Physical Induction: . . . How to Describe . . . Checking Equality of . . . Finding Roots Logic of Scientific Discovery: Optimization How Physical Induction Computing Fixed Points Randomness and . . . Affects What Is Computable Random Sequences . . . Alternative Idea: No . . . Vladik Kreinovich and Olga Kosheleva Home Page University of Texas at El Paso Title Page 500 W. University El Paso, TX 79968, USA ◭◭ ◮◮ vladik@utep.edu, olgak@utep.edu ◭ ◮ http://www.cs.utep.edu/vladik Page 1 of 39 http://www.cs.utep.edu/vladik/olgavita.html Go Back Full Screen Close Quit
Physical Induction: . . . How to Describe . . . 1. Outline Checking Equality of . . . • Most of our knowledge about a physical world comes Finding Roots from physical induction : Optimization Computing Fixed Points – if a hypothesis is confirmed by a sufficient number Randomness and . . . of observations, Random Sequences . . . – we conclude that this hypothesis is universally true. Alternative Idea: No . . . • We show that a natural formalization of this property Home Page affects what is computable. Title Page • We explain how this formalization is related to Kol- ◭◭ ◮◮ mogorov complexity and randomness. ◭ ◮ • We also consider computational consequences of an al- Page 2 of 39 ternative idea also coming from physics: Go Back – that no physical law is absolutely true, Full Screen – that every physical law will sooner or later need to be corrected. Close Quit
Physical Induction: . . . How to Describe . . . 2. Physical Induction: Main Idea Checking Equality of . . . • How do we come up with physical laws? Finding Roots Optimization • Someone formulates a hypothesis. Computing Fixed Points • This hypothesis is tested, and if it confirmed suffi- Randomness and . . . ciently many times. Random Sequences . . . • Then we conclude that this hypothesis is indeed a uni- Alternative Idea: No . . . versal physical law. Home Page • This conclusion is known as physical induction . Title Page • Different physicists may disagree on how many exper- ◭◭ ◮◮ iments we need to become certain. ◭ ◮ • However, most physicists would agree that: Page 3 of 39 – after sufficiently many confirmations, Go Back – the hypothesis should be accepted as a physical law. Full Screen • Example: normal distribution :-) Close Quit
Physical Induction: . . . How to Describe . . . 3. How to Describe Physical Induction in Precise Checking Equality of . . . Terms Finding Roots • Let s denote the state of the world, and let P ( s, i ) indi- Optimization cate that the property P holds in the i -th experiment. Computing Fixed Points Randomness and . . . • In these terms, physical induction means that for every Random Sequences . . . property P , there is an integer N such that: Alternative Idea: No . . . – if the statements P ( s, 1), . . . , P ( s, N ) are all true, Home Page – then the property P holds for all possible experi- Title Page ments – i.e., we have ∀ n P ( s, n ). ◭◭ ◮◮ • This cannot be true for all mathematically possible states: ◭ ◮ we can have P ( s, 1) , . . . , P ( s, N ) and ¬ P ( s, N + 1). Page 4 of 39 • Our understanding of the physicists’ claims is that: Go Back – if we restrict ourselves to physically meaningful states, Full Screen – then physical induction is true. Close Quit
Physical Induction: . . . How to Describe . . . 4. Resulting Definition Checking Equality of . . . • Let S be a set; its elements will be called states of the Finding Roots world. Optimization Computing Fixed Points • Let T ⊆ S be a subset of S . We say that T consists of Randomness and . . . physically meaningful states if: Random Sequences . . . – for every property P , there exists an integer N P Alternative Idea: No . . . such that Home Page – for each state s ∈ T for which P ( s, i ) holds for all Title Page i ≤ N P , we have ∀ n P ( s, n ). ◭◭ ◮◮ • For this definition to be precise, we need to select a ◭ ◮ theory L which is: Page 5 of 39 – rich enough to contain all physicists’ arguments and Go Back – weak enough so that we will be able to formally talk about definability in L . Full Screen Close Quit
Physical Induction: . . . How to Describe . . . 5. Definition: Equivalent Form Checking Equality of . . . • We can reformulate this definition in terms of definable Finding Roots sets , i.e.: Optimization Computing Fixed Points – sets of the type { x : P ( x ) } Randomness and . . . – corresponding to definable properties P ( x ). Random Sequences . . . • Let S be a set; its elements will be called states of the Alternative Idea: No . . . world. Home Page • Let T ⊆ S be a subset of S . We say that T consists of Title Page physically meaningful states if: ◭◭ ◮◮ – for every definable sequence of sets { A n } , there ex- ◭ ◮ ists an integer N A Page 6 of 39 N A ∞ – such that T ∩ � A n = T ∩ � A n . Go Back n =1 n =1 Full Screen Close Quit
Physical Induction: . . . How to Describe . . . 6. Existence Proof Checking Equality of . . . • There are no more than countably many words, so no Finding Roots more than countably many definable sequences. Optimization Computing Fixed Points • For real numbers, we can enumerate all definable se- quence, as { A 1 n } , { A 2 Randomness and . . . n } , . . . Let us pick ε ∈ (0 , 1). Random Sequences . . . ∞ n def • For each k , for the difference sets D k A k A k � n − � = n , Alternative Idea: No . . . n i =1 i =1 Home Page ∞ we have D k n +1 ⊆ D k D k n = ∅ , thus, µ ( D k � n and n ) → 0. Title Page n =1 ≤ 2 − k · ε . D k � � • Hence, there exists n k for which µ ◭◭ ◮◮ n k ∞ ◭ ◮ D k • We then take T = S − � n k . k =1 Page 7 of 39 � ∞ � ∞ ∞ 2 − k · ε = ε < 1, Go Back D k � D k � • Here, µ � ≤ � µ ≤ � n k n k k =1 k =1 k =1 Full Screen and thus, the difference T is non-empty. Close • For this set T , we can take N A k = n k . Quit
Physical Induction: . . . How to Describe . . . 7. From States of the World to Specific Quantities Checking Equality of . . . • Usually, we only have a partial information about a Finding Roots state: we have a definable f-n f : S → X which maps Optimization Computing Fixed Points – every state of the world Randomness and . . . – into the corresponding partial information. Random Sequences . . . • Then the range f ( T ) corresponding to all physically Alternative Idea: No . . . meaningful states has the same property as T : Home Page • Let a set T ⊆ S consist of physically meaningful states, Title Page and let f : S → X be a definable function. ◭◭ ◮◮ • Then, for every definable sequence of subsets B n ⊆ X , ◭ ◮ there exists an integer N B such that Page 8 of 39 N B ∞ Go Back � � f ( T ) ∩ B n = f ( T ) ∩ B n . Full Screen n =1 n =1 Close Quit
Physical Induction: . . . How to Describe . . . 8. Proof Checking Equality of . . . • We want to prove that for some N B , Finding Roots Optimization – if an element x ∈ f ( T ) belongs to the sets B 1 , . . . , B N B , Computing Fixed Points – then x ∈ B n for all n . Randomness and . . . • An arbitrary element x ∈ f ( T ) has the form x = f ( s ) Random Sequences . . . for some s ∈ T . Alternative Idea: No . . . def = f − 1 ( B n ). Home Page • Let us take A n Title Page • Since T consists of physically meaningful states, there exists an appropriate integer N A . ◭◭ ◮◮ def ◭ ◮ • Let us take N B = N A . Page 9 of 39 • By definition of A n , the condition x = f ( s ) ∈ B i im- plies that s ∈ A i ; so we have s ∈ A i for all i ≤ N A . Go Back • Thus, we have s ∈ A n for all n , which implies that Full Screen x = f ( s ) ∈ B n . Q.E.D. Close Quit
Physical Induction: . . . How to Describe . . . 9. Computations with Real Numbers: Reminder Checking Equality of . . . • From the physical viewpoint, real numbers x describe Finding Roots values of different quantities. Optimization Computing Fixed Points • We get values of real numbers by measurements. Randomness and . . . • Measurements are never 100% accurate, so after a mea- Random Sequences . . . surement, we get an approximate value r k of x . Alternative Idea: No . . . • In principle, we can measure x with higher and higher Home Page accuracy. Title Page • So, from the computational viewpoint, a real number ◭◭ ◮◮ is a sequence of rational numbers r k for which, e.g., ◭ ◮ | x − r k | ≤ 2 − k . Page 10 of 39 • By an algorithm processing real numbers, we mean an Go Back algorithm using r k as an “oracle” (subroutine). Full Screen • This is how computations with real numbers are de- fined in computable analysis . Close Quit
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