For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !-continuous iff X is separable. The computational model BX is used to give domain-theoretic proofs of Banach's fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique non-empty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an !-continuous computational model for measure theory on a separable complete metric space X . 1 Introduction In this paper, we establish new connections between the theory of metric spaces and domain theory, the two basic mathematical structures in computer science. For every metric space X, we define a continuous poset (not necessar...

In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of Stoltenberg-Hansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turing-machine based approach which dates back to Grzegorczyk and Lacombe, is used by Pour-El & Richards in their found...

Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1

) F. Alessi, P. Baldan, F. Honsell Dipartimento di Matematica ed Informatica via delle Scienze 208, 33100 Udine, Italy falessi, baldan, honsellg@dimi.uniud.it Abstract. In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2-Stone by the functor MAX, which associates to each object of SFP m the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which "partialize" solutions of a vast class of domain equations in 2-Stone, can be obtained by solving the corresponding equations in SFP m . Furthermore, we compare two classical partializations of the space of Milner's Synchronization Tre...

It is shown that every compact metric space X is homeomorphically embedded in an !-algebraic domain D as the set of minimal limit elements.

In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2-Stone by the functor MAX, which associates to each object of SFP m the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which "partialize" solutions of a vast class of domain equations in 2-Stone, can be obtained by solving the corresponding equations in SFP m . Furthermore, we compare two classical partializations of the space of Milner's Synchronization Trees using SFP domains (see [3], [15]). Using the notion of "rigid" embedding projection pair, we show that the two domains are not isomorphic, thus providing a negative a...

Both preorders and ordinary ultrametric spaces are instances of generalized ultrametric spaces. Every generalized ultrametric space can be isometrically embedded in a (complete) function space by means of an ultrametric version of the categorical Yoneda Lemma. This simple fact gives naturally rise to: 1. a topology for generalized ultrametric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary ultrametric spaces reduces to the ffl-ball topology; 2. a topology for algebraic complete generalized ultrametric spaces generalizing both the Scott topology for arbitrary algebraic complete partial orders and the ffl-ball topology for complete ultrametric spaces. 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (cf., e.g., the recent textbooks [Win93] and [BV95]). Parts of their theory have been developed because of semantic necessity (see, e.g., [SP82] and [AR89]). Generalized ultram...

Abstract. General methods of investigating effectivity on regular Hausdorff (T3) spaces is considered. It is shown that there exists a functor from a category of T3 spaces into a category of domain representations. Using this functor one may look at the subcategory of effective domain representations to get an effectivity theory for T3 spaces. However, this approach seems to be beset by some problems. Instead, a new approach to introducing effectivity to T3 spaces is given. The construction uses effective retractions on effective Scott–Ershov domains. The benefit of the approach is that the numbering of the basis and the numbering of the elements are derived at once. 1

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