Predictions in . . . Predictions in . . . Possible of Objective . . . Possibility of Kolmorogov’s . . . How Can We Define . . . Objective Interval Observation and a . . . Proof Uncertainty in Physics: References Home Page Analysis Title Page Darrell Cheu and Luc Longpr´ e ◭◭ ◮◮ ◭ ◮ Department of Computer Science University of Texas at El Paso Page 1 of 9 500 W. University Ave. El Paso, Texas 79968, USA Go Back emails darrell cheu@Hotmail.com Full Screen longpre@utep.edu Close Quit
1. Predictions in Newtonian Physics Predictions in . . . Predictions in . . . • In Newtonian physics : Possible of Objective . . . Kolmorogov’s . . . – once we know the current state of the system, How Can We Define . . . – we can predict (at least in principle) all the future states of Observation and a . . . this system. Proof • In real life: References – measurements are never absolutely accurate, so we do not Home Page have the exact knowledge of the current state. Title Page • However: ◭◭ ◮◮ – the more accurate our measurements of the current state, the more accurate predictions we can make. ◭ ◮ • The inaccuracy Page 2 of 9 – of the existing knowledge and Go Back – of the resulting predictions Full Screen • can often be described in terms of interval uncertainty. Close Quit
2. Predictions in Quantum Physics Predictions in . . . Predictions in . . . • In quantum physics: Possible of Objective . . . Kolmorogov’s . . . – we cannot predict the exact future state of a system; How Can We Define . . . – we can only predict the probabilities of different future states. Observation and a . . . • According to the modern quantum physics: Proof References – if we know the exact initial state of the world we can uniquely predict these probabilities. Home Page • This means: Title Page – the more accurate our measurements of the current state, the ◭◭ ◮◮ more accurate predictions of probabilities we can make. ◭ ◮ • In practice: Page 3 of 9 – we can often predict the intervals of possible values of the probability. Go Back Full Screen Close Quit
3. Possible of Objective Interval-Valued Probabil- Predictions in . . . Predictions in . . . ities Possible of Objective . . . • It is reasonable to conjecture that: Kolmorogov’s . . . How Can We Define . . . – for some real-life processes, Observation and a . . . – there is no objective probability. Proof References • In other words: – for different subsequences, Home Page – the corresponding frequencies can indeed take different values Title Page from a given interval. ◭◭ ◮◮ • The analysis of such processes is given by Gorban in 2007. ◭ ◮ • How can we go beyond frequencies in this analysis? Page 4 of 9 • A common sense idea: Go Back – if an event has probability 0, – then it cannot happen. Full Screen • This cannot be literally true since every number has probability Close 0, and thus, no number is random. Quit
4. Kolmorogov’s Definition of Randomness Predictions in . . . Predictions in . . . • A common sense idea (reminder): Possible of Objective . . . Kolmorogov’s . . . – if an event has probability 0, How Can We Define . . . – then it cannot happen. Observation and a . . . • Problem: this cannot be literally true. Proof • Reason: References – every number has probability 0, and Home Page – thus, no number is random. Title Page • Idea of Kolmorogov and Martin-L¨ of: we only require that defin- able events of probability 0 do not happen. ◭◭ ◮◮ • Good news: we get a consistent definition of randomness. ◭ ◮ • Reason: Page 5 of 9 – there are only countably many defining texts; Go Back – thus countably many definable events, – the union of countably many events of probability 0 has prob- Full Screen ability 0; Close – thus, we indeed have a consistent definition of a random ob- ject. Quit
5. How Can We Define When an Object is Ran- Predictions in . . . Predictions in . . . dom Possible of Objective . . . • Randomness under a known probability distribution P (reminder): Kolmorogov’s . . . How Can We Define . . . – an object x is random Observation and a . . . – if its does not belong to any definable event E with P ( E ) = 0. Proof References • Meaning: if a (definable) event E has probability 0, then it cannot happen. Home Page • New situation: Title Page – we do not know the probability distribution; ◭◭ ◮◮ – we only know a class P of possible probability distributions. ◭ ◮ • Idea: if a definable event E is guaranteed to have probability 0 (i.e., P ( E ) = 0 for all possible P ) then it cannot happen. Page 6 of 9 • Resulting definition: Go Back – an object x is random Full Screen – if it does not belong to any definable event E for which Close P ( E ) = 0 for all P ∈ P . Quit
6. Observation and a Surprising Result Predictions in . . . Predictions in . . . • Observation: Possible of Objective . . . Kolmorogov’s . . . – if an object x is random w.r.t. some P 0 ∈ P , How Can We Define . . . – then it is also random w.r.t. P . Observation and a . . . • Proof: Proof – let E be a definable event for which P ( E ) = 0 for all P ∈ P ; References – we want to prove that x �∈ E ; Home Page – since P ( E ) = 0 for all P ∈ P and P 0 ∈ P , in particular, P 0 ( E ) = 0; Title Page – since x is P 0 -random, we have x �∈ E ; ◭◭ ◮◮ – the observation is proven. ◭ ◮ • Case: the class P is finite: P = { P 1 , . . . , P n } . Page 7 of 9 • According to observation: for every i , every P i -random object is P -random. Go Back • Natural expectation: there are P -random objects which are not Full Screen P i -random. Close • Surprising result: every P -random object is random with respect to one of the probability measures P i . Quit
7. Proof Predictions in . . . Predictions in . . . • Formulation of the result (reminder): every P -random object is Possible of Objective . . . random with respect to one of the probability measures P i . Kolmorogov’s . . . How Can We Define . . . • Proof: by contradiction: Observation and a . . . – let x be P -random and not random with respect to all P i ; Proof – by definition, P i -random means that x �∈ E for all definable References E with P i ( E ) = 0; Home Page – thus, the fact that x is not P i -random means that there exists an event E i with P i ( E i ) = 0 for which x ∈ E i ; Title Page – since x ∈ E i for all i , the object x belongs to the intersection ◭◭ ◮◮ n def = � E i : x ∈ E ; E i =1 ◭ ◮ – since P i ( E i ) = 0 and E ⊆ E i , we have P i ( E ) = 0; Page 8 of 9 – thus, x belongs to the event E for which P i ( E ) = 0 for all i ; Go Back – this contradicts to our assumption that x is P -random; – the statement is proven. Full Screen • We hope: that this problem does not appear in the more physical Close interval-valued class P . Quit
8. References Predictions in . . . Predictions in . . . (1) Gorban, I.I.; Theory of Hyper-Random Phenomena , Kyiv, Ukrainian Possible of Objective . . . National Academy of Sciences Publ, 2007 (in Russian). Kolmorogov’s . . . How Can We Define . . . • Comment: this book promotes the idea of objective interval- Observation and a . . . valued probabilities. Proof References (2) Li, M., and Vit´ anyi, P.: An Introduction to Kolmogorov Complex- ity and Its Applications , Springer, Berlin-Heidelberg, 1997. Home Page • Comment: this book provides a general introduction to Kolmogorov complexity and randomness. Title Page (3) Kreinovich, V., and Longpr´ e, L.; International Journal on Theo- ◭◭ ◮◮ retical Physics, 1997, Vol. 36, No. 1, pp. 167–176. ◭ ◮ • Comment: this paper contains the ideas that we used in our proof. Page 9 of 9 Go Back Full Screen Close Quit
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