Virtual Center for the Paradigmatic Studies

The aim is for the prepared publication to have theoretical and practical values, essential for the development of the future monograph, and for its direct practical applications at the Virtual Center for Paradigmatic Studies, prepared for the needs of the Faculty of Management Warsaw University of Technology. In order to achieve academic and civic goals – our intention is an open formula of work for the benefit of people studying or significantly interested in the research areas specified in the article.

Table 1 defines three key research areas (A, B, and C) for the Virtual Center for Paradigmatic Studies, which each covers five operational programs, the implementation of which will be based on the processing of paradigmatic data

Paradigmatic data – data intended to work with semantic models and the characterization principle.

A team of seven leading editors was invited to implement individual programs. Some of them (T. Krupa, E. Kulińska, T. Ostrowska, J. Patalas-Maliszewska, M. Wiśniewski) were the authors of the monograph published in 2019: Strategic Risk Management Against Hazards for Safety of Processes in Critical Infrastructures, developed under Management Sciences Series Vol. VIII, published by the Faculty of Management of the Warsaw University of Technology.

The team of leading editors of the planned monograph entitled: The Virtual Paradigmatic Studies of Characterization Theory will be (in alphabetical order): Tadeusz Krupa, Ewa Kulińska, Krzysztof Maj, Teresa Ostrowska, Justyna Patalas-Maliszewska, Tomasz Prokopowicz, Michał Wiśniewski.

The article presents V.A. Gorbatov's characterization theory – the most significant cyberneticist in the last 100 years, a member of the Russian Academy of Sciences, a specialist in solving seemingly unrecognizable problems. Life and modern technologies impose similar challenges on us almost every day, and responsibly, we cannot leave them unanswered.

Gorbatov's characterization principle

The concept of the characterization principle was born in the 1980s in work on the synthesis of logical structures.

The essence of the characterization principle is the flawless adequacy of the functioning of logical calculations and the structure of connections of the logical computational elements that implement them – in situations where their correctness is proven by the theory of the characterization principle and not by simulation.

The characterization principle, the formal description of which is contained in Section 3, takes into account the model of functioning Ψ_a, the model of structure Ψ_b, and forbidden graph figures (Q^A)

Q^A – Forbidden graph figures of type Q^A (Fig. 3).

The essence of the characterization principle is illustrated by an example expressed by the formula of the sequence of sentence variables (SSV)

SSV – sequence of sentence variables, can be built from symbols of evaluating events X_i, X_j, X_k ... or letter evaluation events a, b, c, ... connected by symbols of serial sequences (&) and connected by symbols of concurrent sequences (#), for example, X_i & X_j & X_k ... or a # (b & c) # - respectively.

The sequence of sentence variables consists of sentence variables (SVs) and is treated as an evaluation tool (used to determine the available evaluation events): increasing or decreasing the available potential. This may relate to issues such as prioritization of insurance decisions, construction of a communication network, or the specific example of Model of 5 Projects, presented in Section 3.

The conducted research shows that the theoretical correctness of the model of functioning Ψ_a is not always absolute, which is presented in Section 4.

For the presentation of the above-mentioned evaluation events, which are also the model of structure Ψ_b, the graphic form is used in the form of Hasse diagrams, the use of which is presented in Section 5.

Section 6 is a summary of the existing and further activities in the research areas defined at the beginning.

The formal record of Gorbatov's characterization principle (predicate) is as follows: (1)<Ψa, Ψb, P0(Ψa,Ψb)>< {\Psi _{\rm{a}}},\,{\Psi _{\rm{b}}},\,{{\rm{P}}_0}({\Psi _{\rm{a}}},{\Psi _{\rm{b}}}) > where:

Ψ_a – model of functioning,

The essence of the P₀ predicate is to show the mutual unambiguous (bilateral) mapping of both models.

The model of functioning Ψ_a written in the form of an sequence of sentence variables (SSV) is expressed by the formula: (2)[X11 & X21 &…& Xm1]1#[X12 & X22 &…& Xm2]2#…#[X1n & X2n &…Xmn]n\matrix{{{{\left[ {{{\rm{X}}_{11}}\,\& \,{{\rm{X}}_{21}}\,\& \ldots \& \,{{\rm{X}}_{{\rm{m}}1}}} \right]}_1}\# } \hfill \cr {{{\left[ {{{\rm{X}}_{12}}\,\& \,{{\rm{X}}_{22}}\,\& \ldots \& \,{{\rm{X}}_{{\rm{m}}2}}} \right]}_2}\# \ldots \# } \hfill \cr {{{\left[ {{{\rm{X}}_{1{\rm{n}}}}\,\& \,{{\rm{X}}_{2{\rm{n}}}}\,\& \ldots {{\rm{X}}_{{\rm{mn}}}}} \right]}_{\rm{n}}}} \hfill \cr } or its equivalent semantic graph (Fig. 1)

Figure 1

The model of functioning Ψ_a:

Model of functioning Ψa – a key model of Gorbatov's characterization principle, showing how the model of functioning is created, expressed using letter symbols and connecting symbols, for example, linking sentence variables or letter symbols with symbols connecting #, &, V, and others, which were given appropriate linking interpretation.

The formal description of the characterization principle will be determined using the model of structure Ψ_b presented in the form of Hasse diagrams

Hasse diagram – is a graph model of the SSV built from vertices and connecting them directed (oriented) edges of evaluating events.

Universal SSVs are linked together by operations that perform actions: conjunction serial (&), concurrent serial (#) operations, and extremely complicated strong alternative operations

The concept of “strong alternative” refers to a phenomenon established in the form of a contract concluded between partners for which the SSV can be composed of symbols of evaluation events X_i, X_j, X_k ... or letter evaluation events a, b, c..., which can be used to control serial sequences (&) and symbol-connected concurrent sequences (#), for example, X_i & X_j & X_k ... or a # (b & c) # - respectively.

Symbol X_ij – which means the occurrence of the sentence variable X_i inside the j-th sequence with the possibility of repetition in different SSVs. SSV concurrency also allows for alternation or parallelism.

The value of the variable X_i can be homogeneous objects of the modeled problem (e.g., events, logical variables, identifiers of specific resources, etc.). The & symbol denotes serial linking of sentence variables within each SSV.

The sequence of symbols [X_1j & X_2j &… &X_mj] – is usually taken as the j-th homogeneous of m-element sequence SSV.

# – is a symbol of concurrency separating homogeneous SSV; additional restrictions may be imposed on the serial ordering of the SSV in the functioning models, only in terms of positioning in the SSV (any or a priori position, for example: “only in the first place in each sequence” or in some other predetermined random, descriptive, or deterministically determined).

A particularly important limitation is the inadmissibility of a situation in which an SSV is identical to another SSV or is included in another SSV, regardless of the ordering of the sentence variables in the SSV and regardless of the arrangement of individual SSV among themselves.

At this point, we will consider an example of the SSV called the “Example of the Semantic Model of 5 Projects”, which in the first step is presented in Fig. 1 in the form of a semantic graph.

The terms “project” and “object” are used interchangeably – rather with an indication of the term “project”, because the term “object” allows for a more flexible formulation of the interpretation of the terms used.

The Semantic Model of 5 Projects is used to indicate routine activities used for a relatively simple procedure to build a graph semantic model that unambiguously defines the connections between vertices and edges of the semantic graph model.

Based on the preliminary preparations made, we will try, using a graphic notation, to present a semantic model consisting of five vertices and eight edges in order to fully formalize the graphic form of the constructed semantic model, presented in Fig. 1.

In the example of the Semantic Model of 5 Projects, the carrier M of the SSV model is a fixed set of sentence variables: (3)M={b,c,d,e,f,g}{\rm{M}} = \left\{ {{\rm{b}},{\rm{c}},{\rm{d}},{\rm{e}},{\rm{f}},{\rm{g}}} \right\}

On the set M, five SSVs of the model Ψ_a were defined, corresponding to five concurrent projects (objects):

Projekt 1.: [X₁ & ... & X_m]₁ = [e, g]

Ultimately, we get five Hasse twigs examples for a set of concurrent SSV designs.

Semantic Model of 5 Projects has been correctly defined, which means that:

none of the five projects is repeated, and none is part of another project,

In Fig. 2, we consider the sequence of the initially available evaluation events belonging to Hasse twigs for a set of concurrent SSV.

Figure 2

The essence of individual projects has not changed – while the form of some Hasse twigs makes it impossible to build a uniform Hasse diagram due to unexpected obstacles – some of the twigs have become fragments of the model of functioning Ψ_a, called forbidden graph figures (FGF).

The vertices of forbidden graph figures FGF of type Q^A (Fig. 3) and type Q^B (Fig. 4) correspond to the sentence variables X_i.

Figure 3

Figure 4

In the case of Q^A graph figures, the sequences of sentence variables SSV should be of odd lengths: 3, 5, 7, 2n + 1; in the case of a Q^B graph figure, it appears only in the form of the so-called “snowflake.”

It turns out, however, that the theoretical correctness of the model of functioning Ψ_a is not always sufficient for the correctness of the semantic model and, consequently, for the construction of a model of structure, as illustrated in Fig. 5.

Figure 5

Often, it becomes necessary to correct the model of functioning Ψ_a in accordance with the Gorbatov predicate P₀ (Ψ_a, Ψ_b), consisting in the search for and liquidation of FGF Q^A and Q^B classes.

It is worth noting that when assessing the computational complexity – the number of different combinations of the SSV of the Model of 5 Projects is 288 and may increase exponentially, especially in the case of similarity or overlapping between FGF: |[e,g]|! x|[b,c,d]|! x|[d,e]|! x|[c,e]|! x|[b,c,f]|! =2! x 3! x 2! x 2! x 3! = 288\matrix{{\left| {\left[ {{\rm{e,g}}} \right]} \right|!\,{\rm{x}}\left| {\left[ {{\rm{b,c,d}}} \right]} \right|!\,{\rm{x}}\left| {\left[ {{\rm{d,e}}} \right]} \right|!\,{\rm{x}}\left| {\left[ {{\rm{c,e}}} \right]} \right|!\,{\rm{x}}\left| {\left[ {{\rm{b,c,f}}} \right]} \right|!} \hfill \cr { = 2!\,{\rm{x}}\,{\rm{3!}}\,{\rm{x}}\,{\rm{2!}}\,{\rm{x}}\,{\rm{2!}}\,{\rm{x}}\,{\rm{3!}}\, = \,288} \hfill \cr }

In the case of the Model of 5 Projects, containing five sentence variable SVs, the computational complexity increases to the value of 1,205 = approximately 250 million variants. It is obvious that the number of SSV variants of this size can be troublesome even for terahertz processors.

Based on the example of Model of 5 Projects (SSV) included in Fig. 1, a semantic graph was presented, which will be used to analyze the correctness of the model Ψ_a by identifying forbidden graph figures (FGF) presented in Figs 3 and 4.

The analysis of the graph figures presented in Figs 3 and 4 allows us to check whether the individual FGF does not coincide by way of coincidence. This, in turn, enables, by splitting a single sentence variable (SV), to split the FGF multiple times. The correctness of the model Ψ_a guarantees obtaining the correct Hasse diagram (without FGF).

Unfortunately, among the 288 possible combinations of s, if we do not identify all FGFs and do not split them (i.e., we do not eliminate them), we can – with sweat on the forehead – find out that we will not find a complete Hasse diagram, which will be similar to the diagram in Fig. 6.

The last stage of building the Model of 5 Projects is to present the solution in the form of an SSV diagram and Hasse diagram, similar to Fig. 5 – but correct, thanks to the application of the characterization principle (by liquidation of the FGF.

The SSV set for the Model of 5 Project example, as a result of dividing one sentence variable d into d'' and d', will obtain the form: (5)[g & e]1# [c & b & d″]2 # [d′ & e]3 # [c & e]4 # [c # b # f]5\matrix{{{{\left[ {{\rm{g}}\, \&\,{\rm{e}}} \right]}_1}\,\# \,{{\left[ {{\rm{c}}\, \&\,{\rm{b}}\,{\rm{\, \&\,}}\,{\rm{d}''}} \right]}_2}\,\#\, {{\left[ {{\rm{d'}}\,{\rm{\, \&\,}}\,{\rm{e}}} \right]}_3}\,\#\, } \hfill \cr {{{\left[ {{\rm{c}}\, \&\,{\rm{e}}} \right]}_4}\,\# \,{{\left[ {{\rm{c}}\,\#\, \,{\rm{b}}\,{\rm{\,\#\, f}}} \right]}_5}} \hfill \cr }

Thanks to the application of the characterization principle, the necessity to generate about 300 possible solutions was avoided in order to find out that there is no possibility of obtaining SSV without splitting at least one sentence variable, presented in Fig. 6.

Figure 6

Comparing with the example of Hasse diagram construction shown in Fig. 5 for the Model of 5 Projects - you can see different positions of the vertices and their connections. Hasse twigs are clearly visible.

The Hasse diagram proposal will move from the model of functioning Ψ_a to the model of structure Ψ_b according to Gorbatov's characterization principle.

The building of the model of functioning will follow a plan that includes the following activities: 1)

Specifying the five-letter carrier M = {a, b, c, d, e}. The carrier M, which is small in number, is a condition for the effective implementation of the model of structure Ψ_b, as we will see during the implementation of the semantic model and the following stages.

Figure 7

Figure 8

Fig. 9 illustrated the dynamics of “spinning” graph forbidden figures of Q^A type.

Figure 9

The aim of the article The Virtual Center for Paradigmatic Studies was to present the principle of Gorbatov's characterization theory and its development as a paradigm for many applications. The basic concept was developed in the 1980s in the work on the synthesis of logical structures.

The essence of the principle of characterization theory is flawless isomorphic operation and the structure of adequacy in situations where the correctness of the research is confirmed by theory, not simulation.

The article gives an example of the cyclic mechanism of the characterization principle – this phenomenon requires further research due to its unexpected effects and risks.

The included educational examples mainly cover design issues, educating the academic staff to prepare doctoral and postdoctoral dissertations.

These examples are an attempt to synthesize Hasse diagrams at the Center for Paradigmatic Studies, organized at the Faculty of Management Warsaw University of Technology by Prof. Tadeusz Krupa and his successors.

This subject is also popularized by the international journal Foundations of Management (run by the Faculty of Management Warsaw University of Technology) and a group of leading chairs, habilitated doctors and doctors engaged in independent and advanced research on practical applications, largely inspired by the Faculty of Management at the University of Warsaw.

The author would like to thank the Deans and Employees of the Faculty of Management at the University of Warsaw for many years of support and cooperation.