Active Sets and Unification

An active set is a unifying space being able to act as a “bridge” for transferring information, ideas and results between distinct types of uncertainties and different types of applications. An active set is a set of agents who independently deliver true or false values for a given proposition. It is not a simple vector of logic values for different propositions, the results are a vector but the set is not. The difference between an ordinary set and active set is that the ordinary set has passive elements with values of the attributes defined by an external agent, in the active set any element is an agent that internally defines the value of a given attribute for a passive element. So agents in many cases are in a logic conflict and this generates semantic uncertainty on the evaluation. Criteria and agents are the two variables by which we give different logic values to the same attribute or proposition. In modal logic, given a proposition, we can evaluate the proposition only when we know the worlds where the proposition is located. In epistemic logic any world is an agent that knows that the proposition is true or false. The active set is a set of agents as in epistemic logic but the difference with modal logic is that all the agents (worlds) are not separate but are joined in the evaluation of the given proposition. In active set for one agent and one criteria we have one logic value but for many agents and criteria the evaluation is not true or false but is a matrix of true and false. This matrix is not only a logic evaluation as in the modal logic but give us the conflicting structure of the active set evaluation. The agent multi dimensional space to evaluate active set also includes the Hilbert multidimensional space where is possible to simulate quantum logic gate. New logic operation are possible as fuzzy gate operations and more complex operations as conflicting solving, consensus operations, syntactic inconsistency, semantic inconsistency and knowledge integration. In the space of the agents evaluations morphotronic geometric operations are a new frontier to model new types of computers. In this paper we show the connection between classical, fuzzy, evidence theory and active sets.

The fuzzy set concept is often used in solution of problems in which the initial data is difficult or impossible to represent in the form of specific numbers or sets. Geo-information objects are distinguished by their uncertainty, their characteristics are often vague and have some error. Therefore, in the study of such objects is introduced the concept of "fuzziness" — fuzzy sets, fuzzy logic, linguistic variables, etc. The fuzzy set concept is given in the form of membership function. An ordinary set is a special case of a fuzzy one. If we consider a fuzzy object on the map, for example, a lake that changes its shape depending on the time of year, we can build up for it a characteristic function from two variables (the object’s points coordinates) and put a certain number in accordance with each point of the object. That is, we can describe a fuzzy set using its two-dimensional graphical image. Thus, we obtain an approximate view of a surface z = μ(x, y) in three-dimensional space. Let us now draw planes parallel to the plane. We’ll obtain intersections of our surface with these planes at 0 ≤ z ≤ 1. Let's call them as isolines. By projecting these isolines on the OXY plane, we’ll obtain an image of our fuzzy set with an indication of intermediate values μ(x, y) linked to the set’s points coordinates. So we’ll construct generalized Euler — Venn diagrams which are a generalization of well-known Euler — Venn diagrams for ordinary sets. Let's consider representations of operations on fuzzy sets A a n d B. Th e y u s u a l l y t a k e : μA B = min (μA,μB ), μA B = max (μA,μB ), μA = 1 − μA. Algebraic operations on fuzzy sets are defined as follows: μ A B x μ A x μ B x ( ) = ( ) + ( ) − −μ A (x)μ B (x), μ A B x μ A x μ B x ( ) = ( ) ( ), μ A (x) = 1 − μ A (x). Let's construct for a particular problem a generalized Euler — Venn diagram corresponding to it, and solve subtasks graphically, using operations on fuzzy sets, operations of intersection and integrating of the diagram’s bars.

In 1965, L.A. Zadeh published his famous paper “Fuzzy sets” in Information and Control providing a new mathematical tool which enables us to describe and handle vague or ambiguous notions such as “a set of all real numbers which are much greater than 1,” “a set of beautiful women,” or “the set of tall men.” Since then, fuzzy set theory has been rapidly developed by Zadeh himself and numerous researchers, and an increasing number of successful real applications of this theory in a wide variety of unexpected fields has been appearing. The main idea of fuzzy set theory is quite intuitive and natural: Instead of determining the exact boundaries as in an ordinary set, a fuzzy set allows no sharply defined boundaries because of a generalization of a characteristic function to a membership function.

Hamouda E. H. [J. of Advances in Math. Vol 6 (2014), 973-941] has introduced some universal constructions in the category FTOP , whose objects are the I-fuzzy topological spaces where X is an ordinary set, is a fuzzy set in X and F is a family of fuzzy sets in X satisfying some axioms. In this paper we introduce the dual universal constructions, namely, fuzzy co-products, fuzzy co-equalizers and fuzzy pushouts for I-fuzzy topological spaces. Also we discuss some results concerning all such universal objects.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

A weighted inference engine based on interval-valued fuzzy relational theory

This paper proposes a conceptually simple but expressive framework for handling propositional information stemming from several sources, namely a two-tiered propositional logic augmented with classical modal axioms (BC-logic), a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set-functions called Boolean capacities. We present a theorem-preserving translation of Belnap logic in this setting. As special cases, we can recover previous translations of three-valued logics such as Kleene and Priest logics. Our translation bridges the gap between Belnap logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various inconsistency handling methods.

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

With the rapid development of information technology, multi-sensor information fusion research has been made great progress, currently there are Bayesian test, the voting method, fuzzy sets, neural networks and other methods algorithm being used in multi-sensor information fusion commonly. However, there are many shortcomings which are difficult to overcome in the process of fusion by these methods, so now the international community has already proposed to apply the D-S evidence theory to information fusion. This method does not require the priori probability, and it's very useful when it is used in information fusion, but the information with the conflict can not be effectively fused by using D-S evidence theory. Larger conflict in the evidence, the fusion result is more easily prone to false consensus. Therefore, this paper presents an improved information fusion method based on original D-S evidence theory, and this method can deal with the evidences with conflict effectively. In this paper, the improvement is in the denominator of the D-S evidence theory. When the evidence is purely of conflict, the denominator is zero, the formula does not make sense. This improvement thinking is coupled with sum between the orthogonal sum of all of evidence and product, which is between the conflict factor and the inverse value of the standard deviation of the support degree, and then it subtract maximum difference of the basic probability of the evidence. If the conflict of the evidence is greater, this advantage of this method is more obvious. If there is no conflict in the evidence, the fusion result is consistent with item of original D-S evidence theory. Finally the simulation experiment gives a set of data with more conflict. The comparative data from the experiments show that the improved method of information fusion is essential to improved in resolving conflict issues, and it is effective.

The LNCS journal Transactions on Rough Sets is devoted to the entire spectrum of rough sets related issues, from logical and mathematical foundations, through all aspects of rough set theory and its applications, such as data mining, knowledge discovery, and intelligent information processing, to relations between rough sets and other approaches to uncertainty, vagueness, and incompleteness, such as fuzzy sets and theory of evidence. Volume XVI includes extensions of papers from the Rough Sets and Knowledge Technology Conference which was held in Banff, Canada, in October 2011. In addition this book contains a long paper based on a PhD thesis. The papers cover both theory and applications of rough, fuzzy and near sets. They offer a continuation of a number of research streams which have grown out of the seminal work by Zdzislaw Pawlak during the first decade of the 21st century.

We present a general method that can be used to unify the theoretical results for several new MV-valued fuzzy sets, such as intuitionistic fuzzy sets, neutrosophic fuzzy sets, fuzzy soft sets, or multivalued fuzzy sets and their mutual combinations. The results created in this way have properties identical to analogous results for classic MV-valued fuzzy sets, and it is not necessary to prove these results separately for each new type of fuzzy sets. To illustrate this general method, we will show how two related fuzzy Chang topologies can be derived in these new types of fuzzy sets from general approximation spaces.

On measurements of covering rough sets based on granules and evidence theory

Rough set theory for the interval-valued fuzzy information systems

This paper generalizes the evidence theory and makes it can be used in fuzzy sets in order to make it can be used more effectively. With the fuzzy membership function, this paper presents a method to set up the basic probability assignment function (mass function) in evidence theory, and integrates fuzzy theory and evidence theory effectively. This method brings the evidence theory nearer to practice. This paper presents an improved combination rule to deal with consistent or inconsistent evidences obtained from multiple sources. The improved rule reflects the intersection of focus elements and adapts AND-operation to combine consistent evidences, and allocates the conflicts to vary focus elements according to relevant evidences' creditability. From the theoretic analysis and the results of the numerical examples, the given rule is very rational and effective for both high conflictive evidences and consistent evidences. Especially, the given rule can provide reasonable results with better convergence efficiency than other rules when evidence sources are high conflict.

The introduction of uncertainty theory into the field of decision making provides a solution for the processing of qualitative information. In the field of uncertain information decision making, there are many widely used theoretical methods, such as D-S evidence theory and D number theory. Recently, Yager proposed a method for evaluating the satisfaction of belief structure based on uncertain target values based on D-S evidence theory. However, there is no related research on satisfaction evaluation in D number theory. In this paper, combined with fuzzy set theory, a calculation method of D numbers satisfaction is proposed. Given the element correlation, the decision information is effectively processed by generating a correlation fuzzy membership function. An application example of a manufacturing company selecting a supplier illustrates the effectiveness of the calculation method in this paper.