Need to Supplement . . . How to Describe . . . Resulting Definitions Definitions (cont-d) Adding Possibilistic In General, Many . . . Knowledge to Probabilities Under Possibility . . . Under Possibilistic . . . Makes Many Problems Proof by Quantifier . . . Relation to Possibility . . . Algorithmically Decidable Home Page Title Page Olga Kosheleva and Vladik Kreinovich ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ 500 W. University El Paso, Texas 79968, USA Page 1 of 13 olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Need to Supplement . . . How to Describe . . . 1. Need to Supplement Probabilistic Predictions Resulting Definitions with Possibilistic Information Definitions (cont-d) • Physical laws enable us to predict probabilities p . In General, Many . . . Under Possibility . . . • In general, probability p is a frequency f with which Under Possibilistic . . . an event occurs, but sometimes, f � = p . Proof by Quantifier . . . • Example: due to molecular motion, a cold kettle on a Relation to Possibility . . . cold stove can spontaneously boil with p > 0. 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. Page 2 of 13 • Indeed, if we toss a coin many times N , we can get 2 − N < p 0 , but the result is still possible. Go Back Full Screen • So, to describe physics, we need to supplement proba- bilities with information on what is possible. Close Quit
Need to Supplement . . . How to Describe . . . 2. How to Describe Information about Possibility Resulting Definitions • Let U be the universe of discourse, i.e., in our case, the Definitions (cont-d) set of possible events. In General, Many . . . Under Possibility . . . • We assume that we know the probabilities p ( S ) of dif- Under Possibilistic . . . ferent events S ⊆ U . Proof by Quantifier . . . • From all possible events, the expert select a subset T Relation to Possibility . . . of all events which are possible. Home Page • The main idea that if the probability is very small, Title Page then the corresponding event is not possible. ◭◭ ◮◮ • What is “very small” depends on the situation. ◭ ◮ • Let A 1 ⊇ A 2 ⊇ . . . ⊃ A n ⊇ . . . be a definable sequence Page 3 of 13 of events with p ( A n ) → 0. Go Back • Then for some sufficiently large N , the probability of 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
Need to Supplement . . . How to Describe . . . 3. Resulting Definitions Resulting Definitions • Let U be a set with a probability measure p . Definitions (cont-d) In General, Many . . . • We say that T ⊆ U is a set of possible elements if: Under Possibility . . . • for every definable sequence A n for which Under Possibilistic . . . A n ⊇ A n +1 and p ( A n ) → 0, Proof by Quantifier . . . • there exists N for which T ∩ A N = ∅ . Relation to Possibility . . . • 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 + Page 4 of 13 a 2 · x 2 + . . . can be safely ignored. Go Back • Theoretically, we can have a 2 s.t. | a 2 · x 2 | ≫ | a 1 · x | . Full Screen • However, physicists believe that such a 2 are not phys- Close ically possible. Quit
Need to Supplement . . . How to Describe . . . 4. Definitions (cont-d) Resulting Definitions • Physicists believe that very large values of a 2 are not Definitions (cont-d) physically possible. In General, Many . . . Under Possibility . . . • Here, we have A n = { a 2 : | a 2 | ≥ n } . Under Possibilistic . . . • The physicists’ belief is that for a sufficiently large N , Proof by Quantifier . . . event A N is impossible, i.e., A N ∩ T = ∅ . Relation to Possibility . . . Home Page • Here, ∩ A n = ∅ , so p ( A n ) → 0 for any probability mea- 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 5 of 13 – 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
Need to Supplement . . . How to Describe . . . 5. In General, Many Problems Are Not Algorith- Resulting Definitions mically Decidable Definitions (cont-d) • A simple example is that it is impossible to decide In General, Many . . . whether two computable real numbers are equal or not. Under Possibility . . . Under Possibilistic . . . • What are computable real numbers? Proof by Quantifier . . . • In practice, real numbers come from measurements, Relation to Possibility . . . and measurements are never absolutely accurate. 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 6 of 13 • A real number is called computable if there is a proce- Go Back dure that, given n , returns x n . Full Screen Close Quit
Need to Supplement . . . How to Describe . . . 6. Many Problems Are Not Algorithmically De- Resulting Definitions cidable (cont-d) Definitions (cont-d) • Computing with computable real numbers means that, In General, Many . . . Under Possibility . . . – in addition to usual computational steps, Under Possibilistic . . . – we can also, given n , ask for r n . Proof by Quantifier . . . • Some things can be computed: e.g., given x and y , we Relation to Possibility . . . 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 7 of 13 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
Need to Supplement . . . How to Describe . . . 7. Many Problems Are Not Algorithmically De- Resulting Definitions cidable (cont-d) Definitions (cont-d) ? • The x = y procedure consists of finitely many steps, In General, Many . . . thus it can only ask for finitely many values r n and s n . Under Possibility . . . Under Possibilistic . . . • Let N be the smallest number which is larger than all Proof by Quantifier . . . such requests n . So: Relation to Possibility . . . – if we keep x = 0 and take y ′ = 2 − N � = 0 with 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 Page 8 of 13 computable numbers are equal is not possible. Go Back • Similar negative results are known for many other problems. Full Screen Close Quit
Need to Supplement . . . How to Describe . . . 8. Under Possibility Information, Equality Be- Resulting Definitions comes Decidable: Known Result Definitions (cont-d) • On the set U = I R × I R of all possible pairs of real In General, Many . . . numbers, we have a subset T of possible numbers. Under Possibility . . . Under Possibilistic . . . • In particular, we can consider the following definable def Proof by Quantifier . . . = { ( x, y ) : 0 < | x − y | ≤ 2 − n } . sequence of sets A n Relation to Possibility . . . • One can easily see that A n ⊇ A n +1 for all n and that Home Page ∩ A n = ∅ . Title Page • Thus, there exists a natural number N for which no ◭◭ ◮◮ element s ∈ T belongs to the set A N . ◭ ◮ • This, in turn, means that for every pair ( x, y ) ∈ T , Page 9 of 13 either | x − y | = 0 (i.e., x = y ) or | x − y | > 2 − N . Go Back • So, to check whether x = y or not, it is sufficient to compute both x and y with accuracy 2 − ( N +2) . Full Screen Close Quit
Need to Supplement . . . How to Describe . . . 9. Under Possibilistic Information, Many Prob- Resulting Definitions lems Become Decidable: A New Result Definitions (cont-d) • In terms of sequences r n and s n , equality x = y can be In General, Many . . . described as ∀ n ( | r n − s n | ≤ 2 − ( n − 1) ) . Under Possibility . . . Under Possibilistic . . . • Many properties involving limits, differentiability, etc., Proof by Quantifier . . . can be described by arithmetic formulas Relation to Possibility . . . def Φ = Qn 1 Qn 2 . . . Qn k F ( r 1 , . . . , r ℓ , n 1 , . . . , n k ) . Home Page Title Page • Here, Qn i is ∀ n i or ∃ n i ; r 1 , . . . , r ℓ are sequences. ◭◭ ◮◮ • F is a propositional combination of =’s and � =’s be- ◭ ◮ tween computable rational-valued expressions. Page 10 of 13 • For every Φ, for every set T of possible tuples r = ( r 1 , . . . , r ℓ ), there exists an algorithm that, Go Back – given a tuple r = ( r 1 , . . . , r ℓ ) ∈ T , Full Screen – checks whether Φ is true. Close Quit
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