On the Continuity of Effective Multifunctions

AbstractIf one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed in better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) in (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest and well known theorems such as those of Myhill and Shepherdson, Kreisel, Lacombe and Shoenfield, Ceĭtin, and/or Moschovakis present answers for important special cases. A general, unifying approach has been developed by the present author in [D. Spreen. On effective topological spaces. The Journal of Symbolic Logic, 63 (1998), 185–221. Corrections ibid., 65 (2000), 1917–1918].In this paper the approach is extended to the case of multifunctions. Such functions appear very naturally in applied mathematics, logic and theoretical computer science. Various ways of coding (indexing) sets are discussed and effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system and possess certain witness functions are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Important special cases are discussed where such witnessing functions always exist

On domains witnessing increase in information

[EN] The paper considers algebraic directed-complete partial orders with a semi-regular Scott topology, called regular domains. As is well know, the category of Scott domains and continuous maps is Cartesian closed. This is no longer true, if the domains are required to be regular. Two Cartesian closed subcategories of the regular Scott domains are exhibited: regular dI-domains with stable maps and strongly regular Scott domains with continuous maps. Here a Scott domains is strongly regular if all of its compact open subsets are regular open. In one considers only embeddings of dependent products and sums. Moreover, they are w-cocomplete and their object classes are closed under several constructions used in programming language semantics. It follows that recursive domains equations can be solved and models of typed and untyped lambda calculi can be constructed. Both kinds of domains can be udes in giving meaning to programming language constructs.Spreen, D. (2000). On domains witnessing increase in information. Applied General Topology. 1(1):129-152. https://doi.org/10.4995/agt.2000.13640OJS1291521

Computing with Infinite Objects: the Gray Code Case

Infinite Gray code has been introduced by Tsuiki as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also H. Tsuiki, K. Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates \bS and \bG are defined and the inclusion \bS \subseteq \bG is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version $\S_{2}$ of $S$ and \bG \subseteq \bS_{2} is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version \bG^{*} of \bG and a modification \bS^{*} of \bS_{2} are presented such that \bS^{*} = \bG^{*}. A crucial tool in U. Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that \bS^{*} and \bG^{*} coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals

Representations versus numberings: on the relationship of two computability notions

AbstractThis paper gives an answer to Weihrauch's (Computability, Springer, Berlin, 1987) question whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in general. A condition is introduced and for countably based topological T0-spaces it is shown that exactly the (partial) effective maps meeting the requirement are extendable. For total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed-complete partial orders, in which the extendability is already known. In the first case a similar result holds also for partial effective maps, but not in the second

Foreword

Digitalitzat per Artypla

06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality

From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

04351 Abstracts Collection -- Spatial Representation: Discrete vs. Continuous Computational Models

From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 ``Spatial Representation: Discrete vs. Continuous Computational Models\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available