## 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}

}

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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...