Digital Commons Network^™

When Is A Single "And"-Condition Enough?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, there are several possible decisions. Any general recommendation means specifying, for each possible decision, conditions under which this decision is recommended. In some cases, a single "and"-condition is sufficient: e.g., a condition under which a patient is recommended to take aspirin is that "the patient has a fever and the patient does not have stomach trouble". In other cases, conditions are more complicated. A natural question is: when is a single "and"-condition enough? In this paper, we provide an answer to this question.

Which Fuzzy Implications Operations Are Polynomial? A Theorem Proves That This Can Be Determined By A Finite Set Of Inequalities, Sebastia Massanet, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To adequately represent human reasoning in a computer-based systems, it is desirable to select fuzzy operations that are as close to human reasoning as possible. In general, every real-valued function can be approximated, with any desired accuracy, by polynomials; it is therefore reasonable to use polynomial fuzzy operations as the appropriate approximations. We thus need to select, among all polynomial operations that satisfy corresponding properties -- like associativity -- the ones that best fit the empirical data. The challenge here is that properties like associativity mean satisfying infinitely many constraints (corresponding to infinitely many possible triples of values), while most …

Logical Inference Inevitably Appears: Fuzzy-Based Explanation, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich, Orsolya Csiszar

Departmental Technical Reports (CS)

Many thousands years ago, our primitive ancestors did not have the ability to reason logically and to perform logical inference. This ability appeared later. A natural question is: was this appearance inevitable -- or was this a lucky incident that could have been missed? In this paper, we use fuzzy techniques to provide a possible answer to this question. Our answer is: yes, the appearance of logical inference in inevitable.

Is Fully Explainable Ai Even Possible: Fuzzy-Based Analysis, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main limitations of many current AI-based decision-making systems is that they do not provide any understandable explanations of how they came up with the produced decision. Taking into account that these systems are not perfect, that their decisions are sometimes far from good, the absence of an explanation makes it difficult to separate good decisions from suspicious ones. Because of this, many researchers are working on making AI explainable. In some applications areas -- e.g., in chess -- practitioners get an impression that there is a limit to understandability, that some decisions remain inhuman -- not explainable. …

Selecting The Most Adequate Fuzzy Operation For Explainable Ai: Empirical Fact And Its Possible Theoretical Explanation, Orsolya Csiszar, Gábor Csiszar, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

A reasonable way to make AI results explainable is to approximate the corresponding deep-learning-generated function by a simple expression formed by fuzzy operations. Experiments on real data show that out of all easy-to-compute fuzzy operations, the best approximation is attained if we use an operation a + b − 0.5 ( limited to the interval [0,1]$. In this paper, we provide a possible theoretical explanation for this empirical result.

Interval-Valued And Set-Valued Extensions Of Discrete Fuzzy Logics, Belnap Logic, And Color Optical Computing, Victor L. Timchenko, Yury P. Kondratenko, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been recently shown that in some applications, e.g., in ship navigation near a harbor, it is convenient to use combinations of basic colors -- red, green, and blue -- to represent different fuzzy degrees. In this paper, we provide a natural explanation for the efficiency of this empirical fact: namely, we show that it is reasonable to consider discrete fuzzy logics, it is reasonable to consider their interval-valued and set-valued extensions, and that a set-valued extension of the 3-values logic is naturally equivalent to the use of color combinations.

Why People Tend To Overestimate Joint Probabilities, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in general, people overestimate the probabilities of joint events. In this paper, we provide an explanation for this phenomenon -- as explanation based on Laplace Indeterminacy Principle and Maximum Entropy approach.

How Success In A Task Depends On The Skills Level: Two Uncertainty-Based Justifications Of A Semi-Heuristic Rasch Model, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The more skills a student acquires, the more successful this student is with the corresponding tasks. Empirical data shows that the success in a task grows as a logistic function of skills; this dependence is known as the Rasch model. In this paper, we provide two uncertainty-based justifications for this model: the first justification provides a simple fuzzy-based intuitive explanation for this model, while the second -- more complex one -- explains the exact quantitative behavior of the corresponding dependence.

Complete Description Of Idempotent Hedges In Fuzzy Logic, Jaime Nava

Departmental Technical Reports (CS)

In describing expert knowledge, it is often important to properly take into account hedges} like "very", "somewhat", etc. In particular, fuzzy logic provides a consistent way of describing hedges. For some of the hedges, a repetition changes the meaning: e.g., "very very small" is smaller than "very small". However, other hedges -- like "somewhat" -- are idempotent, in the sense that repeating this hedge twice does not change the meaning. In this paper, we provide a complete description of such idempotent hedges.

Why Trapezoidal And Triangular Membership Functions Work So Well: Towards A Theoretical Explanation, Aditi Barua, Lalitha Snigdha Mudunuri, Olga Kosheleva

Departmental Technical Reports (CS)

In fuzzy logic, an imprecise ("fuzzy") property is described by its membership function μ(x), i.e., by a function which describes, for each real number x, to what degree this real number satisfies the desired property. In principle, membership functions can be of different shape, but in practice, trapezoidal and triangular membership functions are most frequently used. In this paper, we provide an interval-based theoretical explanation for this empirical fact.

Computing With Words: Towards A New Tuple-Based Formalization, Olga Kosheleva, Vladik Kreinovich, Ariel Garcia, Felipe Jovel, Luis A. Torres Escobedo, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

An expert opinion describes his or her opinion about a quantity by using imprecise ("fuzzy") words from a natural language, such as "small", "medium", "large", etc. Each of these words provides a rather crude description of the corresponding quantity. A natural way to refine this description is to assign degrees to which the observed quantity fits each of the selected words. For example, an expert can say that the value is reasonable small, but to some extent it is medium. In this refined description, we represent each quantity by a tuple of the corresponding degrees.

Once we have such a …

Minimization Of Average Sensitivity As A Method Of Selecting Fuzzy Functions And Operations: Successes And Limitations, Riya George, Suresh Subramanian, Alejandro Vega, Olga Kosheleva

Departmental Technical Reports (CS)

Fuzzy logic is an extension of the standard 2-valued logic -- with two possible truth values 0 ("false") and ("true") -- to values (degrees of certainty) represented by arbitrary numbers from the interval [0,1]. One of the main challenges in fuzzy logic is that we need to extend the usual logical operations from the set {0,1} to the entire interval, and there are many possible extensions. One promising technique for selecting a reasonable extension is to take into account that the fuzzy degrees of certainty are themselves only known with uncertainty; so, it makes sense to select an operation which …

Square Root Of "Not": A Major Difference Between Fuzzy And Quantum Logics, Vladik Kreinovich, Ladislav J. Kohout, Eunjin Kim

Departmental Technical Reports (CS)

Many authors have mentioned the similarity between quantum logic and fuzzy logic. In this paper, we show that, in spite of this similarity, these logics are not identical. Specifically, we emphasize that while quantum logic has a special ``square root of not'' operation which is very useful in quantum computing, fuzzy logic lacks such an operation.

Which Fuzzy Logic Is The Best: Pragmatic Approach (And Its Theoretical Analysis), Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In this position paper, we argue that when we are looking for the best fuzzy logic, we should specify in what sense the best, and that we get different fuzzy logics as ``the best'' depending on what optimality criterion we use.

Towards More Realistic (E.G., Non-Associative) And- And Or-Operations In Fuzzy Logic, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

On Efficient Representation Of Expert Knowledge By Fuzzy Logic: Towards An Optimal Combination Of Granularity And Higher-Order Approaches, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

A natural approach to designing an intelligent system is to incorporate expert knowledge into this system. One of the main approaches to translating this knowledge into computer-understandable terms is the approach of fuzzy logic. It has led to many successful applications, but in several aspects, the resulting computer representation is somewhat different from the original expert meaning. Two related approaches have been used to make fuzzy logic more adequate in representing expert reasoning: granularity and higher-order approaches. Each approach is successful in some applications where the other approach did not succeed so well; it is therefore desirable to combine these …

Non-Associative Operations, Bernadette Bouchon-Meunier, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A&B and B&A are the same, the corresponding and-operation should be commutative. Similarly, since A&A means the same as A, we should expect that the and-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of …

Towards More Realistic (E.G., Non-Associative) And- And Or-Operations In Fuzzy Logic, Jesus Martinez, Leopoldo Macias, Ammar Esper, Jesus Chaparro, Vick Alvarado, Scott A. Starks, Vladik Kreinovich

Departmental Technical Reports (CS)

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A&B and B&A are the same, the corresponding and-operation should be commutative. Similarly, since A&A means the same as A, we should expect that the and-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of …

Multi-Criteria Optimization - An Important Foundation Of Fuzzy System Design, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life design situations, there are several different criteria that we want to optimize, and these criteria are often in conflict with each other. Traditionally, such multi-criteria optimization situations are handled in an ad hoc manner, when different conflicting criteria are artificially combined into a single combination objective that is then optimized. The use of unnatural ad hoc tools is clearly not the best way of describing a very natural aspect of human reasoning. Fuzzy logic describes a much more natural way of handling multi-criterion optimization problems: when we cannot maximize each of the original conflicting criteria 100%, we …