Partial Orders are . What We Plan to Do Uncertainty is . Properties of Ordered . Partial Orders for Towards Combining . Representing Uncertainty, Main Result Auxiliary Results Causality, and Decision Proof of the Main Result My Publications Making: General Properties, Home Page Operations, and Algorithms Title Page JJ II Francisco Zapata Department of Computer Science J I University of Texas at El Paso Page1 of 45 500 W. University El Paso, TX 79968, USA Go Back [email protected] Full Screen Close Quit Partial Orders are . What We Plan to Do 1. Partial Orders are Important Uncertainty is . • One of the main objectives of science and engineering Properties of Ordered . is to select the most beneficial decisions. For that: Towards Combining . { we must know people's preferences, Main Result Auxiliary Results { we must have the information about different events Proof of the Main Result (possible consequences of different decisions), and My Publications { since information is never absolutely accurate, we Home Page must have information about uncertainty. Title Page • All these types of information naturally lead to partial orders: JJ II { For preferences, a ≤ b means that b is preferable to J I a. This relation is used in decision theory. Page2 of 45 { For events, a ≤ b means that a can influence b. This Go Back causality relation is used in space-time physics. Full Screen { For uncertain statements, a ≤ b means that a is Close less certain than b (fuzzy logic etc.). Quit Partial Orders are . What We Plan to Do 2. What We Plan to Do Uncertainty is . • In each of the three areas, there is a lot of research Properties of Ordered . about studying the corresponding partial orders. Towards Combining . Main Result • This research has revealed that some ideas are common in all three applications of partial orders. Auxiliary Results Proof of the Main Result • In our research, we plan to analyze: My Publications { general properties, operations, and algorithms Home Page { related to partial orders for representing uncertainty, Title Page causality, and decision making. JJ II • In our analysis, we will be most interested in uncer- J I tainty { the computer-science aspect of partial orders. Page3 of 45 • In our presentation: Go Back { we first give a general outline, Full Screen { then present two results in detail (if time allows). Close Quit Partial Orders are . What We Plan to Do 3. Uncertainty is Ubiquitous in Applications of Partial Orders Uncertainty is . Properties of Ordered . • Uncertainty is explicitly mentioned only in the computer- Towards Combining . science example of partial orders. Main Result • However, uncertainty is ubiquitous in describing our Auxiliary Results knowledge about all three types of partial orders. Proof of the Main Result My Publications • For example, we may want to check what is happening Home Page exactly 1 second after a certain reaction. Title Page • However, in practice, we cannot measure time exactly. JJ II • So, we can only observe an event which is close to b { e.g., that occurs 1 ± 0:001 sec after the reaction. J I Page4 of 45 • In general, we can only guarantee that the observed Go Back event is within a certain neighborhood Ub of the event b. • In decision making, we similarly know the user's pref- Full Screen erences only with some accuracy. Close Quit Partial Orders are . What We Plan to Do 4. Uncertainty-Motivated Experimentally Confirmable Relation Uncertainty is . Properties of Ordered . • Because of the uncertainty: Towards Combining . { the only possibility to experimentally confirm that Main Result a precedes b (e.g., that a can causally influence b) Auxiliary Results Proof of the Main Result { is when for some neighborhood Ub of the event b, My Publications we have a ≤ eb for all eb 2 Ub. Home Page • In topological terms, this \experimentally confirmable” relation a ≺ b means that: Title Page + JJ II { the element b is contained in the future cone Ca = fc : a ≤ cg of the event a J I { together with some neighborhood. Page5 of 45 + • In other words, b belongs to the interior Ka of the Go Back closed cone C+. a Full Screen • Such relation, in which future cones are open, are called Close open. Quit Partial Orders are . What We Plan to Do 5. Uncertainty-Motivated Experimentally Confirmable Relation (cont-d) Uncertainty is . Properties of Ordered . • In usual space-time models: Towards Combining . + Main Result { once we know the open cone Ka , Auxiliary Results { we can reconstruct the original cone C+ as the clo- a Proof of the Main Result sure of K+: C+ = K+. a a a My Publications • A natural question is: vice versa, Home Page { can we uniquely reconstruct an open order Title Page { if we know the corresponding closed order? JJ II • In our paper (Zapata Kreinovich to appear), we show J I that this reconstruction is possible. Page6 of 45 • This result provides a partial solution to a known open Go Back problem. Full Screen Close Quit Partial Orders are . What We Plan to Do 6. From Potentially Experimentally Confirmable (EC) Relation to Actually EC One Uncertainty is . Properties of Ordered . • It is also important to check what can be confirmed Towards Combining . when we only have observations with a given accuracy. Main Result • For example: Auxiliary Results Proof of the Main Result { instead of the knowing the exact time location of My Publications an an event a, Home Page { we only know an event a that preceded a and an Title Page event a that follows a. JJ II • In this case, the only information that we have about the actual event a is that it belongs to the interval J I Page7 of 45 [a; a] def= fa : a ≤ a ≤ ag: Go Back • It is desirable to describe possible relations between Full Screen such intervals. Close Quit Partial Orders are . What We Plan to Do 7. From Potentially Experimentally Confirmable (EC) Relation to Actually EC One (cont-d) Uncertainty is . Properties of Ordered . • It is desirable to describe possible relations between Towards Combining . such intervals. Main Result • Such a description has already been done for intervals Auxiliary Results on the real line. Proof of the Main Result My Publications • The resulting description is known as Allen's algebra. Home Page • In these terms, what we want is to generalize Allen's Title Page algebra to intervals over an arbitrary poset. JJ II • We are currently working on a paper about intervals. J I • Instead of intervals, we can also consider more general Page8 of 45 sets. Go Back • Our preliminary results about general sets are described Full Screen in a paper (Zapata Ramirez et al. 2011). Close Quit Partial Orders are . What We Plan to Do 8. Properties of Ordered Spaces Uncertainty is . • Once a new ordered set is defined, we may be interested Properties of Ordered . in its properties. Towards Combining . • For example, we may want to know when such an order Main Result is a lattice, i.e., when: Auxiliary Results Proof of the Main Result { for every two elements, My Publications { there is the greatest lower bound and the least up- Home Page per bound. Title Page • If this set is not a lattice, we may want to know: JJ II { when the order is a semi-lattice, i.e., e.g., J I { when every two elements have the least upper bound. Page9 of 45 • For the class of all subsets, we prove the lattice prop- erty in (Zapata Ramirez et al. 2011). Go Back • We also describe when special relativity-type ordered Full Screen spaces are lattices (K¨unziet al. 2011). Close Quit Partial Orders are . What We Plan to Do 9. Towards Combining Ordered Spaces: Fuzzy Logic Uncertainty is . • In the traditional 2-valued logic, every statement is Properties of Ordered . either true or false. Towards Combining . Main Result • Thus, the set of possible truth values consists of two elements: true (1) and false (0). Auxiliary Results Proof of the Main Result • Fuzzy logic takes into account that people have differ- My Publications ent degrees of certainty in their statements. Home Page • Traditionally, fuzzy logic uses values from the interval Title Page [0; 1] to describe uncertainty. JJ II • In this interval, the order is total (linear) in the sense J I that for every a; a0 2 [0; 1], either a ≤ a0 or a0 ≤ a. Page 10 of 45 • However, often, partial orders provide a more adequate Go Back description of the expert's degree of confidence. Full Screen Close Quit Partial Orders are . What We Plan to Do 10. Towards General Partial Orders Uncertainty is . • For example, an expert cannot describe her degree of Properties of Ordered . certainty by an exact number. Towards Combining . Main Result • Thus, it makes sense to describe this degree by an in- terval [d; d] of possible numbers. Auxiliary Results Proof of the Main Result • Intervals are only partially ordered; e.g., the intervals My Publications [0:5; 0:5] and [0; 1] are not easy to compare. Home Page • More complex sets of possible degrees are also some- Title Page times useful. JJ II • Not to miss any new options, in this research, we con- J I sider general partially ordered spaces. Page 11 of 45 Go Back Full Screen Close Quit Partial Orders are . What We Plan to Do 11. Need for Product Operations Uncertainty is . • Often, two (or more) experts evaluate a statement S. Properties of Ordered . Towards Combining . • Then, our certainty in S is described by a pair (a1; a2), Main Result where ai 2 Ai is the i-th expert's degree of certainty. Auxiliary Results • To compare such pairs, we must therefore define a par- Proof of the Main Result tial order on the set A1 × A2 of all such pairs.