Physicists assume that . . . A seemingly natural . . . The above . . . Relation to . . . Events with 0 . . . Kolmogorov’s idea: . . . To Properly Reflect Towards Formalization Main result Physicists’ Reasoning about Discussion Auxiliary result Randomness, We Also Need Conclusion Acknowledgments Proof: Part I a Maxitive (Possibility) Proof: Part I (cont-d) Proof: Part II Measure Proof: Part II (cont-d) Title Page Andrei M. Finkelstein ◭◭ ◮◮ Inst. App. Astronomy, Russian Acad. of Sci., St Petersburg ◭ ◮ Olga Kosheleva, Vladik Kreinovich, Scott A. Starks Page 1 of 17 Pan-American Center for Earth & Environ. Stud. University of Texas, El Paso, TX 79968, USA Go Back Hung T. Nguyen Full Screen New Mexico State U., Las Cruces, NM, 88003, USA Close Quit
Physicists assume that . . . A seemingly natural . . . 1. Physicists assume that initial conditions and values The above . . . Relation to . . . of parameters are not abnormal Events with 0 . . . Kolmogorov’s idea: . . . • To a mathematician, the main contents of a physical theory is its equations. Towards Formalization • Not all solutions of the equations have physical sense. Main result Discussion • Ex. 1: Brownian motion comes in one direction; Auxiliary result • Ex. 2: implosion glues shattered pieces into a statue; Conclusion Acknowledgments • Ex. 3: fair coin falls heads 100 times in a row. Proof: Part I • Mathematics: it is possible. Proof: Part I (cont-d) Proof: Part II • Physics (and common sense): it is not possible. Proof: Part II (cont-d) • Our objective: supplement probabilities with a new formalism that more Title Page accurately captures the physicists’ reasoning. ◭◭ ◮◮ ◭ ◮ Page 2 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 2. A seemingly natural formalizations of this idea The above . . . Relation to . . . • Physicists: only “not abnormal” situations are possible. Events with 0 . . . Kolmogorov’s idea: . . . • Natural formalization: idea. If a probability p ( E ) of an event E is small Towards Formalization enough, then this event cannot happen. Main result • Natural formalization: details. There exists the “smallest possible probabil- Discussion ity” p 0 such that: Auxiliary result Conclusion – if the computed probability p of some event is larger than p 0 , then this Acknowledgments event can occur, while Proof: Part I – if the computed probability p is ≤ p 0 , the event cannot occur. Proof: Part I (cont-d) • Example: a fair coin falls heads 100 times with prob. 2 − 100 ; it is impossible Proof: Part II if p 0 ≥ 2 − 100 . Proof: Part II (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 3. The above formalization of the notion of “typical” The above . . . Relation to . . . is not always adequate Events with 0 . . . Kolmogorov’s idea: . . . • Problem: every sequence of heads and tails has exactly the same probability. Towards Formalization • Corollary: if we choose p 0 ≥ 2 − 100 , we will thus exclude all possible sequences Main result of 100 heads and tails as physically impossible. Discussion Auxiliary result • However, anyone can toss a coin 100 times, and this proves that some such Conclusion sequences are physically possible. Acknowledgments • Similar situation: Kyburg’s lottery paradox: Proof: Part I Proof: Part I (cont-d) – in a big (e.g., state-wide) lottery, the probability of winning the Grand Proof: Part II Prize is so small that a reasonable person should not expect it; Proof: Part II (cont-d) – however, some people do win big prizes. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 4. Relation to non-monotonic reasoning The above . . . Relation to . . . • Traditional logic is monotonic: once a statement is derived it remains true. Events with 0 . . . Kolmogorov’s idea: . . . • Expert reasoning is non-monotonic: Towards Formalization Main result – birds normally fly, Discussion – so, if we know only that Sam is a bird, we conclude that Sam flies; Auxiliary result – however, if we learn the new knowledge that Sam is a penguin, we Conclusion conclude that Sam doesn’t fly. Acknowledgments • Non-monotonic reasoning helps resolve the lottery paradox (Poole et al.) Proof: Part I Proof: Part I (cont-d) • Our approach: in fact, what we propose can be viewed as a specific non- Proof: Part II monotonic formalism for describing rare events. Proof: Part II (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 5. Events with 0 probabilities are possible: another The above . . . Relation to . . . explanation for the lottery paradox Events with 0 . . . Kolmogorov’s idea: . . . • Idea: common sense intuition is false, events with small (even 0) probability Towards Formalization are possible. Main result • This idea is promoted by known specialists in foundations of probability: Discussion K. Popper, B. De Finetti, G. Coletti, A. Gilio, R. Scozzafava, W. Spohn, etc. Auxiliary result Conclusion • Out attitude: our objective is to formalize intuition, not to reject it. Acknowledgments • Interesting: both this approach and our approach lead to the same formalism Proof: Part I (of maxitive measures). Proof: Part I (cont-d) Proof: Part II • Conclusion: Maybe there is a deep relation and similarity between the two Proof: Part II (cont-d) approaches. Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 6. Kolmogorov’s idea: use complexity The above . . . Relation to . . . • Problem with the above naive approach: we use the same threshold p 0 for all Events with 0 . . . events. Kolmogorov’s idea: . . . Towards Formalization • Kolmogorov’s idea: the probability threshold t ( E ) below which an event E is Main result dismissed as impossible must depend on the event’s complexity. Discussion • The event E 1 in which we have 100 heads is easy to describe and generate; Auxiliary result so t ( E 1 ) is higher. Conclusion Acknowledgments • If t ( E 1 ) > 2 − 100 then, within this Kolmogorov’s approach, we conclude that Proof: Part I the event E 1 is impossible. Proof: Part I (cont-d) • On the other hand, the event E 2 corresponding to the actual sequence of Proof: Part II heads and tails is much more complicated; so, t ( E 2 ) is lower. Proof: Part II (cont-d) • If t ( E 2 ) < 2 − 100 , we conclude that the event E 2 is possible. Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 7. Towards Formalization The above . . . Relation to . . . • Original idea: an event E is possible if and only its probability p ( E ) exceeds Events with 0 . . . a certain threshold p 0 . Kolmogorov’s idea: . . . Towards Formalization • New idea: Main result – each event E has a “complexity” c ( E ); Discussion Auxiliary result – an event E is possible if and only if p ( E ) > p 0 · c ( E ). Conclusion • Equivalent formulation: E is possible of and only if m ( E ) > p 0 , where Acknowledgments def m ( E ) = p ( E ) /c ( E ) is a “ratio” measure. Proof: Part I Proof: Part I (cont-d) • Standard probability setting: Proof: Part II – Let X be the set of all possible outcomes. Proof: Part II (cont-d) – An event is a subset E of the set X . Title Page – p is a probability measure on a σ -algebra A of sets from X . ◭◭ ◮◮ ◭ ◮ Page 8 of 17 Go Back Full Screen Close Quit
Physicists assume that . . . A seemingly natural . . . 8. Main result The above . . . Relation to . . . • Let T ⊆ X be the set of all outcomes that are actually possible. Events with 0 . . . Kolmogorov’s idea: . . . • An event E is possible ↔ there is a possible outcome that belongs to the set Towards Formalization E , i.e., ↔ E ∩ T � = ∅ . Main result • Definition. A ratio measure is a mapping from A to [0 , ∞ ] s.t. ∀ p 0 > 0 ∃ T ( p 0 ) Discussion for which Auxiliary result ∀ E ∈ A ( m ( E ) > p 0 ↔ E ∩ T ( p 0 ) � = ∅ ) . Conclusion Acknowledgments • Reminder: m is a maxitive ( possibility ) measure if for every family of sets E α Proof: Part I Proof: Part I (cont-d) �� � m E α = sup α m ( E α ) . Proof: Part II α Proof: Part II (cont-d) Title Page • Theorem. A function m ( E ) is a ratio measure if and only if it is a maxitive (possibility) measure. ◭◭ ◮◮ ◭ ◮ Page 9 of 17 Go Back Full Screen Close Quit
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