Representations of Models and Representations of Models and algorithmic properties: algorithmic properties: Results and problems.Results and problems.

S. GoncharovS. Goncharov

Maltsev MeetingMaltsev Meeting Novosibirsk, Novosibirsk, October 11October 11--1414, 20, 201111

Computability theory.Computability theory.

This mathematical theory and This mathematical theory and applications was started by applications was started by

works of works of А.А.TuringTuring, , E.PostE.Post, , A.A. ChurchChurch, , S. Kleene.S. Kleene.

Alan Alan MathisonMathison Turing Turing (June 23, 1912 – June 7, 1954) was a British (June 23, 1912 – June 7, 1954) was a British mathematician, logician, and cryptographer. Turing is often considered to mathematician, logician, and cryptographer. Turing is often considered to

be a father of modern computer science.be a father of modern computer science.The next year will be in England the series of conference in honor of The next year will be in England the series of conference in honor of

Turing.Turing.

Stephen Cole KleeneStephen Cole KleeneUniversal partial recursive functions and universal programming languages.Universal partial recursive functions and universal programming languages.

Godel numberings.Godel numberings.

The The problem of problem of decidability decidability of of arithmetics.arithmetics.

The The computability computability from from definability in definability in PA. PA.

The The competness competness of calculusof calculus

The non-The non-completenescompleteness of PA.s of PA.

Numberings and Constructive Numberings and Constructive models.models.

Constructive algebras.Constructive algebras.

1. General theory of 1. General theory of numbering was numbering was started.started.

2. Basic notions of 2. Basic notions of constructive structures constructive structures on the base of on the base of numberings.numberings.

Computable models and Computable models and programming systems.programming systems.

Computable sets of types of elements, Computable sets of types of elements, computable operations on elements of basic computable operations on elements of basic types and computable relations on its.types and computable relations on its.

Main problems.Main problems.

1. Existence problems for costructive 1. Existence problems for costructive representations.representations.

2. Equivalence for numberings and 2. Equivalence for numberings and constructivizations.constructivizations.

3. Algebraic conditions of computable 3. Algebraic conditions of computable structures.structures.

Boolean algebras, 1970-Boolean algebras, 1970-

The theory of computable boolean algebras The theory of computable boolean algebras and some algorithmic propertiesand some algorithmic properties

With A.Morozov, S.Odintsov, D.Palchunov, With A.Morozov, S.Odintsov, D.Palchunov, V.Vlasov, P.Alaev, D.Drobotun, N.Bagenov. V.Vlasov, P.Alaev, D.Drobotun, N.Bagenov. V.Leont’eva.V.Leont’eva.

Algorithmic properties of Boolean Algorithmic properties of Boolean algebras.algebras.

A.Tarski and Yu. A.Tarski and Yu. Ershov Ershov

1. Existence problems1. Existence problems 2. Autostability2. Autostability 3. Decidability and 3. Decidability and

bounded levels.bounded levels. 4. Hyperarithmetical 4. Hyperarithmetical

levels and Turing levels and Turing degrees of degrees of autostability.autostability.

Computable boolean algebras,1996.Computable boolean algebras,1996.

Decidable models. 1972-Decidable models. 1972-

Existence of strongly constructive models.Existence of strongly constructive models. Autoequivalence of strongly constructive Autoequivalence of strongly constructive

models.models.

Strongly constructive and decidable Strongly constructive and decidable modelsmodels

Yu.L.Ershov, 1968, Yu.L.Ershov, 1968, Lectures in Alma-ata.Lectures in Alma-ata.

L.Harrington, 1973, L.Harrington, 1973, M.Morley, 1975M.Morley, 1975

Computable model theory of Yu. L. Computable model theory of Yu. L. Ershov Ershov

1. Decidable theories and strongly 1. Decidable theories and strongly computable models.computable models.

2. Extensions of computable models.2. Extensions of computable models. 3. Special models and computability.3. Special models and computability. 4. Existence problem for constructive 4. Existence problem for constructive

models and connections with model theory.models and connections with model theory. 5. Constructive representensions of 5. Constructive representensions of

classical algebraic structures and classical algebraic structures and autistability.autistability.

Special models and Decidability. Special models and Decidability.

1. Prime models(S.Goncharov, A.Nurtazin, 1. Prime models(S.Goncharov, A.Nurtazin, L.Harrington)L.Harrington)

2. Saturated models(M.Morley) and Morley 2. Saturated models(M.Morley) and Morley Problem.Problem.

3. Homogeneous models(S.Goncharov, 3. Homogeneous models(S.Goncharov, V.Peretyatkin)V.Peretyatkin)

4. Homogeneous models in decidable 4. Homogeneous models in decidable theories (S.Goncharov)theories (S.Goncharov)

Autostability (A.Nurtazin, K.Kudaibergenov)Autostability (A.Nurtazin, K.Kudaibergenov)

Autostable prime model, 2009.Autostable prime model, 2009.

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

Non-autostable prime model, 2011.Non-autostable prime model, 2011.

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

Morley and Millar-Goncharov Morley and Millar-Goncharov problems.problems.

If M is countable models of a decidable If M is countable models of a decidable Ehrenfeucht theory then this model M has Ehrenfeucht theory then this model M has decidable representation?decidable representation?

If a theory T is decidable and has countably If a theory T is decidable and has countably many countable models then the prime many countable models then the prime model of this theory is decidable?model of this theory is decidable?

Turing degres of autostability relative to Turing degres of autostability relative to strong constructivizations.strong constructivizations.

Constructive models. 1970-Constructive models. 1970-

1. The existence 1. The existence problem of problem of constructive models.constructive models.

2. Autostability for 2. Autostability for algebraic closer of algebraic closer of constructive models.constructive models.

3. Strongly 3. Strongly constructive models constructive models and model theory.and model theory.

Algorithmic dimension of models.Algorithmic dimension of models.

Bounded models.Bounded models. Branching models.Branching models. Classical algebraic structures and autostability and Classical algebraic structures and autostability and

algorithmic dimension.algorithmic dimension. With O.Kudinov, Yu. Ventsov, O.Kudinov, With O.Kudinov, Yu. Ventsov, O.Kudinov,

B.Drobotun, P.Alaev, B. Khoussainov, E.Fokina, B.Drobotun, P.Alaev, B. Khoussainov, E.Fokina, N. Kogabaev, D.Tusupov and my colleagues from N. Kogabaev, D.Tusupov and my colleagues from USA: J.Knight, V.Harizanov, S.Lempp, R.Shore, USA: J.Knight, V.Harizanov, S.Lempp, R.Shore, R.Solomon, C.McCoy, S.Miller, J.Chisholm.R.Solomon, C.McCoy, S.Miller, J.Chisholm.

Ershov Problem:Ershov Problem:Finite algorithmic dimension.Finite algorithmic dimension.

Scott families of computable Scott families of computable categorical models.categorical models.

Algorithmic dimension for theories Algorithmic dimension for theories with special properties and relative with special properties and relative

to hyperarithmetical levels.to hyperarithmetical levels.Series of papers with my collegues: J.Knight, Series of papers with my collegues: J.Knight,

V.Harizanov, E.Fokina, S.Miller, V.Harizanov, E.Fokina, S.Miller, J.Chisholm,S.Lempp, R.Solomon, J.Chisholm,S.Lempp, R.Solomon, B.Khoussainov, R.Shore and …B.Khoussainov, R.Shore and …

Problem.Problem.

If for limit level e we have two not e-If for limit level e we have two not e-autoequivalent constructivization of a model autoequivalent constructivization of a model M is it true that e-Dim(M) is infinite?M is it true that e-Dim(M) is infinite?

Turing degrees of autostability?Turing degrees of autostability?

Handbook of Recursive Mathmatics,Handbook of Recursive Mathmatics,19981998

Constructive models, 2000.Constructive models, 2000.

Computability and Computable Models.Computability and Computable Models.

Eds:D.Gabbay, S.Goncharov,Eds:D.Gabbay, S.Goncharov, M.Zakharyaschev, 2007M.Zakharyaschev, 2007

Numbering Theory,1975-.Numbering Theory,1975-.

A.N.Kolmogorov and A.N.Kolmogorov and V.UspenskiiV.Uspenskii

A.I.MalcevA.I.Malcev H.RodgersH.Rodgers R.FriedbergR.Friedberg Yu.ErshovYu.Ershov

Computable arithmetical numberings Computable arithmetical numberings with A.Sorbi, S.Badaev, S.Podzorov with A.Sorbi, S.Badaev, S.Podzorov 1. The Ershov operator 1. The Ershov operator

of Complitions in of Complitions in arithmetical arithmetical numberings.numberings.

2. Algebraic properties 2. Algebraic properties of Rogers Semilattices of Rogers Semilattices of arithmetical of arithmetical numberings.numberings.

3. Types of 3. Types of isomorphisms.isomorphisms.

Problems for Computable Problems for Computable numberings.numberings.

1. Ershov problem about types of 1. Ershov problem about types of isomorphism for Rodgers semilattices of isomorphism for Rodgers semilattices of computable numberings of finite families of computable numberings of finite families of finite sets.finite sets.

2. Ershov problem of number of minimal 2. Ershov problem of number of minimal computable numberings.computable numberings.

3. The cardinality of Rodgers semilattices of 3. The cardinality of Rodgers semilattices of computable numberings in Ershov computable numberings in Ershov hierarchy.hierarchy.

Computable numberigs of Computable numberigs of computable models and Index sets.computable models and Index sets.

A.Nurtazin, V.Selivanov.A.Nurtazin, V.Selivanov. Universal computable numberings of partial Universal computable numberings of partial

computable models.computable models. Index sets for classes of computable models Index sets for classes of computable models

and classifications problems with J.Knight.and classifications problems with J.Knight. By E.Pavlovskii, E.Fokina.By E.Pavlovskii, E.Fokina.

Computable numberings and Computable numberings and inductive inference.inductive inference.

With K.Ambos-Spies and S.Badaev.With K.Ambos-Spies and S.Badaev.

Problems.Problems.

1. Complexity of autostable models.1. Complexity of autostable models. 2. Complexity of models with finite 2. Complexity of models with finite

algorithmic dimension.algorithmic dimension. 3. Complexity of isomorphisms for models 3. Complexity of isomorphisms for models

with finite dimensions.with finite dimensions. 4. Scott ranks of autostable models.4. Scott ranks of autostable models. 5. Scott ranks of models with finite 5. Scott ranks of models with finite

computable dimension.computable dimension.

Computer Science and mathematical Computer Science and mathematical logic.logic.

Semantic programming: Computability on abstract models and logic programming Semantic programming: Computability on abstract models and logic programming language.language.

Malcev problem for classes with strong homomorphisms and erimorphisms.Malcev problem for classes with strong homomorphisms and erimorphisms.

By M.Korovina, O.Kudinov, A.Morozov, A.Khisamiev, A.Stukachev, V.Puzarenko, By M.Korovina, O.Kudinov, A.Morozov, A.Khisamiev, A.Stukachev, V.Puzarenko, A.Mantsivoda, M.Smoyan, O. Il’icheva and ….A.Mantsivoda, M.Smoyan, O. Il’icheva and ….

Some applications in Bioinformatics with N.A.Kolchanov, P. Demenkov, E.Vityaev, ….Some applications in Bioinformatics with N.A.Kolchanov, P. Demenkov, E.Vityaev, ….

Thanks for attention!Thanks for attention!

http://www.math.nsc.ru/LBRT/http://www.math.nsc.ru/LBRT/logic/persons/gonchar/win.htmllogic/persons/gonchar/win.html