Abstract
This article introduces an approach to the automated generation of special algebras through genetic algorithms. These algorithms can be also used for a broader variety of applications in mathematics. We describe the results of research aiming at automated production of such algebras with the help of evolutionary techniques. Standard approach is not relevant due to the time complexity of the task, which is superexponential. Our research concerning the usage of genetic algorithms enabled the problem to be solvable in reasonable time and we were able to produce finite algebras with special properties called EQ-algebras. EQ-algebras form an alternate truth–value structure for new fuzzy logics. We present the algorithms and special versions of genetic operators suitable for this task. Then we performed experiments with application EQ-Creator are discussed with proper statistical analysis through ANOVA. The genetic approach enables to automatically generate algebras of sufficient extent without superexponential complexity. Our main results include: that elitism is necessary at least for several parent members, a high mutation ratio must be set, optional axioms fulfilment increases computing time significantly, optional properties negatively affect convergence, and colorfulness was defined to prevent trivial solutions (evolution tends to the simplest way of achieving results).
Highlights
Genetic algorithms (GA) are among the basic methods used to solve optimization problems
This paper presents the results of the use of genetic algorithms to search for the structure of truth values in a new class of algebras called EQ-algebras
The example scheme for 10 members m[0]..m[9] sorted by fitness function from best to worst is demonstrated on Figure 4; Weight of optional properties—relative weight of special EQ-algebras requirements—should be significantly lower than essential axioms weight; Notion of colorfulness—required distinct elements incidence in non-trivial positions as operator function values; Colorfulness assures non-trivial EQ-algebras to be generated, e.g., for fuzzy equality when 3 out of 5 required—at least 3 different elements occur as functional values in non-determined cases; colorfulness experimentally needed for Multiplication (⊗) and Fuzzy Equality (∼)—
Summary
Genetic algorithms (GA) are among the basic methods used to solve optimization problems. This paper presents the results of the use of genetic algorithms to search for the structure of truth values in a new class of algebras called EQ-algebras. Development and experiments concerning the usage of these optimized evolutionary techniques are discussed in the article. The computer application working through the methods described in the article (software package EQCreator), which produces EQ-algebras (equality based structures usable as truth values algebras) in a feasible time. This article shows a formulation of the problem at first and a description of EQ-algebras basic principle. We describe in detail our application of evolutionary principles, especially Genetic Algorithms and the implementation of the presented methods in the form of computer application EQCreator [2]. We present experiments demonstrating the time efficiency of the presented evolutionary approach as well as the main issues with an impact on the efficiency of the method
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