A general approach to asymptotic elimination of aggregation functions and generalized quantifiers

Relations between two Boolean attributes derived from data can be quantified by truth functions defined on fourfold tables corresponding to pairs of the attributes. Several classes of such quantifiers (implicational, double implicational, and equivalence ones) with truth values in the unit interval were investigated in the frame of the theory of data mining methods. The definition of double implicational quantifiers is based on the idea of conjunction of both directions implications (similarly for equivalence). In the fuzzy logic theory, there are well-defined classes of fuzzy operators, namely t-norms representing various types of evaluations of fuzzy conjunction (and t-conorms representing fuzzy disjunction). I assert that each t-norm applied to an implicational quantifier gives a double implicational quantifier. Analogously, each t-conorm applied to a double implicational quantifier gives an equivalence quantifier. Logical properties of obtained quantifiers are discussed. The method is illustrated by examples of well-known quantifiers and operators. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 325–334, 2006.

Equilibrium semantics interprets expected payoffs resulting from Nash equilibria of semantics games with imperfect information (Hintikka-Sandu or ) as truth values. In [1] we showed that all rational truth values in the unit interval can be obtained in this manner from games with just 0 and 1 as possible payoffs in any run of the game. Here, we investigate whether the propositional connectives of various fuzzy logics, in particular of Łukasiewicz, Godel, and Product logic, can be interpreted as operators that combine parallel $\mathcal{H}\mathcal{S}{\text{ - games}}$ into single games. It turns out that the interpretation of different connectives calls for definitions of overall payoffs for combined games that operate at different levels of abstraction.

Type-2 operations on finite chains

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 α–C-equivalences

In this paper, we study alethic modal operators in Łukasiewicz’ many-valued logics where truth values are certain rational numbers from the closed unit interval [0,1]. The alethic modal operators necessary and possible are added to Ł3 using Tarski’s idea. The result is a modal logic denoted by Ł\(_{3}^{\mathrm{mod}}\). A formula-equivalency between Ł n and Ł\(_{n}^{\mathrm{mod}}\) is stated for n ≥ 3. Truth value assignments, or valuations, in Łukasiewicz’ n-valued modal logic are considered. The laws of excluded middle and contradiction are considered and found that they are possible in Ł\(_{n}^{\mathrm{mod}}\). Normal modal many-valued systems based on axiom schemes (K), (T), (S4), (B) and (S5) are considered.

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

Knowledge Based System (KBS) is a problem solving approach that makes use of human knowledge in decision strategies. Modeling and representing imperfect human knowledge associated with uncertainty is an important task in KBS development. There are various types of uncertainty, and randomness and fuzziness are among the most important. Handling hybrid uncertainty in one KBS is critical to support real world applications. Knowware System (KWS) is an intelligent tool designed to support application developers in constructing customized hybrid KBS without requiring developers being familiar with relevant intelligent techniques. It is essential for KWS to construct corresponding inference structure in resulting KBS and process the inference with hybrid uncertainty. To fulfill this requirement the extended Truth Value Flow Inference (TVFI) and Interval-Valued Confidence (IVC) have been defined and developed as ambedded mechanisms of KWS, and the hybrid logic has been adopted for the framework of handling hybrid uncertainty.This chapter discusses the IVC for the inference in hybrid KBS constructed by KWS with hierarchical knowledge representation. The knowledge content (precise or imprecise) represented in multiple units of knowledge hierarchy and the confidence obtained during inference process are treated as at two levels separately but simultaneously based on the extended TVFI. With the basic form of IVC, a fuzzy truth value is represented as a fuzzy number defined as a three-parametric triangular type-1 fuzzy set on the unit interval [0, 1]. The inference with both fuzzy truth and probability results an Extended Interval-Valued Confidence (EIVC) which is an interval type-2 fuzzy set on [0, 1] having the probability as an uncertainty measure on the fuzzy truth. The main focus of discussion will be put on interval-valued confidence. To provide the background of discussion, the KWS scheme, the extended TVFI and the concepts of hybrid logic will be briefly introduced.KeywordsUncertaintyRandomnessFuzzinessKnowwareSystemHybridLogicInterval-Valued Confidence

On Selected Properties of Uninorm Petri Nets and Their Application in Modeling Knowledge-Based Systems

A Comparison of J. M. Keynes’s Logical Approach to Probability and Any ‘Objective Bayesian’ Approach to Probability Needs to Incorporate All Five Parts of Keynes’s a Treatise on Probability, Not Just Part I

Fuzzy points are concepts used in mathematics, particularly in fuzzy set theory and related fields, where the membership of a given set is described using a membership function that allocates values in the unit interval [0, 1] rather than assigning binary values (true or false).This idea of a fuzzy point is used to consider the fuzzification of (ordered) filters in ordered BCI-algebras.Consequently, the concept of fuzzy (ordered) filters is introduced, and several related properties are investigated.The relationships between fuzzy filter, fuzzy (ordered) filter, and fuzzy (ordered) subalgebra are discussed, and characteristics of the fuzzy ordered filter are considered.This paper provides the conditions for a fuzzy set to become a fuzzy-ordered filter.We also investigate when a positive subset forms an ordered filter and examine whether the level and q-sets satisfy the same properties.

: This article explains in general terms how some sequential bioassay methods like the stochastic approximation method or the up-and-down method are not in conformity with the likelihood principle. Irrespective of the sampling plan, the bioassay data may be analyzed in terms of the following simple prior probability model for the response probabilities. Let x'1 x'2 ... x'm be the distinct dosage levels used in a bioassay experiment and let P1 P2 ... pm be the corresponding unknown response probabilities. Let U1 = P1 and Ui = (Pi - P(i-1)/(1 - P(i-1), i = 2, 3, ... m. The ui's are regarded as mutually independent random variables taking values in the unit interval. The Pi's then form a Markov chain. The means and the variances of the Pi's are related to those of the ui's in a rather simple fashion. The case where ui' approx Beta(1, lambda sub i) is found to be particularly tractable for the analysis of bioassay data. (Author)

A general algebraic system M is considered with two binary operations. The family of all measurable functions with values in the unit interval is a motivating example. A state is a morphism from M to the unit interval, an observable is a morphism from the family of Borel sets to M. A joint distribution of two observables is constructed. It is applied for the construction of the sum of observables and for a representation of conditional probability.

Investigating equality: The Rényi spectrum

We answer two open questions concerning the existence of universal schemes for classification and regression estimation from stationary ergodic processes. It is shown that no measurable procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in the unit interval. It is further shown that no measurable procedure can produce weakly consistent classification rules from every bivariate stationary ergodic process for which the response variable is binary valued. The results of the paper are derived via reduction arguments and are based in part on recent work concerning density estimaton from ergodic processes.