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Fusions of modal logics revisited
 In Advances in modal logic
, 1998
"... The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of n ..."
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Cited by 44 (7 self)
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The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht &Wolter [10]]. The paper de nes the fusion `l `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics. Given two logical system L1 and L2 it is natural to ask whether the fusion (or join) L1 L2 of them inherits the common properties of both L1 and L2. Let us consider some examples: (i) It is known that the rst order theory of one equivalence relation has the nite model property and is decidable. However, the rst order theory of two equivalence relations does not have the nite model property and is in fact undecidable (see Janiczak [7]). This result shows that even if we know the rst order properties of the individual relations of a theory, there may be no algorithm to determine the purely logical consequences of these properties. (ii) Various positive and negative results are known for joins of term rewriting systems (TRSs) whose vocabularies are disjoint. For example, the join of two TRSs is con uent i the two TRSs are con uent but there are complete TRSs whose join is not complete (see e.g. Klop [8]). In fact, the literature on TRSs shows how useful the study of joins of systems can be. (iii) In contrast to rst order theories the join of two decidable equational theories in disjoint languages is decidable as well. This was proved by Pigozzi in [12]. So we observe interesting di erences between logical systems by investigating the behavior of joins. To form the join of two modal logics (in languages with disjoint sets of modal operators) is { in a sense { a generalization of forming the join of two equational theories in disjoint languages. Namely, it is wellknown that each modal logic corresponds to an equational theory of boolean algebras with operators. So the join of two modal logics corresponds to
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 13 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
On the nonhamiltonian interaction of two rotators
 Eprint (MSRI Archive on Diff. Geom. and Geom. Anal.): dgga/9409004
, 1994
"... Abstract. Classical dynamical equations describing a certain version of the nonHamiltonian interaction of two rotators (Euler tops with completely degenerate inertia tensors) are considered. The simplest case is integrated. It is shown that the dynamics is almost periodic with periods depending on ..."
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Cited by 4 (3 self)
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Abstract. Classical dynamical equations describing a certain version of the nonHamiltonian interaction of two rotators (Euler tops with completely degenerate inertia tensors) are considered. The simplest case is integrated. It is shown that the dynamics is almost periodic with periods depending on the initial data. Classical dynamics of Hamiltonian systems is often described by remarkable algebraic structures such as Lie algebras [1] or their nonlinear generalizations [2]. There is a hope that not less important algebraic objects govern a behaviour of the interacting Hamiltonian systems. It seems that these mathematical objects may be unravelled in a certain formal way. There exist several types of an interaction of Hamiltonian systems: often it has a potential character, sometimes it is ruled by a deformation of the Poisson brackets; however, one of the most intriguing and less explored forms is a nonHamiltonian interaction. In general, it can not be described by deformations of the standard Hamiltonian data (Poisson brackets and Hamiltonians). Sometimes, such interaction is realized by the dependence of the Poisson brackets of one Hamiltonian
Type Inferencing Based on Complete Type Specifications
 Proceedings of the 2nd International Andrei Ershov Memorial Conference, Akademgorodok
, 1995
"... Type specification completeness is a necessary prerequisite for support of object creating formulae in object calculus leading to formation of new types to be integrated into a type lattice containing the types from which they were formed. The paper shows what conditions should be satisfied in order ..."
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Cited by 3 (3 self)
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Type specification completeness is a necessary prerequisite for support of object creating formulae in object calculus leading to formation of new types to be integrated into a type lattice containing the types from which they were formed. The paper shows what conditions should be satisfied in order that the inferred types could be correct and what is the systematic way of integration of these types into the existing type lattice on the basis of a welldefined subtype relation. Ignoring of the specification completeness for type inference may lead to inconsistent results. The paper contributes to clarification of type inferencing operations for the case of complete type specifications. 1 Introduction Abstract data type (ADT) concepts play more and more important role in data models and database systems. Introduced in database world in early 80s as a technique for inclusion of new userdefined types in relational DBMSs [16], ADT provides now a basis for objectoriented data models [10]...
Modeling Facilities for the Componentbased Software Development Method
 In Proceedings of the Third International Workshop ADBIS'96
, 1996
"... Componentbased software development (CBSD) technology uses components as firstclass objects and therefore requires a good understanding of the nature of components. Industrial approaches to CBSD based on interoperability standards (such as OMG CORBA) lack of component semantics in their descriptio ..."
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Cited by 2 (2 self)
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Componentbased software development (CBSD) technology uses components as firstclass objects and therefore requires a good understanding of the nature of components. Industrial approaches to CBSD based on interoperability standards (such as OMG CORBA) lack of component semantics in their descriptional models. In this paper we present an overview of the SYNTHESIS method emerging the CBSD approach by introduction of semantic information to enrich and complement the industrial modeling facilities. The paper contributes to the development of modeling facilities for CBSD focusing on the interoperable systems design. Proper balance of formal and semiformal modeling facilities is demonstrated to cope with the CBSD requirements 1 . 1 Introduction Componentbased software development (CBSD) has become one of the hottest topics in the area of software engineering. CBSD is a promising solution intended to break up large monolithic software systems into interoperable components and thus to m...
On the structure of left and right F, SM and Equasigroups
, 2008
"... Abstract. It is proved that any left Fquasigroup is isomorphic to the direct product of a left Fquasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right Fquasigroups, left and right SM and Equasigroups ..."
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Cited by 2 (0 self)
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Abstract. It is proved that any left Fquasigroup is isomorphic to the direct product of a left Fquasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right Fquasigroups, left and right SM and Equasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple Fquasigroup is a simple group or a simple medial quasigroup. It is proved that any left Fquasigroup is isotopic to the direct product of a group and a left Sloop. Some properties of loop isotopes of Fquasigroups (including Mloops) are pointed out. A left special loop is an isotope of a left Fquasigroup if and only if this loop is isomorphic the direct product of a group and a left Sloop (this is an answer to Belousov “1a ” problem). Any left FESMquasigroup is isotopic to the direct product of an abelian group and a left Sloop (this is an answer to KinyonPhillips 2.8(2) problem). New proofs of some known results on the structure of commutative
On approximation of topological algebraic systems by finite ones
, 2003
"... We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the wellknown definition of a local embedding of an algebraic system A in a class K of ..."
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Cited by 1 (0 self)
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We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the wellknown definition of a local embedding of an algebraic system A in a class K of algebraic systems. According to this definition A is locally embedded in K iff it is a subsystem of an ultraproduct of some systems in K. We obtain a similar characterization of approximation of a locally compact system A by systems in K. We inroduce the bounded formulas of the signature of A and their approximations similar to those introduced by C.W.Henson [8] for Banach spaces. We prove that a positive bounded formula ϕ holds in A if all precise enough approximations of ϕ hold in all precise enough approximations of A. We prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field R can be concedered as computer systems for reals. Thus, it is impossible to construct a computer arithmetic for reals that is an associative ring. 1
On finite approximations of topological algebraic systems
, 2006
"... We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept s ..."
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We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula ϕ holds in A if and only if all precise enough approximations of ϕ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be noncommutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. 1