Min and max are the only continuous ampersand-, V-operations for finite logics

Experts usually express their degrees of belief in their statements by the words of a natural language (like 'maybe', 'perhaps', etc.). If an expert system contains the degrees of beliefs t(A) and t(B) that correspond to the statements A and B, and a user asks this expert system whether 'A&B' is true, then it is necessary to come up with a reasonable estimate for the degree of belief of A&B. The operation that processes t(A) and t(B) into such an estimate t(A&B) is called an &-operation. Many different &-operations have been proposed. Which of them to choose? This can be (in principle) done by interviewing experts and eliciting a &-operation from them, but such a process is very time-consuming and therefore, not always possible. So, usually, to choose a &-operation, the finite set of actually possible degrees of belief is extended to an infinite set (e.g., to an interval (0,1)), define an operation there, and then restrict this operation to the finite set. Only this original finite set is considered. It is shown that a reasonable assumption that an &-operation is continuous (i.e., that gradual change in t(A) and t(B) must lead to a gradual change in t(A&B)), uniquely determines min as an &-operation. Likewise, max is the only continuous V-operation. These results are in good accordance with the experimental analysis of 'and' and 'or' in human beliefs

Uncertainty reasoning in expert systems

Intelligent control is a very successful way to transform the expert's knowledge of the type 'if the velocity is big and the distance from the object is small, hit the brakes and decelerate as fast as possible' into an actual control. To apply this transformation, one must choose appropriate methods for reasoning with uncertainty, i.e., one must: (1) choose the representation for words like 'small', 'big'; (2) choose operations corresponding to 'and' and 'or'; (3) choose a method that transforms the resulting uncertain control recommendations into a precise control strategy. The wrong choice can drastically affect the quality of the resulting control, so the problem of choosing the right procedure is very important. From a mathematical viewpoint these choice problems correspond to non-linear optimization and are therefore extremely difficult. In this project, a new mathematical formalism (based on group theory) is developed that allows us to solve the problem of optimal choice and thus: (1) explain why the existing choices are really the best (in some situations); (2) explain a rather mysterious fact that fuzzy control (i.e., control based on the experts' knowledge) is often better than the control by these same experts; and (3) give choice recommendations for the cases when traditional choices do not work

Strongly transitive fuzzy relations: A more adequate way to describe similarity

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. It is based on translating the standard definition of transitivity and transitive closure into fuzzy terms. This definition works fine, but to some extent it does not fully capture our understanding of transitivity. The reason is that this definition is based on fuzzifying only the positive side of transitivity: if R(a,b) and R(b,c), then R(a,c); but transitivity also includes a negative side: if R(a,b) and not R(a,c), then not R(b,c). In classical logic, this negative statement follows from the standard 'positive' definition of transitivity. In fuzzy logic, this negative part of the transitivity has to be formulated as an additional demand. A strongly transitive fuzzy relation as the one that satisfies both the positive and the negative transitivity demands is defined, the existence of strongly transitive closure is proven, and the relationship between strongly transitive similarity and clustering are found

How to help intelligent systems with different uncertainty representations cooperate with each other

In order to solve a complicated problem one must use the knowledge from different domains. Therefore, if one wants to automatize the solution of these problems, one has to help the knowledge-based systems that correspond to these domains cooperate, that is, communicate facts and conclusions to each other in the process of decision making. One of the main obstacles to such cooperation is the fact that different intelligent systems use different methods of knowledge acquisition and different methods and formalisms for uncertainty representation. So an interface f is needed, 'translating' the values x, y, which represent uncertainty of the experts' knowledge in one system, into the values f(x), f(y) appropriate for another one. The problem of designing such an interface as a mathematical problem is formulated and solved. It is shown that the interface must be fractionally linear: f(x) = (ax + b)/(cx + d)

Space-Related Applications of Intelligent Control: Which Algorithm to Choose? (Theoretical Analysis of the Problem)

For a space mission to be successful it is vitally important to have a good control strategy. For example, with the Space Shuttle it is necessary to guarantee the success and smoothness of docking, the smoothness and fuel efficiency of trajectory control, etc. For an automated planetary mission it is important to control the spacecraft's trajectory, and after that, to control the planetary rover so that it would be operable for the longest possible period of time. In many complicated control situations, traditional methods of control theory are difficult or even impossible to apply. In general, in uncertain situations, where no routine methods are directly applicable, we must rely on the creativity and skill of the human operators. In order to simulate these experts, an intelligent control methodology must be developed. The research objectives of this project were: to analyze existing control techniques; to find out which of these techniques is the best with respect to the basic optimality criteria (stability, smoothness, robustness); and, if for some problems, none of the existing techniques is satisfactory, to design new, better intelligent control techniques

Maximum entropy approach to fuzzy control

For the same expert knowledge, if one uses different &- and V-operations in a fuzzy control methodology, one ends up with different control strategies. Each choice of these operations restricts the set of possible control strategies. Since a wrong choice can lead to a low quality control, it is reasonable to try to loose as few possibilities as possible. This idea is formalized and it is shown that it leads to the choice of min(a + b,1) for V and min(a,b) for &. This choice was tried on NASA Shuttle simulator; it leads to a maximally stable control

How far we are from the complete knowledge: Complexity of knowledge acquisition in Dempster-Shafer approach

When a knowledge base represents the experts' uncertainty, then it is reasonable to ask how far we are from the complete knowledge, that is, how many more questions do we have to ask (to these experts, to nature by means of experimenting, etc) in order to attain the complete knowledge. Of course, since we do not know what the real world is, we cannot get the precise number of questions from the very beginning: it is quite possible, for example, that we ask the right question first and thus guess the real state of the world after the first question. So we have to estimate this number and use this estimate as a natural measure of completeness for a given knowledge base. We give such estimates for Dempster-Shafer formalism. Namely, we show that this average number of questions can be obtained by solving a simple mathematical optimization problem. In principle this characteristic is not always sufficient to express the fact that sometimes we have more knowledge. For example, it has the same value if we have an event with two possible outcomes and nothing else is known, and if there is an additional knowledge that the probability of every outcome is 0.5. We'll show that from the practical viewpoint this is not a problem, because the difference between the necessary number of questions in both cases is practically negligible

From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty (Invited Talk)

Interval computations estimate the uncertainty of the result of data processing in situations in which we only know the upper bounds $Delta$ on the measurement errors. In this case, based on the measurement result $widetilde x$, we can only conclude that the actual (unknown) value $x$ of the desired quantity is in the interval $[widetilde x-Delta,widetilde x+Delta]$. In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. As a result, we often have bounds with excess width. To remedy this problem, in our previous papers, we proposed an extension of interval technique to {it set computations}, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time