13 results
Recursion theoretic papers. Introduction to Part VI
- from PART VI - RECURSION THEORY
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- By Leo A. Harrington, University of California, Berkeley, Theodore A. Slaman, University of California, Berkeley
- Edited by Alexander S. Kechris, California Institute of Technology, Benedikt Löwe, Universiteit van Amsterdam, John R. Steel, University of California, Berkeley
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- Ordinal Definability and Recursion Theory
- Published online:
- 05 December 2015
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- 11 January 2016, pp 349-354
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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The Complexity of Orbits of Computably Enumerable Sets
- Peter A. Cholak, Rodney Downey, Leo A. Harrington
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- Journal:
- Bulletin of Symbolic Logic / Volume 14 / Issue 1 / March 2008
- Published online by Cambridge University Press:
- 15 January 2014, pp. 69-87
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- March 2008
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The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).
Isomorphisms of splits of computably enumerable sets
- Peter A. Cholak, Leo A. Harrington
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- The Journal of Symbolic Logic / Volume 68 / Issue 3 / September 2003
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1044-1064
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- September 2003
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We show that if A and are automorphic via Φ then the structures (A) and () are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.
Definable Encodings in the Computably Enumerable Sets
- Peter A. Cholak, Leo A. Harrington
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- Journal:
- Bulletin of Symbolic Logic / Volume 6 / Issue 2 / June 2000
- Published online by Cambridge University Press:
- 15 January 2014, pp. 185-196
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- June 2000
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The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.
We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called ε.
All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.
We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an infinite non-computable subset). We use A* to denote the equivalent class of A under the ideal of finite sets.
Codable sets and orbits of computably enumerable sets
- Leo Harrington, Robert I. Soare
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- Journal:
- The Journal of Symbolic Logic / Volume 63 / Issue 1 / March 1998
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1-28
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- March 1998
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A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, ε = ({We}eϵω,⊆). We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets).
Here we show first that Q(X) implies that X has a certain “slowness” property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ϵ ε there exists B in the orbit of A such that X ≤TB under relative Turing computability (≤T). We produce B using the -automorphism method we introduced earlier.
Definability, Automorphisms, and Dynamic Properties of Computably Enumerable Sets
- Leo Harrington, Robert I. Soare
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- Journal:
- Bulletin of Symbolic Logic / Volume 2 / Issue 2 / June 1996
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- 15 January 2014, pp. 199-213
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- June 1996
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We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).
Dynamic properties of computably enumerable sets
- Edited by S. B. Cooper, University of Leeds, T. A. Slaman, University of Chicago, S. S. Wainer, University of Leeds
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- Computability, Enumerability, Unsolvability
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- 23 February 2010
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- 11 January 1996, pp 105-122
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Summary
Abstract
A set A ⊆ ω is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let ε denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the definable (especially ε-definable) properties of a c.e. set A to its “information content”, namely its Turing degree, deg(A), under ≤T, the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly ε-definable property Q(A) which guarantees that A is incomplete (A <TK). The property Q(A) is of the form (∃C)[A ⊂mC & Q−(A, C)], where A ⊂mC abbreviates that “A is a major subset of C”, and Q−(A,C) contains the main ingredient for incompleteness.
A dynamic property P(A), such as prompt simplicity, is one which is defined by considering how fast elements elements enter A relative to some simultaneous enumeration of all c.e. sets. If some set in deg(A) is promptly simple then A is prompt and otherwise tardy. We introduce here two new tardiness notions, small-tardy (A, C) and Q-tardy(A, C). We begin by proving that small-tardy(A, C) holds iff A is small in C (A ⊂sC) as defined by Lachlan [1968]. Our main result is that Q-tardy(A, C) holds iff Q−(A,C). Therefore, the dynamic property, Q-tardy(A, C), which is more intuitive and easier to work with than the ε-definable counterpart, Q−(A,C), is exactly equivalent and captures the same incompleteness phenomenon.
Analytic determinacy and 0#
- Leo Harrington
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- Journal:
- The Journal of Symbolic Logic / Volume 43 / Issue 4 / December 1978
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- 12 March 2014, pp. 685-693
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- December 1978
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Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:
Theorem. If analytic games are determined, then x2 exists for all reals x.
This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π11-determinacy (where α − Π11 is the αth level of the difference hierarchy based on − Π11 see [1]). Martin has also shown that the existence of sharps implies < ω2 − Π11-determinacy.
Our method also produces the following:
Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x.
The converse to this theorem had been previously proven by Steel [7], [18].
We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.
For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16].
Throughout this paper we will concern ourselves only with methods for obtaining 0# (rather than x# for all reals x). By relativizing our arguments to each real x, one can produce x2.
Meeting of the Association for Symbolic Logic, Reno, 1976
- Solomon Feferman, Jon Barwise, Leo Harrington
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- Journal:
- The Journal of Symbolic Logic / Volume 42 / Issue 1 / March 1977
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- 12 March 2014, pp. 156-160
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- March 1977
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On Σ1 well-orderings of the universe
- Leo Harrington, Thomas Jech
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- The Journal of Symbolic Logic / Volume 41 / Issue 1 / March 1976
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- 12 March 2014, pp. 167-170
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- March 1976
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The constructible universe L of Gödel [2] has a natural well-ordering < L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.
Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.
The main result of this paper is
Theorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.
The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.
Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.
On characterizing Spector classes
- Leo A. Harrington, Alexander. S. Kechris
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- Journal:
- The Journal of Symbolic Logic / Volume 40 / Issue 1 / March 1975
- Published online by Cambridge University Press:
- 12 March 2014, pp. 19-24
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- March 1975
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We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first essentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise.
The second part of our paper is concerned with finding structural characterizations of classes of relations on the reals in the spirit of Moschovakis [7]. Our main result provides a single abstract characterization for the class of relations on the reals and the 2-envelope of 3E, the first one being valid if projective determinacy is true, the second if V = L is true.
Recursively presentable prime models
- Leo Harrington
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- Journal:
- The Journal of Symbolic Logic / Volume 39 / Issue 2 / June 1974
- Published online by Cambridge University Press:
- 12 March 2014, pp. 305-309
- Print publication:
- June 1974
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It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.
The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.
If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.