P. Taylor. The xed point property in synthetic domain theory. In Proc. 6th IEEE Symposium on Logic in Computer Science, pages 152-160, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....argument using classical logic, cannot be full subcategories of the category of sets. In [38] Dana Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain theory [36, 14, 28, 46, 22, 40, 35, 27, 7]. In this paper, we exploit this idea to obtain algebraically compact categories in a uniform way. Roughly speaking, we start o with a category S of intuitionistic sets that satis es one simple axiom, Axiom 1 of Section 2. From any such category S, we extract a full subcategory of predomains, ....

....slice category C=I and referring to external structure on that category. The remaining goal of this section is to isolate a full subcategory of S to act as a category of predomains. This will require imposing further axioms on C. Many axiomatizations have been suggested for this purpose, see e.g. [36, 14, 28, 46, 22, 40, 35, 27]. Here, we follow [22, 40] As rst proposed in [36] the de nition of predomain is predicated on a notion of partiality. To implement this, we require a distinguished subobject As is a subobject of it classi es a collection of subobjects in C, namely those whose characteristic map ....

P. Taylor. The xed point property in synthetic domain theory. In Proc. 6th IEEE Symposium on Logic in Computer Science, pages 152-160, 1991.

....of s; s 0 : L L, where s(s n ( s n 1 ( s(s n (0) s 0 (s n (0) s n 1 (0) and are the identity otherwise. 1 In looking for a more general semantics of necessity, we have been motivated by the desire to integrate Evaluation Logic with Synthetic Domain Theory SDT (see [Hyl91, Tay91]) rather than Classical Domain Theory. In line with this objective, we have tried to extend the semantics of necessity (proposed in [Mog93] by dropping any additional requirement on T , at the expense of making stronger assumptions on C and M (consistently with SDT ) The main results are: an ....

P. Taylor. The xed point property in synthetic domain theory. In 6th LICS Conf. IEEE, 1991. 11

....This paper proposes an alternative semantics, which drops all additional requirement on monads, at the expense of stronger assumptions on the underlying category. These assumptions are satis ed by any topos, but not by the category of cpos. However, in the setting of Synthetic Domain Theory (see [7, 23]) it is possible to reconcile the needs of Denotational Semantics with those of Logic. 1 Introduction Evaluation logic (EL T ) is a typed predicate logic originally proposed by [18] which is based on the metalanguage ML T for computational monads (see [12] and permits statements about the ....

....Theory. Anyway, when looking for an expressive logic (including FOL) we are forced to leave the category of cpos, as only a fragment of FOL may be interpreted in Cpo. Fortunately, one may reconcile the needs of Denotational Semantics with an expressive logic by moving to the SDT setting (see [7, 23]) De nition 5.1 Given two brations p: C B and q: D B, we say that q is a full re ective sub bration of p ( there is a full and faithful bered functor i: q p with a bered left adjoint r a i. For our purposes the main result of SDT is: there are quasi toposes C (more generally ....

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P. Taylor. The xed point property in synthetic domain theory. In 6th LICS Conf. IEEE, 1991.

....elements, i.e. the equaliser of s; s 0 : L L, where s(s n ( s n 1 ( s(s n (0) s 0 (s n (0) s n 1 (0) and are the identity otherwise. To overcome these problems, we have decided to replace Cpo with the category Rep of replete objects in some model (E ; of SDT (see [Tay91]) This note establishes some basic facts about replete objects in the category Fil of lter spaces (see [Hyl79] namely: SFP Rep, where means full subcategory (see Theorem 2.4) a characterisation of the regular subobjects (in Rep) of an SFP (see Theorem 2.7) the following axiom is ....

....I expect that only three of them are dominances. Proposition 1.2 For any of the embeddings : C 1 C 2 , if 2 C 1 is a dominance and X 2 C 1 , then the set of subobjects of X in C 1 is isomorphic (via ) to the set of subobjects of X in C 2 . 2 2 Replete objects in lter spaces In [Tay91] a model of SDT is a pair (E ; where E is a topos and is a dominance satisfying certain axioms. Here we work in the setting of a quasi topos, and we write for the strong subobjects classi er. We investigate the model (Fil; where is the lter space over 2 s.t. every lter converges to ....

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P. Taylor. The xed point property in synthetic domain theory. In 6th LICS Conf. IEEE, 1991.

....Hyland M.Hyland pmms.cam.ac.uk DPMMS, Univ. of Cambridge 16 Mill Lane CB2 1SB Cambridge, UK E. Moggi moggi disi.unige.it DISI, Univ. di Genova v. Benedetto XV, 3 16132 Genova, Italy January 30, 1995 1 Introduction The replete construction was introduced independently by Hyland and Taylor (see [Hyl91, Tay91]) in the context of Synthetic Domain Theory (SDT ) Given a model of SDT , i.e. a topos E with a classi er t: 1 satisfying certain axioms, the construction was used to de ne a full re ective subcategory of E , the category of replete objects, which is suitable for Denotational Semantics. ....

P. Taylor. The xed point property in synthetic domain theory. In 6th LICS Conf. IEEE, 1991. 18

....each class possibly being classi ed by some object M . 2.13. Remark. Many of the ideas in this paper evolved from synthetic domain theory, a model of which is a topos (with a classi er for all monos) that also has a classi er for recursively enumerable subsets [Ros86, Pho90a, Pho90b, Hyl91, Tay91, FR97, BR98] In this case, is a subsemilattice of Such models exist wherein the full subcategory of replete objects satis es the monadicity property discussed in this paper for , in addition to that for the whole category for [RT97] A distilled account of repleteness and other terminology ....

....by Proposition 2.8) i (b) satis es the Euclidean principle. In this case, a mono i is classi ed (open) i there is some map E that satis es id = i E and the Frobenius law. Theory and Applications of Categories, Vol. 7, No. 15 298 This result partly answers a criticism of [Tay91] that it did not ask for a dominance, since the Euclidean principle is a part of the Phoa principle (Proposition 5.7) although monadicity was not considered there. It does not seem to be possible, in general, to deduce id = i E from the simpler condition that i be mono (with E a i ) ....

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Paul Taylor. The xed point property in synthetic domain theory. In Gilles Kahn, editor, Logic in Computer Science 6, pages 152-160. IEEE, 1991.

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