## Computational Adequacy in an Elementary Topos (1999)

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Venue: | Proceedings CSL ’98, Springer LNCS 1584 |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Simpson99computationaladequacy,

author = {Alex K. Simpson},

title = {Computational Adequacy in an Elementary Topos},

booktitle = {Proceedings CSL ’98, Springer LNCS 1584},

year = {1999},

pages = {323--342},

publisher = {Springer LNCS}

}

### Years of Citing Articles

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### Abstract

. We place simple axioms on an elementary topos which suffice for it to provide a denotational model of call-by-value PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their set-theoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true \Sigma 0 1 -sentences). 1 Introduction One axiomatic approach to domain theory is based on axiomatizing properties of the category of predomains (in which objects need not have a "least" element). Typically, such a category is assumed to be bicartesian closed (although it is not really necessary to require all exponentials) with natural numbers object, allowing the denotations of simple datatypes to be determined by universal properties. It is well known...

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Citation Context ...hat an identical result will hold for other programming languages too. 2 Partiality and Lifting Throughout this paper we assume that E is an elementary topos with a natural numbers object N, see e.g. =-=[13, 17]-=-. We write 0 and 1 for chosen initial and terminal objects respectively, [0; s] : 1 + N - N for the structure map of N as the initial algebra of the endofunctor 1 + (\Gamma) on the topos, and pred : N... |

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Citation Context ...r simple strengthening is necessary in general, although not in the case that Markov's Principle is valid in the topos (Theorem 1). The axioms are sufficient to enable any of the many variants of PCF =-=[22]-=- to be modelled in the topos. Moreover, the closure properties of predomains have the important consequence that datatypes are modelled by their set-theoretic counterparts within the internal logic of... |

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Citation Context ...ll-by-value version with sum and product types. The question we consider is: When is the interpretation of the language in the topos computationally adequate with respect to its operational semantics =-=[8, 32]-=-? We obtain a complete characterisation in terms of the internal logic of the topos. We prove that computational adequacy holds if and only if the topos is 1-consistent (Theorem 2). (A topos is said t... |

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Citation Context ...hat an identical result will hold for other programming languages too. 2 Partiality and Lifting Throughout this paper we assume that E is an elementary topos with a natural numbers object N, see e.g. =-=[13, 17]-=-. We write 0 and 1 for chosen initial and terminal objects respectively, [0; s] : 1 + N - N for the structure map of N as the initial algebra of the endofunctor 1 + (\Gamma) on the topos, and pred : N... |

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Citation Context ...ll-by-value version with sum and product types. The question we consider is: When is the interpretation of the language in the topos computationally adequate with respect to its operational semantics =-=[8, 32]-=-? We obtain a complete characterisation in terms of the internal logic of the topos. We prove that computational adequacy holds if and only if the topos is 1-consistent (Theorem 2). (A topos is said t... |

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Citation Context ...rty of uniformity. Further, the monad determines a category of partial functions, which is arguably the most suitable category for program semantics. The development of this viewpoint can be found in =-=[23, 2, 18, 29, 3]-=-. In recent years it has become apparent that many natural categories of predomains arise as full subcategories of elementary toposes. For example, the category of !-complete partial orders and !-cont... |

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Citation Context ...l natural model furnished by either a Grothendieck or realizability topos is computationally adequate. (A different argument for computational adequacy in the case of realizability toposes appears in =-=[14]-=-.) One may wonder whether in fact any non-trivial model is computationally adequate. As a negative application of Theorem 2, we show that this is not the case. To be precise, we say that a model (E ; ... |

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Citation Context ... suggests that elementary toposes provide a plausible environment for developing a unified account of the many models of domain theory. However, early axiomatizations of domain theory inside toposes, =-=[10, 31, 27, 24]-=-, included axioms that are valid only in restricted classes of models. A general axiomatization, encompassing both parallel and sequential computation, was proposed in [15], where its consequences wer... |

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Citation Context ...e above notation is adopted for its readability. The formally inclined reader will have no problem in translating the expressions we use into an appropriate rigorous logic for partial terms (see e.g. =-=[28]-=-), although our partial terms will always be special ones whose definedness property is a \Sigma -proposition. The operation 1 * (\Gamma) determines, in the obvious way, an endofunctor on E , the lift... |

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Citation Context ... \Sigma arising as a subobject \Sigma - -\Omega (thus \Sigma classifies a family of subobjects, the \Sigma -subobjects, in E). Moreover, we require that \Sigma is a dominance in the sense of Rosolini =-=[26, 3]-=-. Specifically this means that ? : 1 -\Omega factors through the mono \Sigma - -\Omega (i.e. all isomorphisms are \Sigma -subobjects) and that \Sigma -subobjects are closed under composition. Because ... |

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Citation Context ...rty of uniformity. Further, the monad determines a category of partial functions, which is arguably the most suitable category for program semantics. The development of this viewpoint can be found in =-=[23, 2, 18, 29, 3]-=-. In recent years it has become apparent that many natural categories of predomains arise as full subcategories of elementary toposes. For example, the category of !-complete partial orders and !-cont... |

38 |
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Citation Context ...ebra for the monad (L; j; ��) [16]. The morphism �� 0 = oe ffi �� ffi Loe \Gamma1 is a monad algebra LI - I. We write up : I - I for the composite oe ffi j. The result below appears as The=-=orem A.5 of [12], wh-=-ere it is attributed to B'enabou and Jibladze. Proposition 1. For any monad algebra ff : LX - X and any morphism f : X - X, there exists a unique algebra homomorphism h : �� 0 - ff such that f ffi... |

34 |
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Citation Context ... suggests that elementary toposes provide a plausible environment for developing a unified account of the many models of domain theory. However, early axiomatizations of domain theory inside toposes, =-=[10, 31, 27, 24]-=-, included axioms that are valid only in restricted classes of models. A general axiomatization, encompassing both parallel and sequential computation, was proposed in [15], where its consequences wer... |

33 |
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Citation Context ...rty of uniformity. Further, the monad determines a category of partial functions, which is arguably the most suitable category for program semantics. The development of this viewpoint can be found in =-=[23, 2, 18, 29, 3]-=-. In recent years it has become apparent that many natural categories of predomains arise as full subcategories of elementary toposes. For example, the category of !-complete partial orders and !-cont... |

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Citation Context ...gories of predomains are found as full subcategories of realizability toposes, see [15] for examples and references. Certain such examples have been shown to account for phenomena such as effectivity =-=[21, 7]-=- and strong stability [19]. Work in progress by a number of researchers looks likely to establish similar embeddings for categories of games. The wealth of examples suggests that elementary toposes pr... |

21 | A combinatory algebra for sequential functionals of finite type
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(Show Context)
Citation Context ...d as full subcategories of realizability toposes, see [15] for examples and references. Certain such examples have been shown to account for phenomena such as effectivity [21, 7] and strong stability =-=[19]-=-. Work in progress by a number of researchers looks likely to establish similar embeddings for categories of games. The wealth of examples suggests that elementary toposes provide a plausible environm... |

20 |
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Citation Context ...gories of predomains are found as full subcategories of realizability toposes, see [15] for examples and references. Certain such examples have been shown to account for phenomena such as effectivity =-=[21, 7]-=- and strong stability [19]. Work in progress by a number of researchers looks likely to establish similar embeddings for categories of games. The wealth of examples suggests that elementary toposes pr... |

17 | An extension of models of axiomatic domain theory to models of synthetic domain theory
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Citation Context ...tinuous functions is a full reflective subcategory of the Grothendieck topos, H, considered in [6, 5]. More generally, models of Fiore and Plotkin's axioms for domain theory also have such embeddings =-=[4]-=-. Other categories of predomains are found as full subcategories of realizability toposes, see [15] for examples and references. Certain such examples have been shown to account for phenomena such as ... |

15 |
Program Verification in Synthetic Domain Theory
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Citation Context ... suggests that elementary toposes provide a plausible environment for developing a unified account of the many models of domain theory. However, early axiomatizations of domain theory inside toposes, =-=[10, 31, 27, 24]-=-, included axioms that are valid only in restricted classes of models. A general axiomatization, encompassing both parallel and sequential computation, was proposed in [15], where its consequences wer... |

12 |
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Citation Context |

11 | Two models of synthetic domain theory
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Citation Context ...ry). (The adjective "natural" is to emphasise that the natural numbers object is well-complete.) All the realizability examples considered in [15] provide natural models, as does the the mod=-=el H from [6, 5]-=-. Throughout the rest of this paper, unless otherwise stated, we assume that E and \Sigma together form a natural model. 5 Interpreting a Programming Language The axioms we have are sufficient for sim... |

10 | General synthetic domain theory — a logical approach
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(Show Context)
Citation Context ...toposes. This axiomatization has since proved to be applicable also to: Grothendieck toposes [5, 4], models of intuitionistic ZermeloFraenkel set theory [30], and models of intuitionistic type theory =-=[25]-=-. In this paper we consider the same general axiomatic approach in the setting of an elementary topos. Given an elementary topos with a natural numbers object and a distinguished dominance (determinin... |

9 |
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Citation Context ... 5]. More generally, models of Fiore and Plotkin's axioms for domain theory also have such embeddings [4]. Other categories of predomains are found as full subcategories of realizability toposes, see =-=[15]-=- for examples and references. Certain such examples have been shown to account for phenomena such as effectivity [21, 7] and strong stability [19]. Work in progress by a number of researchers looks li... |

8 |
A presentation of the initial lift algebra
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Citation Context ...gma N , and the unique algebra homomorphism from oe to �� \Gamma1 is a mono ' : I - - F. These results were proved in 1995 by the author and Mamuka Jibladze independently. For a published account =-=see [11]. Th-=-e initial algebra of L as a functor, interacts nicely with the monad structure on L. We call a morphism ff : LX - X a monad algebra if it is an EilenbergMoore algebra for the monad (L; j; ��) [16]... |

8 | Axioms and (counter)examples in synthetic domain theory
- Oosten, Simpson
(Show Context)
Citation Context ...) = fix ff (f ), i.e. that fix ff (f) factors through the subobject m : X 0 - - X . 4 Axioms for Synthetic Domain Theory The complete objects are not, in general, closed under the lifting functor, L, =-=[20]-=-. As our category of predomains we take the largest full subcategory of complete objects that is closed under lifting, following [15]. Definition 2. An object X is said to be well-complete if LX is co... |

4 |
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Citation Context |

1 |
The category of cpos from a synthetic viepoint. Presented at MFPS XIII
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Citation Context ...ull subcategories of elementary toposes. For example, the category of !-complete partial orders and !-continuous functions is a full reflective subcategory of the Grothendieck topos, H, considered in =-=[6, 5]-=-. More generally, models of Fiore and Plotkin's axioms for domain theory also have such embeddings [4]. Other categories of predomains are found as full subcategories of realizability toposes, see [15... |

1 |
A note on inconstencies caused by fixpoints in a cartesian closed category
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Citation Context ...onentials) with natural numbers object, allowing the denotations of simple datatypes to be determined by universal properties. It is well known that such a category cannot have a fixed-point operator =-=[9]-=-, but crucially predomains have a lift monad acting on them which plays a critical role in recovering the more familiar category of domains in which the expected fixed-point operator resides. The lift... |

1 |
Notes on Synthetic Domain Theory. Unpublished notes, Available from ftp://ftp.disi.unige.it
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Citation Context |

1 |
Algebraic Compactness in Intuitionistic Set Theory. Presented at PSSL
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Citation Context ...n detail in the specific case of realizability toposes. This axiomatization has since proved to be applicable also to: Grothendieck toposes [5, 4], models of intuitionistic ZermeloFraenkel set theory =-=[30]-=-, and models of intuitionistic type theory [25]. In this paper we consider the same general axiomatic approach in the setting of an elementary topos. Given an elementary topos with a natural numbers o... |