In many practical applications, we are interested in computing the product of given matrices and/or a power of a given matrix. In some cases, the initial matrices are only known with interval uncertainty. It turns out that under this uncertainty, there is a principal difference between the product of two matrices and the product of three (or more) matrices:• on the one hand, it is more or less known that the problems of computing the exact range for the product of two matrices - and for the square of a matrix - are computationally feasible;• on the other hand, we prove that the problems of computing the exact ranges for the product of three matrices - and for the third power of a matrix - are NP-hard.

/pdf/computing-the-cube-of-an-interval-matrix-is-np-hard-25uprahsco.pdf

Computing the cube of an interval matrix is NP-Hard

In many practical situations, the relation between the experts’ degrees of confidence in different related statements is well described by Fuzzy Cognitive Maps (FCM). This empirical success is somewhat puzzling, since from the mathematical viewpoint, each FCM relation corresponds to a simplified one-neuron neural network, and it is well known that to adequately describe relations, we need multiple neurons. In this paper, we show that the empirical success of FCM can be explained if we take into account that human’s subjective opinions follow Miller’s seven plus minus two law.

/pdf/why-fuzzy-cognitive-maps-are-efficient-3ok8k1xyxe.pdf

Why Fuzzy Cognitive Maps Are Efficient

In classical (two-valued) logic, CNF and DNF forms of each propositional formula are equivalent to each other. In fuzzy logic, CNF and DNF forms are not equivalent, they form an interval that contains the fuzzy values of all classically equivalent propositional formulas. If we want to select a single value from this interval, then it is natural to select a linear combination of the interval's endpoints. In particular, we can do that for CNF and DNF forms of "and" and "or", thus designing natural fuzzy analogues of classical "and" and "or" operations. The problem with thus selected "and" and "or" operations is that, contrary to common sense expectations, they are not associative. We show the largest possible value of the corresponding non-associativity is reasonably small and thus, this non-associativity does not make these operations impractical.

/pdf/selecting-a-fuzzy-logic-operation-from-the-dnf-cnf-interval-5efdhnhgyn.pdf

Selecting a fuzzy logic operation from the DNF-CNF interval: how practical are the resulting operations?

A model for reasoning about trust and risk is developed as a result of investigating the representation and processing of beliefs within a type-2 fuzzy logic framework. This model is shown to be more realistic than the Dempster-Shaffer theory and subjective logic because of its lack of restriction to binary propositions. The model also gives an indication of the quality of its implications by allowing best case and worst case analyses. This model is tested using the game theoretic prisoners dilemma problem as well as the hypothetical operation of an Internet store. In both cases the model is shown to give a more realistic modeling environment and deliver more useful information than when using crisp values or type-1 sets in situations where input values and pay-offs are uncertain.

Trust evaluation within a type-2 fuzzy logic framework

One of the most empirically successful tools for studying dependence between different quantities in econometrics is the tool of vine copulas. In this paper, we explain this empirical success by sh...

http://www.cs.utep.edu/vladik/2015/olg15-05.pdf

Why are vine copulas so successful in econometrics

https://scholarworks.utep.edu/cgi/viewcontent.cgi?article=1493&context=cs_techrep

A Realistic (Non-Associative) Logic and a Possible Explanations of 7+-2 Law

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A Realistic (Non-Associative) Logic and a Possible Explanations of 7+-2 Law