Nguyen Cat Ho

Extended hedge algebras and their application to fuzzy logic

This paper continues our investigation on hedge albebras (6). We extend hedge algebras by two additional operations corresponding to infimum and supremum of the so-called concept category of an element x, i.e. the set which is generated from x by means of the hedge operations. It is shown that every extended hedge algebra with a lattice of the primary generators is a lattice. In the symmetrical extended hedge algebras we are able to define negation and implication, called concept-negation and concept- implication. Furthermore, it is proved that there exists an isomorphism from a subaigebra of a symmetrical extended hedge algebra of a linguistic truth variable into the closed unit interval (0, 1), under which the concept-negation and the concept-implication correspond to the negation and a kind of implication in multiple-valued logic based on the unit- interval (0, 1).

An algebraic approach to linguistic hedges in Zadeh's fuzzy logic

The paper addresses the mathematical modelling of domains of linguistic variables, i.e. term-sets of linguistic variables, in order to obtain a suitable algebraic structure for the set of truth values of Zadehs fuzzy logic. We shall give a unified algebraic approach to the natural structure of domains of linguistic variables, which was proposed by Ho and Wechler (Fuzzy Sets and Systems 35 (1990) 281) and, then, by Ho and Nam (Proc. NCST Vietnam 9 (1) (1997) 15; Logic, Algebra and Computer Science, Vol. 46, Banach Center Publications, PWN, Warsaw, 1999, p. 63). In this approach, every linguistic domain can be considered as an algebraic structure called hedge algebra, because properties of its unary operations reflect semantic characteristics of linguistic hedges. Many fundamental properties of refined hedge algebras (RH_algebras) are examined, especially it is shown that every RH_algebra of a linguistic variable with a chain of the primary terms is a distributive lattice. RH_algebras with exactly two distinct primary terms, one being an antonym of the other, will also be investigated and they will be called symmetrical RH_algebras. It is shown that a class of finite symmetrical RH_algebras has a rich enough algebraic structure.

Fuzziness measure on complete hedge algebras and quantifying semantics of terms in linear hedge algebras

In the paper, we shall examine the fuzziness measure (FM) of terms or of complete and linear hedge algebras of a linguistic variable. The notion of semantically quantifying mappings (SQMs) previously examined by the first author will be redefined more generally and a closed relation between the FM of linguistic terms and a family of SQMs with the parameters to be the FM of primary terms and linguistic hedges will be established. A semantics-based topology of hedge algebras and a closed and interesting relation between this topology, the FM and the above family of SQMs will be discovered and examined. An applicability of the FM and SQMs will be shown by an examination of some application examples.

Plain Semi-Post algebras as a poset-based generalization of post algebras and their representability

Semi-Post algebras of any type T being a poset have been introduced and investigated in [CR87a], [CR87b]. Plain Semi-Post algebras are in this paper singled out among semi-Post algebras because of their simplicity, greatest similarity with Post algebras as well as their importance in logics for approximation reasoning ([Ra87a], [Ra87b], [RaEp87]). They are pseudo-Boolean algebras generated in a sense by corresponding Boolean algebras and a poset T. Every element has a unique descending representation by means of elements in a corresponding Boolean algebra and primitive Post constants which form a poset T. An axiomatization and another characterization, subalgebras, homomorphisms, congruences determined by special filters and a representability theory of these algebras, connected with that for Boolean algebras, are the subject of this paper.

Semi-Post algebras

In this paper, semi-Post algebras are introduced and investigated. The generalized Post algebras are subcases of semi-Post algebras. The so called primitive Post constants constitute an arbitrary partially ordered set, not necessarily connected as in the case of the generalized Post algebras examined in [3]. By this generalization, semi-Post products can be defined. It is also shown that the class of all semi-Post algebras is closed under these products and that every semi-Post algebra is a semi-Post product of some generalized Post algebras.

Subalgebras and homomorphisms of semi-Post algebras

Semi-Post algebras have been introduced and investigated in [6]. This paper is devoted to semi-Post subalgebras and homomorphisms. Characterization of semi-Post subalgebras and homomorphisms, relationships between subalgebras and homomorphisms of semi-Post algebras and of generalized Post algebras are examined.

LT -fuzzy sets

The problem of axiomatization of fuzzy algebra is one of the most interesting topics in fuzzy sets theory. There is not as yet a uniform point of view concerning the question what a fuzzy algebra is (cf. [1, 6, 7, 20, 4]) and there are different fuzzy algebras useful in different applications. In this paper a poset-based concept of LT-fuzzy sets is proposed, being a simple modification of L-fuzzy sets (cf. [5]). This new approach permits the development of an axiomatic fuzzy sets algebra based on the theory of plain semi-Post algebras as investigated in [3] and applied to logic for approximation reasoning [14, 15, 16].

ON THE REAL-WORLD-SEMANTICS INTERPRETABILITY OF FUZZY RULE BASED SYSTEMS UNDER FUZZY SET APPROACH AND HEDGE ALGEBRA APPROACH

The study will further discuss the novel real-world-semantics-based approach (RWSapproach) to the interpretability of fuzzy systems proposed in [8] to show that the RWS-interpretability of fuzzy systems in this approach is very essential and practical. It is also analyzed that the usual theories as in mathematics and physics are all RWS-interpretable or, roughly speaking, they are able to model their real-world parts, properly. It is pointed out that though the fuzzy set theory is a great one which is very flexible, has many advantages in the application and obtains numerous achievements, methodologically, it still has an essential shortcoming that it is not RWS-interpretable on the viewpoint of this RWS-approach. To ensure the RWS-interpretability of a fuzzy linguistic system, it is argued that word-domains of variables with their own order-based semantics should be made use and interpreted as a formal bridge to connect the real-world semantics with the constructed fuzzy linguistic system that works on the designed computational semantics of linguistic words. It is initially shown that there exists a formalism based on the theory of hedge algebras to design RWS-interpretable fuzzy systems.

HEDGE-ALGEBRAS-BASED FUZZY CONTROLLER: APPLICATION TO ACTIVE CONTROL OF A FIFTEEN-STORY BUILDING AGAINST EARTHQUAKE

Active control problem of seism-excited civil structures has attracted considerable attention in recent years. In this paper, conventional, hedge-algebras-based and optimal hedge-algebras-based fuzzy controllers, respectively denoted by FC, HAFC and OHAFC, are designed to suppress vibrations of a structure with active tuned mass damper (ATMD) against earthquake. The interested structure is a high-rise building modeled as a fifteen-degree-of-freedom structure system with two type of actuators installed on the first storey and fifteenth storey which has ATMD. The structural system is simulated against the ground accelerations, acting on the base, of the El Centro earthquake in USA on May 18th. The control effects of FC, HAFC and OHAFC are compared via the time history of the storey displacements of the structure.

Complete and Linear Hedge Algebras, Fuzziness Measure of Vague Concepts and Linguistic Hedges and Application

In the paper, we shall examine fuzziness measure of terms in linear and complete hedge algebras of a linguistic variable. A notion of the so‐called semantically quantifying mappings will be redefined more generally and it will be established a closed relationship between fuzziness measure and a class of semantically quantifying mappings defined by a recursive expression with parameters to be fuzziness measure of primary terms and linguistic hedges. An application of fuzziness measure and semantically quantifying mappings in solving fuzzy the multiple conditional reasoning problem will be presented to show an applicability of hedge algebras.