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 .