How to Reconcile Randomness with Conditional . . . Physicists - PowerPoint PPT Presentation

What Sequences of . . . Kolmogorov . . . Prefix Kolmogorov . . . How to Reconcile Randomness with Conditional . . . Physicists’ Belief that Every Theory Is Levin’s Formalization . . . Every Theory Is . . . Approximate: Informal Knowledge Is Resulting Challenge Needed How to Reconcile the . . . Ricardo Alvarez 1 , Nick Sims 2 , Christian Servin 2 , Home Page Martine Ceberio 1 , and Vladik Kreinovich 1 Title Page 1 University of Texas at El Paso 500 W. University, El Paso, TX 79968, USA ◭◭ ◮◮ ralvarezlo@miners.utep.edu, mceberio@utep.edu ◭ ◮ vladik@utep.edu 2 Computer Science and Information Technology Page 1 of 29 Systems Department El Paso Community College (EPCC), 919 Hunter Dr. Go Back El Paso, TX 79915-1908, USA Full Screen nsims2@my.epcc.edu, cservin1@epcc.edu Close Quit

What Sequences of . . . 1. What Sequences of Measurement Results Can Kolmogorov . . . We Expect Prefix Kolmogorov . . . Conditional . . . • The usual analysis of algorithms and computability Levin’s Formalization . . . takes into account what we can compute “from scratch”. Every Theory Is . . . • Namely, we only us the given input and performing Resulting Challenge appropriate computational steps. How to Reconcile the . . . Home Page • In principle, in addition to the given input, we could also use the results of physical experiments. Title Page • We normally do not use these results when performing ◭◭ ◮◮ unrelated computations. ◭ ◮ • We do not expect the results of physical experiments Page 2 of 29 to help us solve unrelated complex problems. Go Back Full Screen Close Quit

What Sequences of . . . 2. What Sequences Measurement Results Can We Kolmogorov . . . Expect (cont-d) Prefix Kolmogorov . . . Conditional . . . • It would be very strange to find, e.g.: Levin’s Formalization . . . – a radioactive material Every Theory Is . . . – whose emissions will solve a given NP-hard problem Resulting Challenge like propositional satisfiability. How to Reconcile the . . . Home Page • We may get some help in solving this problem: Title Page – from other people, ◭◭ ◮◮ – maybe from signals sent to us by an alien civiliza- tion – in short, from someone with intelligence. ◭ ◮ • However, we do not expect such help from an (unintel- Page 3 of 29 ligent) physical process. Go Back • This idea was formalized by L. Levin in terms of Kol- Full Screen mogorov complexity . Close • Let us briefly recall the corresponding formalization. Quit

What Sequences of . . . 3. Kolmogorov Complexity: A Brief Reminder Kolmogorov . . . Prefix Kolmogorov . . . • Computers can process many different types of objects Conditional . . . – arrays, graphs, images, videos, etc. Levin’s Formalization . . . • However, in a computer, every object is represented as Every Theory Is . . . a sequence of 0s and 1s, i.e., as a binary sequence . Resulting Challenge How to Reconcile the . . . • Kolmogorov complexity was invented to formalize an- Home Page other intuitive idea: of a random binary sequence. Title Page • From the purely mathematical viewpoint: ◭◭ ◮◮ – if we consequently flip a coin and write down all ◭ ◮ heads as 1s and all tails as 0s, – then all length- n sequences of 0s and 1s have the Page 4 of 29 exact same probability 2 − n . Go Back • From this viewpoint, they are equally probable and Full Screen thus, seem to have an equal right to be called random. Close Quit

What Sequences of . . . 4. Kolmogorov Complexity (cont-d) Kolmogorov . . . Prefix Kolmogorov . . . • In particular, for a large n : Conditional . . . – a sequence that we will actually get after flipping Levin’s Formalization . . . the coin n times has the same probability as Every Theory Is . . . – the sequence 0101...01, in which the sequence 01 is Resulting Challenge repeated many times. How to Reconcile the . . . Home Page • However, intuitively: Title Page – the seemingly lawless sequence that we will actually ◭◭ ◮◮ get after flipping a coin looks random, while ◭ ◮ – the sequence 0101...01 does not look random – and it would be very surprising if such a sequence Page 5 of 29 indeed appears as a result of flipping a coin. Go Back Full Screen Close Quit

What Sequences of . . . 5. Kolmogorov Complexity (cont-d) Kolmogorov . . . Prefix Kolmogorov . . . • Moreover: Conditional . . . – if something like this sequence will be observed in Levin’s Formalization . . . a casino, Every Theory Is . . . – after a while we will be absolutely sure that cheat- Resulting Challenge ing is taking place, and How to Reconcile the . . . Home Page – that the corresponding slot machine is not truly random (as it is supposed to be). Title Page • Kolmogorov noticed that there is a difference between: ◭◭ ◮◮ ◭ ◮ – sequences which are intuitively random and – sequences which are not intuitively random. Page 6 of 29 Go Back Full Screen Close Quit

What Sequences of . . . 6. Kolmogorov Complexity (cont-d) Kolmogorov . . . Prefix Kolmogorov . . . • The reason why we do not believe that the sequence Conditional . . . 0101...01 is truly random is that: Levin’s Formalization . . . – this sequence can be generated by a reasonably Every Theory Is . . . short program, Resulting Challenge – in which we print 01 in a loop. How to Reconcile the . . . Home Page • On the other hand: Title Page – if we consider a real sequence of 0s and 1s obtained ◭◭ ◮◮ by flipping a coin, ◭ ◮ – we do not expect to find any regularity there, – so the only way to generate this sequence is to print Page 7 of 29 it bit-by-bit. Go Back • The length of such a program is practically equal to Full Screen the length len( x ) of the binary string. Close Quit

What Sequences of . . . 7. Kolmogorov Complexity (cont-d) Kolmogorov . . . Prefix Kolmogorov . . . • To describe this difference in precise terms, Kolmogorov Conditional . . . introduced the notion of Kolmogorov complexity C ( x ). Levin’s Formalization . . . • It is the shortest length of a program (in a fixed pro- Every Theory Is . . . gramming language) that generates the string x . Resulting Challenge How to Reconcile the . . . • In these terms, if C ( x ) ≪ len( x ), this sequence is not Home Page random. Title Page • On the other hand: ◭◭ ◮◮ – if C ( x ) ≈ len( x ) – or, to be more precise, if C ( x ) ≥ ◭ ◮ len( x ) − c for some small c , – then we can say that the sequence x is random. Page 8 of 29 Go Back Full Screen Close Quit

What Sequences of . . . 8. Prefix Kolmogorov Complexity Kolmogorov . . . Prefix Kolmogorov . . . • The above definition of randomness can be made even Conditional . . . more intuitive if we take into account that: Levin’s Formalization . . . – we can flip the coin as many times as we want, so Every Theory Is . . . – the coin-flipping sequence can be extended to any Resulting Challenge length. How to Reconcile the . . . Home Page • The fact that we stopped should not change our opin- Title Page ion on whether this sequence is random or not. ◭◭ ◮◮ • From this viewpoint: ◭ ◮ – if it is not so easy to generate the binary sequence that we have so far, but Page 9 of 29 – much easier to generate its extension, Go Back – then the resulting sequence should still be marked Full Screen as not random. Close Quit

What Sequences of . . . 9. Prefix Kolmogorov Complexity (cont-d) Kolmogorov . . . Prefix Kolmogorov . . . • Generating an extension can be easier than the se- Conditional . . . quence itself; for example: Levin’s Formalization . . . – to generate a sequence 001001...001001 is somewhat Every Theory Is . . . easier than Resulting Challenge – its initial segment 001001...00100 when we need to How to Reconcile the . . . take special care of the last two 0s. Home Page Title Page • So, we need to replace the original Kolmogorov com- plexity C ( x ) with prefix Kolmogorov complexity K ( x ). ◭◭ ◮◮ • K ( x ) is the shortest length of the program that gener- ◭ ◮ ates either x or a sequence starting with x . Page 10 of 29 Go Back Full Screen Close Quit

What Sequences of . . . 10. Conditional Kolmogorov Complexity And the Kolmogorov . . . Notion of Information Prefix Kolmogorov . . . Conditional . . . • We are interested in how using an auxiliary sequence Levin’s Formalization . . . y can potentially affect our computations. Every Theory Is . . . • In principle: Resulting Challenge How to Reconcile the . . . – when we use the bits from the given sequence y , Home Page – we do not need to compute them and thus, Title Page – it may be possible to have a shorter program for computing x . ◭◭ ◮◮ ◭ ◮ • An analogy is that if we already have a program for computing sin( x ), then computing sin 2 ( x ) is fast. Page 11 of 29 Go Back Full Screen Close Quit