Algebraic theory of structures of truth values for fuzzy logic
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1. Introduction to the problem
Classical logic is based on the lattice structure of the simplest non-degenerated lattice L = {0,1}. Fuzzy logics are based mostly on the lattice L = [0,1] equipped with some special functions and the algebraic structure of such tuples (e.g., BL-algebra, MTL-algebra, etc.) heavily influence the corresponding fuzzy logic. Deeper knowledge of algebraic structures of truth degrees is thus an important step towards the description of the relevant fuzzy logics as well as to the possible fields of applications. Related areas of fuzzy approximation and fuzzy modeling also requires deeper understanding of the fuzzy numbers and fuzzy measures (or fuzzy densities) theories.
2. Focus of our research
Our research in this item is focused (not exclusively) on:
- Study various algebraic structures and properties of substructures of these algebraic structures for fuzzy logic, their embedding, the role of irrationality and transcendence.
- Study of behavior of sequences of its objects with respect to various types of metrics; study of fuzzy measures and fuzzy integrals and subsequently of the uncertainty and regularity measures of distribution of the sequences and their interrelations, algebraic and linear independence of the sequences.
- Research and development of number approximation methods and their application in the study of the indeterminacy phenomenon.
3. Description of the main results
In the study of various algebraic structures for fuzzy logics we examined various structures and obtained several interesting and important results. For example, in [1] we have shown the redundancy of some fuzzy logical connectives in special cases such as the standard negation and the Hamacher product conjunction. Lipschitzianity of fuzzy conjunctions was studied in several papers, see [2,3,4,5].
New types of algebraic structures possibly leading to the new types of fuzzy logics were introduced in the next two papers under preparation. In [7] we have introduced and studied fuzzy logics where the residual implication has special Lipschitz property, what correspond to quasi-copulas/copulas as conjunction evaluations. Next, in [31] we continue in the study of implications related to (associative) copulas.
Lattice-based conjunctions and other connectives for fuzzy logics (in Goguen sense) were studied in [21, 23, 34, 35].
Deeper look on preservation of some properties of fuzzy relations under aggregation in [36] enabled us to characterize the aggregation operators preserving the T-transitivity of fuzzy orders and fuzzy equivalence relations.
Some applications of fuzzy set theory and fuzzy logic were shown to be useful in multicriteria decision making in [22].
Linearity of fuzzy integrals is rather restrictive requirement as shown in [24].
New types of fuzzy measures – universal fuzzy measures - were introduced and studied in [25, 26].
In paper [27] the asymptotic behaviour of universal fuzzy measures is studied. Using the method of convergence with respect to an ultrafilter it is described a method producing from a given universal fuzzy measure a measure on the set of all subsets of positive integers which preserves „nice“ properties of the original universal fuzzy measure.
In [9] a system of fuzzy rational numbers is introduced and used to approximate irrational numbers by means of this system. Irrational numbers are classified with respect to this approximation. The obtained results are in correspondence with results in Diophantine Approximation Theory.
In [10] the concept of asymptotic distance of two number sets is introduced. By means of this concept new criteria for series in order that their sums are Liouville numbers (i.e.numbers which are the best approximable by rational numbers) are derived.
In [29] it is analysed the case when the asymptotic distribution function of a block sequence of a given set of positive integers exists. Necessary and sufficient conditions for the occurrence of this case are found and some new relations among various characteristics of distribution and size of sets are derived.
In paper [32] complete relations between values of dispersion and the density of block sequence in the interval (0,1) were derived. Further, strict bounds for possible values of dispersions of certain types of block sequences with respect to characteristics of the structure of the original sequences of positive integers.
In paper [6] a class of densities given by weight [-1, ∞) is studied. It is proved thatÎfunctions f(n) = np , p both the upper and lower densities depend continuously on the value of parameter p > 1. On the other hand, discontinuity of both densities at p = -1 can occur.
In [11] a system of fuzzy rational numbers is used to classify irrational numbers with respect to the quality of their approximation by the given system of rationals. Some properties of these approximations are studied.
In [28] it is presented a construction defining a measure on the complete power set of positive integers from a given universal fuzzy measure and preserving a large class of properties of the original universal fuzzy measure.
In [8] we introduce the so called irrational measure of sequences and prove the irrational measure for special sequences. Several criteria for transcendence of certain infinite series of rational numbers can be found in [13, 14]. If the sequences of positive real numbers converge to infinity very quick then we can define the linearly unrelated sequences. Several criteria of linearly unrelated sequences are included in [18, 19].
Some criteria for irrationality of the sums of infinite series we can find in [14]. Here the numerators are integer part of functions with the convergence to infinity less then polynomials. If the number is well-approximated by rationals then we call such a number Liouville. Similar concept can be defined for sequences. Such criteria we can find in [15]. Some criteria for the irrationality of the sums of infinite series can be found in [16, 17]. In [12] the concept expressible set of infinite sequence is introduced with the criterion when the expressible set has Hausdorf dimension equal zero. This result is proved only for the sequences with rational numbers.
A special algebraic structure called EQ-algebra is introduced in the papers [29], [30]. This is a new algebra developed as a structure of truth valued for the fuzzy type theory (a higher-order fuzzy logic). The EQ-algebra is a lower semilattice ordered monoid extended by a fuzzy equality and fulfilling 5 special properties. Special properties of this algebra are intensively studied.
REFERENCES
[1] CINTULA, P., KLEMENT, E., MESIAR, R., NAVARA, M. Residuated logics based on strict triangular norms with an involutive negation. In Math. Logic Quarterly. 52. vyd. 2006, roč.52, sv.3, s.269-282, ISSN 0942-5616.
[2] GHISELLI-RICCI, R., MESIAR, R. k-Lipschitz strict triangular norms. In EUSFLAT-LFA 2005. 7.9.2005-9.9.2005 Barcelona. Technical University of Catalania, Barcelona, 2005. s. 1307-1312. ISBN 84-7653-872-3.
[3] GHISELLI-RICCI, R., MESIAR, R. On the Lipschitz property of strict triangular norms. Int. J. General Systems 36 (2007) 127-146.
[4] GHISELLI-RICCI, R., MESIAR, R., MESIAROVA, A. Lipschitzianity of triangular subnorms. In IPMU'2006. 2.7.2006-7.7.2006 Paris. Paris : Editions E.D.K., 2006. s. 671-677. ISBN 2-84254-112-X.
[5] GHISELLI-RICCI, R., MESIAR, R., MESIAROVA, A. Lipschitzianity of triangular subnorms. In IPMU'2006. 2.7.2006-7.7.2006 Paris. Paris : Editions E.D.K., 2006. s. 671-677. ISBN 2-84254-112-X.
[6] GIULIANO-ANTONINI, R., GREKOS, G., MIŠÍK, L. On weighted densities, to appear in Czech Math. Journal.
[7] HÁJEK, P. , MESIAR, R. On copulas, quasi-copulas and fuzzy logic. Submitted to SoCo
[8] HANČL, J., FILIP , F. Irrational measure of sequences, Hiroshima Math. J., vol. 35, no. 2, (2005), 183-195.
[9] HANČL, J., MIŠÍK, L., TÓTH, J. Fuzzy rational numbers and approximation of irrationals. To appear in Fuzzy Sets and Systems.
[10] HANČL, J., MIŠÍK, L., TÓTH, J. Asymptotic distance and its application. Submitted.
[11] HANČL, J., MIŠÍK, L., TÓTH, J. Fuzzy rational numbers and approximation of irrationals. In Workshop of the ERCIM working group on Soft Computing. 13.09.2006 15.09.2006 Malaga. University Malaga, 2006, pp. 5-8.
[12] HANČL, J., NAIR, R., ŠUSTEK, J. On the Lebesgue Measure of the Expressible Sets Certain Sequences, Indag. Math., Indag. Math. N.S. vol. 17, no.4, (2006), 567-581.
[13] HANČL, J., RUCKI, P. The Transcendence of certain infinite series, Rocky Mountain J. Math., vol. 35, no. 2, (2005), 531—537.
[14] HANČL, J., RUCKI, P. A note to the transcendence of special infinite series, Math. Slovaca, vol. 56, no.4., (2006), 409—414.
[15] HANČL, J., RUCKI, P. Certain Liouville Series, An. Univ. Ferrara Sci. Math., vol. 52, no. 1, (2006), 45--51.
[16] HANČL, J., RUCKI, P. A Generalization of Sándor's Theorem, Commentarii Mathematici Universitatis Sancti Pauli, vol. 55, no. 2, (2006), 97--111.
[17] HANČL, J., RUCKI, P. , ŠUSTEK, J. A Generalization of Sándor's Theorem Using Iterated Logarithms, Kumamoto J. Math., vol. 19, (2006), 25--36.
[18] HANČL, J., SOBKOVÁ, S. Criteria for rapidly convergent sequences to be linearly unrelated, JP Journal Algebra Num. Theory Appl. vol. 5, no. 2, (2005), 205--219.
[19] HANČL, J., SOBKOVÁ, S. Special linearly unrelated sequences, J. Math. Kyoto Univ., vol. 46, no. 1, (2006), 31--45.
[20] HANČL, J., TIJDEMAN, R. On the irrationality of factorial series, Acta Arith. vol. 118, no. 4, (2005), 383--401.
[21] KLEMENT, E.P., , MESIAR, R., ZEMÁNKOVÁ-MESIAROVÁ, A., SAMINGER, S., Logical connectives for granular computing. In: W.Pedrycz et al., eds., Handbook of Granular Computing, J.Wiley and sons, in press.
[22] MESIAR , R. Fuzzy sets and multicriteria decision making. In 9th Fuzzy Days. 18.9.2006-.9.2006 Dortmund. Berlin : Springer-Verlag, 2006. s. 629-635. ISBN 10 3-540-34780-1.
[23] MESIAR, R., KLEMENT, E., SAMINGER, S. A note on ordinal sum t-norms on bounded lattices. In EUSFLAT-LFA. 7.9.2005-9.9.2005 Barcelona. Barcelona : Technical University of Catalonia, Barcelona, 2005. s. 385-388. ISBN 84-7653-872-3.
[24] MESIAR, R., MESIAROVÁ, A. Fuzzy integrals and linearity. In Int. J. Approximate Reasoning. 2007.
[25] MESIAR, R., MESIAROVÁ, A., VALÁŠKOVÁ, Ľ. Generator-based universal fuzzy measures and integrals. In IPMU'2006. 02.07.2006-07.07.2006 Paris. Paris : Editions E.D.K., 2006. s. 1718-1723. ISBN 2-84254-112-X.
[26] MESIAR, R., MESIAROVÁ, A., VALÁŠKOVÁ, Ľ. Generated universal fuzzy measures. In Modeling Decisions for Artificial Intelligence. LNAI 3885. Berlin : Springer Verlag, 2006. ISBN 10-3-540-32780-0. s. 191-202.
[27] MIŠÍK, L., TÓTH, J. On asymptotic behaviour of universal fuzzy measures. In Kybernetika. 42 (2006), no.3, pp. 379-388.
[28] MIŠÍK, L., TÓTH, J. On asymptotic behaviour of universal fuzzy measures. In Workshop of the ERCIM working group on Soft Computing. 13.09.2006-15.09.2006 Malaga. University, 2006, pp. 1- 4.
[29] NOVÁK, V. EQ-algebras in progress. In IFSA 2007 World Congress. 18. 6. -21. 6. Cancun, Mexico. Cancun: 2007.
[30] NOVÁK, V. EQ-algebras: primary concepts and properties. In Czech-Japan Seminar on Data Analysis & Decision Making under Uncertainty - Ninth Meeting. 18.8.-20.8. Kitakyushu & Nagasaki. Japan : Waseda University, 2006. pp. 219-223.
[31] TÓTH, J., MIŠÍK, L., FILIP, F. On asymptotic distribution function of certain block sequence. Submitted.
[32] TÓTH, J., FILIP, F. On estimations of dispersions of certain dense block sequences. In Tatra Mountains Mathematical Publications. 32 (2005) 51-60.
[33] SAINIO, E., MESIAR, R., TURUNEN, E. A characterization of fuzzy implications generated by generalized quantifiers. In preparation.
[34] SAMINGER, S., KLEMENT, E., MESIAR, R. A note on ordinal sums of t-norms and t-subnorms on bounded lattices. In IPMU'2006. 2.7.2006-7.7.2006 Paris. Paris : Editions E.D.K., 2006. s. 664-670. ISBN 2-84254-112-X.
[35] SAMINGER, S., KLEMENT, E., MESIAR, R. On extensions of triangular norms on bounded lattices. Indag. Math., submitted.
[36] SAMINGER, S., KLEMENT, E., MESIAR, R., BODENHOFER, U. Aggregation of Fuzzy Relations and Preservation of Transitivity. In Lecture Notes in Artifitial Intelligence 4342, TARSKI. Springer-Verlag, Berlin, 2006, pp. 185-206.
Updated: 16. 05. 2018
