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1. Local Definitions In Degree Structures The Turing Jump
Local definitions in degree structures the Turing jump, hyperdegrees and beyond. Richard A. Shore. Source Bull. Symbolic Logic Volume 13, Issue 2 (2007),
http://projecteuclid.org/handle/euclid.bsl/1185803806
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##### Local definitions in degree structures: the Turing jump, hyperdegrees and beyond
Richard A. Shore Source: Bull. Symbolic Logic Volume 13, Issue 2 (2007), 226-239.
##### Bulletin of Symbolic Logic

 2. JSTOR Defining The Turing Jump The study of definability in degree structures, and in the lattice of r.e. sets, has long been a central topic in computability theory. For the Turinghttp://links.jstor.org/sici?sici=1079-8986(200103)7:1<73:DTTJ>2.0.CO;2-C

3. Turing Degrees Of Isomorphism Types Of Algebraic Objects -- Calvert Et Al. 75 (2
The Turing degree spectrum of a countable structure A We show that there are structures with isomorphism types of arbitrary Turing degrees in each of
http://jlms.oxfordjournals.org/cgi/content/abstract/75/2/273
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Journal of the London Mathematical Society Advance Access originally published online on February 2, 2007
Journal of the London Mathematical Society 2007 75(2):273-286; doi:10.1112/jlms/jdl012
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##### Turing degrees of isomorphism types of algebraic objects
Wesley Calvert Valentina Harizanov and Alexandra Shlapentokh Wesley Calvert Department of Mathematics and Statistics
Murray State University Murray
KY 42071
USA
Valentina Harizanov Department of Mathematics
George Washington University Washington DC 20052 USA Alexandra Shlapentokh Department of Mathematics East Carolina University Greenville NC 27858 USA The Turing degree spectrum of a countable structure is the set of all Turing degrees of isomorphic copies of . The Turing degree of the isomorphism type of , is the least Turing degree in its degree spectrum. We show that there are structures with

4. Research And Publications
Abstract Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that
http://qcpages.qc.cuny.edu/~rmiller/research.html
##### Research Interests
I study computability theory, the branch of mathematical logic concerned with finite algorithms and the mathematical problems which such algorithms can or cannot solve. By relativizing, one forms a partial order of the degrees of difficulty (the Turing degrees ) of such problems. Computable model theory, my specialty, applies such techniques to general mathematical structures such as fields, trees, linear orders, groups and graphs. At present my investigations focus particularly on fields, and on the possibility of extending results from them to differential fields. Traditional computable model theory only considers countable structures, but I am currently examining different ways of extending the notions of computable model theory to certain uncountable structures, either by examining the structures locally rather than globally, or by extending the notion of computability to ordinal time to allow computation of functions with larger domains.
##### Publications
Journal of Symbolic Logic
Abstract: Slaman and Wehner have constructed structures which distinguish the computable Turing degree from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable Delta^0_2 degree, but not

 5. Undecidability Results For Low Complexity Degree Structures Undecidability Results For Low Complexity degree structures degrees with respect to polynomial time Turing and manyone reducibility is undecidable.http://portal.acm.org/citation.cfm?id=792314

6. 1. Computability And Randomness Higher Randomness Notions And
Global properties of degree structures. In the Scope of Logic, The Pi3theory of the enumerable Turing degrees is undecidable (with S. Lempp and T.
http://www.cs.auckland.ac.nz/~nies/onlinepapers.html
 1. Computability and Randomness Higher randomness notions and their lowness properties (with Chitat Chong and Liang Yu). Israel J. Math, To appear Eliminating concepts. To appear in Proc. of the IMS workshop on computational aspects of infinity, Singapore. Lowness for Computable Machines (with Rod Downey, Noam Greenberg and Nenad Mikhailovich). To appear in Proc. of the IMS workshop on computational aspects of infinity, Singapore. A lower cone in the wtt degrees of non-integral effective dimension (with Jan Reimann). To appear in Proc. of the IMS workshop on computational aspects of infinity, Singapore. Non-cupping and randomness. Proc. Amer. Math. Soc. 135 (2007), no. 3, 837844. (with Rod Downey, Rebecca Weber, and Liang Yu). Journal of Symbolic logic 71( 3), 2006, pp. 1044-1052. Randomness via effective descriptive set theory (with Greg Hjorth). J. London Math Soc Nies 75 (2): 495-508. Calibrating randomness. Bull. Symb. Logic. 12 no 3 (2006) 411-491 (with Downey, Hirschfeldt and Terwijn). Randomness and computability: Open questions (with Joe Miller). Bull. Symb. Logic. 12 no 3 (2006) 390-410.

algebraic structures of, and decidability of fragments of theories of, degree structures (esp. the c.e. Turing degrees, the d.c.e. and nc.e. Turing degrees
http://www.math.wisc.edu/~lempp/main.html
##### STEFFEN LEMPP
Department of Mathematics
University of Wisconsin

480 Lincoln Drive

...
USA
##### Phone/Fax/Email
Office: +1-608-263-1975
Department: +1-608-263-3054
Fax: +1-608-263-8891
Email: lempp@math.wisc.edu
##### Office and office hours during semester
Office: 517 Van Vleck Hall
Office hours: Mondays, Wednesdays and Fridays, 2:25-3:15 p.m., or email me ( lempp@math.wisc.edu ) to arrange another time.
##### Conferences I plan to attend (and other times I will be out of town)
• May 19-23, 2008:

 8. EULER Record Details Subject, Turing degree structures. MSC, 03D28. Type, Text.Article. Record Source, Zentralblatt MATH 0958.03029. Document Delivery, Online Ordering viahttp://www.emis.de/projects/EULER/detail?ide=1999shordefiturijump&matchno=5&matc

9. Conference On Logic, Computability And Randomness
This talk will primarily be an exposition of the general approaches to proving that the theories of various Turing degree structures are undecidable or even
http://www.dc.uba.ar/people/logic2007/
 Conference on Logic, Computability and Randomness January 10-13, 2007 Buenos Aires, Argentina program committee submission local organizers plenary speakers ... conference address The theme of the conference will be algorithmic randomness and related topics in logic, computability and complexity. The program will consist of invited talks contributed talks 3 introductory courses. NEW! Check here The booklet with the abstracts of invited and contributed talks is available here See the poster of the conference in jpg or ... YPF Foundation , located in the biggest park in Buenos Aires "los bosques de Palermo" (very near the University campus, Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires The meeting is sponsored by the Association for Symbolic Logic There is no registration fee. Student members of the ASL can apply for travel grants (the approval process takes a few weeks). The spirit of the Conference will be the same as earlier meeting in CÃ³rdoba, 2004. The booklet with the abstracts of invited and contributed talks of the earlier meeting is available here If you wish to attend to the conference (and you are not giving a talk) please let us know in advance.

10. 03Dxx
03D28, Other Turing degree structures. 03D30, Other degrees and reducibilities. 03D35, Undecidability and degrees of sets of sentences
http://www.impan.gov.pl/MSC2000/03Dxx.html

 11. MSC 2000 : CC = Other 03D28 Other Turing degree structures Nouveau code MSC 2000 17C50; 17C50 Jordan structures associated with other structures See also 16W10http://www-mathdoc.ujf-grenoble.fr/cgi-bin/msc2000.py?CC=other&L=fr

12. List KWIC DDC22 510 And MSC+ZDM E-N Lexical Connection
turbulent flows control of 76F70 turbulent transport, mixing 76F25 Turing degree structures other 03D28 Turing machines and related notions 03D10
http://www.math.unipd.it/~biblio/kwic/msc-cdd/dml2_11_62.htm
 transitive finite groups # multiply transitive infinite groups # multiply translation planes and spreads translations # miscellaneous volumes of translations # volumes of selected translations of classics # collected or selected works; reprintings or transonic flows transport # nuclear reactor theory; neutron transport processes transport, mixing # turbulent transportation, logistics transportation, multi-index, etc.) # special problems of linear programming ( transversal (matching) theory transversal theory # Helly-type theorems and geometric transversality # general position and transversality # general position and treatment # loop groups and related constructions, group-theoretic treatment # material properties given special treatment of dynamical systems # approximation methods and numerical treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) # synthetic treatment) # other social and behavioral sciences (mathematical trees trees trees # groups acting on trees and graphs # maps of trellis codes) # combined modulation schemes (including triad), algebras for a triple, homology and derived functors for triples # triples (=standard construction, monad or

13. Program On Computation Prospects Of Infinity - IMS
We prove that the degree structures of the d.c.e. and the 3c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey.
http://www.ims.nus.edu.sg/Programs/infinity/abstracts.htm
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##### ~ Abstracts ~
Refuting Downey's conjecture
Steffen Lempp University of Wisconsin, Madison, USA Â« B ack... Seetapun's theorem and related conjectures on the strength of stable Ramsey's theorem for pairs
Carl Jockusch, University of Illinois, Urbana-Champaign, USA Â« B ack... Weak degrees of Pi^0_1 subsets of 2^omega
Stephen G. Simpson, Pennsylvania State University, USA Let P,Q subseteq 2^omega be viewed as mass problems, i.e., "decision problems with more than one solution." We say that the mass problem P is weakly reducible to the mass problem Q if, for every solution Y of Q, there exists a solution X of P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. (Weak degrees are also known as Muchnik degrees.) Let P_w be the set of weak degrees of nonempty Pi^0_1 subsets of 2^omega, partially ordered by weak reducibility. It is easy to see that P_w is a countable distributive lattice. The speaker and others have studied P_w in a series of publications beginning in 1999. Our principal findings are as follows. 1. There is a natural embedding of R_T, the countable semilattice of recursively enumerable Turing degrees, into P_w. This embedding is one-to-one and preserves the partial ordering leq, the semilattice operation v, and the top and bottom elements and 0'. We identify R_T with its image in P_w under this embedding.

14. List KWIC DDC And MSC Lexical Connection
structures ordered 06Fxx structures other finite incidence 51E30 structures other Turing degree 03D28 structures probability theory on algebraic and
http://www.mi.imati.cnr.it/~alberto/dml_11_50.htm
 structure theorems # general structure theory structure theory structure theory structure theory structure theory structure theory structure theory structure theory structure theory # characterization and structure theory # characterization and structure theory # general structure, canonical formalism, Cauchy problems) # Einstein's equations (general structure, classification of commutative topological algebras structure, classification of topological algebras structure, classification theorems structure; Euclidean algorithm; greatest common divisors # multiplicative structured objects in a category (group objects, etc.) structured programming structured surfaces and interfaces, coexistent phases structures # algebraic structures # associated Lie structures # asymptotic results on counting functions for algebraic and topological structures # basic properties of first-order languages and structures # combinatorial complexity of geometric structures # data structures # data structures # deformations of structures # deformations of analytic structures # deformations of complex structures # deformations of special (e.g. CR)

15. B.Sc. Honours - Philosophy - School Of Philosophy And Religious Studies - Univer
Not long after that, the work of Turing and others opened up logical and best be served by the existing B.Sc., B.A., or B.A.(Hons) degree structures.
http://www.phil.canterbury.ac.nz/degrees/phil_bschons.shtml
UC Home Courses Departments Library ... Philosophy and Religious Studies Philosophy and Religious Studies
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Entire University
Philosophy and Religious Studies
##### Contact
sue.howie
@canterbury.ac.nz
Philosophy and Religious Studies
University of Canterbury
Private Bag 4800
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##### The degree
Entry to a B.Sc.(Hons) degree program is at 300 level.
##### Prerequisites
The prerequisites for entry at 300 level to B.Sc.(Hons) are:
• MATH 211 and 212 and either MATH 204 or MATH 217 and MATH 218; and 22 points in Philosophy at level and either PHIL 134 (or MATH 134) or PHIL 246 or PHIL 247.
• ##### Requirements
The requirements at 300 level are:
• 84 points in Mathematics at level, normally including MATH 321, MATH 335, MATH 341, MATH 343; and
• 16. Recent PhD-Theses In Logic
Continuity in degree structures. University of Leeds, 2001. Supervisor S. Barry Cooper. Splittings and Nonsplittings in the Turing degrees.
http://www.cs.bham.ac.uk/~exr/blc/PhDThesis.html
##### Recent PhD-Theses in Logic
• Ruth Hardy, Formal Methods for Control Engineering: A validated decision procedure for Nichols plot analysis, St Andrews University, 2006. Supervisor Roy Dyckhoff. Abstract.
• Mark Jago, Logics for Resource Bounded Agents. University of Nottingham, 2006. Supervisors: Natasha Alechina, Brian Logan, Eros Corazza. Abstract.
• Abstract.
• Dmitry Shkatov, Modal Logics with Existential Modality, Finite-iteration Modality, and Intuitionistic Base: Decidability and Completeness. University of Nottingham, 2005. Supervisor: Natasha Alechina. Abstract.
• David Gabelaia, Topological Semantics and Two-Dimensional Combinations of Modal Logics, King's College London, 2005. Supervisor: Michael Zakharyaschev. Abstract.
• Dickon Lush: Finitary Geometry in Minimal Parts Graphs. University of Bristol, 2004. Supervisor: John Mayberry. Abstract.
• Katie Chicot: Transitivity properties of countable trees. University of Leeds 2004. Supervisor: J.K.Truss
• Roman Kontchakov, Monodic First-Order Temporal Logics: Complexity, Tableaux, and Applications, King's College London, 2004. Supervisor: Michael Zakharyaschev Abstract.

17. Turing Degree - Wikipedia, The Free Encyclopedia
5 Structure of the r.e. Turing degrees; 6 Post s problem and the A great deal of research has been conducted into the structure of the Turing degrees.
http://en.wikipedia.org/wiki/Turing_degree
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##### Turing degree
Jump to: navigation search "Post's problem" redirects here. For the other "Post's problem", see Post's correspondence problem In computer science and mathematical logic , the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. The concept of Turing degree is fundamental in computability theory , where sets of natural numbers are often regarded as decision problems ; the Turing degree of a set tells how difficult it is to solve the decision problem associated with the set. Two sets are Turing equivalent if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent. Furthermore, the Turing degrees are partially ordered so that if the Turing degree of a set X is less than the Turing degree of a set Y then any (noncomputable) procedure that correctly decides whether numbers are in Y can be effectively converted to a procedure that correctly decides whether numbers are in X . It is in this sense that the Turing degree of a set corresponds to its level of algorithmic unsolvability. The Turing degrees were introduced by Stephen Cole Kleene and Emil Leon Post in the 1940s and have been an area of intense research since then.

18. [math/0507128] Turing Degrees Of Isomorphism Types Of Algebraic Objects
The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$.
http://arxiv.org/abs/math/0507128
##### arXiv.orgmath
Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
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##### Title: Turing Degrees of Isomorphism Types of Algebraic Objects
Authors: Wesley Calvert Valentina Harizanov Alexandra Shlapentokh (Submitted on 6 Jul 2005) Abstract: Comments: 17 pages Subjects: Logic (math.LO) ; Commutative Algebra (math.AC); Number Theory (math.NT) Cite as: arXiv:math/0507128v1 [math.LO]
##### Submission history
From: Wesley Calvert [ view email
Wed, 6 Jul 2005 19:26:45 GMT (49kb)
Which authors of this paper are endorsers?
Link back to: arXiv form interface contact

19. Sarah Pingrey | George Washington University
Let A be a computable structure and R be an additional relation on A. The Turing degree spectrum of R on A is the set of all Turing degrees of the images of
http://home.gwu.edu/~spingrey/
 Sarah Pingrey Mathematics Department Monroe 240 2115 G Street, NW Washington, DC 20052 Telephone: 202-994-3983 FAX: 202-994-6760 Email: spingrey@gwu.edu Curriculum Vitae Advisor: Valentina S. Harizanov Office Hours: Mondays 9:15-10:30am, and Tuesdays 9:15-11:00am. Office: Hall of Government 226 Job Dossier Research Partial automorphism semigroups Annals of Pure and Applied Logic Journal of Symbolic Logic 72 (2007), pp. 1003-1018. Presentations Orderings on computable torsion free abelian groups - Invited to speak at the 25th Conference on Knot Theory and its Ramifications, at The George Washington University in Washington, D.C. on 12/9/07. A countable group is computable if its domain is a computable set and its group theoretic operation is computable. We examine complexity of orders on a computable torsion-free abelian (hence orderable) group G, using Turing degrees as a complexity measure. There are continuum many Turing degrees and they form an upper semilattice under Turing reducibility. All computable sets have Turing degree zero. It is easy to see that if G is of rank 1, then G has exactly two orders and they are computable. Solomon showed that if G has a finite rank greater than 1, then G has an order in every Turing degree. On the other hand, if G is of infinite rank, then G does not necessarily have a computable order, as shown by Downey and Kurtz.

20. Proceedings Of The American Mathematical Society
Abstract For a countable structure $\mathcal{A}$ , the (Turing) degree spectrum of $\mathcal{A}$ is the set of all Turing degrees of its isomorphic
http://www.ams.org/proc/2007-135-10/S0002-9939-07-08845-4/home.html
 ISSN 1088-6826 (e) ISSN 0002-9939 (p) Previous issue Table of contents Next issue Articles in press ... Next Article Turing degrees of nonabelian groups Author(s): M. A. Dabkowska; M. K. Dabkowski; V. S. Harizanov; A. S. Sikora Journal: Proc. Amer. Math. Soc. MSC (2000): Primary 03C57, 03D45 Posted: May 14, 2007 Retrieve article in: PDF DVI PostScript Abstract ... Additional information Abstract: For a countable structure , the (Turing) degree spectrum of is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of has the least degree , then we say that is the (Turing) degree of the isomorphism type of . So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees. References: S. I. Adjan

21. IngentaConnect Turing Degrees Of Hypersimple Relations On Computable Structures
Turing degrees of hypersimple relations on computable structures. Author Harizanov V.S.. Source Annals of Pure and Applied Logic, Volume 121, Number 2,
http://www.ingentaconnect.com/content/els/01680072/2003/00000121/00000002/art001
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22. Cornell Math - Antonio Montalban
First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few we show that every countable upper
http://www.math.cornell.edu/People/PhD/2005Aug/montalban.html
Antonio Montalban Ph.D. (2005) Cornell University
##### First Position
Postdoctoral position, University of Chicago
##### Dissertation
Beyond the Arithmetic
Research Area: Logic Abstract: Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees, of course, preserving jump and join; we show that every finite partial ordering labeled with the classes in the generalized high/low hierarchy can be embedded into the Turing degrees; we show that every generalized high degree has the complementation property; and we show that if a Turing degree a is either 1-generic and delta-zero-one, 2-generic and arithmetic, n -REA, or arithmetically generic, then the theory of the partial ordering of the Turing degrees below a is recursively isomorphic to true first order arithmetic. Last modified: November 29, 2005

23. [FOM] Natural R.e. Degrees
PRINCIPAL FINDINGS First, let R_T be the set of r.e. Turing degrees, i.e., natural degrees look like in a degree structure, P_w, which is slightly
http://cs.nyu.edu/pipermail/fom/2005-February/008809.html
##### [FOM] natural r.e. degrees
Stephen G Simpson simpson at math.psu.edu

24. Logic - Department Of Mathematics - University Of Notre Dame
Computability theory concerns computability and complexity, often measured by Turing degree. A set is computable if there is a program for computing its
http://www.nd.edu/~mathwww/research/logic.shtml
 Math Faculty Steven Buechler Peter Cholak Julia F. Knight Sergei Starchenko Philosophy Faculty Timothy Bays Patricia Blanchette Michael Detlefsen Curtis Franks Graduate Students Student Advisor Prerna Bihani Buechler Donald Brower Buechler Chris Porter Cholak Sean Walsh Cholak/Detlefsen Joshua Cole Cholak Logan Axon Cholak Sara Quinn Knight Christina Maher Knight John Walbaum Knight Pantelis Eleftheriou Starchenko Demirhan Tunc Starchenko Jacob Heidenreich was enrolled in a joint Mathematics and Philosophy Ph.D. program. Andrew Arana was the first Ph.D. graduate of this program. For more details contact Julia Knight and/or Michael Detlefsen . For the Philosophy graduate students see the Department of Philosophy home page Some recent mathematical logic Ph.D's (With latest known job information) Ambar Chowdhury, 1992, Buechler, working in business Leefong Low, 1992, Pillay, National Teaching University (Singapore) Zeljko Sokolovic, 1992, Pillay Alan Vlach, 1993, Knight, St. Mary's College (IN) Katrin Tent, 1994, Buechler, University of Wuerzburg John Thurder, 1994, Knight, Eastern Oregon State University

25. Liang Yu: Preprints
We show that there exists no Turing degree which is low for 1genericity and all of WOne approach to understanding the fine structure of initial segment
http://math.nju.edu.cn/~yuliang/preprint.html
 [1]Yue Yang and Liang Yu. On the Definable Ideal Generated by Nonbounding C.E.~Degrees. Journal of Symbolic logic, 70(2005), No.1, 252-270.[ pdf [2]Decheng Ding, Rod Downey, and Liang Yu. The Kolmogorov complexity of random reals. Ann. Pure Appl. Logic 129 (2004), no. 1-3, 163180. [ pdf [4]Decheng Ding and Liang Yu. There is no $SW$-complete c.e. real. J. Symbolic Logic 69 (2004), no. 4, 1163-1170.[ ps [5]Rod Downey and Liang Yu. There are no maximal low d.c.e. degrees. Notre Dame J. Formal Logic 45 (2004), no. 3, 147- 159. [ pdf We prove that there are no maximal low d.c.e. degrees. [6]Liang Yu. Lowness for genericity. Archive for Mathematical Logic 45 (2): 233-238 2006. [ ps We study lowness for genericity. We show that there exists no Turing degree which is low for 1-genericity and all of computably traceable degrees are low for weak 1-genericity. [7]Liang Yu Measure theory aspects of Locally Countable Orderings. Journal of Symbolic logic 71(3), 2006, pp. 958-968. [ pdf [8]Joseph Miller and Liang Yu.On initial segment complexity and degrees of randomness. To appear in Transaction of AMS. [ pdf [9]Rod Downey and Liang Yu. Arithmetical Sacks Forcing. Archive for Mathematical Logic 45(6) 715 - 720 2006. [

 26. The Value Of A College Degree - Computer Science@Everything2.com However, the framework of ideas is without structure and quickly crumbles . It is the tape of the Turing machine that is n in the big O. Anything thathttp://everything2.com/index.pl?node_id=947621

27. DBLP: Valentina S. Harizanov
11 EE, Valentina S. Harizanov Turing degrees of hypersimple relations on computable structures. Ann. Pure Appl. Logic 121(23) 209-226 (2003)
http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/h/Harizanov:Valentina_
##### Valentina S. Harizanov
List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL Guide CiteSeer CSB ... EE Valentina S. Harizanov, Frank Stephan : On the learnability of vector spaces. J. Comput. Syst. Sci. 73 EE Wesley Calvert Douglas Cenzer , Valentina S. Harizanov, Andrei S. Morozov : Effective categoricity of equivalence structures. Ann. Pure Appl. Logic 141 EE Rumen D. Dimitrov , Valentina S. Harizanov, Andrei S. Morozov : Dependence relations in computably rigid computable vector spaces. Ann. Pure Appl. Logic 132 EE Sergei S. Goncharov , Valentina S. Harizanov, Julia F. Knight Charles F. D. McCoy Russell Miller Reed Solomon : Enumerations in computable structure theory. Ann. Pure Appl. Logic 136 EE Valentina S. Harizanov: Turing degrees of hypersimple relations on computable structures. Ann. Pure Appl. Logic 121 EE Sergei S. Goncharov , Valentina S. Harizanov, Julia F. Knight Charles F. D. McCoy : Simple and immune relations on countable structures. Arch. Math. Log. 42 EE Valentina S. Harizanov, Frank Stephan : On the Learnability of Vector Spaces.

28. Department Of Computer Science
their algebraic structure, the degree of information they encode (Turing degree), their automorphisms, and their computational complexity properties.
http://www.cs.uchicago.edu/research/interests
##### Yali Amit
Computer vision, image analysis and speech recognition: Object detection, recognition and model registration algorithms in digital images and acoustic data using hierarchies of templates. Applications to optical character recognition, zip code reading, face detection and recognition, automatic anatomy identification in medical images, detection and recognition of acoustic signals. Computational efficiency is emphasized. Statistical modeling and analysis of the data for understanding the performance of the algorithms. Parallel and biologically plausible neural architectures for implementing these algorithms involving interactions with research on biological visual and acoustic pattern recognition.
##### Laszlo Babai
I work in the fields of theoretical computer science and discrete mathematics; more specifically in computational complexity theory, algorithms, combinatorics, and finite groups, with an emphasis on the interactions between these fields. Asymptotic questions and probabilistic methods are common features in my work in each of these areas. The introduction of Las Vegas algorithms, interactive proofs, holographic proofs (proofs verifiable by spotchecks) are among the conceptual highlights. A recent example: methods of the complexity theories of Boolean circuits and branching programs have been brought to bear on the analysis of a popular random sampling technique in computational group theory.
##### Todd Dupont
My research deals with the analysis, evaluation and construction of numerical methods to approximate the solutions of partial differential equations (PDEs).

29. Scientific Commons Yue Yang
We show that the structure of recursively enumerable degrees is not a A computably enumerable (c.e.) Turing degree is a diamond base if and only if it
http://en.scientificcommons.org/yue_yang
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30. Math: Faculty And Research: Mieczyslaw Dabkowski
of determining the full threedimensional structure of recombination systems. of 3-manifold groups that admit an order of arbitrary Turing degree.
http://www.utdallas.edu/math/faculty/dabkowski.html
##### Department of Mathematical Sciences
School of Natural Sciences and Mathematics ... The University of Texas at Dallas Main Page Navigation Sub Navigation nsm math Faculty and Research
##### Contact Information
Mieczyslaw K Dabkowski
Assistant Professor
ECSN 3.914

mdab@utdallas.edu
##### Mieczyslaw K. Dabkowski, Ph.D.
Assistant Professor
##### Education
Ph. D., George Washington University, 2003
##### Overview
Knot invariants and 3-manifold invariants, applications of topology to biology, recursion theory.
##### Research Interest
Another area of my mathematical research investigations concerns properties of known algebraic invariants of 3-manifolds called fundamental groups. In my research work, I investigated the property of the spaces of orders on 3-manifold groups. This topic of mathematical research has recently gained a considerable attention due to its applications to other areas of 3-dimensional topology. My main contributions to this area include results about existence of left orders on important classes of 3-manifold groups. The results have applications to the important problem of the existence of foliations.
In my current research I study computational properties of the spaces of orders. We showed, in particular, that many classes of 3-manifolds groups admit infinitely many orders that are arbitrarily computationally complex. That is, there are examples of 3-manifold groups that admit an order of arbitrary Turing degree. The spaces of orders on such groups also admit an embedding of the Cantor set, which establishes new connections between topology and recursion theory.

31. Artificial Consciousness
However, I do think McGath overstates the jurisdiction of Turing s ideas. between the degree of complexity of the physical support structure and the
http://folk.uio.no/thomas/po/artificial-consciousness.html
##### ON THE POSSIBILITY OF "ARTIFICIAL CONSCIOUSNESS"
A RESPONSE TO GARY MCGATH'S ARTICLE ON THE TURING TEST
IN OBJECTIVITY VOL. 1 NO. 5 (1993) Thomas Gramstad Objectivity Vol. 1 no. 6 (1993) I enjoyed Gary McGath's clear and concise exposition of the errors and weaknesses of the Turing test. However, I do think McGath overstates the "jurisdiction" of Turing's ideas. That is to say, not all ideas about artificial intelligence presupposes Turing's assumptions, and refuting Turing is not the same as refuting the possibility of artificial intelligence as such. "AWAKENING" AS EMERGENCE McGath claims that having a computer "wake up", like Mike in Heinleins's The Moon is a Harsh Mistress, is no more plausible than a beautiful statue waking up, and that there is no objectively valid reason to consider the former more plausible than the latter. I disagree. There is such a reason. Anyone can observe a proportional relationship between, on the one hand, the increasing complexity of the nervous system in the animal kingdom, and on the other hand, a corresponding incremental increase in their mental or cognitive capacities. From observing this correspondence in many independent and diverse instances (i.e., species) one is justified in concluding that there is a necessary link, a causal connection, between the degree of complexity of the physical support structure and the possibility for, origin of, and degree, scope and intensity of consciousness. This is further supported by the observation that damaging specific parts of a brain damages specific or corresponding parts of the organism's mental or cognitive capacities.

 32. NUI Maynooth > Department Of Computer Science > We present a number of timeefficient small universal Turing ma- chines. PBL into a computer science and software engineering degree structure.http://www.cs.nuim.ie/research/rptcontent.php?repid=11

33. ON $tt$-DEGREES OF RECURSIVELY ENUMERABLE TURING DEGREES
DEGREES OF RECURSIVELY ENUMERABLE Turing DEGREES. G N Kobzev 1979 Math. Semiconductor Science and Technology, Smart Materials and structures
http://www.iop.org/EJ/abstract/0025-5734/35/2/A02
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##### ON -DEGREES OF RECURSIVELY ENUMERABLE TURING DEGREES
G N Kobzev Math. USSR Sb. 173-180 doi:10.1070/SM1979v035n02ABEH001463 PDF (535 KB)
G N Kobzev
Abstract. The main result of the paper asserts that if is a semirecursive -hyperhypersimple set, then for every set with there exists a recursive set such that and . If is recursively enumerable, then . A corollary asserts that if a -degree contains an -maximal semirecursive set, then it is a minimal element in the semilattice of all -degrees.
Bibliography: 9 titles. Mathematics Subject Classification: 03D30, 03D50, 03D25, 03D55 Print publication: Issue 2 (1979)
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Biomedical Materials British Journal of Applied Physics (1950-1967) Chinese Journal of Astronomy and Astrophysics Chinese Journal of Chemical Physics Chinese Physics (2000-2007) Chinese Physics B Chinese Physics Letters Classical and Quantum Gravity Clinical Physics and Physiological Measurement (1980-1992) Communications in Theoretical Physics Distributed Systems Engineering (1993-1999) Environmental Research Letters European Journal of Physics Europhysics Letters (EPL) IOP Conference Series: Earth and Environmental Science Inverse Problems Izvestiya: Mathematics

Perhaps the most commonly used formalization is that of a Turing machine, I have also studied the degreetheoretic structure of the families of sets
http://web.centre.edu/alexmcal/
 Alex M. McAllister Contact Info Teaching Research Publications ... Some Links Contact Information I am an Associate Professor of Mathematics and the Chair of the Mathematics Program at Centre College. I teach courses in the Mathematics, the Computer Science, and the Philosophy Program. I can be reached via e-mail at alexmcal@centre.edu or in my office in 117 Olin Hall during my office hours . You can also reach me via any of the following: Math Department Centre College 600 West Walnut Street Danville, KY 40422-1394 (office) (dept) (fax) (home) Teaching I love teaching - that's one of the main reasons came to Centre College. You can reach course web pages via the following links: Term Course Fall, 2006 MAT 170: Calculus I MAT 380: Real Analysis Spring, 2005 MAT 171: Calculus II MAT 407: Mathematical Logic I am also a big proponent of reading and, in particular, of reading mathematics textbooks. And not just the exercises in the text, but all the ideas and examples and definitions and theorems that come before the exercises. Some of my thoughts and reflections on this topics can be found at: Reading Your Mathematics Textbook.

35. New Zealand Mathematical Societu Newsletter Number 91, August 2004
At that time the goal was to show that there was a MartinPour-El theory of every Turing degree but Rod showed this was impossible.
http://ifs.massey.ac.nz/mathnews/NZMS91/news91a.shtml
 Newsletters Index Centrefolds Index Number 91 August 2004 NEWSLETTER OF THE NEW ZEALAND MATHEMATICAL SOCIETY (INC.) Contents PUBLISHERÂS NOTICE EDITORIAL PRESIDENTÂS COLUMN LOCAL NEWS ... CENTREFOLD Rod Downey NEW COLLEAGUES FEATURES BOOK REVIEWS CONFERENCES ... MATHEMATICAL MINIATURE 24 Blowing our own Google; Dynamical Systems and Numerical Analysis ISSN 0110-0025 CENTREFOLD Professor Rod Downey Since coming to New Zealand in 1986 Rod Downey's career has flourished. He is now one of New Zealand's most prominent mathematicians and is undoubtedly one of the best two or three computability theorists in the world. Rod rapidly rose through the ranks to a Personal Chair in Mathematics at Victoria University in 1995. He currently has over 170 publications. He has a string of awards including the RSNZ Hamilton Award and an NZMS Research Award. He has had numerous major research grants including being a PI on at least three Marsden Grants as well as an AI on several others. He is a director of the NZIMA and the NZMRI. He is a former president of the NZMS, has had a number of graduate students, has very successfully supervised numerous post docs, is an FRSNZ, and no doubt I've missed a number of other things I should have mentioned. Having got that out of the way we can proceed to the good stuff; that is, the human interest and the question of what drives Rod's research. Human interest first. Rod grew up in a working class family in Brisbane, Australia. His father was a bookie; a career in which survival required a sharp mind. Given Rod's current interest in Martingales (essentially betting strategies), it seems the wheel has turned full circle. Rod claims that mathematics is one of the few academic areas in which you can achieve even if you do not come from a cultured background and perhaps he is right, but there is no doubt that his parents regarded his interest in mathematics as eccentric at best. After graduating from Queensland University Rod had to decide between doing a PhD at Monash or managing the bottle shop at the local pub. His parents were keen on the pub. From an economic point of view they were probably right.

36. ÃÃ®Â°ÂºÃÃºÂ¡ÂªÂ¡ÂªÃÃÂ¹ÃºÂ¿ÃÃÂ§ÃÂºÃÃ­Â¼Ã¾ÃÃÂ¾Â¿ÃÃ¹Â¼ÃÃÃ£Â»ÃºÃÃ­Â¼Ã¾
Translate this page structures and hierarchies of the Turing and the enumeration degrees, elementary differences among structures of the Turing degrees, enumeration operators
http://daoshi.kaoyantj.com/2006/02/18/5F570A1866348923.html
 MBAÃÂªÂ¿Â¼ EMBAÃÂ¶ÃÂ¿ MPAÃÂ¶ÃÂ¿ MPHÃÂ¶ÃÂ¿ Angsheng LiÂ¡Â¡Â¡Â¡ Address: Institute of Software, Chinese Academy of Sciences P.O. Box 8718 Beijing, 100080 P.R. CHINA Phone/Fax/Email: Office: 86-10-82625471 Home: 86-10-82522687 Fax: 86-10-82625471 kaoyantj Email: angsheng@gcl.iscas.ac.cnÂ¡Â¡ Information for prospective research students kaoyangj Web of the Institute of Software, Chinese Academy of Sciences Information for students My research interests My primary research interest is computability and computational complexity. Some particular problems I am currently working on include Turing definability in the local Turing degrees, structures and hierarchies of the Turing and the enumeration degrees, elementary differences among structures of the Turing degrees, enumeration operators automorphism of the Turing degrees, computable approximations Courses Computability Theory Computational Complexity My curriculum vitae and bibliography in postscript in pdf format Papers on line Journal papers Strong decomposition theorem of the recursively enumerable degrees (in Chinese), Acta Math. Sinica, A decomposition theorem of 0Â¡Â¯, Science in China, No.10, 1992 (in Chinese) and No.6, 1993.

37. Computability Theory
The ordering of the Turing degrees (and also of the enumeration degrees) is a very complex structure, requiring highly sophisticated and technically
http://www.maths.leeds.ac.uk/~pmt6sbc/research.html
 Computability Theory Research Logic Computability Seminars ... Leeds Alan Turing, Sark, 1931 Classical computability theory originated with the seminal work of Church , Turing, Kleene and Post in the 1930's, and includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, subrecursive hierarchy classifications, computable structures, and complexity theory relating to the preceding. Basic to recent developments in the subject is the theory of relative computability. This classifies real numbers according to the Turing reducibility relation a < b meaning 'a is computable from knowledge of b', and is captured mathematically via the Turing degree structure. (The structure of the enumeration degrees is the analogous structure derived from nondeterministic Turing reducibility.) Thus if a Slaman and Woodin), promise to have far reaching mathematical, scientific and philosophical consequences. There are still very many open questions, both of a technical nature, concerning extensions of what is known about the Turing degrees and their context in the enumeration degrees, and similar questions for other natural reducibility degree structures, and less well-defined questions concerning the scope of relevance of such work.

38. Dr Benedikt Loewe: Recursion Theory (1st Semester 2003/2004)
Structure of the 1degrees and m-degrees. Myhill s Isomorphism Theorem. The upper-semilattice of Turing Degrees. The upper-semilattice of r.e. degrees.
http://staff.science.uva.nl/~bloewe/2003-I-RT.html
 Recursion Theory 2003/2004; 1st Semester Universiteit van Amsterdam Instructor: Dr Benedikt LÃ¶we Vakcode: Time: Thursday 3-5 Place: P.015A First Lecture: September 11th, 2003 (Note that there will be no lecture on September 4th due to the traditional Introductory Boat Trip of the Master of Logic program. Also, there will be no lecture on October 23rd.) Course language: English Intended Audience: MoL students, Mathematics students in their fourth year Prerequisites: This course assumes basic mathematical skills and some knowledge of basic logic. This course will cover the basic notions of computability: Turing machines, recursivity and computer programs. The Equivalence Theorem will lead to Church's Thesis. Having a formalization of computability allows us now to investigate the boundaries of computation: what problems are not computationally solvable? This question will lead us to Turing's Halting Problem and the Decidability. We discuss several notions of recursion theoretic reducibility, their derived degree structures, and mathematical properties of these structures. We shall mostly follow Part A of the textbook Robert I.

39. Book Computability Theory, Maths For Engineers, Lavoisier Publishers
Recent work in computability theory has focused on Turing definability and Degrees,and the Natural Embedding of the Turing Degrees The Structure of De
http://www.lavoisier.fr/notice/gb402380.html
 Search on All Book CD-Rom eBook Software The french leading professional bookseller Description Approximate price Computability theory Author(s) : COOPER S. Barry Publication date : 09-2003 Language : ENGLISH Status : In Print (Delivery time : 12 days) Description Computability theory originated with the seminal work of GÃ¶del, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. Summary Subject areas covered: Mathematics and physics Applied maths and statistics Maths for engineers New search Your basket Information New titles BiblioAlerts E-books Customer services Open an account Ordering non-listed items Order tracking Help Lavoisier.fr

40. Mathematische Logik Und Theoretische Informatik
The structure of Exdegrees is coarser than that of the Turing degrees. Only very restricted learning models like finite learning (Fin) generate the same
http://www.math.uni-heidelberg.de/logic/abstracts.html
##### Arbeitsgruppe Mathematische Logik und Theoretische Informatik
Abstracts der Forschungsberichte
Abstracts of the Research Reports
K. AMBOS-SPIES, D. DING:
Discontinuity of Cappings in the Recursively Enumerable Degrees and Strongly Nonbranching Degrees
October 1993, 33 pages Abstract. We construct an r.e. degree a which possesses a greatest a-minimal pair b0, b1, i.e., r.e. degrees b0 and b1 such that b0,b1 > a, b0 meet b1 = a, and, for any other pair c0 and c1 with these properties, c0 less or equal bi and c1 less or equal b1-i for some i less or equal 1. By extending this result, we show that there are strongly nonbranching degrees which are not strongly noncappable. Finally, by introducing a new genericity concept for r.e. sets, we prove a jump theorem for the strongly nonbranching and strongly noncappable r.e. degrees. A. NIES:
The Model Theory of the Structure of Recursively Enumerable Many-One Degrees
October 1993, 13 pages Abstract. The theory of the r.e. m-degrees has the same computational complexity as true arithmetic. In fact, it is possible to define without parameters a standard model of arithmetic in this degree structure. B. BORCHERT:

41. Logic Colloquium 2006
One of the main goals of computability theory is to understand the shape of the structure of the Turing degrees. Two common ways of studying this is by
http://www.cs.ru.nl/lc2006/invited.html
main invited contributed registration how to get there
##### Invited talks
This page contains the schedule and abstracts of the tutorials, the plenary talks and the special sessions at the Logic Colloquium 2006
##### Schedule
The following facilities will be available in each room:
• Beamer Laptop (for people who did not bring their own laptop: possible file formats .pdf .ps .ppt) Overhead projector Black board or white board
At your convenience you can send a .pdf file with your slides to Jasper Stein ( jasper@cs.ru.nl ) who will have it pre-installed on the presentation computer. Room 1
Room 1
Room 1 Room 1 (Chair: Ralf Schindler) Room 2 (Chair: Michael Rathjen)
• Klaus Aehlig

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