Physicists Assume . . . A Seemingly Natural . . . The Above . . . Relation to . . . Use of Maxitive (Possibility) Events with 0 . . . New Idea Coin Example Measures in Foundations of Main Result: Relation . . . Possible Practical Use . . . Physics and Description of Result Degree of Typicalness Randomness: Case Study Beyond Maxitive . . . Acknowledgments Title Page Andrei M. Finkelstein ◭◭ ◮◮ Institute of Applied Astronomy Russian Academy of Sciences, St Petersburg, Russia ◭ ◮ Olga Kosheleva, Vladik Kreinovich, Scott A. Starks Page 1 of 14 Pan-American Center for Earth & Environ. Stud. Go Back University of Texas, El Paso, TX 79968, USA vladik@cs.utep.edu Full Screen Hung T. Nguyen Close New Mexico State U., Las Cruces, NM, 88003, USA Quit
Physicists Assume . . . 1. Physicists Assume that Initial Conditions and Val- A Seemingly Natural . . . ues of Parameters are Not Abnormal The Above . . . Relation to . . . • To a mathematician, the main contents of a physical theory is its equations. Events with 0 . . . New Idea • Not all solutions of the equations have physical sense. Coin Example • Ex. 1: Brownian motion comes in one direction; Main Result: Relation . . . Possible Practical Use . . . • Ex. 2: implosion glues shattered pieces into a statue; Result • Ex. 3: fair coin falls heads 100 times in a row. Degree of Typicalness Beyond Maxitive . . . • Mathematics: it is possible. Acknowledgments • Physics (and common sense): it is not possible. Title Page • Our objective: supplement probabilities with a new formalism that more ◭◭ ◮◮ accurately captures the physicists’ reasoning. ◭ ◮ Page 2 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 2. A Seemingly Natural Formalizations of This Idea A Seemingly Natural . . . The Above . . . • Physicists: only “not abnormal” situations are possible. Relation to . . . Events with 0 . . . • Natural formalization: idea. If a probability p ( E ) of an event E is small enough, then this event cannot happen. New Idea Coin Example • Natural formalization: details. There exists the “smallest possible probabil- Main Result: Relation . . . ity” p 0 such that: Possible Practical Use . . . – if the computed probability p of some event is larger than p 0 , then this Result event can occur, while Degree of Typicalness Beyond Maxitive . . . – if the computed probability p is ≤ p 0 , the event cannot occur. Acknowledgments • Example: a fair coin falls heads 100 times with prob. 2 − 100 ; it is impossible Title Page if p 0 ≥ 2 − 100 . ◭◭ ◮◮ ◭ ◮ Page 3 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 3. The Above Formalization of the Notion of “Typi- A Seemingly Natural . . . cal” is Not Always Adequate The Above . . . Relation to . . . • Problem: every sequence of heads and tails has exactly the same probability. Events with 0 . . . New Idea • Corollary: if we choose p 0 ≥ 2 − 100 , we will thus exclude all sequences of 100 Coin Example heads and tails. Main Result: Relation . . . • However, anyone can toss a coin 100 times, and this proves that some such Possible Practical Use . . . sequences are physically possible. Result Degree of Typicalness • Similar situation: Kyburg’s lottery paradox: Beyond Maxitive . . . – in a big (e.g., state-wide) lottery, the probability of winning the Grand Acknowledgments Prize is so small that a reasonable person should not expect it; Title Page – however, some people do win big prizes. ◭◭ ◮◮ ◭ ◮ Page 4 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 4. Relation to Non-Monotonic Reasoning A Seemingly Natural . . . The Above . . . • Traditional logic is monotonic: once a statement is derived it remains true. Relation to . . . Events with 0 . . . • Expert reasoning is non-monotonic: New Idea – birds normally fly, Coin Example – so, if we know only that Sam is a bird, we conclude that Sam flies; Main Result: Relation . . . Possible Practical Use . . . – however, if we learn the new knowledge that Sam is a penguin, we conclude that Sam doesn’t fly. Result Degree of Typicalness • Non-monotonic reasoning helps resolve the lottery paradox (Poole et al.) Beyond Maxitive . . . • Our approach: in fact, what we propose can be viewed as a specific non- Acknowledgments monotonic formalism for describing rare events. Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 5. Events with 0 Probabilities are Possible: Another A Seemingly Natural . . . Explanation for the Lottery Paradox The Above . . . Relation to . . . • Idea: common sense intuition is false, events with small (even 0) probability Events with 0 . . . are possible. New Idea Coin Example • This idea is promoted by known specialists in foundations of probability: Main Result: Relation . . . K. Popper, B. De Finetti, G. Coletti, A. Gilio, R. Scozzafava, W. Spohn, etc. Possible Practical Use . . . • Out attitude: our objective is to formalize intuition, not to reject it. Result Degree of Typicalness • Interesting: both this approach and our approach lead to the same formalism Beyond Maxitive . . . (of maxitive measures). Acknowledgments • Conclusion: Maybe there is a deep relation and similarity between the two Title Page approaches. ◭◭ ◮◮ ◭ ◮ Page 6 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 6. New Idea A Seemingly Natural . . . The Above . . . • Example: height: Relation to . . . Events with 0 . . . – if height is ≥ 6 ft, it is still normal; New Idea – if instead of 6 ft, we consider 6 ft 1 in, 6 ft 2 in, etc., then ∃ h 0 s.t. everyone Coin Example taller than h 0 is abnormal; Main Result: Relation . . . – we are not sure what is h 0 , but we are sure such h 0 exists. Possible Practical Use . . . • General description: on the universal set U , we have sets A 1 ⊇ A 2 ⊇ . . . ⊇ Result � A n ⊇ . . . s.t. A n = ∅ . Degree of Typicalness Beyond Maxitive . . . n Acknowledgments • Example: A 1 = people w/height ≥ 6 ft, A 2 = people w/height ≥ 6 ft 1 in, etc. Title Page • A set T ⊆ U is called a set of typical (not abnormal) elements if for every ◭◭ ◮◮ � definable sequence of sets A n for which A n ⊇ A n +1 for all n and A n = ∅ , n ◭ ◮ there exists an integer N for which A N ∩ T = ∅ . Page 7 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 7. Coin Example A Seemingly Natural . . . The Above . . . • Universal set U = { H , T } I N Relation to . . . Events with 0 . . . • Here, A n is the set of all the sequences that start with n heads and have at least one tail. New Idea Coin Example • The sequence { A n } is decreasing and definable, and its intersection is empty. Main Result: Relation . . . Possible Practical Use . . . • Therefore, for every set T of typical elements of U , there exists an integer N for which A N ∩ T = ∅ . Result Degree of Typicalness • This means that if a sequence s ∈ T is not abnormal and starts with N heads, Beyond Maxitive . . . it must consist of heads only. Acknowledgments • In physical terms, it means a random sequence (i.e., a sequence that contains Title Page both heads and tails) cannot start with N heads. ◭◭ ◮◮ • This is exactly what we wanted to formalize. ◭ ◮ Page 8 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 8. Main Result: Relation to Possibility Measures A Seemingly Natural . . . The Above . . . • Idea: to describe a set of typical elements, we ascribe, to each definable Relation to . . . monotonic sequence { A n } , the smallest integer N ( { A n } ) for which Events with 0 . . . New Idea A N ∩ T = ∅ . Coin Example Main Result: Relation . . . • This integer can be viewed as measure of complexity of the sequence: Possible Practical Use . . . – for simple sequences, it is smaller, Result – for more complex sequences, it is larger. Degree of Typicalness Beyond Maxitive . . . • In terms of complexity: an element x ∈ U is typical if and only if for every Acknowledgments definable decreasing sequence { A n } with an empty intersection, x �∈ A N , where N = N ( { A n } ) is the complexity of this sequence. Title Page • Theorem: N ( { A n } ) is a maxitive (possibility) measure, i.e., N ( { A n ∪ B n } ) = ◭◭ ◮◮ max( N ( { A n } ) , N ( { B n } )) . ◭ ◮ Page 9 of 14 Go Back Full Screen Close Quit
Physicists Assume . . . 9. Possible Practical Use of This Idea: When to Stop A Seemingly Natural . . . an Iterative Algorithm The Above . . . Relation to . . . • Situation in numerical mathematics: Events with 0 . . . New Idea – we often know an iterative process whose results x k are known to con- Coin Example verge to the desired solution x , but Main Result: Relation . . . – we do not know when to stop to guarantee that Possible Practical Use . . . Result d X ( x k , x ) ≤ ε. Degree of Typicalness Beyond Maxitive . . . • Heuristic approach: stop when d X ( x k , x k +1 ) ≤ δ for some δ > 0. Acknowledgments • Example: in physics, if 2nd order terms are small, we use the linear expression Title Page as an approximation. ◭◭ ◮◮ ◭ ◮ Page 10 of 14 Go Back Full Screen Close Quit
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