Indistinguishability operators in measurement theory,Part II: Construction of indistinguishability operators based on probability distributions

As a continuation of the work initiated by Demirci, the main subject of this paper is the problem of constructing indistinguishability operators in terms of probability distribution functions and the probability density functions, addressed in Demirci [“Indistinguishability operators in measurement theory, Part I: Conversions of indistinguishability operators with respect to scales”, Int. J. General Systems (2003f) To appear]. In detail, two different approaches have been developed for the solution of this construction problem.     

Maps and isometries between indistinguishability operators

 In this paper, some geometric aspects of indistinguishability operators are studied by using the concept of morphism between them. Among all possible types of morphisms, the paper is focused on the following cases: Maps that transform a T-indistinguishability operator into another of such operators with respect to the same t-norm T and maps that transform a T-indistinguishability operator into another one of such operators with respect to a different t-norm T ^′. The group of isometries of a given T-indistinguishability operator is also studied and it is determined for the case of one-dimensional operators, in particular for the natural indistinguishability operators E _T on [0, 1]. Finally, the indistinguishability operators invariant under translations on the real line are characterized.     

Relevancy transformation operators: Construction methods

This article deals with the construction of relevancy transformation (RET) operators for fuzzy systems. The notion of pseudo‐duality is introduced to obtain new RET operators, and t‐norms, t‐conorms, nullnorms, and uninorms are used in different ways for the same purpose. Finally, several other methods to construct new RET operators from old ones are pointed out. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 155–171, 2006.     

On contra‐symmetry and MPT conditionality in fuzzy logic

This article deals with the N‐contrapositive symmetry of fuzzy implication operators J verifying either Modus Ponens or Modus Tollens inequalities, in a similar and complementary framework to the one in which Fodor (“Contrapositive symmetry of fuzzy implications.” Fuzzy Set Syst 1995;69:141–156) did begin with the subject in fuzzy logic, that is, with the verification of J(a, b) = J(N(b), N(a)) for all a, b in [0,1] and some strong‐negation function N. This property corresponds to the classical p → q = ¬q → ¬p. The aim of this article is to study that property in relation to either Modus Ponens or Modus Tollens meta‐rules of inference when the functions J are taken among those that belong to the usual families of implications in fuzzy logic. That is, the contra‐positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor. Within this study new types of implication functions are introduced and analyzed. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 313–326, 2005.     

Rates of A-statistical convergence of approximating operators

Abstract Using the concepts of rates of statistical convergence we investigate approximation properties of positive linear operators defined on the space C[0,b], 0<b<1, which includes many well-known operators in approximation theory. We also use the modulus of continuity and Lipschitz functions.     

Darboux transformations for factorable Laplace operators

Factorable Laplace operators of the form L = ? _x ? _y + a? _x + b? _y + c, where the coefficients a, b, c are not necessarily constants, are considered. For these operators, the Darboux transformations \(L\xrightarrow{M}L_1\) , M ∈ K[? _x ], defined by the intertwining relation NL = L ₁ M are considered. It is shown that only the following cases are possible: either (1) M ∩ ker? _x + b = {0} and L ₁ is also factorable or (2) M ∩ ker? _x + b contains a nonzero element. We prove that, in both cases, the Darboux transformation can be represented as a product of first-order Darboux transformations. For case (2), the proof is based on the fact that the Darboux transformation of operator L can be reduced to Darboux transformations of first-order operators.     

Generating indistinguishability operators from prototypes

Given a T‐indistinguishability operator E defined between some fuzzy subsets of a universe of discourse X, this paper studies three ways to generate an indistinguishability operator on X compatible with E inspired by the tools used in approximate reasoning and on the duality principle. Of special interest are the cases when the fuzzy subsets determine a partition and a hard‐partition of the universe X. © 2002 Wiley Periodicals, Inc.     

A duality relationship between fuzzy metrics and metrics

Based on the duality relationship between indistinguishability operators and (pseudo-)metrics, we address the problem of establishing whether there is a relationship between the last ones and fuzzy (pseudo-)metrics. We give a positive answer to the posed question. Concretely, we yield a method for generating fuzzy (pseudo-)metrics from (pseudo)-metrics and vice versa. The aforementioned methods involve the use of the pseudo-inverse of the additive generator of a continuous Archimedean t-norm. As a consequence, we get a method to generate non-strong fuzzy (pseudo-)metrics from (pseudo-)metrics. Examples that illustrate the exposed methods are also given. Finally, we show that the classical duality relationship between indistinguishability operators and (pseudo)-metrics can be retrieved as a particular case of our results when continuous Archimedean t-norms are under consideration.     

The structure of decomposable indistinguishability operators

Decomposable fuzzy relations are studied.Symmetric fuzzy relations are proved to be generated by a single fuzzy subset.For Archimedean t-norms, decomposable indistinguishability operators generate special kinds of betweenness relations that characterize them.A new way to generate indistinguishability operators coherent with the underlying ordering structure of the real line is given in the sense that this structure is compatible with the betweenness relation generated by the relation is developed.     

Extending multicriteria decision making by mixing t‐norms and OWA operators

We consider the problem of multicriteria decision making. We indicate how the evaluation of an alternative involves a determination of the degree to which subsets of criteria are satisfied by the alternative. This calculation is based upon an anding of the satisfactions of the individual criteria in the subsets. We consider the possibility of using t‐norms other than the Min for the and operation. Using this generalization we develop an extension of the OWA operators, called the TOWA, which involves a mixing of the t‐norm with the OWA operator. We extend this generalization to other aggregation techniques, the Choquet and Sugeno integrals. We introduce the concept of the Power of a t‐norm to provide an ordering over the t‐norm operators. We look at Power of a number of families of t‐norm. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 453–474, 2005.     

Approximation processes for resolvent operators

We introduce general sequences of linear operators obtained from classical approximation processes which are useful in the approximation of the resolvent operators of the generators of suitable C ₀-semigroups. The main aim is the representation of the resolvent operators in terms of classical approximation operators. Work performed under the auspices of PRIN 2006–07 “Kolmogorov equations” (coordinator G. Da Prato)     

CausalC 0 operators and feedback stability

Linear feedback systems, for which the return differenceT is aC ₀ operator are studied. It is shown that ifT is essentially unitary then such a system is well posed and stable. All strictly causalC ₀ operators for whichI–T ^* T is compact are characterized.     

A majority model in group decision making using QMA–OWA operators

Group decision‐making problems are situations where a number of experts work in a decision process to obtain a final value that is representative of the global opinion. One of the main problems in this context is to design aggregation operators that take into account the individual opinions of the decision makers. One of the most important operators used for synthesizing the individual opinions in a representative value of majority in the OWA operator, where the majority concept used aggregation processes, is modeled using fuzzy logic and linguistic quantifiers. In this work the semantic of majority used in OWA operators is analyzed, and it is shown how its application in group decision‐making problems does not produce representative results of the concept expressed by the quantifier. To solve this type of problem, two aggregation operators, QMA–OWA, are proposed that use two quantification strategies and a quantified normalization process to model the semantic of the linguistic quantifiers in the group decision‐making process. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 193–208, 2006.     

The effect of operators on parsing and evaluation in APL

The importance of function valued functions (operators) in APL is now well recognized, and considerable attention has been paid to the introduction of more general operators into the language. In this article we consider some of the consequences for the implementation of interpreters and compilers for APL.We first consider the difficulties that arise with the introduction of basic valued and operator valued operators. We then discuss various modifications to the syntactic and semantic structure of APL that serve to avoid these difficulties. Abstraction operators as in the λ calculus are also introduced. Finally, we describe an abstract machine for interpreting APL-like expressions which is analogous to the SECD machine used in the λ calculus.     

The axiomatic characterizations on L-fuzzy covering-based approximation operators

Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.     

Non-ambiguity of blind watermarking: a revisit with analytical resolution

Resistance to ambiguity attack is an important requirement for a secure digital rights management (DRM) system. In this paper,we revisit the non-ambiguity of a blind watermarking based on the compu-tational indistinguishability between pseudo random sequence generator (PRSG) sequence ensemble and truly random sequence ensemble. Ambiguity attacker on a watermarking scheme,which uses a PRSG sequence as watermark,is viewed as an attacker who tries to attack a noisy PRSG sequence. We propose and prove the security theorem for binary noisy PRSG sequence and security theorem for gen-eral noisy PRSG sequence. It is shown that with the proper choice of the detection threshold Th = an1/2 (a is a normalized detection threshold; n is the length of a PRSG sequence) and n 1.39×m/a2 (m is the key length),the success probability of an ambiguity attack and the missed detection probability can both be made negligibly small thus non-ambiguity and robustness can be achieved simultaneously for both practical quantization-based and blind spread spectrum (SS) watermarking schemes. These analytical resolutions may be used in designing practical non-invertible watermarking schemes and measuring the non-ambiguity of the schemes.     

Parameterized additive neat OWA operators with different orness levels

The article proposes an extension of the BADD OWA operator—ANOWA (additive neat OWA) operator—and defines its orness measure. Some properties of the weighting function associated with orness level are analyzed. Then two special classes of ANOWA operator with maximum entropy and minimum variance are proposed, and the orness of the BADD OWA operator is discussed. For a given orness level, these ANOWA operators can be uniquely determined. Their aggregation values for any aggregation elements set always monotonically increase with their orness levels. Therefore they can be used as a parameterized aggregation method with orness as its control parameter and to represent the decision maker's preference. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1045–1072, 2006.     

Decidability and Combination Results for Two Notions of Knowledge in Security Protocols

In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually considered: deducibility and indistinguishability. Those notions are well-studied and several decidability results already exist to deal with a variety of equational theories. Most of the existing results are dedicated to specific equational theories and only few results, especially in the case of indistinguishability, have been obtained for equational theories with associative and commutative properties (AC)(\textsf{AC}). In this paper, we show that existing decidability results can be easily combined for any disjoint equational theories: if the deducibility and indistinguishability relations are decidable for two disjoint theories, they are also decidable for their union. We also propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of equational theories involving AC\textsf{AC} operators. As a consequence of these two results, new decidability and complexity results can be obtained for many relevant equational theories.     

Robust stability of Metzler operators under parameter perturbations

In this paper we study how the spectral bound of Metzler operators changes under parameter perturbations. Characterizations of the stability radii of Metzler operator with respect to this type of disturbances are established. The results generalize those obtained in (Vietnam J. Math. 2006; 34 :357–368; Vietnam J. Math. 1998; 26 :147–163). Copyright © 2009 John Wiley & Sons, Ltd.     

Indistinguishability operators in measurement theory,Part I: Conversions of indistinguishability operators with respect to scales

For two given ordinal scales in a measurement process, the present paper investigates how an indistinguishability operator evaluated according to one of these ordinal scales can be converted to another indistinguishability operator w.r.t. the other ordinal scale, and establishes the mathematical base of these conversions under the framework of measurement theory [Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971) Foundations of Measurement, Vol. 1 (Academic Press, San Diego)]. Additionally, this work exposes the rudimentary facts behind the studies in [“Fuzzy Numbers and Equality Relations”, Proc. FUZZ'IEEE 93 (1993) 1298–1301; “Fuzzy Sets and Vague Environments”, Fuzzy Sets and Systems 66 (1994) 207–221; “Fuzzy Control on the Basis of Equality Relations-with an Example from Idle Speed Control”, IEEE Transactions on Fuzzy Systems 3 (1995) 336–350; and “T-partitions of the Real Line Generated by Idempotent Shapes”, Fuzzy Sets and Systems 91 (1997) 177–184], and points out the measurement theoretic derivations of the results in these studies.