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September 1997 - July 2020
Education
September 1992 - June 1997
Publications
Publications (40)
We consider algebras of finite subsets under the assumption that the original algebra is an infinite groupoid. For linear spaces over fields of finite characteristic, we prove that the finite subsets algebra is algorithmically equivalent to the first-order arithmetic. We also generalize this result to arbitrary infinite Abelian groups. As a corolla...
Рассматриваются алгебры конечных подмножеств, когда исходная алгебра является бесконечным группоидом. Доказывается, что для линейных пространств над полями конечной характеристики теория построенной алгебры алгоритмически эквивалентна элементарной арифметике. Далее этот результат обобщается на произвольные бесконечные абелевы группы. В качестве сле...
We study the additive theory of arbitrary figures in linear spaces, that is, the theory of addition extended to sets of vectors. Our main result is the following: if a linear space is infinite, then the additive theory of figures allows to interpret second-order arithmetic and, therefore, has this or higher degree of undecidability. For countably i...
Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted....
The Lambek calculus with the unit can be defined as the atomic theory (algebraic logic) of the class of residuated monoids. This calculus, being a theory of a broader class of algebras than Heyting ones, is weaker than intuitionistic logic. Namely, it lacks structural rules: permutation, contraction, and weakening. We consider two extensions of the...
Early we (with B. N. Karlov) have proved the following claim for the infinite cyclic monoid ℋ. Let exp ℋ be an algebra of finite subsets of ℋ with the same operation, exp ℋ must be a monoid again. So the theory of exp ℋ is equivalent to elementary arithmetic. Thus, the theory of the monoid exp ℋ is undecidable. Here we consider an arbitrary commuta...
This paper is dedicated to studying decidability properties of theories of regular languages with classical operations: union, concatenation, and the Kleene star. The theory with union only is a theory of some Boolean algebra, so it is decidable. We prove that the theory of regular languages with the Kleene star only is decidable. If we use union a...
We investigate concatenation theories for some sets of one-symbol languages. These sets can be the set of all languages, the set of regular languages, or the set of finite languages. We prove that all such theories are undecidable. The last two theories are algorithmically equivalent to elementary arithmetic. The first is equivalent to second order...
We consider unoids consisting of identical non-branching trees
which are connected into an infinite line. We establish that the finite subset algebra admits effective quantifier elimination and it does not depend on the original algebra. So, we have an instance where the finite subset algebra theory is algorithmically simpler than the theory of the...
This paper is dedicated to studying decidability properties of some regular languages theories. We prove that the regular languages theory with the Kleene star only is decidable. If we use union and concatenation simultaneously then the theory becomes both \(\varSigma _1\)- and \(\varPi _1\)-hard over the one-symbol alphabet. Finally, we prove that...
In this paper, we investigate the safety of unary inflationary fixed point operators (IFPoperators). The safety is a computability in finitely many steps. IFP-operators exactly correspond to recursive SQL-queries hence this problem has a value for database theory. The problem appears from the fact that if recursive queries contain universe function...
We continue to investigate a safety of recursive queries (inflationary fix-point operators) over infinite universes. In the previous paper we have proved that models of countable categorical theories are safe. In this paper we propose a necessary and sufficient condition for IFP-safety and obtain a solution of the problem for linear order theories.
We continue the investigation of the expressive power of the language of predicate logic for finite algebraic systems embedded in infinite systems. This investigation stems from papers of M. A. Taitslin, M. Benedikt and L. Libkin, among others. We study the properties of a finite monadic system which can be expressed by formulae if such a system is...
This paper continues investigations into the database of queries of first-order language theory. It is known that for many
decidable theories, the collapse result holds: each locally generic query is equivalent to some restricted query. But, until
now, the problem of effective construction of this query remains almost unexplored. We use earlier res...
We study I-reducible algebraic systems and the theory of I-reducible systems. We show that the lack of an independent formula in a theory is not a necessary condition for the I-reducibility of its models, even for extensions of Presburger arithmetic. In particular, there is an entire class of theories that are extensions of Presburger arithmetic in...
This is a survey of collapse results obtained mainly by members of the Tver State University seminar on the theoretical foundations of computer science. Attention is focused on the relative isolation and pseudo-finite homogeneity properties and universes without the independence property. The Baldwin-Benedikt reducibility theorem is proved for thes...
It has been proved (by S. M. Dudakov and M. A. Taitslin) that the reducibility of some models of a theory implies the second
pseudofinite homogeneity property for this theory. We prove the converse, namely, that any theory with the first or the second
pseudofinite homogeneity property has a reducible model and, therefore, possesses the second isola...
Two methods are used usually for to establish the collapse result for theories. They use the isolation property and the reducibility property. Early it is shown that the reducibility implies the isolation. We prove that these methods are equivalent.
Предлагается пример запроса к базе данных,
который является локально генерическим и может
быть записан с использованием отношений автоматных
систем, но не может быть записан ограниченной формулой.
Тем самым опровергаются гипотезы о том,
что в автоматных структурах соблюдается трансляционная теорема, две гипотезы из работы [11]
о том, что не существ...
Earlier, Belegradek, Stolboushkin, and Taitslin proved that the collapse result holds in the theory of natural numbers with addition, i.e., each locally generic query using addition can be written without it. In this paper, we use the sufficient conditions of the collapse result obtained by Taitslin to prove that it holds in any extensions of the P...
This paper is devoted to the problem of consistency enforcement for logical databases. The enforcement methods we propose compute the minimal real change in a given DB state, which is sufficient to accomplish the input update and preserve the integrity constraints (IC). For IC expressed in the form of a generalized logic program, this problem is pr...
The computational complexity is explored of finding the min- imal real change of a database after an update constrained by a logic program. Quite surprisingly, a polynomial time algorithm is discovered which solves this problem for ground IC in partial interpretations. For- mulated in a "property" form, even under the premise of fixed database sche...
Databases with integrity constraints (IC) are considered. For each DB update, i.e. a set of facts to add and of facts to delete, the IC implies its correct expansion: new facts to add and new facts to delete. Simultaneously, each expanded update induces a correct simplification of the IC. In the limit this sequence of expansions and simplifications...
In this paper we investigate computational complexity of the PERF-consistency and PERF-entailment problems for ground normal logic programs. In [3] it is proved that these problems belong to Σ 2P and II 2P correspondingly. The question of obtaining more accurate results was left as open. We prove that both problems belong to Δ 2P. Lower bounds on t...
To find a minimal real change after an update of a database with integrity constraints (IC) expressed by a generalized logic
program with explicit negation is proven to be a Σstackp
stack2-complete problem. We define a class of operators expanding the input updates correctly with respect to the IC. The particular
monotone expansion operator we desc...
The provability problem for the Horn fragment of linear logic is NP-complete. In this work we investigate various definitions of concurrency proposed by D. A. Archangelsky, M. I. Dekhtyar, E. Kruglov, I. Kh. Musikaev and M. A. Taitslin [Lect. Notes Comput. Sci. 813, 18-22 (1994)] and establish the complexity of the provability problem and the probl...
Ранее было показано, что для арифметики Семенова имеет место коллапс к порядку. В данной работе мы демонстрируем, что в ряде случаев для арифметики Семенова возможна эффективная трансляция формул.
Данная работа продолжает исследования по языкам запросов к базам данных. Ранее установлено, что во многих разрешимых теориях имеет место коллапс к порядку: каждая <-инвариантная формула эквивалентна некоторой <-ограниченной, но вопрос о возможности эффективного нахождения этой формулы почти не исследовался. Используя полученные нами ранее результат...
Работа усиливает результаты, ранее полученные в работах Дехтяря. Показано, что колмогоровская алгоритмическая сложность распознавания начального отрезка длины $n$ $\NP$-трудного по Тьюрингу множества не может быть меньше $O(\lg\lg n)$, если $\NP\neq \P$.
