This article proposes the fuzzy induction method developed by the author as a combination of the provisions of fuzzy mathematics and the theory of fractals, introduces the concept of the degree of recursion of a fuzzy set, describes the incomplete recursion of a set as its fractional dimension for modeling the subject area. As the scope of the proposed method and the knowledge models created on its basis as fuzzy sets, the management of the life cycle of information systems, including the development of use cases and software testing, is considered.
Topicality
In the process of designing and developing, implementing and operating information systems, it is necessary to accumulate and systematize data, information and information that are collected from outside or arise at each stage of the software life cycle. This serves as the necessary information and methodological support for design work and decision-making, and is especially relevant in situations of high uncertainty and in semi-structured environments. The knowledge base formed as a result of the accumulation and systematization of such resources should not only be a source of useful experience gained by the project team in the course of creating an information system, but also the simplest possible means of modeling new visions, methods and algorithms for implementing project tasks. In other words, such a knowledge base is a repository of intellectual capital and, at the same time, a knowledge management tool [3, 10].
The efficiency, usefulness, quality of the knowledge base as a tool correlate with the resource intensity of its maintenance and the effectiveness of knowledge extraction. The simpler and faster the collection and fixation of knowledge in the database and the more pertinent the results of queries to it, the better and more reliable the tool itself [1, 2]. However, discrete methods and structuring tools that are applicable to database management systems, including the normalization of relations in relational databases, do not allow describing or modeling semantic components, interpretations, interval and continuous semantic sets [4, 7, 10]. This requires a methodological approach that generalizes particular cases of finite ontologies and brings the knowledge model closer to the continuity of the description of the subject area of ββthe information system.
Such an approach can be a combination of the provisions of the theory of fuzzy mathematics and the concept of fractal dimension [3, 6]. By optimizing the description of knowledge according to the criterion of the degree of continuity (the size of the discretization step of the description) under conditions of limitation according to the GΓΆdel incompleteness principle (in the information system - the fundamental incompleteness of reasoning, knowledge derived from this system, provided that it is consistent), performing sequential fuzzification (reduction to fuzziness), we obtain a formalized description that displays a certain array of knowledge as fully and coherently as possible, and with which it is possible to perform any operations of information processes - collection, storage, processing and transmission [5, 8, 9].
Definition of fuzzy set recursion
Let X be the set of values ββof some characteristic of the modeled system:
(1)
where n = [N β₯ 3] is the number of values ββof such a characteristic (more than the elementary set (0; 1) - (false; true)).
Let X = B, where B = {a,b,c,β¦,z} is the set of equivalents corresponding element by element to the set of values ββof characteristic X.
Then the fuzzy set , which corresponds to a fuzzy (in the general case) concept that describes the characteristic X, can be represented as:
(2)
where m is the description discretization step, i belongs to N is the step multiplicity.
Accordingly, in order to optimize the model of knowledge about the information system according to the criterion of continuity (softness) of the description, while remaining within the boundaries of the space of incompleteness of reasoning, we introduce fuzzy set recursion degree and get the following representation:
(3)
where - a set corresponding to a fuzzy concept, in the general case, describing the characteristic X more fully than the set , according to the criterion of softness; Re is the degree of description recursion.
It should be taken into account that (reducible to a crisp set) in a special case, if necessary.
Introduction of fractional dimension
For Re = 1, the set is an ordinary fuzzy set of the 2nd degree, including as elements fuzzy sets (or their crisp mappings) that describe all values ββof the characteristic X [1, 2]:
(4)
However, this is a degenerate case, and in the most complete representation, some of the elements can be sets, while the rest are trivial (extremely simple) objects. Therefore, to define such a set, it is necessary to introduce fractional recursion β an analogue of the fractional dimension of space (in this context, the ontology space of a certain subject area) [3, 9].
When Re is fractional, we get the following record :
(5)
where β fuzzy set for value X1, is a fuzzy set for the value X2, etc.
In this case, the recursion becomes essentially fractal, and the sets of descriptions become self-similar.
Defining a set of module functionality
The architecture of an open information system assumes the principle of modularity, which provides the possibility of scaling, replication, adaptability and emergence of the system. Modular construction makes it possible to bring the technological implementation of information processes as close as possible to their natural objective implementation in the real world, to develop the most convenient tools in terms of their functional properties, designed not to replace people, but to effectively help them in knowledge management.
A module is a separate entity of an information system, which may be mandatory or optional for the purposes of the system's existence, but in any case provides a set of functions that is unique within the boundaries of the system.
The whole variety of module functionality can be described by three types of operations: creation (writing new data), editing (changing previously recorded data), deleting (erasing previously recorded data).
Let X be some characteristic of such functionality, then the corresponding set X can be represented as:
(6)
where X1 is creation, X2 is editing, X3 is deleting,
(7)
At the same time, the functionality of any module is such that the creation of data is not self-similar (implemented without recursion - the creation function does not repeat itself), and editing and deletion in the general case can involve both element-by-element implementation (performing an operation on selected elements of data sets) and themselves include operations similar to themselves.
It should be noted that if the operation for the functionality X is not performed in this module (not implemented in the system), then the set corresponding to such an operation is considered as empty.
Thus, to describe the fuzzy concept (statement) βthe module allows you to perform an operation with the corresponding data set for the purposes of the information systemβ, the fuzzy set in the simplest case, it can be represented as:
(8)
Such a set generally has a degree of recursion equal to 1,6(6) and is fractal and fuzzy at the same time.
Preparing usage scenarios and module testing
At the stages of development and operation of an information system, special scenarios are needed that describe the order and content of operations for using the modules according to their functional purpose (use-cases), as well as to check the compliance of the expected and actual results of the modules (testing scenarios). .test-case).
Based on the views outlined above, the process of working on such scenarios can be described as follows.
For the module, a fuzzy set is formed :
(9)
where
β fuzzy set for the operation of creating data on functionality X;
β a fuzzy set for the operation of editing data on the functionality X, while the degree of recursion a (feature nesting) is a natural number and in the trivial case is equal to 1;
β a fuzzy set for the operation of deleting data on the functionality X, while the degree of recursion b (embedding of the function) is a natural number and in the trivial case is equal to 1.
This set describes what exactly (what data objects) are created, edited and/or deleted for any use of the module.
A set of Ux usage scenarios for the X functionality for the module in question is then compiled, each of which describes how why (for what business task) do they create, edit and/or delete data objects described by the set , and in what order:
(10)
where n is the number of use cases for X.
Next, a set of test scenarios Tx on the functionality of X is compiled for each use case of the module under consideration. The test script describes, what data values ββand in what order are used when executing the use case, as well as what result should be obtained:
(11)
where [D] is an array of test data, n is the number of test scenarios for X.
In the described approach, the number of test scenarios is equal to the number of corresponding use cases, which makes it easier to work on their description and updating as the system develops. In addition, such an algorithm can be used to automate the testing of software modules of an information system.
Conclusion
The presented fuzzy induction method can be implemented at different stages of the life cycle of any modular information system, both in order to accumulate the descriptive part of the knowledge base, and in working on scenarios for using and testing modules.
Moreover, fuzzy induction helps to synthesize knowledge based on the obtained fuzzy descriptions like a βcognitive kaleidoscopeβ, in which some elements remain clear and unambiguous, while others, according to the self-similarity rule, are applied the number of times specified in the degree of recursion for each set of known data. Together, the resulting fuzzy sets form a model that can be used both for the purposes of the information system and in the interests of searching for new knowledge in general.
This kind of methodology can be attributed to a peculiar form of "artificial intelligence", taking into account the fact that the synthesized sets should not contradict the principle of incompleteness of reasoning and are designed to help the human intellect, and not replace it.
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Source: habr.com