Solution of the Problem P = L

The problems associated with the construction of polynomial complexity computer programs require new techniques and approaches from mathematicians. One of such approaches is representing some class of polynomial algorithms as a certain class of special logical programs. Goncharov and Sviridenko described a logical programming language L0, where programs inductively are obtained from the set of Δ0-formulas using special terms. In their work, a new idea has been proposed to look at the term as a program. The computational complexity of such programs is polynomial. In the same years, a number of other logical languages with similar properties were created. However, the following question remained: can all polynomial algorithms be described in these languages? It is a long-standing problem, and the method of describing some polynomial algorithm in a not Turing complete logical programming language was not previously clear. In this paper, special types of terms and formulas have been found and added to solve this problem. One of the main contributions is the construction of p-iterative terms that simulate the work of the Turing machine. Using p-iterative terms, the work showed that class P is equal to class L, which extends the programming language L0 with p-iterative terms. Thus, it is shown that L is quite expressive and has no halting problem, which occurs in high-level programming languages. For these reasons, the logical language L can be used to create fast and reliable programs. The main limitation of the language L is that the implementation of algorithms of complexity is not higher than polynomial.

The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

Theoretical foundations of the organization of branches and repetitions in programs in the logic programming language Prolog

Introduction. The organization of branches and repetitions in the context of logical programming is considered by an example of the Prolog language. The fundamental feature of the program in a logical programming language is the fact that a computer must solve a problem by reasoning like a human. Such a program contains a description of objects and relations between them in the language of mathematical logic. At the same time, the software implementation of branching and repetition remains a challenge in the absence of special operators for the indicated constructions in the logical language. The objectives of the study are to identify the most effective ways to solve problems using branching and repetition by means of the logic programming language Prolog, as well as to demonstrate the results obtained by examples of computational problems. Materials and Methods. An analysis of the literature on the subject of the study was carried out. Methods of generalization and systematization of knowledge, of the program testing, and analysis of the program execution were used. Results. Constructions of branching and repetition organization in a Prolog program are proposed. To organize repetitions, various options for completing a recursive cycle when solving problems are given. Discussion and Conclusions. The methods of organizing branches and repetitions in the logic programming language Prolog are considered. All these methods are illustrated by examples of solving computational problems. The results obtained can be used in the further development of the recursive predicates in logical programming languages, as well as in the educational process when studying logical programming in the Prolog language. The examples of programs given in the paper provide using them as a technological basis for programming branches and repetitions in the logic programming language Prolog.

CONCEPTUAL MODEL OF AGROECOLOGICAL PROPERTIES OF LAND. SEQUENCES

Рассмотрен процесс создания последовательностей при описании предметных областей на формально-логическом языке UML. Использование последовательностей основано на понятии «источник данных», введённом авторами на основе предыдущего этапа концептуализации предметной области «агроэкологические свойства земель» – диаграммы классов. В классе начала связи выбирается один из комплектов атрибутов, в классе конца связи – один из методов (запрос), соответствующий этому комплекту. Многократно применяя этот подход при различных значениях атрибутов центрального класса, получается массив данных (в том числе пространственных). Атрибуты являются связующим звеном между создаваемой моделью, методами, потоками данных и запросов системы, так как, с одной стороны, они входят в состав классов, участвующих в сценариях диаграмм последовательностей, а с другой – принадлежат к внешней оболочке модели. На примерах движения информации, необходимой для расчетов гидротермического коэффициента Селянинова и степени проявления эрозии для рабочего участка, построены диаграммы последовательностей «ГидротермическийКоэффициент» и «СтепеньПроявленияЭрозии». Данные для диаграмм последовательностей формируются с помощью геоинформационных систем (географические координаты рабочего участка, цифровая модель рельефа) и справочно-информационного портала «Погода и климат». Предлагаемый подход даёт возможность автоматического построения баз знаний на основе двух концептуальных понятий: «источники данных» и «последовательности». Структурирование и формализация знаний позволяет осуществить переход от набора информации к знаниям и последующему их графическому отображению. Визуализация помогает наглядно отобразить связи между классами, которые могут быть не очевидны. Становится доступной возможность последующей оценки жизнеспособности модели, ее проектирования в симбиозе с использованием инструментов для имитационного моделирования, а также математических методов анализа и обработки информации. Данные диаграммы используются для построения и верификации созданных подсистем в процессе прямого и обратного проектирования аграрной интеллектуальной системы. The process of creating sequences while describing subdicipline in the formal-logical language UML is considered. The sequences usage is based on the concept of a "data source". It was deduced by the authors on the basis of the previous step of subdicipline conceptualization «agroecological lands properties» - class diagrams. In the beginning link's class, one of the attribute set is selected, in the ending class - one of the adequate to this set methods (query). The result of repeated application this approach, with different values of the attributes of the central class, is a database (including spatial data). Attributes mediate the created model, methods, data streams and system requests, as, on the one hand, they are among the classes involved in sequence diagrams scripting, and on the other - belong to the outer shell of the model. Sequences diagrams were constructed by the examples of the information flow necessary for calculating the Selyaninov hydrothermal index and the degree of erosion for the working land area. These diagrams are "HydrothermalIndexQuery" and "ErosionDegreeQuery". Data for sequence diagrams is generated by Geological Information System (geographic coordinates of the working land area, digital terrain model) and the reference-information gateway “Weather and Climate". The proposed approach makes it possible to build knowledge bases with the scope of two concepts: "data sources" and "sequence" automatically. Knowledge structuralizasion and formalization allows produce a shift from collecting information to knowledge and its subsequent graphical image. Visualization helps to demonstrably provide insight into classes' connections that may occur not to be obvious. The possibility of subsequent estimate of model consistency, its creation process using simulation modeling tools, as well as mathematical analysis methods and processing of data becomes more accessible. Diagrams' data is used for sybsystem construction and verification. These parts of a whole system were created in the process of forward and reverse engineering agricultural intelligence system.

Plan recognition in dynamic 2-D environments

We look at the relatively unexplored problem of plan recognition applied to motion in 2-D environments where all moving objects are modelled as circles. Golog is a well-known high level logical language for solving planning problems and specifying agent controllers. Few studies have applied Golog to plan recognition. We use some of the features of this language, but its standard interpreter is adapted to solving plan recognition problems. This thesis makes several other contributions. First, plan recognition procedures are formulated as finite automata and expressed as Golog programs. Second, we elaborate a logical formalism for reasoning about depth and motion from an observer's viewpoint. We not only expand on this situation calculus based formalism, but also apply it to tackle plan recognition problems in the traffic domain. The proposed approach is implemented and thoroughly tested on recognizing simple behaviours such as left turns, right turns, and overtaking.

Plan recognition in dynamic 2-D environments

We look at the relatively unexplored problem of plan recognition applied to motion in 2-D environments where all moving objects are modelled as circles. Golog is a well-known high level logical language for solving planning problems and specifying agent controllers. Few studies have applied Golog to plan recognition. We use some of the features of this language, but its standard interpreter is adapted to solving plan recognition problems. This thesis makes several other contributions. First, plan recognition procedures are formulated as finite automata and expressed as Golog programs. Second, we elaborate a logical formalism for reasoning about depth and motion from an observer's viewpoint. We not only expand on this situation calculus based formalism, but also apply it to tackle plan recognition problems in the traffic domain. The proposed approach is implemented and thoroughly tested on recognizing simple behaviours such as left turns, right turns, and overtaking.

Organic chemistry synthesis problem as artificial intelligence planning.

This thesis formulates organic chemistry synthesis problems as Artificial Intelligence planning problems and uses a combination of techniques developed in the field of planning to solve organic synthesis problems. To this end, a methodology for axiomatizing organic chemistry is developed, which includes axiomatizing molecules and functional groups, as well as two approaches for representing chemical reactions in a logical language amenable to reasoning. A novel algorithm for planning specific to organic chemistry is further developed, based on which a planner capable of identifying 75 functional groups and chemical classes is implemented with a knowledge base of 55 generic chemical reactions. The performance of the planner is empirically evaluated on two sets of benchmark problems and analytically compared with a number of competing algorithms. v

Organic chemistry synthesis problem as artificial intelligence planning.

This thesis formulates organic chemistry synthesis problems as Artificial Intelligence planning problems and uses a combination of techniques developed in the field of planning to solve organic synthesis problems. To this end, a methodology for axiomatizing organic chemistry is developed, which includes axiomatizing molecules and functional groups, as well as two approaches for representing chemical reactions in a logical language amenable to reasoning. A novel algorithm for planning specific to organic chemistry is further developed, based on which a planner capable of identifying 75 functional groups and chemical classes is implemented with a knowledge base of 55 generic chemical reactions. The performance of the planner is empirically evaluated on two sets of benchmark problems and analytically compared with a number of competing algorithms. v

Reasoning about Petri Nets: A Calculus Based on Resolution and Dynamic Logic

Petri Nets are a widely used formalism to deal with concurrent systems. Dynamic Logics (DLs) are a family of modal logics where each modality corresponds to a program. Petri-PDL is a logical language that combines these two approaches: it is a dynamic logic where programs are replaced by Petri Nets. In this work we present a clausal resolution-based calculus for Petri-PDL. Given a Petri-PDL formula, we show how to obtain its translation into a normal form to which a set of resolution-based inference rules are applied. We show that the resulting calculus is sound, complete, and terminating. Some examples of the application of the method are also given.

FROM BOOLE’S LOGIC TO BOOLEAN APPLICATIONS IN COMPUTER SCIENCE

The developments of an algebraic logical language of thoughts by G. Boole are considered using historical and theoretical perspectives. The technical implementations of Boolean logic in combinational circuits and in modern cryptography show strong influences of a 19th century logic on the latest technologies of computing.