Linguistics and uncertainty in intelligent systems

Fuzzy normal forms can be generated with an application of "Normal Form Generation Algorithm" on fuzzy truth tables. This takes place at the third level of knowledge representation, i.e., propositional level. It is shown that at least three distinct sets of normal forms can be generated depending on the axioms one is willing to impose on the propositional fuzzy set and logic theories. All are conjunctive-disjunctive and complement based De Morgan logics with the following three classes of axioms that identify each general class of fuzzy normal forms in order of least to most restrictive set of axioms in the following sense: 1) boundary and monotonicity; 2) boundary, monotonicity, associativity and commutativity; and 3) boundary, monotonicity, associativity, commutativity and idempotency.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

&amp;lt;p&amp;gt;The article presents the results of modeling the solution to the problem of determining the reliability of the components of a multiservice communication network (MCN) based on the theory of fuzzy sets. The main characteristics of the equipment that affect the reliability parameters of the MCN are given. To solve the problem of determining the reliability of MCN components based on the theory of fuzzy sets, a multiservice network is presented in the form of a hierarchical diagram, which shows the main components of each network level. A multiservice network is presented as a parameter of the U function. The reliable state of the MCN depends on the state of the equipment at the corresponding levels. The results of modeling the solution to the problem of determining the reliability of MCN components based on the theory of fuzzy sets are presented using the mathematical apparatus of the theory of fuzzy sets and fuzzy logic in MATLAB fuzzy logic toolbox, fuzzyTECH.&amp;lt;/p&amp;gt;

Chapter 6 - Classes of fuzzy set and logic Theories

Disjunctive and conjunctive normal forms of pseudo-Boolean functions

We consider the problem of efficiently indexing Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) Boolean expressions over a high-dimensional multi-valued attribute space. The goal is to rapidly find the set of Boolean expressions that evaluate to true for a given assignment of values to attributes. A solution to this problem has applications in online advertising (where a Boolean expression represents an advertiser's user targeting requirements, and an assignment of values to attributes represents the characteristics of a user visiting an online page) and in general any publish/subscribe system (where a Boolean expression represents a subscription, and an assignment of values to attributes represents an event). All existing solutions that we are aware of can only index a specialized sub-set of conjunctive and/or disjunctive expressions, and cannot efficiently handle general DNF and CNF expressions (including NOTs) over multi-valued attributes. In this paper, we present a novel solution based on the inverted list data structure that enables us to index arbitrarily complex DNF and CNF Boolean expressions over multi-valued attributes. An interesting aspect of our solution is that, by virtue of leveraging inverted lists traditionally used for ranked information retrieval, we can efficiently return the top-N matching Boolean expressions. This capability enables emerging applications such as ranked publish/subscribe systems [16], where only the top subscriptions that match an event are desired. For example, in online advertising there is a limit on the number of advertisements that can be shown on a given page and only the "best" advertisements can be displayed. We have evaluated our proposed technique based on data from an online advertising application, and the results show a dramatic performance improvement over prior techniques.

The problem is the transformation of a conjunctive normal form (CNF) into a minimized (for the inclusion operator) disjunctive normal form (DNF) and vice versa. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An algorithm built upon the classical Davis and Putnam procedure is presented for calculating, without the explicit minimization for the inclusion, this unionist product.

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

The use of information technology and, in particular, learning management systems, increases the ability of both the teacher and the learner to achieve their goals in the educational process. Such systems provide educational content, help organize and monitor training, collect progress statistics, and can also take into account the individual characteristics of each user of the system. The purpose of this study is to determine the direction of development of modern learning systems and technologies for their implementation. The evolution of learning management systems, the transition to intelligent learning systems, the main stages of such systems were reviewed, the types of learning sequences were analyzed, the transformationinto adaptive learning systems was identified, and the scheme of the system and its mathematical model were presented. Expertise systems, the theory of fuzzy sets and fuzzy logic, cluster analysis, as well as genetic algorithms and artificial neural networks are defined as the mechanisms for implementing the learning systems. An artificial neural network in an adaptive learning system will allow you to create a unique training program that will build on existing knowledge and the level of perception of educational material by students. By formalizing the intellectual processes that both the teacher and the student carry out, it is possible to automate a certain part of the teacher’s functions, reduce the cost of manual labor, which will make it easier to monitor the learning process and also make the learning process more efficient.

The Human Genome Project is the resplendence of the bioinformatics field, especially in health and medicine. It involves research regarding complete nucleotide sequences of the deoxyribonucleic acid (DNA) in human’s chromosome. The primary structure of DNA and ribonucleic acid (RNA) consist of nucleotide construction which became polynucleotide when combined. In reality, genetic research field requires huge biological data, and most of the data are vague with various characteristics. Most of them are incomplete and complex from evolutional, functional, adaptability and other traits. The theory of fuzzy sets and fuzzy logic offers modelling methods in uncertainties and various computational techniques for decision making. This research aims to find similarity, difference, equality and identity between polynucleotide sequences using the concept of fuzzy metric space and fuzzy set theory. The Sadegh-Zadeh fuzzy polynucleotide space (RSZ) is being compared with the Torres and Nieto fuzzy polynucleotide space (RTN) in search of the best approach to analyse polynucleotide sequences. Research methods involve data collection of complete genome sequences for homologous species pairs, construction of the RSZ and RTN models, and data analysis. Outputs from RSZ and RTN are then compared with outputs from the Basic Local Alignment Search Tool (BLAST) for validation purposes from the bioinformatics field. Results show that outputs from both approaches are against each other, and RTN executes outputs that are nearest to the outputs from BLAST. Thus, RTN is the best fuzzy approach to compare complete genome sequences for species pairs.

L.A.Zadeh introduced the concept of fuzzy set theory as the generalization of classical set theory in 1965 and further it has been generalized to intuitionistic fuzzy sets (IFSs) by Atanassov in 1983 to model information by the membership, nonmembership and hesitancy degree more accurately than the theory of fuzzy logic. The notions of intuitionistic fuzzy numbers in different contexts were studied in literature and applied in real life applications. Problems in different fields involving qualitative, quantitative and uncertain information can be modeled better using intutionistic fuzzy numbers introduced in [17] which generalizes intuitionistic fuzzy values [1,7,17], interval valued intuitionistic fuzzy number (IVIFN) [10] than with usual IFNs [5,11,21]. Ranking of fuzzy numbers have started in early seventies in the last century and a complete ranking on the class of fuzzy numbers have achieved by W.Wang and Z.Wang only on 2014. A complete ranking on the class of IVIFNs, using axiomatic set of membership, non membership, vague and precise score functions has been introduced and studied by Geetha et al.[10]. In this paper, a total ordering on the class of IFNs [17] using double upper dense sequence in the interval [0; 1] which generalizes the total ordering on fuzzy numbers (FNs) is proposed and illustrated with examples. Examples are given to show the proposed method on this type of IFN is better than existing methods and this paper will give the better understanding over this new type of IFNs.

Fuzzy logic and fuzzy set theory can be interesting tools for engineers when solving image processing problems. They provide a mathematical model for approximate reasoning, which mimics human reasoning and therefore can be very powerfull. In this paper we review the basic concepts of fuzzy logic and fuzzy set theory, and use these tools for the construction of a filter for random impulse noise removal in image sequences. Experimental results and a comparative study demonstrate the excellent performance of the filter and hence the power of the used tools.

Knowledge Representation Methods for Dairy Decision Support Systems

We describe the problem of mining set valued rules in large relational tables containing categorical attributes taking a finite number of values. Such rules allow for an interval of possible values to be selected for each attribute in condition instead of a single value for association rules, while conclusion contains a projection of the data restricted by the condition onto a target attribute. An example of such a rule might be “if HOUSEHOLDSIZE = {Two OR Tree} AND OCCUPATION={Professional OR Clerical} THEN PAYMENT_METHOD = {CashCheck (Max=249, Sum=4952) OR DebitCard (Max=175, Sum=3021)} WHERE Confidence=85%, Support=10%.}” We use an original conceptional and formal framework for representing multidimensional distribution induced from data by a number of so-called prime disjunctions upper bounding its surface. Each prime disjunction represents a wide multidimensional interval of impossible combinations of attribute values. This original formalism generalises the conventional boolean approach in two directions: (i) finite-valued attributes (instead of only 0 and 1), and (ii) continuous-valued semantics instead of (true and false). In addition, we describe an efficient algorithm, which carries out the generalised dual transformation from possibilistic disjunctive normal form (DNF) representing data into conjunctive normal form (CNF) representing knowledge and thus generates all the most interesting prime disjunctions. Once obtained they can be used to build different forms of rules or for other purposes (prediction, clustering etc.).

An Ontological and Epistemological foundation of Fuzzy Set and Logic Theory is reviewed in comparison to Classical Set and Logic Theory. It is shown that basic equivalences of classical theory breakdown but are re-established as weak equivalences as a containment relation in fuzzy theory. It is also stressed that the law of conservation of information is still upheld within fuzzy theory.

Normal forms for fuzzy logic functions and their approximation ability