**PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES **

**SECTION B: NATURAL, EXACT AND APPLIED SCIENCES **

(ISSN 1407-009X, Language: English)

**Volume 50 (1996) Number 1**

*SPECIALIZED ISSUE: MATHEMATICS/COMPUTER SCIENCE*

**Original Paper: Mathematics/Computer Science**

AUTOMATON REPRESENTATION OF A SEMIGROUP

Ilze Balode & Janis Buls*,

Riga Building Technical College, Gaizina iela 3, Riga, LV 1516 LATVIA,

* Department of Physics and Mathematics, University of Latvia, Raina bulv. 19, Riga, LV
1586 LATVIA.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 1-3. Language: English.

**Abstract:** Automaton representation is offered with partially defined
transformations. Sufficient and necessary conditions for such representation are given.

**Key words:** automaton, partially defined transformation.

1991 MSC: 20M35, 68D30.

**Original Paper: Mathematics. Algebra**

SOME METHODS FOR INVESTIGATING SYSTEMS OF PARAMETRIC POLYNOMIAL EQUATIONS

Aivars Berzins,

University of Latvia, Raina bulv. 19, Riga, LV-1586 LATVIA; e-mail:
aberzins@cclu.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 4-8. Language: English.

**Abstract:** In this paper we prove some theorems and give some algorithms of
systems of polynomial equations with parameters. The first section contains algorithms
which realize operations with polynomial ideals. In the second section we show the
fundamental role of the Jacobi matrix in the radical computing problems. Finally, simple
algorithms for determining a dimension of an ideal under specializations are described in
the third section.

**Key words:** polynomial ideals, algorithm.

1991 MSC: 68Q40, 05A17.

**Original Paper: Mathematics. Algebra**

CORESIDUATED HOMOMORPHISMS BETWEEN IMPLICATIVE SEMILATTICES

Janis Cirulis

Computer Science Department, University of Latvia, Raina bulv. 19, Riga,
LV-1586 LATVIA; e-mail: jc@lanet.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 9-12. Language: English.

**Abstract:** Coresiduated mappings between posets are known also as residual, or
upper, Galois maps. We characterize coresiduated mappings via their ranges and kernel
equivalences, and coresiduated homomorphisms between implicative semilattices via their
ranges and kernel filtres. We also observe that several constructions, known in algebraic
logic, and related to coresiduated Boolean homomorphisms, can be transferred without
essential changes to the field of implicative algebras.

**Key words:** algebraic logic, closure operator, Galois connection, implicative
algebra, quantifier.

1991 MSC: primary, 06A15; secondary, 03G15, 06A12.

**Original Paper: Mathematics. Algebra **

ABSTRACT UNIVERSAL ALGEBRAIC LOGIC. Part I:

A UNIFIED FRAMEWORK OF STRUCTURAL HYPERLOGICS FOR INTEGRATING THE DEDUCTIVE AND
MODEL-THEORETIC APPROACHES

Zinovy Diskin,

Computer Science Department, University of Latvia, Raina bulv. 29, Riga,
LV-1459 LATVIA; e-mail: diskin@frame.riga.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 13-21. Language: English.

**Abstract:** The general goal of the paper is to develop an abstract unifying
framework for algebraizing both deductive and model-theoretic logics in a coherent
fashion. We adjoin theory lattices and quasivarieties of algebras through congruence
closure systems by means of Leibniz and coLeibniz operators. A distinctive feature of our
approach is that we deal with fibrations of these constructs indexed by formula algebras.
In addition, the latter are not assumed to be necessarily free. This machinery enables us
to obtain theorems on compactness and constructivizability of semantically defined logics
in a very general setting.

**Key words:** inferential system, structurality, matrix semantics, quasivariety,
congruence system.

MSC: 036**.

**Original Paper: Mathematics. Algebra**

ABSTRACT UNIVERSAL ALGEBRAIC LOGIC. Part II:

ALGEBRAIZABLE LOGICS AND ALGEBRAIC SEMANTICS (Galois connections, compactness,
constructivizability)

Zinovy Diskin,

Computer Science Department, University of Latvia, Raina bulv. 29, Riga,
LV-1459 LATVIA; e-mail: diskin@frame.riga.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 22-30. Language: English.

**Abstract:** The present paper is an immediate continuation of "Abstract
universal algebraic logic, Part I" (This Journal, pp. 13-21), where a general unified
framework for algebraizing both deductive and semantic logics was introduced and where it
was shown that the notions of structural hyperlogic and matrix quasivariety are dually
equivalent. The goal of Part II is to study, in the general setting, ways from structural
to algebraizable hyperlogics and, dually, from matrix semantics to perfectly algebraic
semantics.

**Key words:** algebraic semantics, quasivariety, Galois connection, compactness.

MSC: 036**.

**Original Paper: Mathematics. Computer Algebra**

CHARACTERISTIC SETS IN UNMIXED IDEALS

Ruvins Lipjanskis,

University of Latvia, Raina bulv. 19, Riga LV-1586, LATVIA.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 31-34. Language: English.

**Abstract:** Solution of the membership problem for unmixed ideals by means of
characteristic sets in exponential time is given.

**Key words:** membership problem, characteristic set.

1991 MSC: 68Q40, 05A17.

**Short Communication: Mathematics. Algebra**

ON ONE CLASS OF LOCALLY FINITE LIE ALGEBRAS

Levs A. Simonjans,

Riga University of Civil Aviation, Lomonosova iela 1, Riga LV-1019,
LATVIA.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 35-36. Language: English.

**Abstract:** It is proved that locally finite Lie algebras, all inner derivations
of which have a finite rank, cannot be simple.

**Key words:** Lie algebras, locally finite algebras, simple algebras.

1991 MSC: 17B65.

**Original Paper: Mathematics. Algebra**

GROUP INVARIANTS OF BOOLEAN FUNCTIONS

Indulis Strazdins,

Riga Technical University, Kalku iela 1, Riga LV-1658, LATVIA; e-mail:
lzalumi@cclu.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 37-41. Language: English.

**Abstract:** Invariants of types of Boolean functions are described with respect to
the fundamental groups of transformations of variables. The problem is connected mainly
with classifications under the linear groups.

**Key words:** Boolean function, transformation group, invariant.

1991 MSC: 06E30, 20B25.

**Original Paper: Mathematics. Algebra**

ON SYMMETRIC S-AUTOMATA OF FUZZILY STRUCTURED SETS

Veniamins Steinbuks,

Department of Applied Mathematics, Riga Technical University, Kalku iela
1, Riga LV-1658, LATVIA.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 42-45. Language: English.

**Abstract:** The problem of determinability up to homeomorphism and
quasihomeomorphism of a fuzzily structured set by means of its associated S-automaton is
discussed.

**Key words:** S-automaton, fuzzy set.

1991 MSC: 20M20.

**Original Paper: Computer Science**

THE COMPLEXITY OF PROBABILISTIC VERSUS DETERMINISTIC FINITE AUTOMATA

Andris Ambainis,

Institute of Mathematics and Computer Science, University of Latvia, Raina
bulv. 29, Riga LV-1459, LATVIA; e-mail: ambainis@cclu.lv

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 46-48. Language: English.

**Abstract:** We show that there exist probabilistic finite automata with isolated
cutpoint and *n* states such that the smallest equivalent deterministic finite
automaton contains OMEGA (2 ^ {n {log log n} / {log n}}) states.

**Key words:** automata theory, complexity of finite automata, probabilistic finite
automata.

1991 MSC: 68Q68, 68Q10.

**Original Paper: Computer Science**

INEVITABLE GAPS BETWEEN THE UPPER AND LOWER COMPLEXITY BOUNDS IN INDUCTIVE INFERENCE

Andris Ambainis, Rusins Freivalds and Juris Smotrovs,

Institute of Mathematics and Computer Science, University of Latvia, Raina
bulv. 29, Riga LV-1459, LATVIA; e-mail: {ambainis, rusins, smotrovs}@cclu.lv.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, pp. 49-54. Language: English.

**Abstract:** We consider the complexity of inductive inference for recursively
enumerable classes of total recursive functions. The complexity is usually understood as
the maximum of mindchanges over the functions defined by the first *n* indices of the
numbering. Instead, we consider the mindchange complexity as the maximum over the first *n*
functions in the numbering (disregarding the repeated functions). Linear upper and lower
bounds for the mindchange complexity are proved. However, there is a big gap between the
bounds for all *n* and for infinitely many *n*.

**Key words:** inductive inference, computational learning theory.

MSC: 68T05

**Science Life**

FIFTY YEARS ANNIVERSARY OF THE ACADEMY OF SCIENCES OF LATVIA, AND MATHEMATICS

Andris Buikis,

Institute of Mathematics, Academy of Sciences and University of Latvia.

*Proc. Latvian Acad. Sci., Section B* (LATVIA, ISSN: 1407-009X), Vol.
**50** (1996), No. 1, p. 55. Language: English.

In Latvia, a Mathematics and Mechanics Institute was planned to be founded in 1940, however, an institute with a slightly different profile, the Physics and Mathematics Institute, Academy of Sciences, Latvia, was founded in 1946. At the time, the Soviet political system "cleansed" itself from the unwanted; hence, the institute director, mathematician Nikolajs Brazma, lost his chair. In 1950, the institute was renamed the Physics Institute, although a small mathematics group continued to work there.

In the second half of the 1940's, in the west, intensive work was going on in the production of computers and their use. Therefore, in the 1950's, work began in this field also in the Soviet Union.

In the end of the 1950's in Latvia, lecturers and students in the Physics and Mathematics Faculty, University of Latvia learned programming. Under the leadership of Docent L. Arins, in 1959 the Computing Centre at the University of Latvia was formed. This was the first scientific institute in Latvia that concentrated in one place mathematics specialists for computing, and many practical mathematics fields were created. Students at the Faculty, who were interested in scientific work in mathematics, after graduation began to work at the computing centre. The curriculum was changed to better prepare students for computing work.

In the 1970's in the Soviet Union, new teaching curricula were formed for computer studies. This type of applied mathematics speciality was begun also in the University of Latvia, and in autumn 1972, I was invited to the Physics and Mathematics Faculty from the Computing Centre to head this new Department. The Department closely cooperated with the Computing Centre scientists, who provided lectures, and the Department further specialized in applied mathematics. In 1976, the Applied Mathematics Department was split into the Mathematical Modelling (Differential Equations and Approximative Methods) and Computer Science (Discrete Mathematics and Programming) Departments.

The studies at the mathematical department were used in filtration theory (modelling of oil and gas technological min<->ing), ecological and drainage problems (ground water pollution and cleaning, drainage system modelling) and magnetic hydrodynamics (for example, aluminium smelting). Most of the conducted work was for institutes outside of Latvia.

Prof. Linards Reizins, at the Physics Institute, Academy of Sciences, led the Mathematics Laboratory, and also lectured in the Differential Equation and Approximative Methods Department. Since the 1970's, he had been attempting to create a Mathematics Institute at the Academy, and in 1988 invited me to the Physics Institute to work toward this goal. There was an idea to form a united University of Latvia and Academy of Sciences Institute, which included scientists from the Physics Institute and the Physics and Mathematics Faculty. The new Institute, was formed with the end of the occupation of Latvia, on 16 May, 1991.

The other, Computer Science, field (Discrete Mathematics and Programming) had grown to a strong Computing Section in the Physics and Mathematics Faculty, under the scientific leadership of Prof. Janis Barzdins and Prof. Martins Rusins Freivalds. Both are internationally known, and successfully combine teaching with science, leading also the Mathematic and Computing Institute (former Computing Centre), University of Latvia. Both theoretical and applied science is carried out, which is used in Latvia and internationally.

In mathematical modelling, applied science is accented, and there exists cooperation with the European Consortium for Mathematics in Industry (ECMI), and with local industries.

In Jurmala, 1995, there was held an ECMI Conference called "Baltic Days: Mathematics for Industry", financed by the European Union.

In the ECMI-1996 Congress in Copenhagen, several Institute workers participated with lectures. This continues the direction spelled out in the Jurmala '1995 conference, to apply mathematical modelling in industry.

The field of mathematics quickly reoriented in the new political and economic situation. Further work must intensify, and comply with European standards.

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