Mathematics and Logic Behind Boolean and Fuzzy Computation

In fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc.

In this chapter, we merge type-II fuzzy logic, and intuitionistic fuzzy logic in a broader way, to develop an intuitionistic type-II fuzzy logic. In intuitionistic type-II fuzzy logic, we deal the uncertainty, concerning with the truth values as well as false values. Intuitionistic type-II fuzzy logic is a stimulus of traditional intuitionistic type-I fuzzy logic in such a way that ambiguity is presented into linguistic variables that can be handled by using the truth grades and false grades with some hesitation margin. On the other hand, the proposed intuitionistic type-II fuzzy logic-based inference system accommodate intuitionistic fuzzy IF–THEN rules, which holds intuitionistic type-II fuzzy sets ( $${\mathrm{IFT}}_{\mathrm{y}}(\mathrm{II}))$$ . We also discussed the applications of proposed system in the various fields including; engineering, medical and agriculture etc. We applied over methodology over a data of lung cancer patients, which consists sixteen medical entities of infected patients. We also gave an example to illustrate our proposed technique.

Traditional fuzzy logic uses real numbers as truth values. This description is not always adequate, so in interval-valued fuzzy logic, we use pairs (t/sup -/, t/sup +/) of real numbers, t/sup -/ /spl les/ t/sup +/, to describe a truth value. To make this description even more adequate, instead of using real numbers to described each value t/sup -/ and t/sup +/, we can use intervals, and thus get fuzzy values which can be described by 4 real numbers each. We can iterate this procedure again and again. The question is: can we get an arbitrary partially ordered set in this manner or an arbitrary lattice? In this paper, we show that although we cannot thus generate arbitrary lattices, we can actually generate an arbitrary partially ordered set in this manner. In this sense, the "intervalization" operation is indeed universal.

Truth values in fuzzy logic are extended into interval truth values {[a, b]: 0≤a≤b≤1} from the numerical truth values of [0, 1]. Next we extend this interval truth values into two directions; one is to n-dimensional case such as [a 1,...,a n ] where a 1≤...≤a n , and the other is to dropping the condition a≤b in [a, b]. We examine the algebraic structures of these truth values on which three kinds of logic operations AND(·), OR(∨), NOT(⊃ are defined, where we interpret (x·y), (x∨y), 0 and (⊃x) as min(x, y), max(x, y), and (1−x), respectively. It is shown that we can find out structures of Kleene algebra and De Morgan algebra in these truth values.

The problem of synthesis of fuzzy control system at a plant with distributed parameters is discussed. An approach to the construction of a fuzzy controller based on a modal representation of the plant with distributed parameters is analyzed. Time modes of spatially distributed signal of mismatch are considered as the linguistic variables. The membership function of fuzzy logic controller output is used as a spatially distributed control signal.

This paper considers the consensual analysis in decision-making (CDM) processes based on fuzzy logic (FL) and interval-valued fuzzy logic (IVFL), providing a CDM-strategy, by exploring the axiomatic properties of fuzzy consensus measures (FCM) via penalty functions. Thus, two models are formalized, FS-FCM and IVFS-FCM. In the former, the fuzzy-valued lattice enables the analysis of fuzzy information for linguistic variables (LV), which is obtained by the aggregation of penalty functions. And, in the latter, the consensus measures of fuzzy sets are aggregated to build a new consensual analysis modeling. Thus, e.g., the cohesion of several terms related to the same LV can be analyzed, and also the coherence between fuzzy sets referring to the lowest and highest projections. Such models decide based on relevance criteria and qualitative assessments, via the selection of alternatives, supporting the corresponding algorithmic strategies: FS-FCM strategy, applied to fuzzy values, and IVFS-FCM strategy, covering fuzzy sets. The Intf-HybridMem approach explores the access patterns to volatile and non-volatile memories related to decision-making in two steps: (i) the FS-FCM strategy explores consensus measures of fuzzy values from membership functions; and (ii) the IVFS-FCM strategy, modeling inaccuracy inherent in input variables, as read/write frequency and access recency, also including the migration recommendation as output, which is validated by evaluations carried out in both proposed strategies.

Fault diagnosis is a challenging work in large communication network. A fault will bring more than one alarms, an alarm may be caused by different faults vice versa. The relationship between faults and alarms are not accurate but fuzzy, which can not be understood using traditional Boolean logic. Also, crisp association rules use sharp partitioning and potentially introduce loss of information. Therefore, fuzzy logic is used to convert numerical attributes into linguistic terms and a fuzzy judge language set is described. We first analyze the meanings of each attribute of network alarms. Then define proper fuzzy membership functions according to the relationship between this field and the root alarm. After that we integrate MDs (membership degrees) and weights of every attribute to construct fuzzy database. At last, we come up with a new fuzzy association rules mining algorithm, which generalizes the popular hierarchy-based algorithm. Simulation experiments on a synthetic network are carried out to prove and validate the accuracy and efficiency of our algorithm with respect to correlation analysis for right positioning root alarms.

In Frege’s theory of judgement, the concept of truth is closely connected with the concept of assertoric force — a connection which justifies us in speaking of a (truth) theory of acknowledgement (Anerkennung). Frege is well aware of the paradoxical nature of his attempts to explicate his views (i.e. to express them in language). The paradox arises because we are concerned here with saying what cannot be (logically) said. Frege thus anticipates Wittgenstein’s view that the categorial discourse in philosophy does not consist of sentences with a truth value, but rather of ‘elucidations’. Two things are central to Frege’s theory of acknowledgement: first, the distinction between the act of judging (pragmatically expressed through the judgement stroke) and the content of judgement; and second, the value-theoretic understanding of the notion of truth values. It will be shown that in developing his own logic, Frege was inspired by traditional logic, and by the Southwest School of Neo-Kantianism in particular.KeywordsHistorical BackgroundPropositional ContentNegative JudgementTraditional LogicConditional SentenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A many‐valued sentential logic with truth values in an injective MV‐algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV‐algebra. In the well‐defined fuzzy sentential logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good‐behaviour, MV‐valued logic can be regarded as a mathematical basis of fuzzy reasoning.

Fuzzy logic is a many-valued logical system with truth values within the subset[0, 1] of real numbers. The set[0, 1] is a totally ordered set, and thus it has limitations in application to various real-life examples. Therefore, generalized many-valued logical systems that can be applied to more various examples of real life need to be studied and their properties investigated. In this paper, we introduce the definitions and properties of lattice implication algebras, Heyting semi-lattices, and DBCK-algebras as algebraic structures that can generalize existing fuzzy logic. We also investigate the fuzzy implications defined in the set[0, 1] and their properties, and propose algebraic structures in which fuzzy logic with such implications can be generalized. To do this, we find out the necessary and sufficient conditions for an algebraic structure[L, →, ] to be DBCK-algebra. Also, we prove that lattice implication algebras and Heyting semilattices are DBCK-algebras, and that the algebraic structure, which is a lattice implication and a Heyting semilattice, is an equivalent notion to Boolean algebra.

We are going to study, from the logical point of view, expressions as “usually ϕ”, “ϕ is probable”, “for many x, ϕ” and similar. We shall handle them both as generalized quantifiers and as modalities. This should not look unnatural since in general modalities can be viewed as hidden quantifiers. We shall try to show how the theory of generalized quantifiers and modal logic can be applied to the above expressions (stressed by Zadeh as items of the specific agenda of fuzzy logic in the narrow sense) and that they admit “classical” logical analysis. We shall offer a logical apparatus, precise definitions, some results and various problems: here much study and development is still necessary. To be able to use and generalize the results and approaches of Boolean logic, we shall have to review and summarize several notions and facts on generalized quantifiers, some modal logics, and also on measures of uncertainty (that will be used to define various quantifiers and modalities). Thus we shall have two preliminary sections in this chapter: Section 1 on generalized quantifiers (with an appendix on uncertainty measures) and Section 2 on modal logics, both sections in the frame of Boolean (two-valued) logic. Generalized quantifiers in fuzzy logic and fuzzy modal logics are studied in Sections 3, 4.

The development of fuzzy logic in the companies mentioned by the authors is meant to automate the plant adaptively. The goal is to reduce downtimes in the plant, and hence, the framework done by the researchers will solve the problems by making the plant intelligent. Fuzzy logic is human concept, potentially applicable to a wide range of processes and tasks that require human intuition and experience. In computer, truth values are either 1 or 0, which correspond to true/false duality. In fuzzy logic, truth is the matter of degree, thus truth-values range between 1 and 0 in a continuous manner. Fuzzy logic is a method for representing information in a way that resembles natural human communication.

In this paper, failure analysis of an unmanned aerial vehicle (UAV) has been conducted through failure mode and effects analysis (FMEA) and finite element (FE) simulation. For FMEA, a system is broken down into subsystems, and possible failures are considered for each subsystem. Each failure is then ranked in 3 categories: severity, occurrence, and difficulty of detection. Each category has 1 of 5 ranks: very low, low, moderate, high, and very high. Since these ranks are linguistic variables that cannot be properly assigned a definitely value, fuzzy logic is applied. Using a membership function and a defuzzification formula, the variables are defuzzified. Grey relational analysis is applied to factor in the relationship between the variables. With these methods applied, the battery failure and rotor arm failure (including motors, ESC & arm structure) were found to be the most critical failures followed by avionics failure and structural failure. For the FE simulation, a 3D model is created. The stress distribution and fatigue life of fuselage are simulated by using the commercial software Comsol. The fuselage was made of Al 7076 T6 with an ultimate tensile strength of 572 MPa. The stress distribution at different loads (5 kg, 50 kg & 75 kg) are simulated. According to the simulation results, the maximum stress occurs at the corner of the fuselage. The fatigue life under different loads were also simulated. The potential failure mode and risk level analysis enables the identification of critical component and prediction of system failures. With FE simulation, the limit of the structure is simulated. Maximum load the structure can support is determined, the simulation of fatigue life cycles enables the prediction of structure failures.

Selection of appropriate Software Development Life Cycle(s) can increase projects success. Depending on the selected SDLC (Software Development Life Cycle), one can decrease development time/cost, minimize overhead and risk exposure, manage uncertainty, improve quality, promote client relations, and provide better project tracking and control. Despite the benefits of using suitable SDLC(s), it is generally difficult to select the most appropriate one. There is not enough information in the literature about the criteria and how to take them into account to select appropriate SDLC(s). Frequently used SDLCs and their comparative properties were elicited from literature, Internet and experts according to the criteria to select appropriate SDLC(s). A Fuzzy Logic (FL) system was developed according to the elicited knowledge about the criteria and their affect on selecting suitable SDLCs in various cases. Fuzzy input/output variables, membership functions and fuzzy rules were defined and generated according to the elicited knowledge. The FL system was tested for various inputs and improved according to the test results.

The use of fuzzy sets to represent extensions of predicates has as a consequence that the truth of a predicate belongs to the interval [0,1]. In this case the underlying logic is a many-valued one in which the law of excluded middle does not hold. This is due to the presence of a truth-value which expresses ignorance about whether an object has a property or not without rejecting the possibility that it might have this property. This is exactly the type of knowledge used in non-monotonic reasoning systems which allows a fact to be asserted as true by default. We capitalize on the natural existence of such a truth value when interpreting fuzzy predicates and propose a formalization of a fuzzy non-monotonic logic. We start by extending fuzzy logic with two connectives M and L where Ma reads as “it may be the case that α is true” and Lα reads as “it is the case that ± is true. In addition, a default operator D is added where D± is interpreted as “a is true by default”. The logic has an intuitive model theoretic semantics without any appeal to the use of a fixpoint semantics for the default operator. The semantics is based on the notion of preferential entailment, where a set of sentences Γ preferentially entails a sentence α, if and only if a preferred set of the models of Γ are models of α. The logic also belongs to the class of cumulative non-monotonic formalisms which are a subject of current interest.