Complexity for partial computable functions over computable Polish spaces

2016 ◽  
Vol 28 (3) ◽  
pp. 429-447 ◽  
Author(s):  
MARGARITA KOROVINA ◽  
OLEG KUDINOV
Keyword(s):  
Crucial Role ◽  
Polish Space ◽  
Computable Function ◽  
Topological Spaces ◽  
Computable Numbering ◽  
Polish Spaces ◽  
The Real ◽  
Computable Functions

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

scholarly journals Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

scholarly journals Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

2021 ◽  
Vol 7 (1) ◽  
pp. 1224-1240
Author(s):  
Sang-Eon Han ◽  
◽  
Saeid Jafari ◽  
Jeong Min Kang ◽  
Sik Lee ◽  
...  
Keyword(s):  
Crucial Role ◽  
Dimensional Space ◽  
Digital Topology ◽  
Topological Spaces ◽  
Khalimsky Topology

<abstract><p>The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.</p></abstract>

On a theorem of Fleischer

1986 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
G. Mehta
Keyword(s):  
Ordered Set ◽  
Separation Theorem ◽  
Order Topology ◽  
Topological Spaces ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractFleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

scholarly journals Denjoy, Demuth and density

2014 ◽  
Vol 14 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Rupert Hölzl ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Computable Function ◽  
Density Theorem ◽  
Positive Density ◽  
Random Real ◽  
Open Problems ◽  
Covering Problems ◽  
Closed Class ◽  
Random Reals ◽  
Upper And Lower Derivatives ◽  
Computable Functions

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

Computational speed-up by effective operators

1972 ◽  
Vol 37 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Albert R. Meyer ◽  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
Turing Machines ◽  
Complexity Measures ◽  
Computational Speed ◽  
Partial Recursive Functions ◽  
Speed Up ◽  
Recursive Equations ◽  
Computable Functions ◽  
Effective Operators

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing(1) domain (φi) = domain (Φi), and(2) the predicate [Φi(x) = m] is recursive in i, x and m.Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

1.—On Measurable Selections

1974 ◽  
Vol 72 (1) ◽  
pp. 1-7 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Measure Space ◽  
Hausdorff Space ◽  
Polish Space ◽  
Measurable Selection ◽  
Selection Theorem ◽  
Polish Spaces ◽  
Measurable Multifunction ◽  
Selection Theorems ◽  
Suitable Measure ◽  
Separable Metric Space

SynopsisMeasurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

The sequential topology on is not regular

2009 ◽  
Vol 19 (5) ◽  
pp. 943-957 ◽  
Author(s):  
MATTHIAS SCHRÖDER
Keyword(s):  
Open Subset ◽  
Open Problem ◽  
Polish Space ◽  
Topological Spaces ◽  
Computable Analysis ◽  
Topological Properties ◽  
Compact Open Topology ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Continuous Functionals

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

scholarly journals Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

2014 ◽  
Vol 25 (7) ◽  
pp. 1490-1519 ◽  
Author(s):  
VERÓNICA BECHER ◽  
SERGE GRIGORIEFF
Keyword(s):  
Topological Spaces ◽  
Descriptive Set Theory ◽  
Stationary Strategy ◽  
Approximation Spaces ◽  
Polish Spaces ◽  
Proper Subclass ◽  
Continuous Domains ◽  
Algebraic Domains ◽  
Descriptive Set ◽  
Work Done

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

Pakistan’s First Successful Launch of a Real Estate Investment Trust-Dolmen City (REIT)—(Shariah Compliant Rental REIT Scheme)

2018 ◽  
Vol 15 (2) ◽  
pp. 129-146
Author(s):  
Fazal Jawad Seyyed ◽  
Salman Khan ◽  
Yasir Mir ◽  
Zeeshan Amir
Keyword(s):  
Real Estate ◽  
Crucial Role ◽  
Real Estate Investment Trust ◽  
Real Estate Investment ◽  
The Real ◽  
The Real Estate ◽  
Real Estate Sector ◽  
Legal Challenges ◽  
Successful Launch ◽  
The Way

It was the start of November 2015. Muhammad Ejaz, the CEO of Arif Habib Dolmen REIT Management Limited (AHDRML), was preparing for a presentation to the Board of AHDRML for the following week. The presentation was to recount the story of Dolmen City REIT (DCR), launched a few months back in June 2015, highlighting the regulatory and legal challenges faced during the process and many lingering issues still confronting this nascent sector. Ejaz realized that the group, as a leading player in the sector, had a crucial role to play in lobbying for further changes in the regulation to pave the way for future launches. More importantly, Ejaz wanted a nod from the Board for launch of a different REIT structure in 2016 to capitalize on the immense opportunity in the real estate sector of Pakistan.

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