Partial Orders for Towards Combining . . . Representing - PowerPoint PPT Presentation

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 ◭◭ ◮◮ Francisco Zapata ◭ ◮ Department of Computer Science University of Texas at El Paso Page 1 of 45 500 W. University El Paso, TX 79968, USA Go Back fazg74@gmail.com 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 . . . Main Result – we must know people’s preferences, 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: ◭ ◮ – For preferences, a ≤ b means that b is preferable to a . This relation is used in decision theory. Page 2 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 Auxiliary Results in all three applications of partial orders. Proof of the Main Result • In our research, we plan to analyze: My Publications Home Page – general properties, operations, and algorithms – related to partial orders for representing uncertainty, Title Page causality, and decision making. ◭◭ ◮◮ • In our analysis, we will be most interested in uncer- ◭ ◮ tainty – the computer-science aspect of partial orders. Page 3 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 Uncertainty is . . . Partial Orders Properties of Ordered . . . • Uncertainty is explicitly mentioned only in the computer- Towards Combining . . . science example of partial orders. Main Result Auxiliary Results • However, uncertainty is ubiquitous in describing our Proof of the Main Result knowledge about all three types of partial orders. 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. ◭◭ ◮◮ • So, we can only observe an event which is close to b – ◭ ◮ e.g., that occurs 1 ± 0 . 001 sec after the reaction. Page 4 of 45 • In general, we can only guarantee that the observed event is within a certain neighborhood U b of the event b . Go Back Full Screen • In decision making, we similarly know the user’s pref- erences only with some accuracy. Close Quit

Partial Orders are . . . What We Plan to Do 4. Uncertainty-Motivated Experimentally Uncertainty is . . . Confirmable Relation Properties of Ordered . . . • Because of the uncertainty: Towards Combining . . . Main Result – the only possibility to experimentally confirm that Auxiliary Results a precedes b (e.g., that a can causally influence b ) Proof of the Main Result – is when for some neighborhood U b of the event b , we have a ≤ � b for all � My Publications b ∈ U b . Home Page • In topological terms, this “experimentally confirmable” Title Page relation a ≺ b means that: ◭◭ ◮◮ – the element b is contained in the future cone C + a = { c : a ≤ c } of the event a ◭ ◮ – together with some neighborhood. Page 5 of 45 • In other words, b belongs to the interior K + a 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 Uncertainty is . . . Confirmable Relation (cont-d) Properties of Ordered . . . • In usual space-time models: Towards Combining . . . Main Result – once we know the open cone K + a , Auxiliary Results – we can reconstruct the original cone C + a as the clo- Proof of the Main Result sure of K + a : C + a = K + 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? ◭◭ ◮◮ • In our paper (Zapata Kreinovich to appear), we show ◭ ◮ that this reconstruction is possible. Page 6 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 Uncertainty is . . . (EC) Relation to Actually EC One 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 Auxiliary Results • For example: 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 . ◭◭ ◮◮ • In this case, the only information that we have about ◭ ◮ the actual event a is that it belongs to the interval Page 7 of 45 def [ a, a ] = { a : a ≤ a ≤ a } . 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 Uncertainty is . . . (EC) Relation to Actually EC One (cont-d) Properties of Ordered . . . • It is desirable to describe possible relations between Towards Combining . . . such intervals. Main Result Auxiliary Results • Such a description has already been done for intervals Proof of the Main Result on the real line. 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. ◭◭ ◮◮ • We are currently working on a paper about intervals. ◭ ◮ • Instead of intervals, we can also consider more general Page 8 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 . . . Main Result • For example, we may want to know when such an order Auxiliary Results is a lattice, i.e., when: 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: ◭◭ ◮◮ – when the order is a semi-lattice , i.e., e.g., ◭ ◮ – when every two elements have the least upper bound. Page 9 of 45 • For the class of all subsets, we prove the lattice prop- Go Back erty in (Zapata Ramirez et al. 2011). Full Screen • We also describe when special relativity-type ordered spaces are lattices (K¨ unzi et 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 Auxiliary Results elements: true (1) and false (0). 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. ◭◭ ◮◮ • In this interval, the order is total ( linear ) in the sense ◭ ◮ that for every a, a ′ ∈ [0 , 1], either a ≤ a ′ or a ′ ≤ 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