Interval-Valued Degrees of Belief: Applications of Interval Computations to Expert Systems and Intelligent Control

“…However, often, partial orders provide a more adequate description of the expert's degree of confidence. For example, since an expert cannot describe her degree of certainty by an exact number, it makes sense to describe this degree by an interval [d, d] of possible numbers [7], [9] -and intervals are only partially ordered; e.g., the intervals [0.5, 0.5] and [0, 1] are not easy to compare.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

Zapata

Kosheleva

Villaverde

2011

2011 Annual Meeting of the North American Fuzzy Information Processing Society

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Abstract-In this paper, we describe how checking whether a given property F is true for a product A1 × A2 of partially ordered spaces can be reduced to checking several related properties of the original spaces Ai. I. FORMULATION OF THE PROBLEMDegrees of certainty: from {0, 1} to [0, 1] to general partially ordered sets. In the traditional 2-valued logic, every statement is either true or false. Thus, the set of possible truth values consists of two elements: true (1) and false (0).Fuzzy logic takes into account that people have different degrees of certainty in their statements; see, e.g., [2], [10].Traditionally, fuzzy logic uses values from the interval [0, 1] to describe uncertainty. In this interval, the order is total (linear) in the sense that for every two elements a, a More complex sets of possible degrees are also sometimes useful. Not to miss any new options, in this paper, we consider general partially ordered spaces.Need for product operations. Often, two (or more) experts evaluate a statement S. Then, our certainty in S is described by a pair (a 1 , a 2 ), where a i ∈ A i is the i-th expert's degree of certainty. To compare such pairs, we must therefore define a partial order on the set A 1 × A 2 of all such pairs.Two examples of product operations. One example of a partial order on A 1 × A 2 is a Cartesian product:This product corresponds to a cautious approach, when our confidence in S ′ is higher than in S if and only if it is higher for both experts.Another example is a lexicographic product:This product corresponds to the case when we have the absolute confidence in the first expert; then, we only use the opinion of the second expert when, to the first expert, the degrees of certainty are indistinguishable. We can have other product operations in which the relation between the pairs (a 1 , a 2 ) and (a A natural question. Once a product is defined, it is reasonable to ask when the resulting partially ordered set A 1 × A 2 it satisfies a certain property: is it a total order? is it a lattice order? etc. It is desirable to have some criteria that would transform the question about the product space into questions about related properties of component spaces.Some such criteria are known (see, e.g., [13] and references therein). For example:• A Cartesian product is a total order if and only if one of the components is a total order, and the other consists of a single element.• A lexicographic product is a total order if and only if both components are totally ordered.What we do in this paper. In this paper, we provide a general algorithm that reduced the question whether a certain property is satisfied for a product to several properties of component spaces. Applications beyond orders. Our algorithm does not use the fact that the original relations are orders (i.e., transitive antisymmetric relations). Thus, our algorithm is applicable to a general case when we have an arbitrary binary relationequivalence, similarity, etc. Moreover, this algorithm can be

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“…It is worth mentioning that there exist alternative justifications of Hurwicz criterion; see, e.g., [6,8].…”

Section: Proposition 1 For Every Closed Shift-and Scale-invariant Inmentioning

confidence: 99%

On Decision Making under Interval Uncertainty: A New Justification of Hurwicz Optimism-Pessimism Approach and its Use in Group Decision Making

Huynh

Nakamori

et al. 2009

2009 39th International Symposium on Multiple-Valued Logic

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Interval-Valued Degrees of Belief: Applications of Interval Computations to Expert Systems and Intelligent Control

Cited by 87 publications

References 123 publications

Membership functions representing a number vs. representing a set: Proof of unique reconstruction

Membership functions representing a number vs. representing a set: Proof of unique reconstruction

How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

On Decision Making under Interval Uncertainty: A New Justification of Hurwicz Optimism-Pessimism Approach and its Use in Group Decision Making

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