DOCSLIB.ORG

A Topological and Domain Theoretical Study of Total Computable Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Topological and Domain Theoretical Study of Total Computable Functions FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE CENTRE OF EXACT AND EARTH SCIENCES DEPARTMENT OF INFORMATICS AND APPLIED MATHEMATICS GRADUATE PROGRAM IN SYSTEMS AND COMPUTATION A topological and domain theoretical study of total computable functions Claudio Andr´esCallejas Olgu´ın Natal{RN July, 2016 Claudio Andr´esCallejas Olgu´ın A topological and domain theoretical study of total computable functions Doctoral thesis presented to the Graduate Program in Systems and Computation in the Department of Informatics and Applied Mathematics of the Federal University of Rio Grande do Norte as a partial fulfilment for obtaining the degree of Doctor in Computer Science. Line of research: Foundations of Computation Supervisor: Benjam´ınRen´eCallejas Bedregal, PhD Natal{RN July, 2016 Catalogação da Publicação na Fonte. UFRN / SISBI / Biblioteca Setorial Centro de Ciências Exatas e da Terra – CCET. Callejas Olguín, Claudio Andrés. A topological and domain theoretical study of total computable functions / Claudio Andrés Callejas Olguín. - Natal, 2016. 71 f. Advisor: PhD Benjamín René Callejas Bedregal. Thesis (Doctorate degree) – Federal University of Rio Grande do Norte. Centre of Exact and Earth Sciences. Departament of Informatics and Applied Mathematics. Graduate Program in Systems and Computation. 1. Topology – Computer theory – Thesis. 2. Computability – Thesis. 3. Total computable functions – Thesis. 4. Topological spaces – Thesis. 5. Domain theory – Thesis. 6. Algebraic quasi-domains – Thesis. 7. Scott topology – Thesis. I. Bedregal, Benjamín René Callejas. II. Title. RN/UF/BSE-CCET CDU: 004:515.1 To Rafaela, my parents, my brothers and my uncle Benjam´ın. Acknowledgement I am truly thankful to my girlfriend Rafaela without whose permanent encour- agement and support given during the whole path of this doctorate I would not have been able to finish this thesis. I am deeply grateful to my mother Ana Maria Roxana Olgu´ınNegr´onfor her encouragement and for never stop believing that I was up to the task. I am also sincerely thankful to my uncle and supervisor professor Dr. Ben- jam´ınRen´eCallejas Bedregal for his clever ideas and guidance. No one else could had explored more my capacities than him. I am also deeply thankful to my father professor Dr. Roberto Callejas Bedregal who gave me the most stunning lectures in mathematics of my whole life. I have the privilege to have in my family two of the scientist that I admire most. Their examples as how to live science has been with me way before I initiated my graduate school. I really thanks for the valued recommendations and corrections made by the members of the examination board: Dr. Benjam´ınRen´eCallejas Bedregal, Dr. Regivan Hugo Nunes Santiago, Dr. Fagner Lemos de Santana, Dr. Jorge Petrucio Viana and Dr. Wilson Rosa de Oliveira Junior. I am thankful to Flaulles, Adriano, Jo~aoDaniel, Evelyn and Sanderson for being my classmate colleges. I am also thankful to Marcos Tindo for his friendship and appropriate help in critical moments. I am also thankful to my college workers in the Federal University Rural do Semi-Arido´ who freed me of my teaching duties during the last three semesters of my doctorate. Abstract Topologically the set of total computable functions has been studied only as a subspace of a Baire space. Where the topology of this Baire space is the induced topology of a Scott topology for the partial functions (not necessarily computable). In this thesis a novel topology on the index set of the set of total computable functions is built and proved that it is not homeomorphic to the aforementioned subspace of the presented Baire space. There is an important undecidable subset of the set of total computable functions called the set of regular computable functions that receives particular attention in this thesis. In order to make a topological study of this set a whole theoretical apparatus is constructed. After presenting the state of the art of generalised domain theory, a novel generalisation of algebraic domains coined as algebraic quasi- domains is introduced. With a suited partial-order an algebraic quasi-domain is built for the set of total computable functions. Through the Scott topology associated with this algebraic quasi-domain a necessary condition for the regular computable functions is obtained. It is proved that this later topology is not homeomorphic to the previously mentioned subspace of the presented Baire space. As a byproduct a Scott topology for the set of total functions (not necessarily computable) is introduced. It is proved that this later topology is not homeomorphic to the presented Baire space. It is also proved that the Scott topologies in the set of total functions and in the subset of total computable functions have the set of total functions with finite support as a common dense set. Analogously it is proved that the topology in the index set of the set of total computable functions has as a dense set the indexes corresponding to a computable enumeration without repetition of the set of total functions with finite support. Resumo O conjunto de fun¸c~oestotais comput´aveis somente tem sido estudado topo- logicamente como um subespa¸code um espa¸code Baire. Onde a topologia deste espa¸code Baire ´ea topologia induzida de uma topologia de Scott para as fun¸c~oesparciais (n~aonecessariamente comput´aveis). Nesta tese constr´oi-se uma topologia original no conjunto de ´ındicesdas fun¸c~oestotais comput´aveis e demonstra-se que ela n~ao´ehomeomorfa com o subespa¸codo espa¸code Baire que foi mencionado. H´aum subconjunto indec´ıdivel importante no conjunto de fun¸c~oestotais comput´aveis chamado \o conjunto de fun¸c~oescomput´aveis regulares", que re- cebe aten¸c~aoespecial nesta tese. Com a finalidade de fazer um estudo topol´ogico deste conjunto constr´oi-setodo um aparato te´orico.Ap´osapresentar o estado da arte da teoria dos dom´ıniosgeneralizada introduz-se uma generaliza¸c~aoorig- inal dos dom´ıniosalg´ebricosnomeados como \quase-dom´ıniosalg´ebricos".Com uma ordem parcial adequada, constr´oi-seum quase-dom´ınio alg´ebricopara o conjunto de fun¸c~oescomput´aveis totais. Por meio da topologia de Scott asso- ciada a esse quase-dom´ınioalg´ebrico,obt´em-seuma condi¸c~aonecess´ariapara as fun¸c~oescomput´aveis regulares. Fica provado que esta ´ultimatopologia n~ao ´ehomeomorfa com o subespa¸copreviamente mencionado do espa¸code Baire apresentado. Como subproduto, introduz-se uma topologia de Scott para o con- junto de fun¸c~oestotais (n~aonecessariamente comput´aveis). Fica provado que esta ´ultimatopologia n~ao´ehomeomorfa com o espa¸code Baire apresentado. Fica tamb´emprovado que as topologias de Scott no conjunto de fun¸c~oesto- tais e no subconjunto de fun¸c~oestotais comput´aveis t^emo conjunto de fun¸c~oes totais com suporte finito como conjunto denso comum. Analogamente, fica provado que a topologia no conjunto ´ındicedo conjunto de fun¸c~oestotais com- put´aveis tem como conjunto denso os´ındicescorrespondentes a uma enumera¸c~ao comput´avel sem repeti¸c~aodo conjunto de fun¸c~oestotais com suporte infinito. List of symbols ! : Set of non-negative integers ' : Extended equality 0 : Zero function ! !c : Set of total computable functions Pc : Set of partial computable functions AnB : Set of all elements in the set A but not in the set B A : Complement of the A ≤m : Many-one reducible relation P(X) : Power set of the set X (X; T ) : Topological space with set X and topology T T (B) : Topology induced by basis B C(A) : Closure of the set A F(A) : Boundary of the set A =∼ : Homeomorphism B(a; r) : Ball with centre a and radius r T (d) : Topology induced by the metric d K(D) : Set of compact elements of domain theoretical structure D approx(x) : Set of all compact elements a such that a v x Cx : Set of all elements y in the set C such that y v x P : Set of partial functions on ! PS : Set of partial functions on the non-empty set S σ; : Nowhere define partial function on the non-empty set S PS : First-order structure PS = (PS; σ;; ≤) with extensional ordering ≤ '; : Nowhere define partial function on ! P : First-order structure P = (P;';; ≤) with extensional ordering ≤ S (S?) : Set of all total functions from the non-empty set S to the flat cpo S? S ?S : Constant function in (S?) that maps every x 2 S to ? S S S (S?) : First-order structure (S?) = ((S?) ; ?S; 4) with pointwise order 4 ! (!?) : Set of all total functions from ! to the flat cpo !? ! ?! : Constant function in (!?) that maps every x 2 ! to ? ! ! ! (!?) : First-order structure (!?) = ((!?) ; ?!; 4) with pointwise order 4 (P; T (B)) : Topological space with basis B Q ( On; Tπ) : Product space obtained from the topological spaces f(On; Tn)gn2! n2! C : First-order structure C = (P; 6) !! : Set of total functions on ! ! (! ; Tind) : Subspace of (P; T (B)) ! (! ; Tprod) : Product space obtained from the topological spaces f(!n; τn)gn2! ! (! ; T (dTot)) : Metric space ! (! ; T (du)) : Ultrametric spaces ! ! (!c ; Trel) : Subspace of (! ; Tind) ! (!c ; T (dTotC)) : Metric space supp('e) : Support of partial computable function 'e vP : Preorder on Pc (Pc; T (BP )) : Topological space with basis BP # x : Principal ideal ! ! : Index set of ! I!c c Pe : Derivation with index e ≡n : n-equivalence [ ! ] : Quotient set of ! I!c n I!c [e] : Element in [ ! ] n I!c n ([ ! ] ; T (d )) : Metric space I!c n n m λn : Connecting map from [I!! ]m to [I!! ]n Q c c ( [ ! ] ; τ ) : Product space obtained from the topological spaces f([ ! ] ; T (d ))g I!c n prod I!c n n n2! n2! ([I!! ]1; T1) : Inverse limit space c Q σ : Function from ! to [ ! ] I!c I!c n n2! T : Coarser topology on ! turning σ continuous σ I!c T : Initial topology on ! initial I!c supp(Pe) : Support of derivation Pe of total computable function M(D) : Set of maximal elements of a set D !! : First-order structure !! = (!!; 0; v) ! ! ! !c : First-order structure !c = (!c ; 0; v) ! T!! : Scott topology associated with ! ! T ! : Scott topology associated with ! !c c Contents 1 Introduction 3 2 Background 7 2.1 A Background on Computability .
Load more

Recommended publications
  • On the Topological Aspects of the Theory of Represented Spaces∗
    On the topological aspects of the theory of represented spaces∗ Arno Pauly Clare College University of Cambridge, United Kingdom [email protected] Represented spaces form the general setting for the study of computability derived from Turing ma- chines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by ESCARDO´ ,SCHRODER¨ , and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property. 1 Introduction ∗ Just as numberings provide a tool to transfer computability from N (or f0;1g ) to all sorts of countable structures; representations provide a means to introduce computability on structures of the cardinality of the continuum based on computability on Cantor space f0;1gN. This is of course essential for com- n m k n m putable analysis [66], dealing with spaces such as R, C (R ;R ), C (R ;R ) or general Hilbert spaces [8]. Computable measure theory (e.g. [65, 58, 44, 30, 21]), or computability on the set of countable ordinals [39, 48] likewise rely on representations as foundation. Essentially, by equipping a set with a representation, we arrive at a represented space – to some extent1 the most general structure carrying a notion of computability that is derived from Turing ma- chines2.
    [Show full text]
  • Arxiv:1703.04075V2 [Cs.LO] 14 Mar 2017 Osbc Ote15sadfcsso H Leri Operation Called Algebraic So the the Al Is the 10]
    COMPUTABLE STRUCTURES ON TOPOLOGICAL MANIFOLDS MARCELO A. AGUILAR AND RODOLFO CONDE Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Ciudad Universitaria, M´exico D.F. 04510, M´exico e-mail address: [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Ciudad Universitaria, M´exico D.F. 04510, M´exico e-mail address: [email protected] Abstract. We propose a definition of computable manifold by introducing computabil- ity as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL manifolds respec- tively. Using the framework of computable topology and Type-2 theory of effectivity, we develop computable versions of all the basic concepts needed to define manifolds, like com- putable atlases and (computably) compatible computable atlases. We prove that given a computable atlas Φ defined on a set M, we can construct a computable topological space (M,τΦ, βΦ,νΦ), where τΦ is the topology on M induced by Φ and that the equivalence class of this computable space characterizes the computable structure determined by Φ. The concept of computable submanifold is also investigated. We show that any compact computable manifold which satisfies a computable version of the T2-separation axiom, can be embedded as a computable submanifold of some euclidean space Rq, with a computable embedding, where Rq is equipped with its usual topology and some canonical computable encoding of all open rational balls. 1. Introduction Computability theory over continuous structures began formally in 1936 with the landmark paper of Alan Turing [1] where he defined the notion of a single computable real number: x ∈ R is computable if its decimal expansion can be calculated in the discrete sense, that arXiv:1703.04075v2 [cs.LO] 14 Mar 2017 is, output by a Turing machine.
    [Show full text]
  • Computable Analysis with Applications to Dynamic Systems
    Mathematical Structures in Computer Science (2020), 30, pp. 173–233 doi:10.1017/S096012952000002X ARTICLE Computable analysis with applications to dynamic systems Pieter Collins∗ Department of Data Science and Knowledge Engineering, Maastricht University, Maastricht, The Netherlands ∗Corresponding author. Email: [email protected] (Received 22 December 2018; revised 4 December 2019; accepted 7 January 2020) Abstract Numerical computation is traditionally performed using floating-point arithmetic and truncated forms of infinite series, a methodology which allows for efficient computation at the cost of some accuracy. For most applications, these errors are entirely acceptable and the numerical results are considered trustwor- thy, but for some operations, we may want to have guarantees that the numerical results are correct, or explicit bounds on the errors. To obtain rigorous calculations, floating-point arithmetic is usually replaced by interval arithmetic and truncation errors are explicitly contained in the result. We may then ask the question of which mathematical operations can be implemented in a way in which the exact result can be approximated to arbitrary known accuracy by a numerical algorithm. This is the subject of computable analysis and forms a theoretical underpinning of rigorous numerical computation. The aim of this article is to provide a straightforward introduction to this subject that is powerful enough to answer questions arising in dynamic system theory. Keywords: Computable analysis; Turing machine; dynamic system 1. Introduction In this paper, we develop a theory of computation for continuous mathematics. The aim of this paper is to give an exposition which is explicitly based on Turing machine model of computation, is powerful enough to study real computational problems arising in practice (taken from the areas of dynamical systems and control theory), yet is as straightforward and direct as possible, and uses terminology which is natural in classical topology and analysis.
    [Show full text]
  • A Lambda Calculus for Real Analysis
    Journal of Logic & Analysis 2:5 (2010) 1–115 1 ISSN 1759-9008 A lambda calculus for real analysis PAUL TAYLOR Abstract: Abstract Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function “hovers” near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD.
    [Show full text]
  • Computable Functional Analysis and Probabilistic Computation with Applications to Numerical Analysis
    Computable Functional Analysis and Probabilistic Computability Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt von Dipl.-Math. Volker Bosserhoff am 2. Oktober 2008 Tag der mündlichen Prüfung: 18. Dezember 2008 Vorsitzender der Kommission: Prof. Dr. Markus Siegle 1. Berichterstatter: Prof. Dr. Peter Hertling 2. Berichterstatter: Prof. Dr. Vasco Brattka 3. Berichterstatter: Prof. Dr. Cornelius Greither 4. Berichterstatter: Prof. DDr. Stefan Schäffler Universität der Bundeswehr München Fakultät für Informatik 2 实事求是 邓小平 „Be practical and realistic.“ Deng Xiaoping 4 Contents 1 Introduction & Overview9 2 Representation-Based Computable Analysis 23 2.1 Computability and naming systems........................ 23 2.1.1 Computability on N and N! ....................... 23 2.1.2 Computability via naming systems.................... 24 2.1.3 Some constructions with naming systems................ 24 2.1.4 Decidability and enumerability via multi-naming systems....... 27 2.2 Computing on topological spaces......................... 28 2.2.1 Spaces with continuous representations are Lindelöf.......... 28 2.2.2 Admissibility............................... 28 2.2.3 Computable T0-spaces.......................... 29 2.2.4 Representations of R ........................... 33 2.3 Computing on metric and regular spaces..................... 33 2.3.1 Computable metric spaces........................ 33 2.3.2 Computably regular spaces and metrization............... 35 2.4 Computing on Banach spaces........................... 36 2.4.1 Computable normed and Banach spaces................. 37 2.4.2 Effective independence.......................... 38 5 6 CONTENTS 2.4.3 Representations of the space of bounded linear operators........ 42 2.4.4 Computable Hilbert spaces........................ 43 2.5 Computing on locally finite Borel measures................... 43 2.5.1 The standard representations....................... 43 2.5.2 The strong representation........................
    [Show full text]
  • Computable Analysis with Applications to Dynamic Systems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CWI's Institutional Repository Computable Analysis with Applications to Dynamic Systems P.J. Collins MAC-1002 Centrum Wiskunde & Informatica Centrum Wiskunde & Informatica (CWI) is the national research institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organisation for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Probability, Networks and Algorithms (PNA) Software Engineering (SEN) Modelling, Analysis and Computing (MAC) Information Systems (INS) Copyright © 2010, Centrum Wiskunde & Informatica P.O. Box 94079, 1090 GB Amsterdam (NL) Science Park 123, 1098 XG Amsterdam (NL) Telephone +31 20 592 9333 Telefax +31 20 592 4199 ISSN 1386-3703 Computable Analysis with Applications to Dynamic Systems Pieter Collins 19 May 2010 Abstract In this article we develop a theory of computation for continuous math- ematics. The theory is based on earlier developments of computable analysis, especially that of the school of Weihrauch, and is presented as a model of intuitionistic type theory. Every effort has been made to keep the presentation as simple yet general as possible. The core sub- ject matter of Turing computability and computable analysis should be accessible to non-specialists with a solid background in general topol- ogy and analysis, but important technical results are also proved. To show the potential use of the theory, some simple applications are given to dynamical systems and control theory.
    [Show full text]
  • Foundations for Computable Topology
    Foundations for Computable Topology Paul Taylor 8 April 2009 Abstract Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpi´nskispace, which plays a key role as the classifier for both open and closed subspaces. In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory. The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0; 1] in ASD is compact. \In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl.) \Point-set topology is a disease from which the human race will soon recover." (Henri Poincar´e,quoted in Comic Sections by Des MacHale.) \Logic is the hygiene that the mathematician practises to keep his ideas healthy and strong." (Hermann Weyl, quoted in American Mathematical Monthly, volume 100, p 93.) \Mathematics is the queen of the sciences and number theory is the queen of mathematics." (Carl Friedrich Gauss) \Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories." (Pierre-Simon Laplace, quoted in Mathematical Maxims and Minims by N.
    [Show full text]
  • The Realizability Approach to Computable Analysis and Topology
    The Realizability Approach to Computable Analysis and Topology Andrej Bauer September 2000 CMU-CS-00-164 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee Dana S. Scott, chair Steven M. Awodey Lenore Blum Abbas Edalat Frank Pfenning Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy This research was sponsored by the National Science Foundation (NSF) under Grant No. CCR-9217549 and the NAFSA Association of International Educators through a generous fellowship. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. Keywords: computability, realizability, modest sets, constructive and computable analysis and topology To disertacijo posveˇcammojemu oˇcetu, ki je moj prvi in najvplivnejˇsiuˇciteljmatematike. I dedicate this dissertation to my father, who is my first and most influential teacher of mathematics. Abstract In this dissertation, I explore aspects of computable analysis and topology in the frame- work of relative realizability. The computational models are partial combinatory alge- bras with subalgebras of computable elements, out of which categories of modest sets are constructed. The internal logic of these categories is suitable for developing a theory of computable analysis and topology, because it is equipped with a computability pred- icate and it supports many constructions needed in topology and analysis. In addition, a number of previously studied approaches to computable topology and analysis are special cases of the general theory of modest sets. In the first part of the dissertation, I present categories of modest sets and axiomatize their internal logic, including the computability predicate.
    [Show full text]