Physicists Usually . . . Kolmogorov’s . . . New Definition of . . . Extending Algorithmic New Definition of . . . Need to Extend . . . Randomness to Natural Extension of . . . the Algebraic Approach Consistency Result: . . . Consistency: Proof to Quantum Physics: Acknowledgments Title Page Kolmogorov Complexity ◭◭ ◮◮ and Quantum Logics ◭ ◮ Page 1 of 24 Vladik Kreinovich Department of Computer Science Go Back University of Texas, El Paso, TX 79968, USA Full Screen vladik@utep.edu Close Quit
Physicists Usually . . . 1. Physicists Usually Assume that Events with a Very Kolmogorov’s . . . Small Probability Cannot Occur New Definition of . . . New Definition of . . . • Known phenomenon: Brownian motion. Need to Extend . . . • In principle: due to Brownian motion, a kettle placed Natural Extension of . . . on a cold stove can start boiling. Consistency Result: . . . • The probability of this event is positive but very small. Consistency: Proof Acknowledgments • A mathematician would say that this event is possible Title Page but rare. ◭◭ ◮◮ • A physicist would say that this event is simply not ◭ ◮ possible. Page 2 of 24 • It is desirable: to formalize this intuition of physicists. Go Back Full Screen Close Quit
Physicists Usually . . . 2. Kolmogorov’s Definition of Algorithmic Random- Kolmogorov’s . . . ness New Definition of . . . New Definition of . . . • Kolmogorov: proposed a new definition of a random Need to Extend . . . sequence, a definition that separates Natural Extension of . . . – physically random binary sequences, e.g.: Consistency Result: . . . ∗ sequences that appear in coin flipping experi- Consistency: Proof ments, Acknowledgments ∗ sequences that appear in quantum measurements Title Page – from sequence that follow some pattern. ◭◭ ◮◮ • Intuitively: if a sequence s is random, it satisfies all the ◭ ◮ probability laws. Page 3 of 24 • What is a probability law: a statement S which is true Go Back with probability 1: P ( S ) = 1. Full Screen • Conclusion: to prove that a sequence is not random, Close we must show that it does not satisfy one of these laws. Quit
Physicists Usually . . . 3. Kolmogorov’s Definition of Algorithmic Random- Kolmogorov’s . . . ness (cont-d) New Definition of . . . New Definition of . . . • Reminder: a sequence s is not random if it does not Need to Extend . . . satisfy one of the probability laws S . Natural Extension of . . . • Equivalent statement: s is not random if s ∈ C for a Consistency Result: . . . (definable) set C (= − S ) with P ( C ) = 0. Consistency: Proof • Resulting definition (Kolmogorov, Martin-L¨ of): s is Acknowledgments random if s �∈ C for all definable C with P ( C ) = 0. Title Page ◭◭ ◮◮ • Consistency proof: ◭ ◮ – Every definable set C is defined by a finite sequence of symbols (its definition). Page 4 of 24 – Since there are countably many sequences of sym- Go Back bols, there are countably many definable sets C . Full Screen – So, the complement −R to the class R of all ran- Close dom sequences also has probability 0. Quit
Physicists Usually . . . 4. Towards a More Physically Adequate Versions of Kolmogorov’s . . . Kolmogorov Randomness New Definition of . . . New Definition of . . . • Problem: the 1960s Kolmogorov’s definition only ex- Need to Extend . . . plains why events with probability 0 do not happen. Natural Extension of . . . • What we need: formalize the physicists’ intuition that Consistency Result: . . . events with very small probability cannot happen. Consistency: Proof • Seemingly natural formalization: there exists the “small- Acknowledgments est possible probability” p 0 such that: Title Page ◭◭ ◮◮ – if the computed probability p of some event is larger than p 0 , then this event can occur, while ◭ ◮ – if the computed probability p is ≤ p 0 , the event Page 5 of 24 cannot occur. Go Back • Example: a fair coin falls heads 100 times with prob. Full Screen 2 − 100 ; it is impossible if p 0 ≥ 2 − 100 . Close Quit
Physicists Usually . . . 5. The Above Formalization of Randomness is Not Kolmogorov’s . . . Always Adequate New Definition of . . . New Definition of . . . • Problem: every sequence of heads and tails has exactly Need to Extend . . . the same probability. Natural Extension of . . . • Corollary: if we choose p 0 ≥ 2 − 100 , we will thus exclude Consistency Result: . . . all sequences of 100 heads and tails. Consistency: Proof • However, anyone can toss a coin 100 times. Acknowledgments Title Page • This proves that some such sequences are physically ◭◭ ◮◮ possible. ◭ ◮ • Similar situation: Kyburg’s lottery paradox: Page 6 of 24 – in a big (e.g., state-wide) lottery, the probability of winning the Grand Prize is very small; Go Back – a reasonable person should not expect to win; Full Screen – however, some people do win big prizes. Close Quit
Physicists Usually . . . 6. New Definition of Randomness Kolmogorov’s . . . New Definition of . . . • Example: height: New Definition of . . . – if height is ≥ 6 ft, it is still normal; Need to Extend . . . – if instead of 6 ft, we consider 6 ft 1 in, 6 ft 2 in, etc., Natural Extension of . . . then ∃ h 0 s.t. everyone taller than h 0 is abnormal; Consistency Result: . . . – we are not sure what is h 0 , but we are sure such h 0 Consistency: Proof exists. Acknowledgments Title Page • General description: on the universal set U , we have ◭◭ ◮◮ sets A 1 ⊇ A 2 ⊇ . . . ⊇ A n ⊇ . . . s.t. P ( ∩ A n ) = 0. ◭ ◮ • Example: A 1 = people w/height ≥ 6 ft, A 2 = people w/height ≥ 6 ft 1 in, etc. Page 7 of 24 • A set R ⊆ U is called a set of random elements if Go Back Full Screen ∀ definable sequence of sets A n for which A n ⊇ A n +1 for all n and P ( ∩ A n ) = 0, ∃ N for which A N ∩ R = ∅ . Close Quit
Physicists Usually . . . 7. Definable: Mathematical Comment Kolmogorov’s . . . New Definition of . . . • What is definable: New Definition of . . . – let L be a theory, Need to Extend . . . – let P ( x ) be a formula from the language of the the- Natural Extension of . . . ory L , with one free variable x Consistency Result: . . . – so that the set { x | P ( x ) } is defined in L . Consistency: Proof Acknowledgments We will then call the set { x | P ( x ) } L -definable . Title Page • How to deal with definable sets: ◭◭ ◮◮ – Our objective is to be able to make mathematical ◭ ◮ statements about L -definable sets. Page 8 of 24 – Thus, we must have a stronger theory M in which Go Back the class of all L -definable sets is a countable set. Full Screen – One can prove that such M always exists. Close Quit
Physicists Usually . . . 8. Coin Example Kolmogorov’s . . . New Definition of . . . • Universal set U = { H , T } I N New Definition of . . . • Here, A n is the set of all the sequences that start with Need to Extend . . . n heads. Natural Extension of . . . • The sequence { A n } is decreasing and definable, and its Consistency Result: . . . intersection has probability 0. Consistency: Proof Acknowledgments • Therefore, for every set R of random elements of U , Title Page there exists an integer N for which A N ∩ R = ∅ . ◭◭ ◮◮ • This means that if a sequence s ∈ R is random and ◭ ◮ starts with N heads, it must consist of heads only. Page 9 of 24 • In physical terms: it means that Go Back a random sequence cannot start with N heads. Full Screen • This is exactly what we wanted to formalize. Close Quit
Physicists Usually . . . 9. From Random to Typical (Not Abnormal) Kolmogorov’s . . . New Definition of . . . • Fact: not all solutions to the physical equations are New Definition of . . . physically meaningful. Need to Extend . . . • Example 1: when a cup breaks into pieces, the corre- Natural Extension of . . . sponding trajectories of molecules make physical sense. Consistency Result: . . . • Example 2: when we reverse all the velocities, we get Consistency: Proof pieces assembling themselves into a cup. Acknowledgments Title Page • Physical fact: this is physically impossible. ◭◭ ◮◮ • Mathematical fact: the reverse process satisfies all the ◭ ◮ original (T-invariant) equations. Page 10 of 24 • Physicist’s explanation: the reversed process is non- physical since its initial conditions are “degenerate”. Go Back Full Screen • Clarification: once we modify the initial conditions even slightly, the pieces will no longer get together. Close Quit
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