Representations of Models and
algorithmic properties:
Results and problems.
S. Goncharov
Maltsev Meeting Novosibirsk,
October 11-14, 2011
Computability theory.
This mathematical theory and
applications was started by
works of А.Turing, E.Post, A.
Church, S. Kleene.
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.
Stephen Cole Kleene
Universal partial recursive functions and universal programming languages.
Godel numberings.
The
problem of
decidability
of
arithmetics.
The
competness
of calculus
The
computability
from
definability in
PA.
The noncompletenes
s of PA.
Numberings and Constructive
models.
Constructive algebras.
 1. General theory of
numbering was
started.
 2. Basic notions of
constructive structures
on the base of
numberings.
Computable models and
programming systems.
 Computable sets of types of elements,
computable operations on elements of basic
types and computable relations on its.
Main problems.
 1. Existence problems for costructive
representations.
 2. Equivalence for numberings and
constructivizations.
 3. Algebraic conditions of computable
structures.
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.
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.
Computable boolean algebras,1996.
Decidable models. 1972 Existence of strongly constructive models.
 Autoequivalence of strongly constructive
models.
Strongly constructive and decidable
models
 Yu.L.Ershov, 1968,
Lectures in Alma-ata.
 L.Harrington, 1973,
M.Morley, 1975
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.
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)
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
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.
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.
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.
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.
Ershov Problem:
Finite algorithmic dimension.
Scott families of computable
categorical models.
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 …
Problem.
 If for limit level e we have two not eautoequivalent constructivization of a model
M is it true that e-Dim(M) is infinite?
 Turing degrees of autostability?
Handbook of Recursive Mathmatics,
1998
Constructive models, 2000.
Computability and Computable Models.
Eds:D.Gabbay, S.Goncharov,
M.Zakharyaschev, 2007
Numbering Theory,1975-.
 A.N.Kolmogorov and
V.Uspenskii
 A.I.Malcev
 H.Rodgers
 R.Friedberg
 Yu.Ershov
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.
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.
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.
Computable numberings and
inductive inference.
 With K.Ambos-Spies and S.Badaev.
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.
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, ….
Thanks for attention!
http://www.math.nsc.ru/LBRT/logic
/persons/gonchar/win.html