was the idea of =-=[15]-=-. The intuition of the reasoning about the model is the following. Even though types in the model (4) are pairs (f r , f d ), when reasoning about parametricity, we will just consider the f d part of ... |

39 |
Categorical Logic and Type Theory, volume 141
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(Show Context)
Citation Context ...structures (see [16] for a discussion of adjunctions versus monads as models of dual intuitionistic / linear lambda calculi). The reader is referred to [5] or [19] for details on PILLY models, and to =-=[11]-=- for background on the theory of fibrations. The model described in this section will be modified to a parametric PILLY -model in Section 4. We now begin the detailed description of the model. Conside... |

35 | Operational properties of Lily, a polymorphic linear lambda calculus with recursion
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(Show Context)
Citation Context ... is presented constructing parametric LAPL-structures out of a large class of models of PILLY . In recent work Simpson and Rosolini [24] have constructed an interpretation of a strict version of Lily =-=[3]-=- — a language that we shall call Lily strict — based on Synthetic Domain Theory (SDT), and show the interpretation adequate. The interpretation uses a class of domains in an intuitionistic set theory,... |

32 |
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Citation Context ...mΣ needed for L to be a strong symmetric monoidal functor. Corollary 2.5. The adjunction (1) is symmetric monoidal. Dom⊥ �� L ⊥ U �� Dom Proof. This is a consequence of a general theorem due to Kelly =-=[13]-=- stating that an adjunction of symmetric monoidal functors is a symmetric monoidal adjunction if and only if the left adjoint is strong. See also [19, Theorem 1.4]. Corollary 2.6. The functor LU : Dom... |

25 |
Elementary axioms for categories of classes
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(Show Context)
Citation Context ...reader how these issues may be resolved. As given model of SDT, we will assume that we have a category of classes satisfying the axioms of Joyal and Moerdijk’s algebraic set theory [12] as refined in =-=[26]-=- with the notion of classic structure on a regular category with a universe and a small natural numbers object. We further assume that the collections of morphisms between any two objects of the categ... |

22 | Categorical models for Abadi-Plotkin’s Logic for Parametricity - Birkedal, Møgelberg - 2010 |

20 | A uniform approach to domain theory in realizability models
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Citation Context ...es. This way, the fibrations in Lemma 3.6 below are defined externally. The example mentioned in [24] of modelling synthetic domain theory in a realizability topos by taking the well-complete objects =-=[14]-=- as the collection of predomains still provide models as these embed into categories of classes as described in [27]. The rest of this section is devoted to showing that the category Dom⊥ has a linear... |

20 |
Type theory and recursion (extended abstract
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(Show Context)
Citation Context ...to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1 Introduction It was first realized by Plotkin =-=[21, 20]-=- that PILLY , a polymorphic type theory with linear as well as intuitionistic variables and fixed points, is a suitable metalanguage for the combination of parametric polymorphism and recursion. Plotk... |

18 | Reflexive graphs and parametric polymorphism
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(Show Context)
Citation Context ...s in LAPL-structures. In [6] a concrete domain-theoretic LAPL-structure based on admissible pers over a reflexive domain is constructed, and in [18] a parametric completion process along the lines of =-=[23]-=- is presented constructing parametric LAPL-structures out of a large class of models of PILLY . In recent work Simpson and Rosolini [24] have constructed an interpretation of a strict version of Lily ... |

14 | Relating categorical semantics for intuitionistic linear logic. Applied Categorical Structures 13(1
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- 2005
(Show Context)
Citation Context ...PILLY model in the first sense and D is the closure of the coKleisli category for F G under fibred products. The reason for this is that the category D plays an important role in LAPL-structures (see =-=[16]-=- for a discussion of adjunctions versus monads as models of dual intuitionistic / linear lambda calculi). The reader is referred to [5] or [19] for details on PILLY models, and to [11] for background ... |

12 |
order type theory and recursion. Notes for a talk at the Scott Fest
- Second
- 1993
(Show Context)
Citation Context ...to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1 Introduction It was first realized by Plotkin =-=[21, 20]-=- that PILLY , a polymorphic type theory with linear as well as intuitionistic variables and fixed points, is a suitable metalanguage for the combination of parametric polymorphism and recursion. Plotk... |

12 |
A logic for parametric polymorphism. In Typed lambda calculi and applications
- Plotkin, Abadi
- 1993
(Show Context)
Citation Context ...g polymorphism rather than the traditional limit-colimit construction. Recently the first two authors together with R.L. Petersen have presented a variant of Abadi & Plotkin’s logic for parametricity =-=[22]-=- suitable for reasoning about parametricity in PILLY and defined the categorical notion of parametric LAPLstructure (Linear Abadi-Plotkin Logic), which are models of the logic [7, 5].Using Plotkin’s c... |

10 | Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness
- Rosolini, Simpson
- 2004
(Show Context)
Citation Context ... and in [18] a parametric completion process along the lines of [23] is presented constructing parametric LAPL-structures out of a large class of models of PILLY . In recent work Simpson and Rosolini =-=[24]-=- have constructed an interpretation of a strict version of Lily [3] — a language that we shall call Lily strict — based on Synthetic Domain Theory (SDT), and show the interpretation adequate. The inte... |

9 | Domain-theoretical models of parametric polymorphism
- Birkedal, Møgelberg, et al.
(Show Context)
Citation Context ...cal notion of parametric LAPLstructure (Linear Abadi-Plotkin Logic), which are models of the logic [7, 5].Using Plotkin’s constructions one can solve recursive domain equations in LAPL-structures. In =-=[6]-=- a concrete domain-theoretic LAPL-structure based on admissible pers over a reflexive domain is constructed, and in [18] a parametric completion process along the lines of [23] is presented constructi... |

9 |
Parametric completion for models of polymorphic intuitionistic / linear lambda calculus
- Møgelberg
- 2005
(Show Context)
Citation Context ...’s constructions one can solve recursive domain equations in LAPL-structures. In [6] a concrete domain-theoretic LAPL-structure based on admissible pers over a reflexive domain is constructed, and in =-=[18]-=- a parametric completion process along the lines of [23] is presented constructing parametric LAPL-structures out of a large class of models of PILLY . In recent work Simpson and Rosolini [24] have co... |

9 | Computational adequacy in an elementary topos
- Simpson
- 1999
(Show Context)
Citation Context ... in loc. cit., this means that given a model of SDT, Theorem 6.2 holds in the ”real world” (not just as seen from 28the given model of SDT) iff the given model of SDT is 1-consistent in the sense of =-=[25, 27]-=-: any sentence of the form ∃n: N. φ(n), for φ a primitive recursive predicate,—a Σ0 1-sentence—is true in the model iff there exists (in the external sense) a natural number n such that φ(n) is true. ... |

8 | Computational adequacy for recursive types in models of intuitionistic set theory
- Simpson
- 2004
(Show Context)
Citation Context ...ynthetic domain theory in a realizability topos by taking the well-complete objects [14] as the collection of predomains still provide models as these embed into categories of classes as described in =-=[27]-=-. The rest of this section is devoted to showing that the category Dom⊥ has a linear category structure, in the sense of [1], i.e. a symmetric monoidal closed category (SMCC) structure with a symmetri... |

7 |
Categorical models of PILL
- Møgelberg, Birkedal, et al.
- 2005
(Show Context)
Citation Context ...tegory D plays an important role in LAPL-structures (see [16] for a discussion of adjunctions versus monads as models of dual intuitionistic / linear lambda calculi). The reader is referred to [5] or =-=[19]-=- for details on PILLY models, and to [11] for background on the theory of fibrations. The model described in this section will be modified to a parametric PILLY -model in Section 4. We now begin the d... |

1 |
and Valeria de Paiva. Linear λ-calculus and categorical models revisited
- Benton, Bierman, et al.
- 1992
(Show Context)
Citation Context ...heorem 1.4]. Corollary 2.6. The functor LU : Dom⊥ → Dom⊥ extends the SMCC structure on the category of domains to a linear category structure. Proof. This follows from Corollary 2.5 by a theorem from =-=[2]-=-, see also [19, Proposition 1.14]. 3 The domains fibration In this section we construct a PILLY model based on the linear category structure on Dom⊥. A PILLY model is a fibred linear category E → B (a... |

1 |
Category theoretic models of linear
- Birkedal, Møgelberg, et al.
(Show Context)
Citation Context ...c for parametricity [22] suitable for reasoning about parametricity in PILLY and defined the categorical notion of parametric LAPLstructure (Linear Abadi-Plotkin Logic), which are models of the logic =-=[7, 5]-=-.Using Plotkin’s constructions one can solve recursive domain equations in LAPL-structures. In [6] a concrete domain-theoretic LAPL-structure based on admissible pers over a reflexive domain is constr... |