Beyond Constructive . . . Taking Into Account . . . Need to Supplement . . . How Transition from Purely How to Describe What . . . Under Possibility . . . Constructive Mathematics How to Take into . . . Solving NP-Complete . . . to Physics-Motivated No Physical Theory Is . . . Intuitionistic Mathematics Main Result Home Page Affects Decidability: Title Page An Important Facet ◭◭ ◮◮ of Mints’s Legacy ◭ ◮ Page 1 of 63 Olga Kosheleva and Vladik Kreinovich Go Back University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Beyond Constructive . . . 1. Constructive Mathematics Taking Into Account . . . Need to Supplement . . . • Many processes from the physical world are described How to Describe What . . . by mathematical equations. Under Possibility . . . • Traditional (non-constructive) mathematics can help How to Take into . . . us prove the existence of a solution to given the equa- Solving NP-Complete . . . tions. No Physical Theory Is . . . • However, existence proofs are often non-constructive : Main Result Home Page they do not help us compute the solution. Title Page • Moreover, in traditional mathematics, it is not easy even to describe the existence of an algorithm. ◭◭ ◮◮ • So logicians invented constructive mathematics , where ◭ ◮ ∃ x means that we have an algorithm for constructing x . Page 2 of 63 • Then, the not-necessarily-constructive existence is de- Go Back scribed, e.g., by ¬¬∃ x . Full Screen Close Quit
Beyond Constructive . . . 2. Beyond Constructive Mathematics Taking Into Account . . . Need to Supplement . . . • In constructive mathematics, only constructive objects How to Describe What . . . are possible. Under Possibility . . . • For applications, this is a serious limitation: non- How to Take into . . . computable objects are possible. Solving NP-Complete . . . • For example, data may come come from a random pro- No Physical Theory Is . . . cess – like quantum measurement. Main Result Home Page • This limitation was one of the main motivations for Title Page G. Mints to consider: ◭◭ ◮◮ – a more general intuitionistic-style constructive mathematics, ◭ ◮ – where non-computable objects are allowed. Page 3 of 63 • In this talk, we study the relation between physics and Go Back the corresponding version of constructive mathematics. Full Screen Close Quit
Beyond Constructive . . . Part I Taking Into Account . . . Taking Into Account that We Need to Supplement . . . How to Describe What . . . Process Physical Data Under Possibility . . . How to Take into . . . Solving NP-Complete . . . No Physical Theory Is . . . Main Result Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 63 Go Back Full Screen Close Quit
Beyond Constructive . . . 3. Need to Supplement Probabilistic Information Taking Into Account . . . with Information re What Is Possible Need to Supplement . . . How to Describe What . . . • Physical laws enable us to predict probabilities p . Under Possibility . . . • In general, probability p is a frequency f with which How to Take into . . . an event occurs, but sometimes, f � = p . Solving NP-Complete . . . • Example: due to molecular motion, a cold kettle on a No Physical Theory Is . . . cold stove can spontaneously boil with p > 0. Main Result Home Page • However, most physicists believe that this event is sim- Title Page ply not possible. ◭◭ ◮◮ • This impossibility cannot be described by claiming that for some p 0 , events with p ≤ p 0 are not possible. ◭ ◮ • Indeed, if we toss a coin many times N , we can get Page 5 of 63 2 − N < p 0 , but the result is still possible. Go Back • So, to describe physics, we need to supplement proba- Full Screen bilities with information on what is possible. Close Quit
Beyond Constructive . . . 4. How to Describe What Is Possible Taking Into Account . . . Need to Supplement . . . • Let U be the set of possible events. How to Describe What . . . • We assume that we know the probabilities p ( S ) of dif- Under Possibility . . . ferent events S ⊆ U . How to Take into . . . Solving NP-Complete . . . • From all possible events, the expert select a subset T No Physical Theory Is . . . of all events which are possible. Main Result • The main idea that if the probability is very small, Home Page then the corresponding event is not possible. Title Page • What is “very small” depends on the situation. ◭◭ ◮◮ • Let A 1 ⊇ A 2 ⊇ . . . ⊃ A n ⊇ . . . be a definable sequence ◭ ◮ of events with p ( A n ) → 0. Page 6 of 63 • Then for some sufficiently large N , the probability of Go Back the corresponding event A N becomes very small. Full Screen • Thus, the event A N is not impossible, i.e., T ∩ A N = ∅ . Close Quit
Beyond Constructive . . . 5. Resulting Definitions Taking Into Account . . . Need to Supplement . . . • Let U be a set with a probability measure p . How to Describe What . . . • We say that T ⊆ U is a set of possible elements if: Under Possibility . . . • for every definable sequence A n for which How to Take into . . . A n ⊇ A n +1 and p ( A n ) → 0, Solving NP-Complete . . . No Physical Theory Is . . . • there exists N for which T ∩ A N = ∅ . Main Result • Physicists uses a similar argument even when do not Home Page know probabilities. Title Page • For example, they usually claim that: ◭◭ ◮◮ – when x is small, ◭ ◮ – quadratic terms in Taylor expansion a 0 + a 1 · x + a 2 · x 2 + . . . can be safely ignored. Page 7 of 63 • Theoretically, we can have a 2 s.t. | a 2 · x 2 | ≫ | a 1 · x | . Go Back Full Screen • However, physicists believe that such a 2 are not phys- ically possible. Close Quit
Beyond Constructive . . . 6. Definitions (cont-d) Taking Into Account . . . Need to Supplement . . . • Physicists believe that very large values of a 2 are not How to Describe What . . . physically possible. Under Possibility . . . • Here, we have A n = { a 2 : | a 2 | ≥ n } . How to Take into . . . Solving NP-Complete . . . • The physicists’ belief is that for a sufficiently large N , No Physical Theory Is . . . event A N is impossible, i.e., A N ∩ T = ∅ . Main Result • Here, ∩ A n = ∅ , so p ( A n ) → 0 for any probability mea- Home Page sure p . Title Page • There are other similar conclusions, so we arrive at the ◭◭ ◮◮ following definition. ◭ ◮ • We say that T ⊆ U is a set of possible elements if: Page 8 of 63 – for every definable sequence A n for which Go Back A n ⊇ A n +1 and ∩ A n = ∅ , Full Screen – there exists N for which T ∩ A N = ∅ . Close Quit
Beyond Constructive . . . 7. In General, Many Problems Are Not Algorith- Taking Into Account . . . mically Decidable Need to Supplement . . . How to Describe What . . . • A simple example is that it is impossible to decide Under Possibility . . . whether two computable real numbers are equal or not. How to Take into . . . • What are computable real numbers? Solving NP-Complete . . . • In practice, real numbers come from measurements, No Physical Theory Is . . . and measurements are never absolutely accurate. Main Result Home Page • In principle, we can measure a real number x with Title Page higher and higher accuracy. ◭◭ ◮◮ • For any n , we can measure x with accuracy 2 − n , and get a rational r n for which | x − r n | ≤ 2 − n . ◭ ◮ Page 9 of 63 • A real number is called computable if there is a proce- dure that, given n , returns x n . Go Back Full Screen Close Quit
Beyond Constructive . . . 8. Many Problems Are Not Algorithmically De- Taking Into Account . . . cidable (cont-d) Need to Supplement . . . How to Describe What . . . • Computing with computable real numbers means that, Under Possibility . . . – in addition to usual computational steps, How to Take into . . . – we can also, given n , ask for r n . Solving NP-Complete . . . No Physical Theory Is . . . • Some things can be computed: e.g., given x and y , we Main Result can compute z = x + y . Home Page • However, it is not possible to algorithmically check Title Page whether x = y . ◭◭ ◮◮ • Indeed, suppose that this was possible. ◭ ◮ • Then, for x = y = 0 with r n = s n = 0 for all n , our Page 10 of 63 procedure will return “yes”. Go Back • This procedure consists of finitely many steps, thus it Full Screen can only ask for finitely many values r n and s n . Close Quit
Beyond Constructive . . . 9. Many Problems Are Not Algorithmically De- Taking Into Account . . . cidable (cont-d) Need to Supplement . . . How to Describe What . . . ? • The x = y procedure consists of finitely many steps, Under Possibility . . . thus it can only ask for finitely many values r n and s n . How to Take into . . . • Let N be the smallest number which is larger than all Solving NP-Complete . . . such requests n . So: No Physical Theory Is . . . – if we keep x = 0 and take y ′ = 2 − N � = 0 with Main Result Home Page s ′ 1 = . . . = s ′ N − 1 = 0 and s ′ N = s ′ N +1 = . . . = 2 − N , Title Page – our procedure will not notice the difference and mistakenly return “yes”. ◭◭ ◮◮ • This proves that a procedure for checking whether two ◭ ◮ computable numbers are equal is not possible. Page 11 of 63 • Similar negative results are known for many other Go Back problems. Full Screen Close Quit
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