Partial orders for representing uncertainty, causality and decision making: General properties, operations, and algorithms

One of the main objectives of science and engineering is to help people select the most beneficial decisions. To make these decisions, we must know people\u27s preferences, we must have the information about different possible consequences of different decisions. Since information is never absolutely accurate and precise, we must also have information about the degree of certainty of different parts on information. All these types of information naturally lead to partial orders: • For preferences, a [special characters omitted] b means that b is preferable to a. This relation is used in decision theory. • For events, a [special characters omitted] b means that a can influence b. This causality relation is one of the fundamental notions of physics, especially of physics of space-time. • For uncertain statements, a [special characters omitted] b means that a is less certain than b. This relation is used in logics describing uncertainty, such as fuzzy logic. In each of these areas, there is abundant research about studying the corresponding partial orders. This research has revealed that some ideas are common in all three applications of partial orders. In this dissertation, we analyze general properties, operations, and algorithms related to partial orders for representing uncertainty, causality, and decision making, with a special emphasis on uncertainty. Under uncertainty, instead of a single partial order, we have a class C of possible partial orders. In such situations, it makes sense to ask when it is possible that a precedes b (i.e., when a precedes b according to some of these orders), and when it is necessary that a precedes b (i.e., when a precedes b according to all these orders). In Chapter 2, we give a general characterization of such possible order and necessary order relations. In Chapter 3, we consider a special case of such a situation, when different partial orders result from measurements with different accuracy. In this case, we can distinguish between the original ( closed ) partial order [special characters omitted] and the open partial order relation a [special characters omitted] b meaning that a [special characters omitted] b and we can verify this based on measurements (i.e., a [special characters omitted] b˜ for all b˜ from some neighborhood of b). It has been proven that once we know the open order, then we can uniquely reconstruct the closed order. Whether it is possible, vice versa, to reconstruct the open order from the closed one was an open problem. In Chapter 3, we prove that, under reasonable conditions, such a reconstruction is indeed possible. In Chapter 4, we move from potentially detectable (measurable) orders to orders which can be detected for a given accuracy. A typical example is when we only know the lower bound a and the upper bound ā for an object a; in this case, we only know that a belongs to the interval [a, ā]. In Chapter 4, we describe all possible relations between such intervals. Once an order is defined, we are interested in its properties , e.g., whether the order is a lattice. For special-relativity-type partial orders, a new necessary and sufficient criterion for being a lattice is described in Chapter 5. In many practical applications, we need to combine different partial orders. In Chapter 6, we describe all possible combination operations, and in Chapter 7, we provide a general algorithm that reduces the analysis of properties of such combined spaces to properties of individual partially ordered spaces