DOCSLIB.ORG

The Turing Universe in the Context of Enumeration Reducibility

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Turing Universe in the Context of Enumeration Reducibility The Turing universe in the context of enumeration reducibility Mariya I. Soskova ? Sofia University and University of California, Berkeley A fundamental goal of computability theory is to understand the way that objects relate to each other in terms of their information content. We wish to understand the relative information content between sets of natural numbers, how one subset of the natural numbers Y can be used to specify another one X. This specification can be computational, or arithmetic, or even by the application of a countable sequence of Borel operations. Each notion in the spectrum gives rise to a different model of relative computability. Which of these models best reflects the real world computation is under question. The most widely used and studied model is the one based on computation: a set of natural numbers A is Turing reducible to set of natural numbers B if there is an effective procedure, by which given a natural number n and using the answers to finitely many membership questions to the oracle B we can correctly decide whether or not n is a member of A. By identifying sets that can be reduced to each other we obtain the partial order of the Turing degrees. Computable sets have least information content and form the least Turing degree 0T . There is a natural way to combine the information content of two sets of natural numbers, A and B are combined into A⊕B = f2n j n 2 Ag[f2n + 1 j n 2 Bg, so that every pair of Turing degrees has a least upper bound. Finally, using a relativization of the halting problem, for every Turing degree dT (A) we can find a degree which is strictly more complex, namely dT (KA), where KA is the halting set, relativized to A, which we call the jump of the Turing degree dT (A), and denote 0 by dT (A) . Thus the structure of the Turing degrees is an upper semi-lattice 0 with least element and jump operation: DT = (DT ; ≤T ; 0T ; _; ). In this article we will examine the structure of the Turing degrees in the context, provided by a slightly weaker form of reducibility, based on the notion of enumeration rather than computation. This reducibility was introduced by Friedberg and Rogers [6] in 1959. A set of natural numbers A is enumeration reducible to a set of natural numbers B, if we can effectively transform any enumeration of the set B into an enumeration of the set A. There is a very close relationship between Turing reducibility and enumeration reducibility. To see this recall that an equivalent way of saying that A ≤T B is to say that both A and the complement of A can be enumerated using oracle B, or in other words A ⊕ A is computably enumerable in B. Going deeper into the definition of c.e. in, we can say that there is a c.e. set W , whose members are pairs hn; ui ? Supported by a BNSF grant No. DMU 03/07/12.12.2011, by Sofia University Science Fund and by a Marie Curie international outgoing fellowship STRIDE (298471) within the 7th European Community Framework Programme of a natural number and a code for a finite set Du, so that A ⊕ A can be represented as the set n j 9u(hn; ui 2 W & Du ⊆ B ⊕ B) . In this definition the oracle set B ⊕ B and the reduced set A ⊕ A have a very specific structure: they both contain all of the positive information about a set in their even part and all the negative information about the same set in their odd part. If we drop this requirement on the structure of the reduced set, we obtain a definition of the relation c.e. in. If we consider a more general form of this definition, by dropping the structural requirements on both the oracle and reduced set, we obtain a definition of enumeration reducibility. Definition 1. Let A and B be sets of natural numbers. A ≤e B if and only if there is a c.e. set W , such that A = W (B) = fx j 9u(hx; ui 2 W ^ Du ⊆ B)g. The set W will be called an enumeration operator. From this analysis we immediately obtain the following relationship: Proposition 1. Let A and B be sets of natural numbers. 1. A is c.e. in B if and only if A ≤e B ⊕ B. 2. A ≤T B if and only if A ⊕ A ≤e B ⊕ B. Enumeration reducibility carries its own structure. By identifying sets that are enumeration reducible to each other, i.e. enumeration equivalent, we obtain the set of enumeration degrees De. This is a partial order, where de(A) ≤e de(B) if and only if A ≤e B. The least element 0e consists of all computably enumerable sets. The way in which we obtained a least upper bound for Turing degrees, gives a least upper bound in the enumeration degrees, de(A ⊕ B) = de(A) _ de(B). Thus De = (De; ≤e; 0e; _) is as well an upper semi-lattice. Proposition 1 yields a standard embedding ι : DT ! De of the Turing degrees into the enumeration degrees, which preserves the order and the least upper bound. This embedding is defined by ι(dT (A)) = de(A ⊕ A). The image of the Turing degrees under this embedding is the set T OT of total enumera- tion degrees. The substructure T OT of the total enumeration degrees plays a central role in the study of the enumeration degrees. Selman [21] showed that enumeration reducibility can be characterized by the following: Theorem 1. Let A; B ⊆ !. A ≤e B if and only if fX j B is c.e. in Xg ⊆ fX j A is c.e. in Xg : Translated in terms of enumeration reducibility using the first part of Proposition 1, this means that every enumeration degree is entirely characterized by the set of total degrees above it. In particular the set of total enumeration degrees is an automorphism base for the structure of the enumeration degrees. We are still missing one ingredient for a complete analogy between the two structures: a jump operation, that agrees with the Turing jump under the stan- dard embedding. The candidate for an analog of the halting set, which first e comes to mind is KA = fhn; ei j n 2 We(A)g. A closer look at this set shows e that this first choice is not satisfactory, as KA is of the same enumeration degree as A. This is not too surprising, in the Turing case the reason that the halting set is not computable comes from the fact that the complement of the halting set diagonalizes against all possible computations. In contrast to the Turing degrees, the enumeration degrees are not closed under complement. This is why, to get a e jump operation, which actually jumps up, we need to use the complement of KA. 0 e e Thus we define A = KA ⊕ KA. We will call sets A, such that A ≡e A ⊕ A, total sets. Total sets are always members of total degrees. Examples of total sets are graphs of total functions, from where the term total originates, and as we just saw - jumps of sets, by definition. In the reverse direction we have the following jump inversion theorem by Soskov [28]. Theorem 2. For every enumeration degree x there exists a total enumeration 0 0 degree a, such that x ≤e a and x = a . It is not hard to see that this definition of the enumeration jump meets our requirements - it agrees with the Turing jump under the standard embedding ι. Thus the structure of the enumeration degrees provides a richer context in which we can study the structure of the Turing degrees. In this article we will argue that there are cases in which this approach is useful, shedding light on phenomena that can be observed, but not well explained by viewing the structure of the Turing degrees alone. 1 Computable model theory Our first example comes from a theorem by Coles, Downey and Slaman [3]. For every set of natural numbers A, consider the set C(A) = fX j A is c.e. in Xg. It follows from a result by Richter [19] that sets of this form do not always have a member of least Turing degree. Coles, Downey and Slaman show that if you instead look at C(A)0 = fX0 j A is c.e. in Xg, then this set always has a member of least Turing degree. Theorem 3. For every sets A the set: C(A)0 = fX0 j A is c.e. in Xg has a member of least degree. They call this degree the least jump enumeration of A. The proof constructs this least jump enumeration using forcing with finite conditions. The motivation for this result comes from computable model theory, and the notion of degree spectrum, used to characterize different structures. Fix a countable relational structure A = (N;R1 :::Rk). The degree spectrum of A, denoted by DST (A), is the set of Turing degrees of the diagrams of structures ∼ B = A. If DST (A) has a least member, it is the (Turing) degree of A. Sometimes the spectrum of a structure does not behave nicely, and it is useful and more informative to consider the jump spectrum of a structure. The jump spectrum 0 0 0 of A is DST (A) = fd j d 2 DST (A)g. If DST (A) has a least member, it is the (Turing) jump degree of A.
Load more

Recommended publications
  • Relationships Between Computability-Theoretic Properties of Problems
    RELATIONSHIPS BETWEEN COMPUTABILITY-THEORETIC PROPERTIES OF PROBLEMS ROD DOWNEY, NOAM GREENBERG, MATTHEW HARRISON-TRAINOR, LUDOVIC PATEY, AND DAN TURETSKY Abstract. A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solu- tion relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set A and a computable instance of a problem P, to find a solution relative to which A is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent 0 to admit avoidance of 1 cone, of ! cones, of 1 hyperimmunity or of 1 non-Σ1 definition. We also prove that the hierarchies of preservation of hyperimmunity 0 and non-Σ1 definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting. 1. Introduction In this article, we classify computability-theoretic preservation properties studied in reverse mathematics, namely cone avoidance, preservation of hyperimmunities, 0 preservation of non-Σ1 definitions, among others. Many of these preservation prop- erties have already been separated using natural problems in reverse mathematics { that is, there is a natural problem which is known to admit preservation of one property but not preservation of the other.
    [Show full text]
  • Co~Nparison of Polynomial- Time Reducibilities Richard Ladner University of Washington Seattle, Wash. Nanay Lynch University Of
    Co~nparison of Polynomial- Time Reducibilities interesting relationships. Richard Ladner University of Washington I. Introduction Seattle, Wash. Computation-resource-bounded reducibil- Nanay Lynch University of Southern California ities play a role in the theory of computational Los Angeles, Cal. complexity which is analogous to, and perhaps AlaD Selman Florida State University as important as, the various kinds of effective Tallahas see, Fla. reducibilities used in recursive function theory. Abstract Just as the effective reducibilities are used to Comparison of the polynomial-time-bounded classify problems according to their degrees reducibilities introduced by Cook [1] and of unsolvability, [9] space- and time- bou6ded t<arp [4] leads naturally to the definition of reducibilities may be used to classify problems several intermediate truth-table reducibilities. according to their complexity level. We give definitions and comparisons for these Tne most fruitful resource-bounded reducibilities ; we note, in particular, that reducibilities thus far have been the polynomial- all reducibilities of this type which do not time-bounded reducibilities of Cook [I] and have obvious implication relationships are I<arp [4], corresponding respectively to in fact distinct in a strong sense. Proofs Turing and many-one reducibilities in recursive are by simultaneous diagonalization and function theory. Other resource-bounded encoding constructions. reducibilities have been defined and used as well Work of Meyer and Stockmeyer [7] and [3] [5] [6] [8]; they differ from Cook's or Karp's Gill [Z] then leads u.,~ to define nondeterministic only in the bound on time or space allowed versions of all of our reducibilities. Although for the reduction, and thus they also correspond many of the definitions degenerate, comparison to Turing or many-one reducibility.
    [Show full text]
  • {\Alpha} Degrees As an Automorphism Base for the {\Alpha}-Enumeration
    α degrees as an automorphism base for the α-enumeration degrees Dávid Natingga February 13, 2019 Abstract Selman’s Theorem [10] in classical Computability Theory gives a characterization of the enumeration re- ducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: A ≤e B ⇐⇒ ∀X[X ≡e X ⊕ X ∧ B ≤e X ⊕ X =⇒ A ≤e X ⊕ .X]. This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the α-Computability Theory to obtain the full generalization when α is a regular cardinal and partial results for a general admissible ordinal α. 1 α-Computability Theory α-Computability Theory is the study of the definability theory over Gödel’s Lα where α is an admissible ordinal. One can think of equivalent definitions on Turing machines with a transfinite tape and time [3] [4] [5] [6] or on generalized register machines [7]. Recommended references for this section are [9], [1], [8] and [2]. Classical Computability Theory is α-Computability Theory where α = ω. 1.1 Gödel’s Constructible Universe Definition 1.1. (Gödel’s Constructible Universe) Define Gödel’s constructible universe as L := Sβ∈Ord Lβ where γ,δ ∈ Ord, δ is a limit ordinal and: L0 := ∅, arXiv:1812.11308v2 [math.LO] 12 Feb 2019 Lγ+1 := Def(Lγ ) := {x|x ⊆ Lγ and x is first-order definable over Lγ}, Lδ = Sγ<δ Lγ. 1.2 Admissibility Definition 1.2. (Admissible ordinal[1]) An ordinal α is Σ1 admissible (admissible for short) iff α is a limit ordinal and Lα satisfies Σ1-collection: ∀φ(x, y) ∈ Σ1(Lα).Lα |= ∀u[∀x ∈ u∃y.φ(x, y) =⇒ ∃z∀x ∈ u∃y ∈ z.φ(x, y)] where Lα is the α-th level of the Gödel’s Constructible Hierarchy (definition 1.1).
    [Show full text]
  • Symmetric Enumeration Reducibility
    SYMMETRIC ENUMERATION REDUCIBILITY CHARLES M. HARRIS DEPARTMENT OF MATHEMATICS, LEEDS, UK. Abstract. Symmetric Enumeration reducibility ( ≤se ) is a subrelation of Enu- meration reducibility (≤e) in which both the positive and negative informa- tion content of sets is compared. In contrast with Turing reducibility (≤T) however, the positive and negative parts of this relation are separate. A basic classification of ≤se in terms of standard reducibilities is carried out and it is shown that the natural embedding of the Turing degrees into the Enumera- tion degrees easily translates to this context. A generalisation of the relativised Arithmetical Hierarchy is achieved by replacing the relation c.e. in by ≤e and ≤T by ≤se in the underlying framework of the latter. 1. Introduction The structure of the relativised Arithmetical Hierarchy is dictated by the relation computably enumerable in (c.e. in) in the sense that a set belongs to level n + 1 of the hierarchy if either it or its complement is c.e. in some set belonging to level n. This of course means that each level is the union of a positive (Σ) class and a negative (Π) class. The intersection (∆) of these classes comprises, by definition, those sets B such that B is Turing reducible to (i.e. B and B are both c.e. in) a set in the antecedent level. A natural question to ask is whether one can refine the framework of the rela- tivised Arithmetical Hierarchy to the context of Enumeration Reducibility. In other words, can one define a similar hierarchy in which the levels are dictated by the relation enumeration reducible to (≤e) and which is identical to the Arithmetical Hierarchy when relativised to graphs of characteristic functions (i.e.
    [Show full text]
  • Enumeration Order Equivalency
    Enumeration Order Equivalency Ali Akbar Safilian (AmirKabir University of Technology, Tehran, Iran [email protected] ) Farzad Didehvar (AmirKabir University of Technology, Tehran, Iran [email protected] ) Let ̻ ɛ ͈ be an infinite recursively enumerable set. There are some total computable functions ͜: ͈ Ŵ ̻ connected with a Turing machine such that ̻ Ɣ ʜ͜ʚ1ʛ, ͜ʚ2ʛ,… ʝ. Definition 2.1 (Listing) A listing of an infinite r.e. set ̻ ɛ ͈ is a bijective computable function ͚: ͈ Ŵ ̻. As mentioned above we can connect every Turing machine with a listing uniquely. Namely, each listing shows the enumeration order of the elements enumerated by the related Turing machine. In the following, we define a reduction on listings and sets based on enumeration orders. Definition 2.2 (Enumeration Order Reducibility on Listings and sets) 1. For listings ͚, ͛: ͈ Ŵ ͈ we say ͚ is “Enumeration order reducible” to ͛ and write ͚ ƙ * ͛, if and only if, Ǣ͝ Ɨ ͞, (͚ʚ͝ʛ Ƙ ͚ʚ͞ʛ ƴ ͛ʚ͝ʛ Ƙ ͛ʚ͞ʛʛ. 2. For r.e. sets ̻, ̼ ɛ ͈, we say ̻ is “Enumeration order reducible” to ̼ and write ̻ ƙ * ̼, if and only if, for any listing ͛ of ̼ there exist some computable function ̀ (from the listings of ̼ to the listings of ̻) such that ̀ ʚ͛ʛ ƙ * ͛. 3. For two listings ͚, ͛ (r.e. sets ̻, ̼), we say ͚ is “Enumeration order equivalent” to ͛ and write ͚ ȸ * ͛ (̻ ȸ * ̼) ,if and only if, both ͚ ƙ * ͛ and ͛ ƙ * ͚ (̻ ƙ * ̼ and ̼ ƙ * ̻) Let ̻ be an r.e.
    [Show full text]
  • On Sets Polynomially Enumerable by Iteration*
    Theoretical Computer Science 80 ( 1991 ) 2(13-225 203 Elsevier On sets polynomially enumerable by iteration* Lane A. Hemachandra** Department oJ Computer Science. Univer.~iO. q/' Roche~ter. Rochester. N Y 14627. U.S.A. Albrecht Hoene and Dirk Siefkes Fachhereich Inlormalik. Technische Unit,er.sitiit Berlin. D.1000 Berlin I0. Fed. Rep. Germany Paul Young*** Department of Computer Science and Engineering, FR-35, Unirer~ity O! Washington, Seattle, WA 98195, U.S.A. Abxtracl Hemachandra, L.A.. A. Hoene, D. Siefkes and P. Young. On sets polynomially enumerable by iteration, Theoretical Computer Science 80 ( 1991 ) 203-225. Sets whose members are enumerated by some Turin$ machine are called recursively enumerable. We define a set to be po(rnomially enumerahle hy iteration if its members are e~cientlr enumerated by iterated application of some Turin8 machine. We prove that many complex sets--including all exponential-time complete s¢:.,, all NP-complete sets yet obtained by direct construction, and the complements of all such sets--are polynomially enumen'able by iteration. These results follow from more general results. In fact. we show that all recursi~,ely enumerabl¢ sets that are ['.,,-sel.freducihle are polynomially enumerable by iteration, and that all recursive sets that are ~'.,,-wl[reducihle i.ire bi-enumerable. We also ,how thai when Ihe ~ ~'.,-self-reduction is via a function whose inverse is computable in polynomial time, then the above results hold with the polynomial enumeration gi,,en by a function whose inverse is computable in pol.~nomial time. In the final section of the paper we show that ato NP.complete set can be iteratively enumerated in lexicographically increasing order unless the polynomial time hierarchy collapses to N P.
    [Show full text]
  • Some Results on Enumeration Reductibility
    SOME RESULTS ENUMERATION REDUCIBILITY by LANCE GUTTERIDGE B.Sc., University of British Columbia, 1967 M.Sc., University of British Columbia, 1969 A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREZWNTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics LANCE GUTTERIDGE 1971 SIMON FR9SER 'VNIVERS ITY AUGUST 1971 APPROVAL Name : Lance Gutteridge Degree : Doctor of Philosophy Title of Dissertation: Some results on enumeration reducibility 1 Examining Committee: 7 , --L-- - I ' r , Chairman: N. R. Reilly ,' A. H. Lachlan Senior Supervisor A. R. Freedman - .- - G. E. Sacks External Examiner, Professor, Massachusetts Institute of Technology Cambridge, Mass. Date ~pproved: September 24, 1971 ABSTRACT Enumeration reducibility was defined by Friedberg and Rogers in 1959. Medvedev showed that there are partial degrees which are not total. Rogers in his book Theory of Recursive Functions and Effective Computability gives all the basic results and definitions concerning enumeration reducibility and the partial degrees. He mentions in this book that the existence of a minimal partial degree is an open problem. In this thesis it is shown that there are no minimal partial degrees. This leads naturally to the conjecture that the partial degrees are dense. This thesis leaves this question unanswered, but it is shown that there are no degrees minimal above a total degree, and there are at most countably many degrees minimal above a non-total degree. J. W. Case has proved several results about the partial degrees. He conjectured that there is no set in a total degree whose complement is in a non-total degree. In this thesis that conjecture is disproved.
    [Show full text]
  • Introduction to Enumeration Reducibility
    Introduction to Enumeration Reducibility Charles M. Harris Department Of Mathematics University of Leeds Computability Theory Seminar, Autumn 2009 (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 1 / 34 Outline 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 2 / 34 Outline 3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator 4 Embedding DT into De Total Degrees Definition of the Embedding Quasi-minimal Enumeration Degrees (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 3 / 34 Outline 5 Further Properties of De Exact Pairs and Branching Density and Minimality Next Week (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 4 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 5 / 34 What is enumeration reducibility? Definition (Intuitive) A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A. Definition (Formal) A≤e B if there exists a c.e. set W such that for all x 2 ! x 2 A iff 9u [ hx; ui 2 W & Du ⊆ B ] This is written A = ΦW (B). (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 6 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 7 / 34 Examples of sets W and X such that x 2 A iff 9D ( hx; Di 2 W & D ⊆ X ) A≤e A via the c.e.
    [Show full text]
  • Symmetric Enumeration Reducibility
    Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Symmetric Enumeration Reducibility Charles M. Harris Department Of Mathematics University of Leeds Computability Theory Seminar, Autumn 2005 Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 3 DT and Dse Characteristic Degrees Definition of the Embedding Quasi-minimal Degrees 4 Jump Operators The Enumeration Jump The Symmetric Enumeration Jump Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 3 DT and Dse Characteristic Degrees Definition of the Embedding Quasi-minimal Degrees 4 Jump Operators The Enumeration Jump The Symmetric Enumeration Jump Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 5 Some Properties of Dse The c.e. and co-c.e. degrees Minimal and Exact Pairs Next Week Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Subsection Guide 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is enumeration reducibility? Definition (Intuitive) A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A.
    [Show full text]
  • Enumeration Degrees
    Decidability and Definability in the 0 Σ2-Enumeration Degrees By Thomas F. Kent A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the UNIVERSITY OF WISCONSIN – MADISON 2005 i Abstract Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider con- text in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the fact that there are several applications of enumeration reducibility in computable mathematics, until recently relatively little research had been done in this area. In Chapter 2 of my thesis, I show that the ∀∃∀-fragment of the first order 0 theory of the Σ2-enumeration degrees is undecidable. I then show how this result actually demonstrates that the ∀∃∀-theory of any substructure of the enumeration 0 degrees which contains the ∆2-degrees is undecidable. In Chapter 3, I present current research that Andrea Sorbi and I are engaged in, in regards to classifying 0 properties of non-splitting Σ2-degrees. In particular I give proofs that there is a 0 0 properly Σ2-enumeration degree and that every ∆2-enumeration degree bounds a 0 non-splitting ∆2-degree. Advisor: Prof. Steffen Lempp ii Acknowledgements I am grateful to Steffen Lempp, my thesis advisor, for all the time, effort, and patience that he put in on my behalf. His insight and suggestions have been of great worth to me, both in and out of my research.
    [Show full text]
  • A. Definitions of Reductions and Complexity Classes, and Notation List
    A. Definitions of Reductions and Complexity Classes, and N otation List A.I Reductions Below, in naming the reductions, "polynomial-time" is implicit unless ot her­ wise noted, e.g., "many-one reduction" means "many-one polynomial-time reduction." This list sorts among t he :Sb first by b, and in the case of identical b's, bya. :S -y - Gamma reduction. -A :S -y B if there is a nondeterministic polynomial-time machine N such that (a) for each st ring x, N(x) has at least one accepting path, and (b) for each string x and each accepting path p of N(x) it holds that: x E A if and only if the output of N(x) on path p is an element of B. :S ~ - Many-one reduction. -A :S~ B if (3f E FP)(Vx)[x E A {=} f(x) EB] . :S~ o s - (Globally) Positive Turing reduction. -A :S~o s B if there is a deterministic Turing machine Mand a polyno­ mial q such that 1. (VD)[runtimeMD(x):s q(lxl)], B 2. A = L(M ), and 3. (VC, D)[C ç D =} L(MC) ç L(MD)]. :S:08 - Locally positive Turing reduction. -A :S:os B if there is a deterministic polynomial-time Turing machine M such that 1. A = L(MB), 2. (VC)[L(M BUC) :2 L(MB)] , and 3. (VC)[L(M B- C) ç L(MB)] . :ST - Exponential-time Turing reduction. -A :ST B if A E EB. If A E EB via an exponential-time machine, M, having the property that on each input of length n it holds that M makes at most g(n) queries to its orade, we write A :S~ ( n ) - T B.
    [Show full text]
  • Enumeration Reducibility and Computable Structure Theory
    ENUMERATION REDUCIBILITY AND COMPUTABLE STRUCTURE THEORY ALEXANDRA A. SOSKOVA AND MARIYA I. SOSKOVA 1. Introduction In classical computability theory the main underlying structure is that of the natural numbers or equivalently a structure consisting of some constructive objects, such as words in a finite alphabet. In the 1960's computability theorists saw it as a challenge to extend the notion of computable to arbitrary structure. The resulting subfield of computability theory is commonly referred to as computability on abstract structures. One approach towards this is the theory of computability in admissible sets of the hereditarily finite superstructure HF(A) over a structure A. The development of computability on ordinals was initiated by Kreisel and Sacks [43, 42], who investigated computability notions on the first incomputable ordinal, and then further developed by Kripke and Platek [44, 58] on arbitrary admissible ordinals and by Barwise [6], who considered admissible sets with urelements. The notion of Σ-definability on HF(A), introduced and studied by Ershov [16, 17] and his students Goncharov, Morozov, Puzarenko, Stukachev, Korovina, etc., is a model of nondeterministic computability on A. A survey of results on HF-computability and on abstract computability based on the notion of Σ-definability can be found in [18, 95]. Montague [53] took a model theoretic approach to generalized computability theory, considering computability as definability in higher order logics. The approach towards abstract computability that ultimately lead to the results discussed in this article starts with searching for ways in which one can identify abstract computability on a structure internally. Let A be an arbitrary abstract structure.
    [Show full text]