Masaharu Mizumoto

Some properties of fuzzy sets of type 2

The concept of fuzzy sets of type 2 has been defined by L. A. Zadeh as an extension of ordinary fuzzy sets. The fuzzy set of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade) of which is a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊔, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzy grades form a distributive lattice under ⊔ and ⊓. Moreover, the algebraic properties of fuzzy grades under the operations and which are slightly different from ⊔ and ⊓, respectively, are briefly discussed.

Some considerations on fuzzy automata

In this paper, we show: The threshold of fuzzy automata can be changed arbitrarily. The fuzzy sets of input sequences characterized by fuzzy automata constitute a distributive lattice, and the complement of the fuzzy set can be characterized by an optimistic fuzzy automaton.

fstds System: A fuzzy-set manipulation system

This paper describes an implementation of a system for fuzzy sets manipulation which is based on fstds (Fuzzy-Set-Theoretic Data Structure), an extended version of Childss stds (Set-Theoretic Data Structure). The fstds language is considered as a fuzzy-set-theoretically oriented language which can deal, for example, with ordinary sets, ordinary relations, fuzzy sets, fuzzy relations, L-fuzzy sets, level-m fuzzy sets and type-n fuzzy sets. The system consists of an interpreter, a collection of fuzzy-set operations and the data structure, fstds, for representing fuzzy sets. fstds is made up of eight areas, namely, the fuzzy-set area, fuzzy-set representation area, grade area, grade-tuple area, element area, element-tuple area, fuzzy-set name area and fuzzy-set operator name area. The fstds system, in which 52 fuzzy-set operations are available, is implemented in fortran, and is currently running on a FACOM 230-45S computer.

FUZZY ROBOT CONTROLS

Abstract A fuzzy program is defined as an ordered sequence of fuzzy instructions. In the execution of a fuzzy program, fuzzy instructions are translated into machine instructions by the use of MAX-METHOD and backtracking. To show how a fuzzy program is executed, a simulation system is exploited which controls a simple inchworm robot. A few examples of computer simulations are presented which deal with the behavior of a stranger (or a robot) searching for his destination in a modeled town under simple fuzzy instructions.

Web grammars and several graphs

This paper is concerned with the class of web grammars introduced by Pfaltz Rosenfeld and Montanari. In this paper, we show that context-sensitive web grammar cannot erase arcs, and monotone context-sensitive web grammar can erase arcs but cannot erase any vertices and they satisfy the condition |@a|@?|@b| in the rules @a@?@b. Then some hierarchical results hold, when grammars are normal and nonnormal. Normal grammars have rules that each vertex to be rewritten has exactly one image in the right member of the rule; nonnormal ones have rules that some vertices have two more images. Also, it is shown that there exists a complete grammar which generates some types of Eulerian graphs, line graphs and 3-connected graphs.

General formulation of formal grammars

Abstract By extracting the basic properties common to the formal grammars appeared in existing literatures, we develop a general formulation of formal grammars. We define a pseudo grammar and derive from it the well-known probabilistic, fuzzy grammars and so on. Moreover, several interesting grammars such as ⊔∗ grammars, ⊔ ⊓ grammars, ⊔ ⊓ grammars, composite B -fuzzy grammars, and mixed fuzzy grammars, which have never appeared in any other papers before, are derived.

N-fold fuzzy grammars

A new form of fuzzy grammar, which is called an n-fold fuzzy grammar, is defined and some of its properties are investigated. The n-fold fuzzy grammars are a generalization of fuzzy grammars defined by Lee and Zadeh, where the grade of application of the rule to be used next is conditioned by the n(⩾1) rules used before in a derivation. The n-fold fuzzy grammars whose rules are of context-free form can be shown to generate context-sensitive languages by setting a threshold appropriately. Fuzzy grammars by Lee and Zadeh with context-free rules, however, cannot generate context-sensitive languages. As to n-fold fuzzy grammars, we mainly focused our attention on n-fold fuzzy grammars with type 3 rules as a preliminary step.

On the Property of Single Input Rule Modules Connected Type Fuzzy Reasoning Method

Yubazaki et al. proposed single input rule modules connected type fuzzy reasoning method (SIRMs method) whose final output is obtained by summarizing the product of the importance degree and the inference result from single input fuzzy rule module. This paper clarifies the relationship between the simplified reasoning method and SIRMs method, and shows that SIRMs method can be transformed into simplified reasoning method, but not vice versa.

Various kinds of automata with weights

By extracting the basic properties common to the automata appeared in existing literatures, we develop a general formulation of automata with ?weights?. We define a pseudoautomaton and derive from it the well-known deterministic automaton, nondeterministic automaton, probabilistic automaton, fuzzy automaton, and so on. Moreover, several interesting automata such as l-semigroup automaton, lattice automaton, dual lattice automaton, mixed boolean automaton, semiring automaton, ring automaton and field automaton which have never appeared in any other paper before are derived.

L-fuzzy grammars

Abstract An L -fuzzy grammar is defined by assigning the element of lattice to the rewriting rules of a formal grammar. According to the kind of lattice, say, distributive lattice, lattice-ordered group, and lattice-ordered monoid, two type of L -fuzzy grammars are defined. It is shown that some context-sensitive languages can be generated by type 3 ∗-L- fuzzy grammars with cut points. It is also shown that for type 2 L -fuzzy grammars, Chomsky and Greibach normal form can be constructed as an extension of corresponding notion in the theory of formal grammars.