44 NAW 5/19 nr. 1 maart 2018 An introduction to (algorithmic) randomness Rutger Kuyper Rutger Kuyper School of Mathematics and Statistics Victoria University of Wellington, New Zealand [email protected] Research Stieltjes Prize 2015 An introduction to (algorithmic) randomness In 2015 Rutger Kuyper has been awarded the Stieltjes Prize, which recognises the best The law of large numbers PhD in Mathematics in the Netherlands. The prize was awarded for his thesis entitled Com- Recall the law of large numbers from prob- putability, Probability and Logic, which he completed at the Radboud University Nijmegen. ability theory: After receiving his PhD he became a research fellow at the Victoria University of Wellington. If we flip a coin infinitely many times, and In this article he describes his research at Victoria, which focuses on computability theory, we denote by h the number of times we with an emphasis on algorithmic randomness. n see heads amongst the firstn flips, then with probability 1, What does it mean for a sequence to be does not seem to be random any longer. random? You probably have an intuitive So, how do we actually get an exam- hn 1 lim n = 2 . idea of what a random sequence is, but ple of a random sequence? Something that n " 3 how do we formalise this in a mathe- we can write down easily follows a pattern Our standard example of a random se- matically rigorous way? To motivate what and is therefore not random by any rea- quence, x4, was obtained by flipping a coin comes next, let us look at the (initial digits sonable definition of randomness. The key like this. Thus, our sequence x4 will (proba- of) a few infinite sequences of zeroes and to obtaining a random sequence is using a bly) satisfy the following property: ones: probabilistic method to generate it. To ob- tain x , we flipped a coin, denoting heads Definition 1. Given an infinite sequence x = 00000000000000000000f 4 1 by 1 and tails by 0. With a coin flip having x of zeroes and ones, if we let #xnQ be x = 01010101010101010101f 2 an equal probability to land on heads or the number of ones amongst the first n x3 = 11001001000011111101f tails, there should not be any easy way of elements of the sequence, we say that x f x4 = 00100101110101011101 predicting what x4 is going to look like, in satisfies the law of large numbers if the sense that even if we know the first Take a break and consider for a moment: #xnQ 1 37 elements of the sequence, there is no lim = . which of these sequences would you con- n " 3 n 2 sider to be random sequences? way of knowing whether the 38th element is going to be a zero or a one. Thus, x More informally, x satisfies the law of Would you consider x1 to be random? 4 Probably not, since it only contains zeroes, should be considered random. large numbers if, when we look far enough no ones, and therefore does not look very So, we now have three examples of in the sequence, roughly half of the ele- sequences that should not be considered ments are zeroes, and roughly half of the random. What about x2? This time it has both zeroes and ones, but the sequence random, and one example of something elements are ones. Intuitively a random follows a very clear pattern and therefore that should be considered random. Again, sequence should not be biased towards again should not be considered to be a we would like to have a mathematical- zeroes or ones, so a random sequence ly rigorous definition of what exactly it should satisfy the law of large numbers. random sequence. On the other hand, x3 looks much more random, since there does means to be random, and we now have We could therefore try to use this as a defi- not seem to be a clear pattern. However, three examples of sequences that should nition of randomness. if we were to write r in binary, we would fail this definition, and one example of Let us go back to our examples at the see that something that should pass it. We will use beginning. The non-random sequence x5 this as a starting point and try to build does not satisfy the law of large numbers, r = 11.001001000011111101f our mathematical definition from the which is a good sign. However, x2 does and if we compare this to x3, suddenly x3 ground up. satisfy it, while we decided this sequence Rutger Kuyper An introduction to (algorithmic) randomness NAW 5/19 nr. 1 maart 2018 45 should not be called random. Thus, we need a stronger definition. Normal numbers Let us have another look at the sequence x2 = 01010101010101010101f This sequence satisfies the law of large numbers, because the digits 0 and 1 both appear roughly half the time. Let us con- sider what happens if we consider blocks of two digits in x2. We see that x2 contains the blocks: 01,,,,,,,,10 01 10 01 10 01 10 01,f In other words, we never see the other two blocks of length 2: 00 and 11. Again, thinking about coin flips, if one flips a coin twice and records both the results, then the four possibilities (heads, heads), (tails, tails), (heads, tails) and (tails, heads) 1 all occur with probability 4 . Thus, in a ran- dom sequence we do not just want each digit to occur roughly half the time, but we also want each block of two digits to occur roughly one quarter of the time. Generalis- ing this to blocks of arbitrary length we get to the following definition. Definition 2. Let x be an infinite sequence of zeroes and ones. We say that x is nor- mal (in base 2) if for every positive nat- ural number k, and every block b of k many zeroes or ones, if we let Nx(,bn,) be the number of times the block occurs b Rutger Kuyper in New Zealand amongst the firstn digits of x, then Nx(,bn,) 1 There is a pattern in this sequence that sitions at which we know for sure the digit lim n = b . n " 3 2 might not be immediately obvious: we write is a one. In particular, if we were to restrict the natural numbers 0123,,,,f in binary, x to W, i.e. we throw away all the digits (The notion of normality — in arbitrary bas- 5 and form the sequence x by concate- at positions not in W, we get a sequence es — was introduced by Borel, when we 5 nating all these binary numbers together that only contains ones and therefore talk about normality we mean the special in order. The sequence x we obtain this no longer even satisfies the law of large case of normality in base 2.) 5 way is called Champernowne’s sequence. numbers. Because x does not satisfy the law of 1 This sequence is known to be normal, so Again, if we think about a sequence of large numbers it is certainly not normal, and would be random if we took normality as coin tosses, if we were, for example, to x is not normal either as argued above. 2 our definition of randomness. However, it ignore all the even coin tosses and only re- What about x ? Perhaps surprisingly, it is 3 does not seem very reasonable to consider cord the odd ones, the resulting sequence not known whether the sequence x , which 3 x to be random. After all, it was generated should still satisfy the law of large num- we obtained from the binary expansion of 5 using such an easy pattern and therefore it bers. Thus, we would like to say that a r, is normal, so we do not know wheth- is very easy to compute the digits of this sequence is random if every infinite sub- er this definition is able to determine the sequence. sequence, that is a sequence obtained by nonrandomness of x . On the other hand, 3 removing all but infinitely many digits, sat- the sequence x obtained by flipping coins 4 Church stochasticity isfies the law of large numbers. is (with probability 1) normal. Going back to Champernowne’s sequence Unfortunately, that is too strong a prop- Does this mean we are done? Let us x , why exactly is it non-random? We no- erty: such sequences do not exist. Indeed, consider the following sequence: 5 ticed that it is very easy to reconstruct the every infinite sequence contains either x5 = 01110 110010111011110001001 pattern used to definex 5. In particular, it is infinitely many zeroes or infinitely many 101010111100f very easy to give an infinite setW of po- ones, so we can always find an infinite 46 NAW 5/19 nr. 1 maart 2018 An introduction to (algorithmic) randomness Rutger Kuyper subsequence which only contains zeroes This notion, called Martin-Löf randomness, mathematically rigorous way? Of course or ones! How do we resolve this? was originally introduced by Martin-Löf as that is an informal statement that cannot Until now, we have been talking about an abstracting of the computable selection be formalised, let alone proven. However, randomness without talking about algo- functions defined above. One of the things the fact that we do not know of any coun- rithmic randomness, and this is the point that makes this definition so robust is that terexamples and have many equivalent at which the algorithmic part of the title there are many equivalent definitions of definitions of Martin-Löf randomness seem comes into play.