Vladik Kreinovich

System reliability: a case when fuzzy logic enhances probability theory's ability to deal with real-world problems

In his paper Probability theory needs an infusion of fuzzy logic to enhance its ability to deal with real-world problems, Zadeh explains how probability theory needs fuzzy logic. We give an example of a real-world problem for which such an infusion is indeed successful: the problem of system reliability.

Propagation of Interval and Probabilistic Uncertainty in Cyberinfrastructure-related Data Processing and Data Fusion

On various examples ranging from geosciences to environmental sciences, thisbook explains how to generate an adequate description of uncertainty, how to justifysemiheuristic algorithms for processing uncertainty, and how to make these algorithmsmore computationally efficient. It explains in what sense the existing approach touncertainty as a combination of random and systematic components is only anapproximation, presents a more adequate three-component model with an additionalperiodic error component, and explains how uncertainty propagation techniques canbe extended to this model. The book provides a justification for a practically efficientheuristic technique (based on fuzzy decision-making). It explains how the computationalcomplexity of uncertainty processing can be reduced. The book also shows how totake into account that in real life, the information about uncertainty is often onlypartially known, and, on several practical examples, explains how to extract the missinginformation about uncertainty from the available data.

How to Make A Proof of Halting Problem More Convincing: A Pedagogical Remark

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data. A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs. To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction. However, this auxiliary program is usually very artificial. So, a natural question arises: what if we only require that the halt-checker work for reasonable programs? In this paper, we show that even with such a restriction, haltcheckers are not possible – and thus, we make a proof of halting problem more convincing for students. 1 Formulation of the Problem Halting problem: reminder. A computer science degree means acquiring both the practical skills needed to design and program software and the theoretical knowledge describing which computational tasks are possible and which are not. Different programs include different examples of problems for which no computational solution is possible, but all of them include – with proof – the very first example of such a problem: the halting problem, according to which no algorithm is possible that, given a program p and data d, always checks whether p halts on d; see, e.g., [2]. Some textbooks describe this result in terms of Turing machines but, in our opinion, this result is much clearer to students when it is described in terms of programs – i.e., something with which are very familiar – rather than in terms of Turing machines, a new concept that they have just learned in the corresponding theoretical course and with which they are not yet very familiar. Let us therefore concentrate on the formulation of this result in terms of programs.

Why Zipf's law: a symmetry-based explanation

Abstract In many practical situations, we have probability distributions for which, for large values of the corresponding quantity x, the probability density has the form ρ(x) ∼ x−α for some α > 0. While, in principle, we have laws corresponding to different α, most frequently, we encounter situations – first described by Zipf for linguistics – when α ≈ 1. The fact that Zipf’s has appeared frequently in many different situations seems to indicate that there must be some fundamental reason behind this law. In this paper, we provide a possible explanation.