=Paper=
{{Paper
|id=Vol-2648/paper6
|storemode=property
|title=Knowledge Flows Processes at Multidimensional Intelligent Systems
|pdfUrl=https://ceur-ws.org/Vol-2648/paper6.pdf
|volume=Vol-2648
|authors=Konstantin Kostenko
}}
==Knowledge Flows Processes at Multidimensional Intelligent Systems==
https://ceur-ws.org/Vol-2648/paper6.pdf
Knowledge Flows Processes at Multidimensional Intelligent
Systems
Konstantin Kostenko
Kuban State University, Krasnodar, Russia
Abstract
The multidimensional intelligent systems formal structure is proposed. It reflects knowledge
aspects, developed within multiple subject areas, and al-lows simulating different knowledge
representation and processing models created at these areas. The four-dimensional intelligent
systems architecture is considered. Unified knowledge representation format of abstract
semantic hierarchies’ formalisms allows performing formal analysis and achieving acceptable
homogeneity for intelligent systems’ structural and functional components. Such components
adaptation to weakly formalized knowledge models attributes allows creating intelligent
systems prototypes that suggest further models’ transforming into applied system descriptions.
Homomorphisms and homomorphic extensions used for modeling knowledge synthesis
processes. Algebraic combinations of knowledge processing operations form basis for
developing the goals realization templates at subject domain intelligent system. These
templates composed as sequences of diagrams, assigned to intelligent systems structural
components. Multidimensional model-ling processes at intelligent systems is implemented as
knowledge transfers between such systems’ components and knowledge transforming within
components. Abstract knowledge processing templates describe the subject areas goals
implementation schemes by diagrams, assigned to intelligent systems architecture components.
Knowledge processing operations’ formal specifications have being applied as such modeling
parameters. Ontologies used as foundation for processes’ diagrams and templates formal
descriptions. This allows creating tools for intelligent systems templates exhaustive processing
performed by operations of templates’ analysis and control.
Keywords1
Intelligent system, subject area, knowledge representation formalism, cognitive goal, cognitive
synthesis, ontology, knowledge processing operation, knowledge-processing diagram
1. Introduction
Mathematical aspects of artificial intelligence concern on investigating and developing the special
formal systems. These systems bases on fundamental invariants of abstract mathematics areas. They
demonstrate possibility for abstract mathematical entities applications within intelligent system unified
and universal components. Such structure realized by variable models adaptation proposed by different
knowledge areas that deals with knowledge representation and processing [1-4]. Coordinated
knowledge areas attributes representation at intelligent system integrated model allows these attributes
joint applying.
Appropriate complete and formalized models creating and exploring is an urgent task of a modern
knowledge-based society. The existing diversity of such models develops extensively and is highly
heterogeneous. The latter feature is explained by the existence of a tendency to the advantage of
specialists' empirical ideas as the basis for artificial intelligence modeling. Intelligent systems general
aspects' formal describing within the framework of existing mathematical theories allow us to
investigate these aspects by theories' separate entities.
Russian Advances in Artificial Intelligence: selected contributions to the Russian Conference on Artificial intelligence (RCAI 2020), October
10-16, 2020, Moscow, Russia
EMAIL: kostenko@kubsu.ru (K. Kostenko)
ORCID: 0000-0002-9851-2455 (K. Kostenko)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
These entities' selected properties are used to model knowledge representation structures and their
processing algorithms in accordance with subject area's existing experience. The systematic
formalization and theoretical study for knowledge aspects' variety existed at different areas had
undertaken within M. Burgin's work [5]. Many significant invariants related to knowledge concept
accumulated and formalized in this work. Nevertheless, uniform formal system based on these concepts
remains unrealized.
Such systems' initial model constructing should begin with knowledge representation formalisms'
abstract invariants. They are homomorphic images for knowledge structures and operations within
applied knowledge models. That allows homomorphic extensions' applying for generating the existing
knowledge representation models. Investigating the essential elements of abstract knowledge structural
representation and structures processing operations is possible for that case. This model's extension
based on formalization such knowledge structures and knowledge processing aspects that developed
outside mathematics. Aspects' values sets form a multidimensional abstract knowledge space structures.
It is convenient for developing such spaces' theory and modeling the intelligent systems' variety founded
on application areas peculiarities.
2. Knowledge representation formalisms
The unified and universal approach to abstract knowledge representation implements the class of
knowledge representation formalisms definition. Every formalism is a system ℑ =( M , DM , , ) with
enumerable sets of abstract knowledge M and knowledge fragments DM , where M ⊆ DM is solvable
in DM . Special empty knowledge Λ belongs to M and is useful in many ways. Computable
composition operation : DM × DM → DM applied for knowledge constructing as knowledge fragments
combination. Let us suppose that composition operations is injection in cases when its meanings are
nonempty. Binary relation on knowledge fragments ⊆ DM × DM is solvable and interpreted as
knowledge content inclusion. The knowledge fragments compositions structure describes knowledge
algebraic structure. Such structure used as knowledge representation format by knowledge processing
algorithms. Such knowledge structure represented by binary tree with leaves marked by elementary
knowledge and other vertices – by operation of composition. Relation of content inclusion is basic for
creating knowledge properties definitions [7].
Formalism of semantic hierarchies is special knowledge representation formalism. It operates with
knowledge representation structures based on binary tree format. Every such formalism defined as pair
H = ( M , d ) , where M − enumerable set of abstract knowledge called configurations and d is pair of
computable mappings (ε ,ψ ) , with decomposition ε : M → M × M that divide every z ∈ M onto two
parts, andψ : M → R . Last mapping defines solvable semantic relation satisfied between ε ( z )
elements. Set of accepted relations R is enumerable with solvable property of relations’ inclusion.
Configuration z ∈ M called elementary and considered as undividable if ε ( z ) = ( Λ, Λ ) . The
configuration structural representation has format of binary tree with elementary configurations
assigned to leaves and internal vertices marked by semantic relations that holds for configurations,
represented by left and right subtrees of given vertices. The uniform format of knowledge representation
formalisms implementing to semantic hierarchies formalisms, applies the set of elementary knowledge
extended by set R . Besides that, it is possible to add such specifications for configurations fragments
composition operation and con-figurations’ inclusion relation that provide keeping properties when
transitions implemented from configurations’ algebraic structures to binaries trees that rep-resent
configurations. This property allows applying formalisms of semantic hierarchies’ as universal unified
base for abstract knowledge representation at considered intelligent systems architecture [7].
The configurations’ tracing relation implements general knowledge inclusion concept for semantic
hierarchies’ formalisms. It based on existence of binary trees vertices isotonic mappings, where binary
trees vertices that connected by mapping have comparable assigned meanings. Class of possible
isotonic map-pings splits on subclasses, with mappings of compressing and stretching as important
cases. These classes supply intelligent systems, based on knowledge representation formalisms, with
special tools for processing knowledge as hierarchical semantic structures.
3. Concept of multidimensional intelligent systems structure
We shall consider multidimensional intelligent systems components architecture, based on
independent aspects of knowledge representation and processing, called knowledge dimensions. This
allows defining intelligent systems structures by combining from predetermined components. The
components general structure depends on knowledge representation separate aspects meanings
represented by certain sets. Such properties extracted as result of analysis the transdisciplinary
knowledge representations models. They allow constructing separate intelligent systems with
components structures selected by examining subject domain properties. Every knowledge dimension
settled by finite set of possible knowledge property meanings. They define components properties
induced by this dimension. The intuitive dimensions sets serve to be the uniform platform for intelligent
systems formal analysis and constructing. The dimension of knowledge abstractness attribute follows
to K. Stanovich model [2]. It used three abstractness attribute meanings (quants) that define reflexive,
algorithmic and abstract minds for intelligent systems components’ classification. These quants
distinguish the levels for thinking abstractness representation [2, 8]. Three additional intelligent systems
dimensions natural for extending abstractness dimension. They reflect properties of knowledge
atomizing, grade and timing. The introduced dimensions and their quantification define intelligent
systems cellular structure (see Figure 1).
Figure 1: The intelligent systems unified four-dimension cellular architecture
Three added dimensions extend intelligent system components cellular structure. Every new
dimension has own quantization by three possible meanings (full-image, partially structured and
completely atomized – for knowledge atomization dimension; superficial, acceptable and exhaustive –
for knowledge grade dimension; past, present, future – for knowledge timing dimension). The
intelligent systems’ last two dimensions are new. They extend thinking processes modeling possibilities
in comparison with models based on two dimensions [4]. The obtained complete intelligent system
structure consists of 81 separate components. Coordinates assigned as quants for intelligent systems
separate dimensions identify the structure components. Every component allows addressing by
dimensions’ quants meanings as four elements sequence ( abstractness, atomization, grade, timing ) .
This sequence defines component’s unified properties and roles for attached knowledge processing
operations with predefined operational and subject domain semantics.
The empty meaning ( Λ ) for separate dimension measurement describes situation of removal the
dimension from intelligent system cellular structure. This allows selecting only necessary system’s
dimensions when intelligent system with certain properties is constructed. Selected sets of adopted
dimensions and dimensions’ quants allow introduce intelligent systems classification. It based on such
systems properties supported for chosen components sets. In such a way it is possible to define class of
expert systems as one-dimensioned intelligent systems based on knowledge atomization dimension with
fixed abstractness dimension meaning equals to «algorithmic» ( B ). Figure 2 represents the intelligent
systems’ four-dimension structure that defines class of expert systems. It composed by components of
knowledge base, tasks solving processes and professional tasks' solutions (specialists’ experience).
Figure 2: The experts systems unified components structure
Expert systems main components, presented on Figure 2, are based on knowledge atomization
dimension with fixed algorithmic level of abstractness. Other dimensions are insignificant for tasks
solving processes. Chosen components described by following cellular structure components sequence
enumerated from left to right as. (α , B, Λ, Λ ) , ( β , B, Λ, Λ ) and (γ , B, Λ , Λ ) or briefly (α , B ) , ( β , B ) and
(γ , B ) .
The scale of proposed components specifications with simple cells basic structure has deep applied
modeling possibilities. Dimensions of abstractness and atomizing allow professional activity modeling
at given subject domain with the identical level of specialist’s experience. The grade dimension
applications allow distinguishing possible meanings for property of knowledge grade and use this
dimension at adaptive learning systems with special possibilities for implementing the multilevel
education [10]. The timing dimension allows introducing into intelligent systems unified architecture
components that model knowledge processing for solving the subject domain problems that are time
dependent.
Separate component realizes independent set of knowledge processing flows relevant to applied
business processes associated with that component. The component memory represented as complex
knowledge structure synthesized by execution the component’s business processes. Independent
components interactions performed as knowledge transferring and implemented by crossing
architecture dimensions with changing quants meanings. Uniform knowledge algebraic structures’
format defined as universal for all intelligent system components con-tent representation allows develop
uniform collection of abstract operations proposed to be the base for knowledge processing
specification.
Applied intelligent system designing technology supposes intelligent systems constructed as several
linked universal components of universal cell structure. Simple abstract restrictions on constructing
models are concerned on decreasing number of dimensions under consideration and sets of separate
dimensions' quants. Cellular structure of intelligent system components allows apply abstract
mathematical systems' invariants for such systems properties formal studying. These invariants suggest
their simultaneous combined applying as formalizations the invariants provided in cognitive
psychology, linguistics and philosophy for thinking processes and knowledge representation. External
concepts insertion into mathematical model allows concepts formal aspects specification with further
applying for abstract mathematical model extension continue up to applied intelligent systems
prototypes.
Every intelligent system components' structure associated with certain knowledge representation
formalism. The compliance is necessary for component formal description and external informal and
semi-structured knowledge and thinking models invariants. Components’ internal life cycles realized
independently from other intelligent system model components. The knowledge processing possibility
demand components’ unified knowledge bases developing.
4. Knowledge transformation and transferring morphisms
Operations of abstract knowledge processing modelled by morphisms for se-mantic hierarchies
formalisms. They form morphisms classes’ hierarchy based on morphisms types variety used within
mathematics different areas. These classes adapted to unified knowledge representation format of
semantic hierarchies formalisms. They define abstract ground for modeling the knowledge processing
by morphisms combinations. Besides that, the knowledge processing morphisms propose basic
classification by ways accepted in mathematics. It includes morphisms of knowledge selecting and
transforming morphisms that model function-al invariants for sets and structures transforming,
algebraic computations and logic inferences as well as knowledge topological properties based
knowledge processing. All these morphisms simulate unary knowledge processing operations with one
exception for configuration direct sum (analogous to composition operation at knowledge
representation formalism) that is binary.
The uniform set of knowledge classes is demanded for morphisms' domains and ranges. These
classes family needs defining in such a way that every morphisms class has certain class’ domain and
range. Such domains and ranges called morphisms’ bases.
Following configurations sets generate well calibrated morphisms bases family as sufficient for
intelligent systems and processes versatile mathematic exploration: unstructured knowledge ( M ),
knowledge unstructured fragments ( DM ), knowledge of given depth ( M k ) with M 0 and M 1 as
elementary and simple knowledge subclasses, knowledge semantic hierarchies ( Σ ), knowledge ordered
and unordered series (sets) ( S and S ), knowledge neighborhoods of different radiuses and selection
predicates described at [9, 11 and 12]. These classes reflect the mathematical models entities adaptations
to uniform knowledge representation structures and experience of intelligent systems multidimensional
structures designing. Union M 0 ∪ M 1 has useful interpretation of abstract ontologies. It settled by
putting in compliance between elementary knowledge with individuals and simple knowledge with
relations between individuals. This implies appearance the new abstract knowledge universal aspects
interpreted as based on individuals and relations at subject domains' ontologies. The knowledge
composition operation ( ), for knowledge representation formalisms, and configuration direct sum
( ⊕ ), for semantic hierarchies’ case, useful for creating special knowledge processing operations bases.
This allows define operations with several parameters and different parameters domains as unary by
parameters' bases compositions (direct sums). Simple convenient format for such composition looks
like (...( B1 B2 ) ...) Bk or (...( B1 ⊕ B2 ) ⊕ ...) ⊕ Bk where B1 , B2 ,..., Bk – considered parameters’
bases.
Example of morphisms’ classes’ hierarchy founded on functional invariants within fundamental
mathematical models presented below (see Figure 3). Every class has exact formal definition and
integrates abstract morphisms adapted to intelligent systems mathematical models.
Figure 1: Morphisms hierarchy for knowledge representation formalisms
Detailed knowledge processing morphisms’ hierarchy description presented at [9]. Knowledge
processing operations offered by given classes realized as morphisms' adaptations to corresponding
abstract or applied knowledge domains. Special schemes are necessary for describing the morphisms
combining. They describe the complex knowledge processing operations. Following morphisms’
properties are important for morphisms adaptation and combining. They represented by concepts of
knowledge based homomorphism and homomorphic extension.
Definition. Mapping h : B 1 → B 2 , where B 1 , B 2 – bases of intelligent systems’ mathematical
model, is called a homomorphism iff
1) ∀z1 , z2 ∈ DM ( z1 z2 → h( z1 ) h( z2 )) ;
1
2) h( z1 ) h( z2 )) .
∀z1 , z2 ∈ DM ( h( z1 z2 ) =
1
Symbols and of last definition represent knowledge representation formalisms’ inclusion
relation and composition operation.
−
Definition. Mapping h : B1 → B2 , where B 1 , B 2 – bases of intelligent systems’ mathematical model,
is called a homomorphic extension iff such homomorphism h + : B 2 → B 1 exists, that
∀z ∈ B 1 ( h + h − ( z ) =z ) .
Possibility for modeling the cross-dimensions intelligent system architecture components
knowledge transferring without general knowledge properties loosing shows significance of abstract
knowledge homomorphisms (homomorphic extensions). Such knowledge transferring are based either
on compressing the knowledge structures by appropriate unified knowledge structures tracings or
knowledge endomorphisms, as knowledge extensions coordinated by the same tracings [11].
Morphisms types proposed above allow exploring the abstract model morphisms combinations by tools
adopted within mathematics. Homomorphic extensions' applied significance defined by possibility of
morphisms' consequent specifications up to knowledge processing operation exact descriptions that
make models of concrete intelligent systems.
Exact definitions for morphisms classes implemented by formal mathematical expressions. They
specify morphisms as mappings with given domains and ranges with predicates satisfied for such
mappings. Abstract expressions represent knowledge family referenced to abstract mind level for
intelligent systems' abstractness dimension.
Descriptions of knowledge processing scenarios contain references to separate morphisms' classes.
Every such class describes stage of goals realization process. Special diagrams present abstract
scenarios. They contain the knowledge processing flows functional descriptions. Separate process
diagram looks like an oriented graph with vertices marked by morphisms classes. The graph edges
describe sequences of morphisms' possible compositions.
This simulates the subject domain tasks solving at intelligent system structural components as
realized by sequences of abstract operations performed over abstract knowledge. Such diagrams initiate
describing knowledge processing general aspects as first stage of applied intelligent system
functionality modeling. Special diagrams describe knowledge processing flows within separate
intelligent systems components. Stages of such diagrams gradual developing from abstract ones to
detailed knowledge based processes. These stages presented by Figure 4.
Figure 4: Levels for knowledge processes diagrams descriptions
The first level of established structure accumulates diagrams for subject domain’s goals
implementation. These diagrams define sequences of activities presented by references on knowledge
processing operations classes. Diagrams, placed on presented structure's second level, extended by
operations domains and ranges vertices. They present operations inputs and outputs. Domains and
ranges may be settled automatically if classes established on first level predetermined as bases for
morphisms' classes presented at diagrams. If morphism classes precisely defined, then class vertices for
domains and ranges of operations inserted into diagrams automatically. Special cases exist when
morphisms' domains and ranges defined as morphisms’ bases compositions (direct sums). Other
automatically checked condition for second level's diagrams relates to inclusion of diagrams'
morphisms' domains and ranges in order of their each after other following. The third level of diagrams'
intends for placing describing the diagrams’ edges properties. The last ones represented by conditional
expressions necessary for control the knowledge flow process through diagram’s vertices. The
conditions roles supply modeling different schemes of diagrams implementations as morphisms'
sequential performing: sequential, parallel, conditional, added with possibility of processes suspension
or halting [7].
The forth level contains diagrams represent completely defined applied knowledge flows processes
for intelligent systems models. Such diagrams constructed by sequences of transformations initiated by
certain diagram on first level. These transformations based on diagrams' changing operations that
reduce uncertainness in diagrams' vertices and edges descriptions. Simple form of unique morphism
selecting consists in assigning fixed operations to separate diagram’s functional vertices given by
computing algorithms or algebraic expressions considered as formulas for morphisms' meanings
computations. Morphisms’ specifications that assigned to diagrams' functional vertices, gradually
detailed by adding new elements to morphisms' classes formal descriptions.
The constructed diagrams have hierarchical structure with functional vertices that represented by
diagrams are allowed under certain conditions. Additional conditions clarify sub-diagrams' external
connections to diagrams vertices linked with functional vertices replaced by sub-diagrams. Several
situations possible at that case. Their uniform descriptions may be concerned on properties of content
placed into external input and output vertices for given vertex that replaced by diagram. Last condition
processed by special procedure that control vertices’ replacement correctness. Diagrams used for
functional vertices replacing developed separately. Their complete descriptions implemented by
sequences of diagrams transformations and presented in multilevel structure of diagrams' constructing
processes. If diagram's functional vertex replaced by diagram then input and output vertices of that
vertex linked to additional correspondence conditions for inserted diagram’s input and output vertices.
The sequence of transforming operations performed one after another generates diagrams’
developing chain. Chains’ general structure looks like a rooted tree with arbitrary finite sets of child
vertices for any internal vertex of the tree Every such tree accumulates different ways of given initial
diagram transformations into its specifications related to different diagrams' presentation levels.
Diagrams' tree make it possible to integrate specialists’ experience of professional tasks solving.
The root vertex of such a tree specifies initial abstract diagram. The tree’s other vertices marked by
descriptions that define used diagrams’ transforming operations. Diagrams’ developing processes
modelled by chains of diagrams sequentially designed by operations assigned to diagrams’ tree vertices.
These vertices form ways in the tree with root as processes' starting vertex. The diagrams tree leaves
specify unique scenarios of tasks solving within intelligent system component's model. The diagrams
defined by trees' leaves and their specifications saved at intelligent systems components’ memory. They
use applied intelligent system for implementing of actual problems’ solving processes at given subject
domain.
Every diagram relates to certain component at intelligent system unified structure. Templates for
diagrams interconnections describe knowledge flows processes that cross dimensions of intelligent
systems unified structure. These flows used for subject domain problems solving, when that knowledge
processing performed by operations related to different intelligent systems’ components. Templates
deal with general models for knowledge flows and realized as cross-dimensions knowledge transferring
based on formats for components connections. Every template’s vertices linked with components and
subject domain tasks solved at these components. It supposed that component’ tasks names linked to
diagrams of tasks’ solving described within components’ ontologies. Such ontologies uniform abstract
structure considered in this paper consist of four general subareas: component’s concepts, properties,
tasks and methods. These subareas at component’s ontologies allow representing necessary knowledge
containing entities and providing these entities applying by component’s agents. These agents perform
controlling the component’s life cycles implementation.
5. Knowledge flows and processes ontology
Processes of templates and diagrams step-by-step constructing for given subject area founded on
system of knowledge that reflects systematic view on templates and diagrams realization stages within
intelligent system direct development. These stages entities integrated by knowledge maps format
define special knowledge flows processes ontology. This ontology relates to entire intelligent system
and used for integrated describing the knowledge flows templates. This ontology allows implementing
by model based on formalism of semantic hierarchies. It uses sets of elementary and simple knowledge
for knowledge flow processes descriptions decompositions. The considered formalism's other
knowledge are synthesized by intelligent system agents when knowledge flows modelling performed.
Ontology offered and implemented by knowledge map. It contains several separate areas. Every area
reflects certain aspect of intelligent systems’ knowledge flows processes formalized descriptions. The
knowledge flows processes ontology easily constructed by linking appropriate classes at these areas.
This ontology compliant with intelligent system unified model entities.
The knowledge maps format belongs to class of semantic networks formats. Maps are convenient
for subject domains’ ontologies modeling by entities that easily transformed into formats adapted to
semantics of descriptive logics. Discussed ontology is made of elementary knowledge entities
(individuals). Individuals' properties represented at ontology by special classes with binary relations
between classes used as basis for knowledge synthesis. The same names for relations between different
pairs of classes allowed in knowledge maps.
Two types of entities possible for elementary knowledge presentation: symbolic names and formal
mathematical expressions. Special semantic hierarchies represent names' and expressions' content with
demanded accuracy. These hierarchies used as elements of knowledge flow processes. Semantic
hierarchies' format used also for knowledge map integrated presenting. The knowledge map special
areas serve as knowledge base about the intelligent system dimensions' types and dimensions' quantized
meanings. We begin with area for knowledge flow processes ontology area for intelligent systems’
goals and components. Possible variant of knowledge map for this area presented below (see Figure 5).
Figure 5: Intelligent system components ontology area
The dimensions’ names and dimensions’ quants classes defined by dimensions’ quants' sets linked
there by relation has. Dimensions names and their quantized meanings putted into knowledge map as
two separate rectangles with the cut-off corner (see Figure 5). The dimension quants' meanings ordered
by relation next. Intelligent system components' positions uniformly described by components’ position
into considered four dimensions' components' model. Components' goals class contains names for
intelligent system goals functionally correspondent to components. This correspondence modelled by
relation has. Goals' class additional ordering realized by is a part of relation. The discussed knowledge
map area other classes are intended for goals’ actuality estimating. These classes' elements used for
recognizing goals that need activating and contain conditions of goals activating.
Separate conditions represented as mathematical expressions. They linked with goals and additional
parameters introduced for subject domains content measurable attributes. Such attributes meanings
reflect situations essential parameters that possible at subject area. The considered knowledge classes
form ontology's separate fragment. It includes homogenous classes of completely atomized knowledge
representations for knowledge flows processes content based on mathematical expressions and entities'
names. This area supports modeling the cognitive operations of knowledge understanding,
remembering, applying, ranging, analysis and synthesis [1, 4, and 10].
Templates' and diagrams' ontology main subarea accumulates simple knowledge that allow generate
knowledge flows processes' full formal descriptions. Knowledge flows processes algorithmic modeling
is similar to thinking processes over mathematical and linguistic content. Let us consider knowledge
map example that is possible for such subarea (see Figure 6).
Figure 6: Subject Domain Templates’ Ontology Area
Ontology's subarea description presented by Figure 6. It integrates subject domain experience about
knowledge flows processes structures and knowledge transfer between components. Considered
ontology area contains new class Subject area goals' templates. Templates linked externally with
subject area goals class. They initiate every template as network composed of Templates’ components
class’ elements. They associated with templates implementing stages at intelligent systems separate
components [8]. Every template description defines knowledge flows processes as abstract intelligent
system components sequences. Template described as components' sequences includes specifying the
knowledge processing diagrams that selected from diagrams’ sets assigned to components. Every
intelligent system's template component identify unique diagram for knowledge processing within this
intelligent system component.
Templates' components sequences, implementing performed by intermediate class of Transferred
knowledge descriptions. Last class elements are mathematical expressions that describe representation
formats for knowledge that transferred between template's linked components. Every such expression
specifies transferred knowledge algebraic structure as delivered from next component relation first
element to the second one. Such structures look as diagrams' operations ranges (domains) compositions.
These compositions defined separately and supplied with correspondence that define knowledge
transfers between diagrams' operations' domains and ranges.
The considered templates and diagrams ontology last subarea accumulate knowledge about separate
diagrams' structures (see Figure 7). The Diagrams class within given knowledge map is the same as at
Figure 6. Every separate diagram belongs to only one intelligent system’s component. It associated with
this component by relation uses. The diagrams' consecutive transformations represented by transformed
by relation. Last relation establishes sequences of diagrams' variants (see Figure 4).
This relation serves for developing the diagrams' structures transforming tools. It supply diagrams
with developing possibility expressed as diagrams linked by intermediate Transformations Descriptions
class elements. The considered ontology allows extension by additional classes used for remembering
the formal descriptions of diagrams' transforming operations. That imply possibility of mathematical
expressions special processing that supply intelligent systems by additional tools for diagrams and
templates' control.
Figure 7: The diagrams knowledge map
The diagrams’ names class (Diagrams) presents complete diagrams set. The class elements used as
roots vertices at diagrams’ descriptions and synthesized of their elements' descriptions. Diagrams
transformations linked by relations: extended into and transformed into. Entities of diagrams' structures
remembered by their names at class Diagrams Elements. Separate diagram’s structure described by its
elements sequences, connected by next relation (elements following one after another). The relation
initiated by used for describing sequences' initial elements attaching to diagrams. The Diagram's
Element Types (condition, operation or operation base) assigned by relation has. This relation applied
for linking Diagram's Elements with class Diagrams’ Elements Types entities. Ontology knowledge
map describes diagrams’ elements with different type’s meanings by correspondent ways. When
element is a condition then it described by mathematical formula that represents condition relevant with
considered expression role. Such role assigned as condition property and represented by appropriate
Expressions Roles class entity. Conditional expression supposes own ways for correspondent formulas
processing within diagrams’ analysis algorithms.
The next examples for conditional expression’s roles are possible: independent, disjunctive, blocking
or accumulative. Conditions roles allow modeling the sequences of performed operations as
implementations controlled by attributes. Operations diagrams’ elements properties presented by
elements’ references with Knowledge Processing Operations class entities. The last class contains
names for operations. It linked with Operations Bases class by relations domain and range with names
of bases assigned to appropriate operations. Operations names linking to correspondent mathematical
expressions defined by relation presented by. These expressions used for describing the operations’
formal properties.
The Algorithms class initiates computable procedures descriptions intended to different applications’
cases. Algorithms can either compute operations and mathematical expressions meanings or analyze
their properties. The Algorithms Roles class entities represent algorithms' applications ways and cases.
If operation has effective algorithm for realization then such operation name also linked with
appropriate operation realization algorithm by applying the relation realized by. When some algorithm
realizes certain operation formal description then it joined with mathematical expression by applied for
relation.
6. Final conclusions
The intelligent systems’ explored model based on components' universal and unified structure. It
deals with searching the borders for intelligent systems' effective mathematical foundations. They
reflects theories of thinking processes and memory structures offered within different knowledge areas
(cognitive science, linguistics, systems engineering and philosophy). These areas invariants’ formal
descriptions allow applying the mathematics possibilities for creating new technologies of intelligent
systems designing.
Two invariants considered as basic for intelligent systems theory. The first one connected with class
of knowledge representation formalisms. This invariant's importance is determined by existing formal
models’ variety that develops extensively and quickly, without serious mathematical exploring the
knowledge representation formalism’s concept. Formalisms of semantic hierarchies propose unified
universal formats for modeling the knowledge representation and processing. They allows integrating
and joint applying for knowledge aspects studied at different subject domains. The second proposed
invariant relates to intelligent systems’ homogenous mathematical structures. The submitted paper's
concept of knowledge flows processes was inspired by K. Stanovich idea of knowledge processing
levels with one universal dimension at intelligent systems' structures [2]. The two-dimension intelligent
systems structure analysis preceded four-dimensional systems’ structure developing. It revealed
templates and diagrams’ importance for knowledge flows processes modeling. This fact explains
knowledge flows templates and diagrams significance for the initial stages' of intelligent systems
designing technology.
Acknowledgements
The reported study was funded by RFBR, and administration of Krasnodar territory grant project
number № 19-41-230008 and by RFBR grant project number № 20-01-00289.
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