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Representations of Models and algorithmic properties: Results and problems.

Representations of Models and algorithmic properties: Results and problems. S. Goncharov Maltsev Meeting Novosibirsk, October 11 - 14 , 20 11. Computability theory. This mathematical theory and applications was started by works of А. Turing , E.Post , A. Church , S. Kleene.

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Representations of Models and algorithmic properties: Results and problems.

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  1. Representations of Models and algorithmic properties: Results and problems. S. Goncharov Maltsev MeetingNovosibirsk, October 11-14, 2011

  2. Computability theory. This mathematical theory and applications was started by works of А.Turing, E.Post, A.Church, S. Kleene.

  3. Alan Mathison Turing (June 23, 1912 – June 7, 1954) was a British mathematician, logician, and cryptographer. Turing is often considered to be a father of modern computer science.The next year will be in England the series of conference in honor of Turing.

  4. Stephen Cole KleeneUniversal partial recursive functions and universal programming languages.

  5. Godel numberings.

  6. Numberings and Constructive models.

  7. Constructive algebras. • 1. General theory of numbering was started. • 2. Basic notions of constructive structures on the base of numberings.

  8. Computable models and programming systems. • Computable sets of types of elements, computable operations on elements of basic types and computable relations on its.

  9. Main problems. • 1. Existence problems for costructive representations. • 2. Equivalence for numberings and constructivizations. • 3. Algebraic conditions of computable structures.

  10. Boolean algebras, 1970- • The theory of computable boolean algebras and some algorithmic properties • With A.Morozov, S.Odintsov, D.Palchunov, V.Vlasov, P.Alaev, D.Drobotun, N.Bagenov. V.Leont’eva.

  11. Algorithmic properties of Boolean algebras. • A.Tarski and Yu. Ershov • 1. Existence problems • 2. Autostability • 3. Decidability and bounded levels. • 4. Hyperarithmetical levels and Turing degrees of autostability.

  12. Computable boolean algebras,1996.

  13. Decidable models. 1972- • Existence of strongly constructive models. • Autoequivalence of strongly constructive models.

  14. Strongly constructive and decidable models • Yu.L.Ershov, 1968, Lectures in Alma-ata. • L.Harrington, 1973, M.Morley, 1975

  15. Computable model theory of Yu. L. Ershov • 1. Decidable theories and strongly computable models. • 2. Extensions of computable models. • 3. Special models and computability. • 4. Existence problem for constructive models and connections with model theory. • 5. Constructive representensions of classical algebraic structures and autistability.

  16. Special models and Decidability. • 1. Prime models(S.Goncharov, A.Nurtazin, L.Harrington) • 2. Saturated models(M.Morley) and Morley Problem. • 3. Homogeneous models(S.Goncharov, V.Peretyatkin) • 4. Homogeneous models in decidable theories (S.Goncharov) • Autostability (A.Nurtazin, K.Kudaibergenov)

  17. Autostable prime model, 2009. • Theorem 1. If the theory totally transcendent and decidable and prime model is not autostable relative to strong constructivizations then any almost prime decidable model is not autostable relative to strong constructivizations

  18. Non-autostable prime model, 2011. • Theorem 2. There exists complete Ehrenfeucht theory with non-autostable relative to strong constructivizations prime model but with autostable relative to strong constructivizations some almost prime model.

  19. Morley and Millar-Goncharov problems. • If M is countable models of a decidable Ehrenfeucht theory then this model M has decidable representation? • If a theory T is decidable and has countably many countable models then the prime model of this theory is decidable? • Turing degres of autostability relative to strong constructivizations.

  20. Constructive models. 1970- • 1. The existence problem of constructive models. • 2. Autostability for algebraic closer of constructive models. • 3. Strongly constructive models and model theory.

  21. Algorithmic dimension of models. • Bounded models. • Branching models. • Classical algebraic structures and autostability and algorithmic dimension. • With O.Kudinov, Yu. Ventsov, O.Kudinov, B.Drobotun, P.Alaev, B. Khoussainov, E.Fokina, N. Kogabaev, D.Tusupov and my colleagues from USA: J.Knight, V.Harizanov, S.Lempp, R.Shore, R.Solomon, C.McCoy, S.Miller, J.Chisholm.

  22. Ershov Problem:Finite algorithmic dimension.

  23. Scott families of computable categorical models.

  24. Algorithmic dimension for theories with special properties and relative to hyperarithmetical levels. Series of papers with my collegues: J.Knight, V.Harizanov, E.Fokina, S.Miller, J.Chisholm,S.Lempp, R.Solomon, B.Khoussainov, R.Shore and …

  25. Problem. • If for limit level e we have two not e-autoequivalent constructivization of a model M is it true that e-Dim(M) is infinite? • Turing degrees of autostability?

  26. Handbook of Recursive Mathmatics,1998

  27. Constructive models, 2000.

  28. Computability and Computable Models.Eds:D.Gabbay, S.Goncharov,M.Zakharyaschev, 2007

  29. Numbering Theory,1975-. • A.N.Kolmogorov and V.Uspenskii • A.I.Malcev • H.Rodgers • R.Friedberg • Yu.Ershov

  30. Computable arithmetical numberings with A.Sorbi, S.Badaev, S.Podzorov • 1. The Ershov operator of Complitions in arithmetical numberings. • 2. Algebraic properties of Rogers Semilattices of arithmetical numberings. • 3. Types of isomorphisms.

  31. Problems for Computable numberings. • 1. Ershov problem about types of isomorphism for Rodgers semilattices of computable numberings of finite families of finite sets. • 2. Ershov problem of number of minimal computable numberings. • 3. The cardinality of Rodgers semilattices of computable numberings in Ershov hierarchy.

  32. Computable numberigs of computable models and Index sets. • A.Nurtazin, V.Selivanov. • Universal computable numberings of partial computable models. • Index sets for classes of computable models and classifications problems with J.Knight. • By E.Pavlovskii, E.Fokina.

  33. Computable numberings and inductive inference. • With K.Ambos-Spies and S.Badaev.

  34. Problems. • 1. Complexity of autostable models. • 2. Complexity of models with finite algorithmic dimension. • 3. Complexity of isomorphisms for models with finite dimensions. • 4. Scott ranks of autostable models. • 5. Scott ranks of models with finite computable dimension.

  35. Computer Science and mathematical logic. • Semantic programming: Computability on abstract models and logic programming language. • Malcev problem for classes with strong homomorphisms and erimorphisms. • By M.Korovina, O.Kudinov, A.Morozov, A.Khisamiev, A.Stukachev, V.Puzarenko, A.Mantsivoda, M.Smoyan, O. Il’icheva and …. • Some applications in Bioinformatics with N.A.Kolchanov, P. Demenkov, E.Vityaev, ….

  36. Thanks for attention! http://www.math.nsc.ru/LBRT/logic/persons/gonchar/win.html

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