Existence of strongly proper dyadic subbases

We consider a topological space with its subbase which induces a coding for each point. Every second-countable Hausdorff space has a subbase that is the union of countably many pairs of disjoint open subsets. A dyadic subbase is such a subbase with a fixed enumeration. If a dyadic subbase is given, then we obtain a domain representation of the given space. The properness and the strong properness of dyadic subbases have been studied, and it is known that every strongly proper dyadic subbase induces an admissible domain representation regardless of its enumeration. We show that every locally compact separable metric space has a strongly proper dyadic subbase.Comment: 11 page

Domain Representations Induced by Dyadic Subbases

We study domain representations induced by dyadic subbases and show that a proper dyadic subbase S of a second-countable regular space X induces an embedding of X in the set of minimal limit elements of a subdomain D of $\{0,1,\perp\}\omega$. In particular, if X is compact, then X is a retract of the set of limit elements of D

A rich hierarchy of functionals of finite types

We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the so called Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric space can be isometrically embedded into it. We discuss why the Urysohn space may be considered as the universal model of possibly infinitary outputs of algorithms. The main result is that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over the Urysohn space. As a preparation for this, we prove an effective density theorem that is also of independent interest.Comment: 21 page

Domain Representable Spaces Defined by Strictly Positive Induction

Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological spaces. A $qcb_{0}$ space is a topological space characterized by its strong representability over domains. In this paper, we study strictly positive inductive definitions for $qcb_{0}$ spaces by means of domain representations, i.e. we show that there exists a canonical fixed point of every strictly positive operation on $qcb_{0}$ spaces.Comment: 48 pages. Accepted for publication in Logical Methods in Computer Scienc

Interval Domains and Computable Sequences: A Case Study of Domain Reductions

The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions. The effectivity theory induced by these variations is not stable, and this paper investigates some of the rich structure found. We follow Mostowski (On computable sequences. Fund. Math., 44, 37-51) and use computable sequences to exhibit this structure

Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications

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

A Stream Calculus of Bottomed Sequences for Real Number Computation

AbstractA calculus XPCF of 1⊥-sequences, which are infinite sequences of {0,1,⊥} with at most one copy of bottom, is proposed and investigated. It has applications in real number computation in that the unit interval I is topologically embedded in the set Σ⊥,1ω of 1⊥-sequences and a real function on I can be written as a program which inputs and outputs 1⊥-sequences. In XPCF, one defines a function on Σ⊥,1ω only by specifying its behaviors for the cases that the first digit is 0 and 1. Then, its value for a sequence starting with a bottom is calculated by taking the meet of the values for the sequences obtained by filling the bottom with 0 and 1. The validity of the reduction rule of this calculus is justified by the adequacy theorem to a domain-theoretic semantics. Some example programs including addition and multiplication are shown. Expressive powers of XPCF and related languages are also investigated

Effectiveness in RPL, with Applications to Continuous Logic

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; show that provability degree of a formula w.r.t. a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an $L^p$ Banach lattice) is decidable

Turing machines on represented sets, a model of computation for Analysis

We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real function or a probability measure, for example. The model is based on TTE, the representation approach for computable analysis. As a main result we prove that the functions that are computable via given representations are closed under GTM programming. This generalizes the well known fact that these functions are closed under composition. The theorem allows to speak about objects themselves instead of names in algorithms and proofs. By using GTMs for specifying algorithms, many proofs become more rigorous and also simpler and more transparent since the GTM model is very simple and allows to apply well-known techniques from Turing machine theory. We also show how finite or infinite sequences as names can be replaced by sets (generalized representations) on which computability is already defined via representations. This allows further simplification of proofs. All of this is done for multi-functions, which are essential in Computable Analysis, and multi-representations, which often allow more elegant formulations. As a byproduct we show that the computable functions on finite and infinite sequences of symbols are closed under programming with GTMs. We conclude with examples of application