## A Categorical Analysis of Multi-Level Languages (Extended Abstract) (1999)

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Citations: | 10 - 7 self |

### BibTeX

@TECHREPORT{Benaissa99acategorical,

author = {Zine El-abidine Benaissa and Eugenio Moggi and Walid Taha and Tim Sheard},

title = {A Categorical Analysis of Multi-Level Languages (Extended Abstract)},

institution = {},

year = {1999}

}

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### Abstract

We propose categorical models for fl, 2 , MetaML, and AIM. First, we focus on the underlying logical modalities and the interactions between them, then we investigate the interactions between logical modalities and computational monads. We give two examples of categorical model: one simpler but with some limitations, the other more complex but able to model all features of AIM.

### Citations

766 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...ng x in front of s. ffl We write n+ for n + 1 and n\Gamma for n \Gamma 1. ffl We use Haskell's notation dofx i / e i ; eg and ret e for monads, instead of the notation let x i ( e i in e and [e] from =-=[8]-=-. If op: Q i A i ! MB, we write op: Q i MA i ! MB for its monadic extension, i.e. op(u i ) \Delta = dofx i / u i ; op(x i )g. 2 Multi-Level Languages We begin by describing the syntax and type systems... |

253 | Multi-stage programming with explicit annotations
- Taha, Sheard
- 1997
(Show Context)
Citation Context ...terpretation is the analogue of the interpretation of the simply typed -calculus in a CCC. However, we are interested in multi-level programming languages, like Mini-ML 2 Mini-ML fl , and MetaML (see =-=[5, 4, 13]-=-), where logical modalities coexist with computational effects. In this section we define a CBV monadic interpretation of AIM in an AIM-model equipped with a strong monad (see [8]). Definition 5.1 A m... |

197 | A modal analysis of staged computation - Davies, Pfenning - 1996 |

101 | A mixed linear and non-linear logic: Proofs, terms and models
- Benton
- 1995
(Show Context)
Citation Context ...ced above. An FP-adjunction is a special case of a symmetric monoidal adjunction, which has been used to give an elegant definition of what is a categorical model for intuitionistic linear logic (see =-=[1, 2, 3]-=-). We recall some properties of FP-adjunctions (and FPfunctors) , which will be exploited in the sequel. Proposition 3.3 If C is a CCC and D ae - ? oe F C is an FP-adjunction, then D is an exponential... |

98 | What is a categorical model of intuitionistic linear logic?’, Typed lambda calculi and applications
- Bierman
- 1995
(Show Context)
Citation Context ...ced above. An FP-adjunction is a special case of a symmetric monoidal adjunction, which has been used to give an elegant definition of what is a categorical model for intuitionistic linear logic (see =-=[1, 2, 3]-=-). We recall some properties of FP-adjunctions (and FPfunctors) , which will be exploited in the sequel. Proposition 3.3 If C is a CCC and D ae - ? oe F C is an FP-adjunction, then D is an exponential... |

84 | A temporal-logic approach to binding-time analysis
- Davies
- 1996
(Show Context)
Citation Context ... systems (multi-level typed calculi), combination of logics (modal and temporal). 1 Introduction This paper proposes a categorical semantics for multilevel languages likesfl ,s2 , MetaML and AIM (see =-=[4, 5, 12, 11]-=-). Developing such a semantics has a number of benefits, including: ffl Suggesting simplifications and extensions. We have already simplified the type system of MetaML and proposed an extension with c... |

35 | Linear logic, monads and the lambda calculus
- Benton, Wadler
- 1996
(Show Context)
Citation Context ...ced above. An FP-adjunction is a special case of a symmetric monoidal adjunction, which has been used to give an elegant definition of what is a categorical model for intuitionistic linear logic (see =-=[1, 2, 3]-=-). We recall some properties of FP-adjunctions (and FPfunctors) , which will be exploited in the sequel. Proposition 3.3 If C is a CCC and D ae - ? oe F C is an FP-adjunction, then D is an exponential... |

34 | MultiStage Programming: Axiomatization and Type Safety (Extended Abstract
- Taha, Benaissa, et al.
- 1998
(Show Context)
Citation Context ... systems (multi-level typed calculi), combination of logics (modal and temporal). 1 Introduction This paper proposes a categorical semantics for multilevel languages likesfl ,s2 , MetaML and AIM (see =-=[4, 5, 12, 11]-=-). Developing such a semantics has a number of benefits, including: ffl Suggesting simplifications and extensions. We have already simplified the type system of MetaML and proposed an extension with c... |

29 | A.: A computational interpretation of modal proofs
- Martini, Masini
- 1996
(Show Context)
Citation Context ...eatures are important for pragmatic reasons. ffl AIM [11], revising and extending MetaML with a closed code type for expressivity and modularity. 2 andsfl already have clean, logical foundations (see =-=[4, 5, 7, 6]-=-): there is a Curry-Howard isomorphism betweensfl and linear time temporal logic, and between 2 and modal logic S4. MetaML had no such foundations, nor the formal hygiene they often promote. Indeed, M... |

18 | A categorical account of two-level languages
- Moggi
- 1997
(Show Context)
Citation Context ... had a complex type system and a number of ad hoc restrictions (see [12]), which demanded deeper investigation and possibly simplification. Starting from the categorical account of twolevel languages =-=[9]-=-, we arrive at a number of results for multi-level languages: ffl We analyze, from a categorical point of view, the logical modalities and how they interact. Borrowing ideas from the work by Benton an... |

15 | Functor categories and two-level languages
- Moggi
- 1998
(Show Context)
Citation Context ... entails observational equivalence (where code inspection is not among the allowed observations). However, we expect such results to hold, and their proof should exploit Kripke logical relations (see =-=[10]-=-). ffl Explaining multi-level languages in terms of more primitive concepts, namely logical modalities (in the sense that the modalities are characterized by universal properties) and computational mo... |

10 |
An idealized MetaML: Simpler, and more expressive (includes proofs
- Moggi, Taha, et al.
- 1998
(Show Context)
Citation Context ... systems (multi-level typed calculi), combination of logics (modal and temporal). 1 Introduction This paper proposes a categorical semantics for multilevel languages likesfl ,s2 , MetaML and AIM (see =-=[4, 5, 12, 11]-=-). Developing such a semantics has a number of benefits, including: ffl Suggesting simplifications and extensions. We have already simplified the type system of MetaML and proposed an extension with c... |

9 |
2-sequent calculus: Intuitionism and natural deduction
- Masini
- 1993
(Show Context)
Citation Context ...eatures are important for pragmatic reasons. ffl AIM [11], revising and extending MetaML with a closed code type for expressivity and modularity. 2 andsfl already have clean, logical foundations (see =-=[4, 5, 7, 6]-=-): there is a Curry-Howard isomorphism betweensfl and linear time temporal logic, and between 2 and modal logic S4. MetaML had no such foundations, nor the formal hygiene they often promote. Indeed, M... |

6 |
A categorical account oftwo-level languages
- Moggi
- 1997
(Show Context)
Citation Context ... had a complex type system and a number of ad hoc restrictions (see [12]), which demanded deeper investigation and possibly simpli - cation. Starting from the categorical account oftwolevel languages =-=[9]-=-, we arrive atanumber of results for multi-level languages: We analyze, from a categorical point of view, the logical modalities and how they interact. Borrowing ideas from the work by Benton and othe... |