## Axioms and (Counter)examples in Synthetic Domain Theory (1998)

Venue: | Annals of Pure and Applied Logic |

Citations: | 8 - 7 self |

### BibTeX

@INPROCEEDINGS{Oosten98axiomsand,

author = {Jaap Van Oosten and Alex K. Simpson},

title = {Axioms and (Counter)examples in Synthetic Domain Theory},

booktitle = {Annals of Pure and Applied Logic},

year = {1998},

pages = {200--0}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of this paper, we believe that it would also be

### Citations

198 |
Sheaves in geometry and logic. A first introduction to topos theory. Universitext
- Lane, Moerdijk
- 1994
(Show Context)
Citation Context ... not well-complete. This provides a counterexample to the converse of the first implication of 1.23. We begin by introducing notation for sites and sheaves over them. Full definitions can be found in =-=[16]-=-. Let C be any small category. We use letters A; B; : : : for objects of C, and Greek letters '; #; : : : for morphisms. We write b C for the category of presheaves on C. Given a presheaf F , an eleme... |

95 | Algebraically complete categories - Freyd - 1990 |

82 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ...reflective category of sets [29] and provide yet another respectable notion of predomain. Well-complete regular \Sigma-posets also arise very naturally in models. For example, in the effective topos, =-=[7]-=-, the well-complete regular \Sigma-posets are (equivalent to) the complete extensional PERs of [6]. Even more strikingly, in Fiore and Rosolini's topos H [3, 4], the category of wellcomplete regular \... |

53 |
First steps in synthetic domain theory
- Hyland
- 1991
(Show Context)
Citation Context ...reatment, one first needs to fix on a version of intuitionistic set theory. In this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in =-=[23, 8, 26]-=-. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of... |

47 |
Continuity and effectiveness in topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...reatment, one first needs to fix on a version of intuitionistic set theory. In this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in =-=[23, 8, 26]-=-. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of... |

43 | New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic
- Crole, Pitts
- 1990
(Show Context)
Citation Context ...(well)complete objects of Eff . In WC, the object F carries both the initial algebra and final coalgebra structures for L, and they are each other's inverse (F is a fixed-point object in the sense of =-=[1]). No-=-w, in Eff , F is the internal limit of 1 / L(1) / L 2 (1) / : : : whereas I 0 is the internal colimit of 0 ! L(0) ! L 2 (0) ! : : : This shows, that F , although a fixed-point object, is not "ind... |

40 |
Algebraic Set Theory
- Moerdijk, Joyal
(Show Context)
Citation Context ...eorem 1.4, another initiality property of I can be derived. This is a general fact about initial algebras for functors which have a monad structure, discovered by B'enabou and Jibladze, and proved in =-=[13]-=-. Let (L; i; ) be a monad on a category. A strict L-algebra with successor is a structure (X; L(X) a ! X;X g ! X) such that a is a strict L-algebra structure, and g is arbitrary. A morphism of strict ... |

38 | The fixed point property in synthetic domain theory
- Taylor
- 1991
(Show Context)
Citation Context ...ks for \Sigma to satisfy a certain "completeness" property. In the presence of this axiom alone, it is possible to identify a number of different notions of predomain. Amongst these, the rep=-=lete sets [8, 30]-=-, and the well-complete sets [15, 22, 26] form two extreme choices. The former form the smallest full reflective subcategory of sets containing \Sigma, and the latter form what appears to be the large... |

25 | The discrete objects in the effective topos
- Hyland, Rosolini
- 1990
(Show Context)
Citation Context ...embly X is modest if it is a regular image of a regular subobject of N . X is discrete, if X is internally orthogonal to r(2), that is if the diagonal: X ! X r(2) is an isomorphism. It is a result of =-=[10]-=- that in the effective topos, for separated objects, these notions coincide (even fiberwise for families of separated objects indexed by a separated object; obviously these notions can and really shou... |

23 |
Extensional PERs
- Freyd, Mulry, et al.
- 1990
(Show Context)
Citation Context ...tor which classifies \Sigma-partial functions. In general (always?), the containments between the categories are proper. From examining the topos models of Axiom 1 that have been investigated to date =-=[20, 6, 15, 3, 4]-=- one may extrapolate another axiom: \Sigma is a ::-separated set (our Axiom 2). Although hard to motivate conceptually, by permitting classical forms of reasoning about \Sigma-properties, the axiom ha... |

22 |
On a topological topos
- Johnstone
- 1979
(Show Context)
Citation Context ...d notion of regular \Sigma-poset (corresponding to the extensional objects in [6, 8, 15]) is useful. Such objects arise very naturally in certain models. For example, in Johnstone's topological topos =-=[12]-=-, the regular \Sigma-posets are exactly the sequential T 0 topological spaces (Mat'ias Menni, private communication). In general, the well-complete regular \Sigma-posets form a full reflective categor... |

22 | A combinatory algebra for sequential functionals of finite type
- Oosten
- 1999
(Show Context)
Citation Context ...axioms, look rather special. The existence of a "parallel termination test" on \Sigma rules out models of SDT based on "sequential" partial combinatory algebras (such as, e.g., the=-= ones considered in [17]-=-, [14] and [15]; [4] gives a non-realizability model 18 where nonetheless Axiom 4 fails); and inasmuch one is interested in models of sequential computation, Axiom 4 cannot be recommended. Axiom 4 8oe... |

22 |
Effective Domains and Intrinsic Structure, in
- Phoa
- 1990
(Show Context)
Citation Context ...tor which classifies \Sigma-partial functions. In general (always?), the containments between the categories are proper. From examining the topos models of Axiom 1 that have been investigated to date =-=[20, 6, 15, 3, 4]-=- one may extrapolate another axiom: \Sigma is a ::-separated set (our Axiom 2). Although hard to motivate conceptually, by permitting classical forms of reasoning about \Sigma-properties, the axiom ha... |

21 | A uniform approach to domain theory in realizability models
- Longley, Simpson
- 1997
(Show Context)
Citation Context .... Following [23], one first identifies a set, \Sigma, of "termination properties" (which one can often think of as classifying an abstract notion of "semidecidable property " or &q=-=uot;open subset"). As in [15], our main-=- axiom, Axiom 1, asks for \Sigma to satisfy a certain "completeness" property. In the presence of this axiom alone, it is possible to identify a number of different notions of predomain. Amo... |

21 |
Relating theories of the -calculus
- Scott
- 1980
(Show Context)
Citation Context ... runs into trouble with classical set theory. For example, there are precious few sets with the property that any endofunction on them has a fixed point. Remarkably however, as Dana Scott observed in =-=[25]-=-, such inconsistencies do not arise if intuitionistic set theory is used instead. For this reason, Scott proposed that intuitionistic set theory might provide an intuitive and powerful framework for d... |

17 | An extension of models of axiomatic domain theory to models of synthetic domain theory
- Fiore, Plotkin
- 1996
(Show Context)
Citation Context ...y in codes for trackings of f , which gives the desired operation: (I 0 ) I ! (I 0 ) F . There is a point about theorem 3.3 which deserves to be made, in particular in connection with the research in =-=[2]-=-. Let WC denote the category of (well)complete objects of Eff . In WC, the object F carries both the initial algebra and final coalgebra structures for L, and they are each other's inverse (F is a fix... |

15 |
Program Verification in Synthetic Domain Theory
- Reus
- 1995
(Show Context)
Citation Context ...?), hence in \Sigma. So fi 2 L(X) and we have A(P ) , 9x:X:x 2 fi P (x) whence A = l X (fi). We prove now the main theorem of this subsection. In itself, the theorem is not new (it was also proved in =-=[21]-=-); what is new is that our proof requires nothing more than Axioms 1 and 2. In fact, we find it rather surprising that in this generality, the property of being a regular \Sigma-poset suffices to rest... |

14 | Modified realizability toposes and strong normalization proofs
- Hyland, Ong
(Show Context)
Citation Context ...ursive function /, but then / would enumerate all total recursive functions; which runs into a familiar diagonal argument. The logic of Mod and the tripos underlying it is described in [18] (see also =-=[9]-=-). Salient features are the following principles: IP (:A ! 9n:N:B) ! 9n:N:(:A ! B) CT 8f :N N 9e:N8x:N9y:N:(T (e; x; y)sU (y) = f(x)) AC-N 8n:N9x:X:R(n; x) ! 9f :X N 8n:N:R(n; f(n)) which are true in ... |

13 |
Domain Theory in Realizability Toposes
- Phoa
- 1990
(Show Context)
Citation Context ...rder on X, the object X I is seen as the "object of chains in X", and sometimes 17 (as in [8]) the desire was expressed to do away with the \Sigma-order altogether (this, by the way, in cont=-=rast with [21, 19, 20]-=-, where the \Sigma-order is taken as basic and the notion of completeness is defined as: having lubs of N -chains for the \Sigma-order. One should however note that, in order to prove the desirable pr... |

12 | Two models of synthetic domain theory
- Fiore, Rosolini
- 1997
(Show Context)
Citation Context ...tor which classifies \Sigma-partial functions. In general (always?), the containments between the categories are proper. From examining the topos models of Axiom 1 that have been investigated to date =-=[20, 6, 15, 3, 4]-=- one may extrapolate another axiom: \Sigma is a ::-separated set (our Axiom 2). Although hard to motivate conceptually, by permitting classical forms of reasoning about \Sigma-properties, the axiom ha... |

11 | General synthetic domain theory - a logical approach
- Reus, Streicher
- 1999
(Show Context)
Citation Context ...r I. Moerdijk, which employs the first author, and enabled the second author to visit Utrecht in the period February--June 1998. 1 possible to use an (impredicative) intuitionistic type theory, as in =-=[22]-=-, or even intuitionistic Zermelo-Fraenkel set theory ([27]), without changing the nature of the mathematics (only the metamathematics). An interesting challenge would be to attempt an axiomatic develo... |

11 |
Investigations into intensional type theory. Habilitiation Thesis
- Streicher
- 1993
(Show Context)
Citation Context ...at so long as we restrict to ::-separated objects, we don't have to bother about 0. 21 Definition 2.1 The category ModAss of modified assemblies (not to be confused with those of Thomas Streicher, in =-=[28]-=-!) is the following: Objects are triples (X; j\Deltaj X ; PX ) where X is a set, j\Deltaj X : X ! P(IN) a function assigning to each x 2 X a nonempty set jxj X of realizers of x, and PX ` IN a nonempt... |

8 |
A presentation of the initial lift algebra
- Jibladze
- 1997
(Show Context)
Citation Context ... in fact, F is a retract of \Sigma N , by the map ' 7! n:N: V kn '(k). L has also an initial algebra (L(I) oe ! I). I can be constructed as the least L-subalgebra of (L(F ) oe ! F ). Mamuka Jibladze (=-=[11]-=-) has given a beautiful formula for I: I = f/ 2 F j 8OE:\Omega :(8n:N:((/(n) ) OE) ) OE)) ) OEg Note that oe : L(F ) ! F restricts to oe : L(I) ! I which is the L-algebra structure on I. There are sev... |

5 | Categories and effective computation - Rosolini - 1987 |

2 |
Oosten, The modified realizability topos
- van
- 1997
(Show Context)
Citation Context ... closed under N-indexed joins in\Omega (existential quantification over N ) then every complete object is well-complete (Proposition 1.31). In Section 2, we consider the Modified Realizability Topos (=-=[18]-=-), as our first new model of SDT. We show that, under an appropriate choice of dominance, Axioms 1--4 are satisfied. This model allows us to obtain a number of independence results. Firstly, the Scott... |

1 |
Computational Adequacy in an Elementary Topos. Presented at CSL
- Simpson
- 1998
(Show Context)
Citation Context ...reatment, one first needs to fix on a version of intuitionistic set theory. In this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in =-=[23, 8, 26]-=-. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of... |

1 |
Domain Theory in Intuitionistic Set Theory
- Simpson
- 1998
(Show Context)
Citation Context ... the second author to visit Utrecht in the period February--June 1998. 1 possible to use an (impredicative) intuitionistic type theory, as in [22], or even intuitionistic Zermelo-Fraenkel set theory (=-=[27]-=-), without changing the nature of the mathematics (only the metamathematics). An interesting challenge would be to attempt an axiomatic development in a predicative type theory. It seems that the best... |

1 |
Inductive construction of repletion. Applied Categorical Structures
- Streicher
- 1997
(Show Context)
Citation Context ...ory of sets supporting an adequate treatment of recursion. Although it is not known if well-complete sets form a reflective subcategory in general, their restriction to well-complete \Sigma-posets do =-=[29]-=-. These form a category intermediate between the replete and well-complete sets. Each of these three notions provides a complete full subcategory of sets closed under important domaintheoretic constru... |