Physically Meaningful . . . Known Negative Results From the Physicists’ . . . How Physicists Argue Negative Results How to Formalize the . . . of Computable Analysis How to Formalize the . . . On “Not Abnormal” . . . Disappear If We Restrict Another Physically . . . When We Restrict . . . Ourselves to Random Home Page (Or, More Generally, Title Page Typical) Inputs ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 31 Department of Computer Science Go Back University of Texas, El Paso, TX 79968, USA vladik@utep.edu Full Screen Close Quit
Physically Meaningful . . . Known Negative Results 1. Physically Meaningful Computations with Real From the Physicists’ . . . Numbers: a Brief Reminder How Physicists Argue • In practice, many quantities such as weight, speed, etc., How to Formalize the . . . are characterized by real numbers. How to Formalize the . . . On “Not Abnormal” . . . • To get information about the corresponding value x , Another Physically . . . we perform a measurement, and get a value � x . When We Restrict . . . • Measurements are never absolute accurate. Home Page • We usually also know the upper bound ∆ on the the Title Page def measurement error ∆ x = � x − x : | x − � x | ≤ ∆. ◭◭ ◮◮ • To fully characterize a value x , we must measure it with ◭ ◮ a higher and higher accuracy, e.g., 2 − n with n = 0 , 1 , . . . Page 2 of 31 • So, we get a sequence of rational numbers r n for which | x − r n | ≤ 2 − n . Go Back Full Screen • Such sequences represent real numbers in computable analysis. Close Quit
Physically Meaningful . . . Known Negative Results 2. Known Negative Results From the Physicists’ . . . • No algorithm is possible that, given two numbers x and How Physicists Argue y , would check whether x = y . How to Formalize the . . . How to Formalize the . . . • Similarly, we can define a computable function f ( x ) On “Not Abnormal” . . . from real numbers to real numbers as a mapping that: Another Physically . . . – given an integer n , a rational number x m and its When We Restrict . . . accuracy 2 − m , Home Page – produces y n which is 2 − n -close to all values f ( x ) with d ( x, x m ) ≤ 2 − m (or nothing) Title Page ◭◭ ◮◮ so that for every x and for each desired accuracy n , there is an m for which a y n is produced. ◭ ◮ • We can similarly define a computable function f ( x ) on Page 3 of 31 a computable compact set K . Go Back • No algorithm is possible that, given f , returns x s.t. Full Screen f ( x ) = max y ∈ K f ( y ). (The max itself is computable.) Close Quit
Physically Meaningful . . . Known Negative Results 3. From the Physicists’ Viewpoint, These Nega- From the Physicists’ . . . tive Results Seem Rather Theoretical How Physicists Argue • In mathematics, if two numbers coincide up to 13 dig- How to Formalize the . . . its, they may still turn to be different. How to Formalize the . . . On “Not Abnormal” . . . • For example, they may be 1 and 1 + 10 − 100 . Another Physically . . . • In physics, if two quantities coincide up to a very high When We Restrict . . . accuracy, it is a good indication that they are equal: Home Page – if an experimentally value is very close to the the- Title Page oretical prediction, ◭◭ ◮◮ – this means that this theory is (triumphantly) true. ◭ ◮ • This is how General Relativity was confirmed. Page 4 of 31 • This is how physicists realized that light is formed of Go Back electromagnetic waves: their speeds are very close. Full Screen Close Quit
Physically Meaningful . . . Known Negative Results 4. How Physicists Argue From the Physicists’ . . . • In math, if two numbers coincide up to 13 digits, they How Physicists Argue may still turn to be different: e.g., 1 and 1 + 10 − 100 . How to Formalize the . . . How to Formalize the . . . • In physics, if two quantities coincide up to a very high On “Not Abnormal” . . . accuracy, it is a good indication that they are equal. Another Physically . . . • A typical physicist argument is that: When We Restrict . . . – while numbers like 1 + 10 − 100 (or c · (1 + 10 − 100 )) Home Page are, in principle, possible, Title Page – they are abnormal (not typical ). ◭◭ ◮◮ • In physics, second order terms like a · ∆ x 2 of the Taylor ◭ ◮ series can be ignored if ∆ x is small, since: Page 5 of 31 – while abnormally high values of a (e.g., a = 10 40 ) Go Back are mathematically possible, Full Screen – typical (= not abnormal) values appearing in phys- ical equations are usually of reasonable size. Close Quit
Physically Meaningful . . . Known Negative Results 5. How to Formalize the Physicist’s Intuition of From the Physicists’ . . . Typical (Not Abnormal): Main Idea How Physicists Argue • To some physicist, all the values of a coefficient a above How to Formalize the . . . 10 are abnormal. How to Formalize the . . . On “Not Abnormal” . . . • To another one, who is more cautious, all the values Another Physically . . . above 10 000 are abnormal. When We Restrict . . . • For every physicist, there is a value n such that all Home Page value above n are abnormal. Title Page • This argument can be generalized as a following prop- ◭◭ ◮◮ erty of the set T of all typical elements. ◭ ◮ • Suppose that we have a monotonically decreasing se- quence of sets A 1 ⊇ A 2 ⊇ . . . for which � Page 6 of 31 A n = ∅ . n Go Back • In the above example, A n is the set of all numbers ≥ n . Full Screen • Then, there exists an integer N for which T ∩ A N = ∅ . Close Quit
Physically Meaningful . . . Known Negative Results 6. How to Formalize the Physicist’s Intuition of From the Physicists’ . . . Typical (Not Abnormal): Resulting Definition How Physicists Argue • Definition. We thus say that T is a set of typical How to Formalize the . . . elements if: How to Formalize the . . . On “Not Abnormal” . . . – for every definable decreasing sequence { A n } for which � Another Physically . . . A n = ∅ , n When We Restrict . . . – there exists an N for which T ∩ A N = ∅ . Home Page • Comment. Of course, to make this definition precise, Title Page ◭◭ ◮◮ – we must restrict definability to a subset of proper- ties, ◭ ◮ – so that the resulting notion of definability will be Page 7 of 31 defined in ZFC itself. Go Back Full Screen Close Quit
Physically Meaningful . . . Known Negative Results 7. Kolmogorov’s Definition of Algorithmic Ran- From the Physicists’ . . . domness How Physicists Argue • Kolmogorov: proposed a new definition of a random How to Formalize the . . . sequence, a definition that separates How to Formalize the . . . On “Not Abnormal” . . . – physically random binary sequences, e.g.: Another Physically . . . ∗ sequences that appear in coin flipping experi- When We Restrict . . . ments, Home Page ∗ 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 8 of 31 • What is a probability law: a statement S which is true with probability 1: P ( S ) = 1. Go Back • Conclusion: to prove that a sequence is not random, Full Screen we must show that it does not satisfy one of these laws. Close Quit
Physically Meaningful . . . Known Negative Results 8. Kolmogorov’s Definition of Algorithmic Ran- From the Physicists’ . . . domness (cont-d) How Physicists Argue • Reminder: a sequence s is not random if it does not How to Formalize the . . . satisfy one of the probability laws S . How to Formalize the . . . On “Not Abnormal” . . . • Equivalent statement: s is not random if s ∈ C for a Another Physically . . . (definable) set C (= − S ) with P ( C ) = 0. When We Restrict . . . • Resulting definition (Kolmogorov, Martin-L¨ of): s is Home Page 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 9 of 31 – Since there are countably many sequences of sym- Go Back bols, there are countably many definable sets C . – So, the complement −R to the class R of all ran- Full Screen dom sequences also has probability 0. Close Quit
Physically Meaningful . . . Known Negative Results 9. Towards a More Physically Adequate Versions From the Physicists’ . . . of Kolmogorov Randomness How Physicists Argue • Problem: the 1960s Kolmogorov’s definition only ex- How to Formalize the . . . plains why events with probability 0 do not happen. How to Formalize the . . . On “Not Abnormal” . . . • What we need: formalize the physicists’ intuition that Another Physically . . . events with very small probability cannot happen. When We Restrict . . . • Seemingly natural formalization: there exists the “small- Home Page 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 cannot occur. Page 10 of 31 Go Back • Example: a fair coin falls heads 100 times with prob. 2 − 100 ; it is impossible if p 0 ≥ 2 − 100 . Full Screen Close Quit
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