and Randomness

Rod Downey and Denis R. Hirschfeldt

Historical Roots How could a binary string representing a sequence of 푛 Von Mises. Around 1930, Kolmogorov and others found- coin tosses be random, when all strings of length 푛 have −푛 ed the of , basing it on measure theory. the same probability of 2 for a fair coin? Probability theory is concerned with the distribution of Less well known than the work of Kolmogorov are early outcomes in sample spaces. It does not seek to give any attempts to answer this kind of question by providing no- meaning to the notion of an individual object, such as tions of randomness for individual objects. The modern a single real number or binary string, being random, but theory of algorithmic randomness realizes this goal. One way rather studies the expected values of random variables. to develop this theory is based on the idea that an object is random if it passes all relevant “randomness tests.” For Rod Downey is a professor of and the deputy head of the School example, by the law of , for a random real 푋, of Mathematics and at Victoria University Wellington. His email we would expect the number of 1’s in the binary expansion address is [email protected]. 1 of 푋 to have limiting frequency 2 . (That is, writing 푋(푗) Denis R. Hirschfeldt is a professor of mathematics at the University of Chicago. for the 푗th bit of this expansion, we would expect to have His email address is [email protected]. |{푗<푛∶푋(푗)=1}| 1 lim푛→∞ 푛 = 2 .) Indeed, we would expect 푋 For permission to reprint this article, please contact: to be normal to base 2, meaning that for any binary string [email protected]. 휎 of length 푘, the occurrences of 휎 in the binary expan- DOI: https://doi.org/10.1090/noti1905

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1001 sion of 푋 should have limiting frequency 2−푘. Since base expected number of steps of a sorting should be representation should not affect randomness, we would the same as that for a single algorithmically random input. expect 푋 to be normal in this sense no matter what base We could also be more fine-grained and seek to understand it were written in, so that in base 푏 the limiting frequency exactly “how much” randomness is needed for certain typ- would be 푏−푘 for a string 휎 of length 푘. Thus 푋 should ical behaviors to arise. (See the section on “Some Applica- be what is known as absolutely normal. tions.”) The idea of normality, which goes back to Borel (1909), As we will discuss, algorithmic randomness also goes was extended by von Mises (1919), who suggested the fol- hand in hand with other parts of algorithmic information lowing definition of randomness for individual binary se- theory, such as , and has ties with quences.1 A selection is an increasing function 푓 ∶ notions such as Shannon entropy and fractal dimension. ℕ → ℕ. We think of 푓(푖) as the 푖th place selected in form- Some basic . In the 1930s, Church, ing a subsequence of a given sequence. (For the defini- Gödel, Kleene, Post, and most famously Turing (1937) tion of normality above, where we consider the entire se- gave equivalent mathematical definitions capturing the in- quence, 푓(푖) = 푖.) Von Mises suggested that a sequence tuitive notion of a , leading to the 푎0푎1 … should be random if any selected subsequence Church-Turing Thesis, which can be taken as asserting that 푎푓(0)푎푓(1) … is normal. a function (from ℕ to ℕ, say) is computable if and only There is, of course, an obvious problem with this ap- if it can be computed by a . (This defini- proach. For any sequence 푋 with infinitely many 1’s we tion can easily be transferred to other objects of countable could let 푓 select the positions where 1’s occur, and 푋 mathematics. For instance, we think of infinite binary se- would fail the test determined by 푓. However, it does not quences as functions ℕ → {0, 1}, and identify sets of nat- seem reasonable to be able to choose the testing places ural numbers with their characteristic functions.) Nowa- after selecting an 푋. The question is then: What kinds days, we can equivalently regard a function as computable of selection functions should be allowed, to capture the if we can write code to compute it in any given general- intuition that we ought not to be able to sample from a purpose (assuming the language random sequence and get the wrong frequencies? It is can address unlimited memory). It has also become clear reasonable to regard prediction as a computational pro- that can be treated as data, and hence that cess, and hence restrict ourselves to computable selection there is a , i.e., there is a listing functions. Indeed, this suggestion was eventually made by Φ0,Φ1,… of all Turing machines and a single algorithm Church (1940), though von Mises’ work predates the defi- that, on input ⟨푒, 푛⟩ computes the result Φ푒(푛) of running 2 nition of computable function, so he did not have a good Φ푒 on input 푛. way to make his definition mathematically precise. It is important to note that a Turing machine might As we will see, von Mises’ approach had a more sig- not halt on a given input, and hence the functions com- nificant flaw, but we can build on its fundamental idea: puted by Turing machines are in general partial. Indeed, Imagine that we are judges deciding whether a sequence as Turing showed, the “Does the 푒th Tur- 푋 should count as random. If 푋 passes all tests we can ing machine halt on input 푛?” is algorithmically unsolv- (in principle) devise given our computational power, then able. Church and Turing famously showed that Hilbert’s we should regard 푋 as random since, as far as we are con- (the for first-order cerned, 푋 has all the expected properties of a random ob- ) is unsolvable, in Turing’s case by showing that the ject. We will use this intuition and the apparatus of com- halting problem can be coded into first-order logic. Many putability and complexity theory to describe notions of al- other problems have since been shown to be algorithmi- gorithmic randomness. cally unsolvable by similar means. Aside from the intrinsic interest of such an approach, We write Φ푒(푛)↓ to mean that the machine Φ푒 eventu- ′ it leads to useful mathematical tools. Many processes in ally halts on input 푛. Then ∅ = {⟨푒, 푛⟩ ∶ Φ푒(푛)↓} is a mathematics are computable, and the expected behavior representing the halting problem. This set is an exam- of such a process should align itself with the behavior ob- ple of a noncomputable (c.e.) set, tained by providing it with an algorithmically random in- 2The realization that such universal machines are possible helped lead to the development put. Hence, instead of having to analyze the relevant dis- of modern . Previously, machines had been purpose-built for given tasks. In a tribution and its statistics, we can simply argue about the 1947 lecture on his design for the Automated Engine, Turing said, “The special machine may be called the universal machine; it works in the following quite simple man- behavior of the process on a single input. For instance, the ner. When we have decided what machine we wish to imitate we punch a description of it on the tape of the universal machine . . . The universal machine has only to keep looking at this description in order to find out what it should do at each stage. Thus the complexity of 1Due to space limitations, we omit historical citations and those found in the books the machine to be imitated is concentrated in the tape and does not appear in the universal Downey and Hirschfeldt [9], Li and Vit´anyi [18], or Nies [24] from the list of references. machine proper in any way. . . . [D]igital computing machines such as the ACE . . . are In some sections below, we also cite secondary sources where additional references can be in fact practical versions of the universal machine.” From our contemporary point of view, found. it may be difficult to imagine how novel this idea was.

1002 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 which means that the set can be listed (not necessarily in large numbers. But which ones? Ville suggested ones - numerical order) by some algorithm. flecting the law of iterated logarithms, which would take Another important notion is that of Turing reducibility care of his specific example. But how could we know that (which we define for sets of natural numbers but issim- further examples along these lines—i.e., sequences satis- ilarly defined for functions), where 퐴 is Turing reducible fying both von Mises’ and Ville’s tests, yet failing to have to 퐵, written as 퐴 ≤T 퐵, if there is an algorithm for com- some other property we expect of random sequences— puting 퐴 when given access to 퐵. That is, the algorithm would not arise? is allowed access to answers to questions of the form “Is The situation was finally clarified in the 1960s by 푛 in 퐵?” during its execution. This notion can be formal- Martin-Löf (1966). In probability theory, “typicality” is ized using Turing machines with oracle tapes. If 퐴 ≤T 퐵, quantified using measure theory, leading to the intuition then we regard 퐴 as no more complicated than 퐵 from a that random objects should avoid null sets. Martin-Löf no- computability-theoretic perspective. We also say that 퐴 is ticed that tests like von Mises’ and Ville’s can be thought of 퐵-computable or computable relative to 퐵. Turing reducibil- as effectively null sets. His idea was that, instead of consid- ity naturally leads to an equivalence relation, where 퐴 and ering specific tests based on particular statistical laws, we

퐵 are Turing equivalent if 퐴 ≤T 퐵 and 퐵 ≤T 퐴. The (Tur- should consider all possible tests corresponding to some ing) degree of 퐴 is its equivalence under this notion. precisely defined notion of effectively null set. The restric- (There are several other notions of reducibility and result- tion to such a notion gets around the problem that no se- ing degree structures in computability theory, but Turing quence can avoid being in every null set. reducibility is the central one.) To give Martin-Löf’s definition, we work for convenience In general, the process of allowing access to an oracle in in Cantor space 2휔, whose elements are infinite binary se- our algorithms is known as relativization. As in the unrel- quences. (We associate a real number with its binary ex- 퐵 퐵 ativized case, we can list the Turing machines Φ0 ,Φ1 ,… pansion, thought of as a sequence, so we will also obtain a ′ 퐵 with oracle 퐵, and let 퐵 = {⟨푒, 푛⟩ ∶ Φ푒 (푛)↓} be the rel- definition of algorithmic randomness for reals. The choice ativization of the halting problem to 퐵. This set is called of base is not important. For example, all of the notions the (Turing) jump of 퐵. The jump operation taking 퐵 to of randomness we consider are enough to ensure absolute 퐵′ is very important in computability theory, one reason normality.) The basic open sets of Cantor space are the 휔 being that 퐵′ is the most complicated set that is still c.e. ones of the form [휎] = {푋 ∈ 2 ∶ 푋 extends 휎} for <휔 <휔 relative to 퐵, i.e., 퐵′ is c.e. relative to 퐵 and every set that is 휎 ∈ 2 , where 2 is the set of finite binary strings. c.e. relative to 퐵 is 퐵′-computable. There are several other The uniform measure 휆 on this space is obtained by defin- −|휎| important classes of sets that can be defined in terms of ing 휆([휎]) = 2 . We say that a sequence 푇0, 푇1,… of ′ ′ 휔 the jump. For instance, 퐴 is low if 퐴 ≤T ∅ and high open sets in 2 is uniformly c.e. if there is a c.e. set 퐺 ⊆ ″ ′ ″ ′ ′ <휔 if ∅ ≤T 퐴 (where ∅ = (∅ ) ). Low sets are in cer- ℕ × 2 such that 푇푛 = ⋃{[휎] ∶ (푛, 휎) ∈ 퐺}. tain ways “close to computable,” while high ones partake ∅′ Definition 2. A Martin-Löf test is a sequence 푇0, 푇1,… of of some of the power of as an oracle. These properties −푛 are invariant under Turing equivalence, and hence are also uniformly c.e. open sets such that 휆(푇푛) ≤ 2 . A se- properties of Turing degrees. quence 푋 passes this test if 푋 ∉ ⋂푛 푇푛. A sequence is Martin-Löf random (ML-random) if it passes all Martin-Löf Martin-Löf randomness. As mentioned above, Church tests. suggested that a sequence should count as algorithmically random if it is random in the sense of von Mises with se- The intersection of a Martin-Löf test is our notion of lection functions restricted to the computable ones. How- effectively null set. Since there are only countably many ever, in 1939, Ville showed that von Mises’ approach can- Martin-Löf tests, and each determines a null set in the clas- not work in its original form, no matter what countable sical sense, the collection of ML-random sequences has collection of selection functions we choose. Let 푋 ↾ 푛 de- measure 1. It can be shown that Martin-Löf tests include note the first 푛 bits of the binary sequence 푋. all the ones proposed by von Mises and Ville, in Church’s 1 (Ville (1939)). For any countable collection of computability-theoretic versions. Indeed they include all selection functions, there is a sequence 푋 that passes all von tests that are “computably performable,” which avoids the Mises tests associated with these functions, such that for every problem of having to adaptively introduce more tests as 푛, there are more 0’s than 1’s in 푋 ↾ 푛. more Ville-like sequences are found. Martin-Löf’s effectivization of measure theory allowed Clearly, Ville’s sequence cannot be regarded as random in him to consider the laws a random sequence should obey any reasonable sense. from an abstract point of view, leading to a mathemati- We could try to repair von Mises’ definition by adding cally robust definition. As Jack Lutz said in a talk atthe 7th further tests, reflecting statistical laws beyond the law of Conference on Computability, Complexity, and Randomness

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1003 (Cambridge, 2012), “Placing computability constraints on martingale is that initially we might have no idea what to a nonconstructive theory like Lebesgue measure seems a bet on some string 휎, but as we learn more about the uni- priori to weaken the theory, but it may strengthen the the- verse, we might discover that 휎 seems more unlikely to ory for some purposes. This vision is crucial for present- be an initial segment of a random sequence, and are then day investigations of individual random sequences, dimen- prepared to bet more of our capital on it. sions of individual sequences, measure and category in The coder’s approach builds on the idea that a random complexity classes, etc.” string should have no short descriptions. For example, in The three approaches. ML-randomness can be thought describing 010101 … (1000 times) by the brief descrip- of as the statistician’s approach to defining algorithmic ran- tion “print 01 1000 times,” we are using regularities in domness, based on the intuition that random sequences this string to compress it. For a more complicated string, 휋 should avoid having statistically rare properties. There are say the first 2000 bits of the binary expansion of 푒 , the two other major approaches: regularities may be harder to perceive, but are still there • The gambler’s approach: random sequences should and can still lead to compression. A random string should be unpredictable. have no such exploitable regularities (i.e., regularities that • The coder’s approach: random sequences should are not present in most strings), so the shortest way to de- not have regularities that allow us to compress the scribe it should be basically to write it out in full. This idea information they contain. can be formalized using the well-known concept of Kol- mogorov complexity. We can think of a Turing machine The gambler’s approach may be the most immediately <휔 푀 with inputs and outputs in 2 as a description system. intuitive one to the average person. It was formalized in If 푀(휏) = 휎 then 휏 is a description of 휎 relative to this the computability-theoretic setting by Schnorr (1971), us- description system. The Kolmogorov complexity 퐶푀(휎) of ing the idea that we should not be able to make arbitrar- 휎 relative to 푀 is the length of the shortest 휏 such that ily much money when betting on the bits of a random se- 푀(휏) = 휎. We can then take a universal Turing machine quence. The following notion is a simple special case of 푈, which emulates any given Turing machine with at most the notion of a martingale from probability theory. (See [9, a constant increase in the size of programs, and define the Section 6.3.4] for further discussion of the relationship be- (plain) Kolmogorov complexity of 휎 as 퐶(휎) = 퐶푈(휎). The tween these concepts.) value of 퐶(휎) depends on 푈, but only up to an additive Definition 3. A martingale is a function 푓 ∶ 2<휔 → ℝ≥0 constant independent of 휎. We think of a string as ran- such that dom if its Kolmogorov complexity is close to its length. 푓(휎0) + 푓(휎1) 푋 푓(휎) = For an infinite sequence , a natural guess would be 2 that 푋 should be considered random if every initial seg- for all 휎. We say that 푓 succeeds on 푋 if lim sup푛→∞ 푓(푋 ↾ ment of 푋 is incompressible in this sense, i.e., if 퐶(푋 ↾ 푛) = ∞. 푛) ≥ 푛 − 푂(1). However, plain Kolmogorov complexity is not quite the right notion here, because the information We think of 푓 as the capital we have when betting on in a description 휏 consists not only of the bits of 휏, but the bits of a binary sequence according to a particular bet- also its length, which can provide another log |휏| many ting strategy. The displayed equation ensures that the bet- 2 bits of information. Indeed, Martin-Löf (see [18]) showed ting is fair. Success then means that we can make arbitrar- that it is not possible to have 퐶(푋 ↾ 푛) ≥ 푛 − 푂(1): ily much money when betting on 푋, which should not Given a long string 휌, we can write 휌 = 휎휏휈, where |휏| happen if 푋 is random. By considering martingales with is the position of 휎 in the length-lexicographic ordering varying levels of effectivity, we get various notions of algo- of 2<휔. Consider the Turing machine 푀 that, on input 휂, rithmic randomness, including ML-randomness itself, as determines the |휂|th string 휉 in the length-lexicographic it turns out. ordering of 2<휔 and outputs 휉휂. Then 푁(휏) = 휎휏. For For example, 푋 is computably random if no computable any sequence 푋 and any 푘, this process allows us to com- martingale succeeds on it, and polynomial-time random if press some initial segment of 푋 by more than 푘 many bits. no polynomial-time computable martingale succeeds on There are several ways to get around this problem by it. (For the purposes of defining these notions we can think modifying the definition of Kolmogorov complexity. The of the martingale as rational-valued.) Schnorr (1971) best-known one is to use prefix-free codes, that is, to re- showed that 푋 is ML-random iff no left-c.e. martingale suc- strict ourselves to machines 푀 such that if 푀(휏) is defined ceeds on it, where a function 푓 ∶ 2<휔 → ℝ≥0 is left-c.e. if (i.e., if the machine eventually halts on input 휏) and 휇 is it is computably approximable from below, i.e., there is a proper extension of 휏, then 푀(휇) is not defined. There a computable function 푔 ∶ 2휔 × ℕ → ℚ≥0 such that are universal prefix-free machines, and we can take such 푔(휎, 푛) ≤ 푔(휎, 푛 + 1) for all 휎 and 푛, and 푓(휎) = a machine 푈 and define the prefix-free Kolmogorov complex- lim푛→∞ 푔(휎, 푛) for all 휎. One way to think of a left-c.e.

1004 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 ity of 휎 as 퐾(휎) = 퐶푈(휎). The roots of this notion can complexity to give a proof of a version of Gödel’s First be found in the work of Levin, Chaitin, and Schnorr, and Incompleteness Theorem, by showing that for any suffi- in a certain sense—like the notion of Kolmogorov com- ciently strong, computably axiomatizable, consistent the- plexity more generally—even earlier in that of Solomonoff ory 푇, there is a number 푐 such that 푇 cannot prove that (see [9,18]). As shown by Schnorr (see Chaitin (1975)), it 퐶(휎) > 푐 for any given string 휎 (which also follows by is indeed the case that 푋 is Martin-Löf random if and only interpreting an earlier result of Barzdins; see [18, Section if 퐾(푋 ↾ 푛) ≥ 푛 − 푂(1). 2.7]). More recently, Kritchman and Raz [17] used these There are other varieties of Kolmogorov complexity, but methods to give a proof of the Second Incompleteness The- 퐶 and 퐾 are the main ones. For applications, it often orem as well.3 As we will see below, a more recent line does not matter which variety is used. The following sur- of research has used notions of effective dimension based prising result establishes a fairly precise relationship be- on partial randomness to give new proofs of classical theo- tween 퐶 and 퐾. Let 퐶(1)(휎) = 퐶(휎) and 퐶(푛+1)(휎) = rems in ergodic theory and obtain new results in geometric 퐶(퐶(푛)(휎)). measure theory.

Theorem 4 (Solovay (1975)). 퐾(휎) = 퐶(휎)+퐶(2)(휎)± Some Interactions with Computability 푂(퐶(3)(휎)), and this result is tight in that we cannot extend Halting . A first question we might ask is it to 퐶(4)(휎). how to generate “natural” examples of algorithmically ran- dom reals. A classic example is Chaitin’s halting probabil- There is a vast body of research on Kolmogorov com- ity. Let 푈 be a universal prefix-free machine and let plexity and its applications. We will discuss some of these −|휎| applications below; much more on the topic can be found Ω = ∑ 2 . in Li and Vit´anyi [18]. 푈(휎)↓ This number is the measure of the set of sequences 푋 such Goals that 푈 halts on some initial segment of 푋, which we can There are several ways to explore the ideas introduced interpret as the halting probability of 푈, and was shown above. First, there are natural internal questions, such as: by Chaitin (1975) to be ML-random (where, as mentioned How do the various levels of algorithmic randomness in- above, we identify Ω with its binary expansion, thought of terrelate? How do calibrations of randomness relate to the as an infinite binary sequence). hierarchies of computability and complexity theory, and For any prefix-free machine 푀 in place of 푈 we can sim- to relative computability? How should we calibrate par- ilarly define a halting probability. In some ways, halting tial randomness? Can a source of partial (algorithmic) probabilities are the analogs of computably enumerable randomness be amplified into a source that is fully ran- sets in the theory of algorithmic randomness. Every halt- dom, or at least more random? The books Downey and ing probability 훼 is a left-c.e. real, meaning that there is Hirschfeldt [9] and Nies [24] cover material along these a computable increasing sequence of rationals converging lines up to about 2010. to it. Calude, Hertling, Khoussainov, and Wang (1998) We can also consider applications. Mathematics has showed that every left-c.e. real is the halting probability of many that involve “almost everywhere” behav- some prefix-free machine. ior. Natural examples come from ergodic theory, analysis, We should perhaps write Ω푈 instead of Ω, to stress the geometric measure theory, and even . Be- dependence of its particular value on the choice of uni- havior that occurs almost everywhere should occur at suf- versal machine, but the fundamental properties of Ω do ficiently random points. Using notions from algorithmic not depend on this choice, much as those of the halting randomness, we can explore exactly how much randomness problem do not depend on the specific choice of enumer- is needed in a given case. For example, the set of reals at ation of Turing machines. In particular, Kuceraˇ and Sla- which an increasing function is differentiable is null. How man (2001) showed that every left-c.e. real is reducible to complicated is this null set, and hence, what level of algo- every Ω푈 up to a strong notion of reducibility known as rithmic randomness is necessary for a real to avoid it (as- Solovay reducibility, and hence all such Ω푈’s are equiva- suming the function is itself computable in some sense)? lent modulo this notion. (The situation is analogous to Is Martin-Löf randomness the right notion here? that of versions of the halting problem, where the relevant We can also use the idea of assigning levels of random- notion is known as 1-reducibility.) ness to individual objects to prove new theorems or give 3 simpler proofs of known ones. Early examples of this Other recent work has explored the effect of adding asserting the incompressibility of certain strings in a probabilistic way. Bienvenu, Romashchenko, Shen, Taveneaux, and method tended to use Kolmogorov complexity and what Vermeeren [4] have shown that this kind of procedure does not help to prove new interest- ing theorems, but that the situation changes if we take into account the sizes of the proofs: is called the “.” For instance, randomly chosen axioms (in a sense made precise in their paper) can help to make proofs Chaitin (1971) (see also [17]) famously used Kolmogorov much shorter under the reasonable complexity-theoretic assumption that NP ≠ PSPACE.

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1005 Left-c.e. and right-c.e. reals (those of the form 1 − 훼 for a sequence is in that class is to increase the complexity of a left-c.e. 훼) occur naturally in mathematics. Braverman our tests by relativizing them to noncomputable oracles. and Yampolsky [7] showed that they arise in connection It turns out that iterates of the are particu- with Julia sets, and there is a striking example in symbolic larly natural oracles to use. Let ∅(0) = ∅ and ∅(푛+1) = dynamics: A 푑-dimensional subshift of finite type is a cer- (∅(푛))′. We say that 푋 is 푛-random if it passes all Martin- 푑 tain kind of collection of 퐴-colorings of ℤ , where 퐴 is a Löf tests relativized to ∅(푛−1). Thus the 1-random , defined by local rules (basically saying that cer- sequences are just the ML-random ones, while the 2- tain coloring patterns are illegal) invariant under the shift random ones are the ones that are ML-random relative to action the halting problem. These sequences have low compu- 푑 (푆푔푥)(ℎ) = 푥(ℎ + 푔) for 푔, ℎ ∈ ℤ푑 and 푥 ∈ 퐴ℤ . tational power in several ways. For instance, they cannot compute any noncomputable c.e. set, and in fact the fol- Its (topological) entropy is an important invariant measuring lowing holds. the asymptotic growth in the number of legal colorings of finite regions. It has been known for some time thaten- Theorem 8 (Kurtz (1981)). If 푋 is 2-random and 푌 is com- tropies of subshifts of finite type for dimensions 푑 ≥ 2 are putable relative both to ∅′ and to 푋, then 푌 is computable. in general not computable, but the following result gives A precise relationship between tests and the dichotomy a precise characterization. mentioned above was established by Franklin and Ng [11]. Theorem 5 (Hochman and Meyerovitch [15]). The values In general, among ML-random sequences, - of entropies of subshifts of finite type over ℤ푑 for 푑 ≥ 2 are al power (or “useful information”) is inversely proportion- exactly the nonnegative right-c.e. reals. al to level of randomness. The following is one of many results attesting to this heuristic. Algorithmic randomness and relative computability.

Solovay reducibility is stronger than Turing reducibility, so Theorem 9 (Miller and Yu (2008)). Let 푋 ≤T 푌. If 푋 is ′ Ω can compute the halting problem ∅ . Indeed Ω and ML-random and 푌 is 푛-random, then 푋 is also 푛-random. ∅′ are Turing equivalent, and in fact Ω can be seen as a “highly compressed” version of ∅′. Other computability- There are many other interesting levels of algorithmic theoretically powerful ML-random sequences can be ob- randomness. Schnorr (1971) argued that his martingale tained from the following remarkable result. characterization of ML-randomness shows that this is an intrinsically computably enumerable rather than computable Theorem 6 (G´acs(1986), Kuceraˇ (1985)). For every 푋 there notion, and defined a notion now called Schnorr random- is an ML-random 푌 such that 푋 ≤T 푌. ness, which is like the notion of computable randomness This theorem and the Turing equivalence of Ω with ∅′ mentioned below Definition 3 but with an extra effective- do not seem to accord with our intuition that random sets ness condition on the rate of success of martingales. He should have low “useful information.” This phenomenon also showed that 푋 is Schnorr random iff it passes all can be explained by results showing that, for certain pur- Martin-Löf tests 푇0, 푇1,… such that the measures 휆(푇푛) poses, the benchmark set by ML-randomness is too low. A are uniformly computable (i.e., the function 푛 ↦ 휆(푇푛) set 퐴 has PA degree if it can compute a {0, 1}-valued func- is computable in the sense of the section on “Analysis and tion 푓 with 푓(푛) ≠ Φ푛(푛) for all 푛. (The reason for the ergodic theory” below). It follows immediately from their name is that this property is equivalent to being able to definitions in terms of martingales that ML-randomness compute a completion of Peano Arithmetic.) Such a func- implies computable randomness, which in turn implies tion can be seen as a weak version of the halting problem, Schnorr randomness. It is more difficult to prove that none but while ∅′ has PA degree, there are sets of PA degree that of these implications can be reversed. In fact, these levels are low, in the sense of the section on “Some basic com- of randomness are close enough that they agree for sets putability theory,” and hence are far less powerful than that are somewhat close to computable, as shown by the ∅′. following result, where highness is as defined in the sec- tion on “Some basic computability theory.” Theorem 7 (Stephan (2006)). If an ML-random sequence has PA degree then it computes ∅′. Theorem 10 (Nies, Stephan, and Terwijn (2005)). Every high contains a set that is computably random Thus there are two kinds of ML-random sequences. but not ML-random and a set that is Schnorr random but not Ones that are complicated enough to somehow “simulate” computably random. This fact is tight, however, because every randomness, and “truly random” ones that are much nonhigh Schnorr random set is ML-random. weaker. It is known that the class of sequences that can compute ∅′ has measure 0, so almost all ML-random se- As we will discuss, various notions of algorithmic ran- quences are in the second class. One way to ensure that domness arise naturally in applications.

1006 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 Randomness-theoretic weakness. As mentioned above, where 퐾퐴 is the relativization of prefix-free Kolmogorov 푋 is ML-random iff 퐾(푋 ↾ 푛) ≥ 푛 − 푂(1), i.e., 푋’s ini- complexity to 퐴. tial segments have very high complexity. There are similar 4. 퐴 is low for ML-randomness, meaning that 퐴 does characterizations of other notions of algorithmic random- not have any derandomization power as an oracle, i.e., any ness, as well as of notions arising in other parts of com- ML-random set remains ML-random when this notion is putability theory, in terms of high initial segment complex- relativized to 퐴. ity. For instance, Downey and Griffiths (2004) showed that 푋 is Schnorr random iff 퐶푀(푋 ↾ 푛) ≥ 푛 − 푂(1) for every prefix-free machine 푀 with computable halting There are now more than a dozen other characteriza- 퐾 probability, while Kjos-Hanssen, Merkle, and Stephan tions of -triviality. Some appear in [9, 24], and several (2006) showed that 푋 can compute a diagonally noncom- others have emerged more recently. These have been used to solve several problems in algorithmic randomness and putable function, that is, a function ℎ with ℎ(푒) ≠ Φ푒(푒) for all 푒, iff there is an 푋-computable function 푓 such that related areas. Lowness classes have also been found for 퐶(푋 ↾ 푓(푛)) ≥ 푛 for all 푛. But what if the initial seg- other randomness notions. For Schnorr randomness, for ments of a sequence have low complexity? Such sequences instance, lowness can be characterized using notions of have played an important role in the theory of algorithmic traceability related to concepts in , as first ex- randomness, beginning with the following information- plored by Terwijn and Zambella (2001). theoretic characterization of computability. Some Applications Theorem 11 (Chaitin (1976)). 퐶(푋 ↾ 푛) ≤ 퐶(푛)+푂(1) Incompressibility and information content. This article iff 푋 is computable. focuses on algorithmic randomness for infinite objects, but we should mention that there have been many applica- It is also true that if 푋 is computable then 퐾(푋 ↾ 푛) ≤ tions of Kolmogorov complexity under the collective ti- 퐾(푛) + 푂(1). Chaitin (1977) considered sequences with tle of the incompressibility method, based on the observa- this property, which are now called 퐾-trivial. He showed tion that algorithmically random strings should exhibit ′ that every 퐾-trivial sequence is ∅ -computable, and asked typical behavior for computable processes. For example, whether they are all in fact computable. Solovay (1975) this method can be used to give average running times answered this question by constructing a noncomputable for sorting, by showing that if the outcome is not what 퐾-trivial sequence. we would expect then we can compress a random input. The class of 퐾-trivials has several remarkable proper- See Li and Vit´anyi [18, Chapter 6] for applications of this ties. It is a naturally definable countable class, contained technique to areas as diverse as combinatorics, formal lan- in the class of low sets (as defined in the section on “Some guages, compact routing, and circuit complexity, among basic computability theory,” where we identify a set with others. Many results originally proved using Shannon en- its characteristic function, thought of as a sequence), but tropy or related methods also have proofs using Kolmogor- with stronger closure properties. (In technical terms, it is ov complexity. For example, Messner and Thierauf [22] what is known as a Turing ideal.) Post’s problem asked gave a constructive proof of the Lov´aszLocal Lemma us- whether there are computably enumerable sets that are nei- ing Kolmogorov complexity. ther computable nor Turing equivalent to the halting prob- Other applications come from the observation that lem. Its solution in the 1950s by Friedberg and Much- in some sense Kolmogorov complexity provides an nik introduced the somewhat complex priority method, “absolute” measure of the intrinsic complexity of a string. which has played a central technical role in computability We can define a notion of conditional Kolmogorov com- theory since then. Downey, Hirschfeldt, Nies, and Stephan plexity 퐶(휎 ∣ 휏) of a string 휎 given another string 휏. (2003) showed that 퐾-triviality can be used to give a sim- Then, for example, 퐶(휎 ∣ 휎) = 푂(1), and 휎 is “inde- ple priority-free solution to Post’s problem. pendent of 휏” if 퐶(휎 ∣ 휏) = 퐶(휎) ± 푂(1). Researchers Most significantly, there are many natural notions of comparing two sequences 휎, 휏 representing, say, two DNA randomness-theoretic weakness that turn out to be equiv- sequences, or two phylogenetic trees, or two languages, or alent to 퐾-triviality. two pieces of music, have invented many distance metrics, such as the maximum parsimony distance on phylogenetic Theorem 12 (Nies (2005), Nies and Hirschfeldt for trees, but it is also natural to use a content-neutral mea- (1) → (3) ). The following are equivalent. sure of “information distance” like max{퐶(휎 ∣ 휏), 퐶(휏 ∣ 1. 퐴 is 퐾-trivial. 휎)}. There have been some attempts to make this work in 2. 퐴 is computable relative to some c.e. 퐾-trivial set. practice for solving classification problems, though results 3. 퐴 is low for 퐾, meaning that 퐴 has no compression have so far been mixed. Of course, 퐶 is not computable, power as an oracle. i.e., that 퐾퐴(휎) ≥ 퐾(휎) − 푂(1), but it can be replaced in applications by measures derived

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1007 푛 from practical compression algorithms. See [18, Sections dorff dimension of 퐸 ⊆ ℝ by dimH(퐸), and its classical

8.3 and 8.4]. packing dimension by dimp(퐸). Effective dimensions. If 푋 = 푥 푥 … is random, then we 0 1 Theorem 14 (Lutz and Lutz [19]). might expect a sequence such as 푥000푥100푥200 … to be 1 퐴 “ -random.” Making precise sense of the idea of partial al- dimH(퐸) = min sup dim (푋). 3 퐴⊆ℕ 푋∈퐸 gorithmic randomness has led to significant applications. 퐴 Hausdorff used work of Carath´eodoryon 푠-dimensional dimp(퐸) = min sup Dim (푋). 퐴⊆ℕ measures to generalize the notion of dimension to possi- 푋∈퐸 bly nonintegral values, leading to concepts such as Haus- For certain well-behaved sets 퐸, relativization is actu- dorff dimension and packing dimension. Much like algo- ally not needed, and the classical dimension of 퐸 is the rithmic randomness can make sense of the idea of individ- supremum of the effective dimensions of its points. In the ual reals being random, notions of partial algorithmic ran- general case, it is of course not immediately clear that the domness can be used to assign dimensions to individual minima mentioned in Theorem 14 should exist, but they reals. do. Thus, for example, to prove a lower bound of 훼 for

The measure-theoretic approach, in which we for dimH(퐸) it suffices to prove that, for each 휖 > 0 and each instance replace the uniform measure 휆 on 2휔 by a gen- 퐴, the set 퐸 contains a point 푋 with dim퐴(푋) > 훼 − 휖. eralized notion assigning the value 2−푠|휎| to [휎] (where In several applications, this turns out to be eas- 0 < 푠 ≤ 1), was translated by Lutz (2000, 2003) into a ier than ones directly involving classical dimension. This notion of 푠-gale, where the fairness condition of a martin- fact is somewhat surprising given the need to relativize to gale is replaced by 푓(휎) = 2−푠(푓(휎0) + 푓(휎1)). We arbitrary oracles, but in practice this issue has so far turned can view 푠-gales as modeling betting in a hostile environ- out not to be an obstacle. ment (an idea due to Lutz), where “inflation” is acting so For example, Lutz and Stull [21] obtained a new lower that not winning means that we automatically lose money. bound on the Hausdorff dimension of generalized sets of Roughly speaking, the effective fractal dimension of a se- Furstenberg type; Lutz [20] showed that a fundamental in- quence is then determined by the most hostile environ- tersection formula, due in the Borel case to Kahane and ment in which we can still make money betting on this Mattila, is true for arbitrary sets; and Lutz and Lutz [19] sequence. gave a new proof of the two-dimensional case (originally Mayordomo (2002) and Athreya, Hitchcock, Lutz, and proved by Davies) of the well-known Kakeya conjecture, Mayordomo (2007) found equivalent formulations in which states that, for all 푛 ≥ 2, if a of ℝ푛 has lines terms of Kolmogorov complexity, which we take as defi- of length 1 in all directions, then it has Hausdorff dimen- nitions. (Here it does not matter whether we use plain or sion 푛. prefix-free Kolmogorov complexity.) There had been earlier applications of effective dimen- sion, for instance in symbolic dynamics, whose iterative Definition 13. Let 푋 ∈ 2휔. The effective Hausdorff dimen- processes are naturally algorithmic. For example, Simp- sion of 푋 is son [25] generalized a result of Furstenberg as follows. Let 퐾(푋 ↾ 푛) 푑 푑 퐺 dim(푋) = lim inf . 퐴 be finite and 퐺 be either ℕ or ℤ . A closed set 푋 ⊆ 퐴 푛→∞ 푛 is a subshift if it is closed under the shift action of 퐺 on 퐴퐺 The effective packing dimension of 푋 is (see the section on “Halting probabilities”). 퐾(푋 ↾ 푛) Theorem 15 (Simpson [25]). Let 퐴 be finite and 퐺 be either Dim(푋) = lim sup . 푑 푑 퐺 푛→∞ 푛 ℕ or ℤ . If 푋 ⊆ 퐴 is a subshift then the topological entropy 푋 It is not hard to extend these definitions to elements of of is equal both to its classical Hausdorff dimension and to the ℝ푛, yielding effective dimensions between 0 and 푛. They supremum of the effective Hausdorff dimensions of its elements. can also be relativized to any oracle 퐴 to obtain the ef- In currently unpublished work, Day has used effective 퐴 fective Hausdorff and packing dimensions dim (푋) and methods to give a new proof of the Kolmogorov-Sinai the- Dim퐴(푋) of 푋 relative to 퐴. orem on entropies of Bernoulli shifts. It is of course not immediately obvious why these no- There are other applications of sequences of high effec- tions are effectivizations of Hausdorff and packing dimen- tive dimension, for instance ones involving the interesting sion, but crucial evidence of their correctness is provided class of shift complex sequences. While initial segments of by point to set principles, which allow us to express the di- ML-random sequences have high Kolmogorov complexity, mensions of sets of reals in terms of the effective dimen- not all segments of such sequences do. Random sequences sions of their elements. The most recent and powerful of must contain arbitrarily long strings of consecutive 0’s, for these is the following, where we denote the classical Haus- example. However, for any 휖 > 0 there are 휖-shift complex

1008 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 sequences 푌 such that for any string 휎 of consecutive bits an 푋-computable 푌 such that Dim(푌) > 1 − 휖. (In fact, 푌 of 푌, we have 퐾(휎) ≥ (1 − 휖)|휎| − 푂(1). These se- can be taken to be equivalent to 푋 via polynomial-time reduc- quences can be used to create tilings with properties such tions.) as certain kinds of pattern-avoidance, and have found uses in symbolic dynamics. See for instance Durand, Levin, and For effective Hausdorff dimension, the situation is Shen (2008) and Durand, Romashchenko, and Shen [10]. quite different. Typically, the way we obtain an 푋 with 1 Randomness amplification. Many practical algorithms dim(푋) = 2 , say, is to start with an ML-random sequence use random seeds. For example, the important Polynomial and somehow “mess it up,” for example by making ev- Identity Testing (PIT) problem takes as input a polynomial ery other bit a 0. This kind of process is reversible, in 푃(푥1, … , 푥푛) with coefficients from a large finite field and the sense that it is easy to obtain an 푋-computable ML- determines whether it is identically 0. Many practical prob- random. However, Miller [23] showed that it is possible to lems can be solved using a to this problem. obtain sequences of fractional effective Hausdorff dimen- There is a natural fast algorithm to solve it randomly: Take sion that permit no randomness amplification at all. a random sequence of values for the variables. If the poly- nomial is not 0 on these values, “no” is the correct answer. Theorem 19 (Miller [23]). There is an 푋 such that dim(푋) = 1 1 Otherwise, the probability that the answer is “yes” is very 2 and if 푌 ≤T 푋 then dim(푌) ≤ 2 . high. It is conjectured that PIT has a polynomial-time de- terministic algorithm,4 but no such algorithm is known. That is, effective Hausdorff dimension cannot in general Thus it is important to have good sources of random- 1 be amplified. (In this theorem, the specific value 2 is only ness. Some (including Turing) have believed that random- an example.) Greenberg and Miller [13] also showed that ness can be obtained from physical sources, and there are there is an 푋 such that dim(푋) = 1 and 푋 does not com- now commercial devices claiming to do so. At a more the- pute any ML-random sequences. Interestingly, Zimand oretical level, we might ask questions such as: (2010) showed that for two sequences 푋 and 푌 of nonzero 1. Can a weak source of randomness always be amplified effective Hausdorff dimension that are in a certain techni- into a better one? cal sense sufficiently independent, 푋 and 푌 together can 2. Can we in fact always recover full randomness from compute a sequence of effective Hausdorff dimension 1. partial randomness? In some attractive recent work, it has been shown that 3. Are random sources truly useful as computational re- there is a sense in which the intuition that every sequence sources? of effective Hausdorff dimension 1 is close to an ML- In our context, we can consider precise versions of such random sequence is correct. The following is a simplified questions by taking randomness to mean algorithmic ran- version of the full statement, which quantifies how much domness, and taking all reduction processes to be com- randomness can be extracted at the cost of altering a se- putable ones. One way to interpret the first two questions quence on a set of density 0. Here 퐴 ⊆ ℕ has (asymptotic) |퐴↾푛| then is to think of partial randomness as having nonzero density 0 if lim푛→∞ 푛 = 0. effective dimension. For example, for packing dimension, we have the following negative results. Theorem 20 (Greenberg, Miller, Shen, and Westrick [14]). If dim(푋) = 1 then there is an ML-random 푌 such that Theorem 16 (Downey and Greenberg (2008)). There is an {푛 ∶ 푋(푛) ≠ 푌(푛)} has density 0. 푋 such that Dim(푋) = 1 and 푋 computes no ML-random sequence. (This 푋 can be built to be of minimal degree, which The third question above is whether sources of random- means that every 푋- is either computable or has ness can be useful oracles. Here we are thinking in terms of the same Turing degree as 푋. It is known that such an 푋 cannot complexity rather than just computability, so results such compute an ML-random sequence.) as Theorem 6 are not directly relevant. Allender and oth- Theorem 17 (Conidis [8])). There is an 푋 such that ers have initiated a program to investigate the speedups Dim(푋) > 0 and 푋 computes no 푌 with Dim(푌) = 1. that are possible when random sources are queried effi- On the other hand, we also have the following strong ciently. Let 푅 be the set of all random finite binary strings positive result. for either plain or prefix-free Kolmogorov complexity (e.g., 푅 = {푥 ∶ 퐶(푥) ≥ |푥|} 풞 풞푅 Theorem 18 (Fortnow, Hitchcock, Pavan, Vinochandran, ). For a , let 푅 and Wang (2006)). If 휖 > 0 and Dim(푋) > 0 then there is denote the relativization of this class to . So, for instance, for the class P of polynomial-time computable functions, 푅 4This conjecture comes from the fact that PIT belongs to a complexity class known as BPP, P is the class of functions that can be computed in poly- which is widely believed to equal the complexity class P of polynomial-time solvable prob- 푅 lems, since Impagliazzo and Wigderson showed in the late 1990s that if the well-known nomial time with as an oracle. (For references to the Satisfiability problem is as hard as generally believed, then indeed BPP = P. articles in this and the following theorem, see [1].)

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1009 Theorem 21 (Buhrman, Fortnow, Kouck´y, and Loff (2010); Theorem 24 (Brattka, Miller, and Nies [5], also Demuth Allender, Buhrman, Kouck´y, van Melkebeek, and Ronneb- (75, see [5]) for (2)). urger (2006); Allender, Buhrman, and Kouck´y(2006)). 1. The reals at which every computable increasing function 1. PSPACE ⊆ P푅. ℝ → ℝ is differentiable are exactly the computably random 2. NEXP ⊆ NP푅. ones. 푅 3. BPP ⊆ Ptt (where the latter is the class of functions 2. The reals at which every computable function ℝ → ℝ that are reducible to 푅 in polynomial time via truth-table of bounded variation is differentiable are exactly the ML- reductions, a more restrictive notion of reduction than Tur- random ones. ing reduction). Ergodic theory is another area that has been studied The choice of universal machine does have some effect from this point of view. A measure-preserving transfor- on efficient , but we can quantify over alluni- mation 푇 on a probability space is ergodic if all measur- versal machines. In the result below, 푈 ranges over uni- able 퐸 such that 푇−1(퐸) = 퐸 have measure 1 versal prefix-free machines, and 푅퐾푈 is the set of random or 0. Notice that this is an “almost everywhere” defini- strings relative to Kolmogorov complexity defined using tion. We can make this setting computable (and many sys- 푈. tems arising from physics will be computable). One way Theorem 22 (Allender, Friedman, and Gasarch (2013); to proceed is to work in Cantor space without loss of gen- Cai, Downey, Epstein, Lempp, and Miller (2014)). erality, since Hoyrup and Rojas [16] showed that any com- putable metric space with a computable probability mea- 푅퐾푈 1. ⋂푈 Ptt ⊆ PSPACE. sure is isomorphic to this space in an effective measure- 푅퐾 2. ⋂푈 NP 푈 ⊆ EXPSPACE. theoretic sense. Then we can specify a computable trans- We can also say that sufficiently random oracles will al- formation 푇 as a computable of computable partial <휔 <휔 ways accelerate some computations in the following sense. maps 푇푛 ∶ 2 → 2 with certain coherence condi- Say that 푋 is low for speed if for any computable set 퐴 tions. We can also transfer definitions like that of ML- and any function 푡 such that 퐴 can be computed in time randomness to computable probability spaces other than 푡(푛) using 푋 as an oracle, there is a polynomial 푝 such Cantor space. that 퐴 can be computed (with no oracle) in time bounded The following is an illustrative result. A classic theorem by 푝(푡(푛)). That is, 푋 does not significantly accelerate of Poincar´eis that if 푇 is measure-preserving, then for all 퐸 푛 any computation of a computable set. Bayer and Slaman of positive measure and almost all 푥, we have 푇 (푥) ∈ 퐸 (see [3]) constructed noncomputable sets that are low for for infinitely many 푛. For a class 풞 of measurable subsets, speed, but these cannot be very random. 푥 is a Poincar´epoint for 푇 with respect to 풞 if for every 퐸 ∈ 풞 of positive measure, 푇푛(푥) ∈ 퐸 for infinitely many Theorem 23 (Bienvenu and Downey [3]). If 푋 is Schnorr 푛. An effectively closed set is one whose complement can be random, then it is not low for speed, and this fact is witnessed specified as a computably enumerable union of basic open by an exponential-time computable set 퐴. sets. Analysis and ergodic theory. is an Theorem 25 (Bienvenu, Day, Mezhirov, and Shen [2]). Let area that has developed tools for thinking about comput- 푇 be a computable ergodic transformation on a computable ability of objects like real-valued functions by taking ad- probability space. Every ML-random of this space is vantage of separability. Say that a sequence of rationals −푛 a Poincar´epoint for the class of effectively closed sets. 푞0, 푞1,… converges fast to 푥 if |푥 − 푞푛| ≤ 2 for all 푛. A function 푓 ∶ ℝ → ℝ is (Type 2) computable if there is In general, the condition that the element be ML- an algorithm Φ that, for every 푥 ∈ ℝ and every sequence random is not just sufficient but necessary, even in asim- 푞0, 푞1,… that converges fast to 푥, if Φ is given 푞0, 푞1,… as ple case like the shift operator on Cantor space. an oracle, then it can compute a sequence that converges The non-ergodic case has also been analyzed, by Frank- fast to 푓(푥). We can extend this definition to similar sep- lin and Towsner [12], who also studied the Birkhoff er- arable spaces. We can also relativize it, and it is then not godic theorem. In these and several other cases, similar difficult to see that a function is continuous iff itiscom- correspondences with various notions of algorithmic ran- putable relative to some oracle, basically because to define domness have been found. While many theorems of er- a continuous function we need only to specify its action godic theory have been analyzed in this way, including the on a countable collection of balls. Birkhoff, Poincar´e, and von Neumann ergodic theorems, Mathematics is replete with results concerning almost some, like Furstenberg’s ergodic theorem, have yet to be everywhere behavior, and algorithmic randomness allows understood from this point of view. us to turn such results into “quantitative” ones like the fol- Regarding the physical of some of the lowing. work in this area, Braverman, Grigo, and Rojas [6] have

1010 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 obtained results that they argue show that, while random References noise makes predicting the short-term behavior of a sys- [1] Allender E. The complexity of complexity. Computability tem difficult, it may in fact allow prediction to be easier in and complexity: Springer, Cham; 2017:79–94. MR3629714 the long term. [2] Bienvenu L, Day A, Mezhirov I, Shen A. Ergodic-type char- acterizations of algorithmic randomness. Programs, proofs, Normality revisited. Borel’s notion of normality, with processes: Springer, Berlin; 2010:49–58. https://doi. which we began our discussion, is a very weak kind of org/10.1007/978-3-642-13962-8_6 MR2678114 randomness. Polynomial-time randomness implies abso- [3] Bienvenu L, Downey R. On low for speed oracles. lute normality, and Schnorr and Stimm (1971/72) showed 35th Symposium on Theoretical Aspects of Sci- that a sequence is normal to a given base iff it satisfies a ence: Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern; martingale-based notion of randomness defined using cer- 2018:15:1–15:13. MR3779296 tain finite state automata, a much weaker model of compu- [4] Bienvenu L, Romashchenko A, Shen A, Taveneaux A, Ver- tation than Turing machines. Building examples of abso- meeren S. The axiomatic power of Kolmogorov complexity, lutely normal numbers is another matter, as Borel already Ann. Pure Appl. Logic, no. 9 (165):1380–1402, 2014, DOI 10.1016/j.apal.2014.04.009. MR3210074 noted. While it is conjectured that 푒, 휋, and all irrational [5] Brattka V, Miller JS, Nies A. Randomness and differentia- algebraic numbers such as √2 are absolutely normal, none bility, Trans. Amer. Math. Soc., no. 1 (368):581–605, 2016. of these have been proved to be normal to any base. In https://doi.org/10.1090/tran/6484. MR3413875 his unpublished manuscript “A note on normal numbers,” [6] Braverman M, Grigo A, Rojas C. Noise vs computational believed to have been written in 1938, Turing built a com- intractability in dynamics. Proceedings of the 3rd Innovations putable absolutely normal real, which is in a sense the clos- in Theoretical Conference: ACM, New York; est we have come so far to obtaining an explicitly-described 2012:128–141. MR3388383 absolutely normal real. (His construction was not pub- [7] Braverman M, Yampolsky M. Computability of Julia lished until his Collected Works in 1992, and there was sets, Mosc. Math. J., no. 2 (8):185–231, 399, 2008, DOI 10.17323/1609-4514-2008-8-2-185-231 (English, with uncertainty as to its correctness until Becher, Figueira, and English and Russian summaries). MR2462435 Picchi (2007) reconstructed and completed it, correcting [8] Conidis CJ. A real of strictly positive effective packing di- 5 minor errors. ) There is a sense in which Turing antici- mension that does not compute a real of effective pack- pated Martin-Löf’s idea of looking at a large collection of ing dimension one, J. Symbolic Logic, no. 2 (77):447–474, effective tests, in this case ones sufficiently strong to ensure 2012, DOI 10.2178/jsl/1333566632. MR2963016 that a real is normal for all bases, but sufficiently weak to al- [9] Downey RG, Hirschfeldt DR. Algorithmic randomness and low some computable sequence to pass them all. He took complexity, Theory and Applications of Computability, advantage of the correlations between blocks of digits in Springer, New York, 2010. MR2732288 expansions of the same real in different bases. [10] Durand B, Romashchenko A, Shen A. Fixed-point tile sets and their applications, J. Comput. System Sci., This approach can also be thought of in terms of effec- no. 3 (78):731–764, 2012, DOI 10.1016/j.jcss.2011.11.001. tive martingales, and its point of view has brought about MR2900032 a great deal of progress in our understanding of normality [11] Franklin JNY, Ng KM. Difference randomness, Proc. recently. For instance, Becher, Heiber, and Slaman (2013) Amer. Math. Soc., no. 1 (139):345–360, 2011, DOI showed that absolutely normal numbers can be 10.1090/S0002-9939-2010-10513-0. MR2729096 constructed in low-level polynomial time, and Lutz and [12] Franklin JNY, Towsner H. Randomness and non-ergodic Mayordomo (arXiv:1611.05911) constructed them in systems, Mosc. Math. J., no. 4 (14):711–744, 827, 2014, “nearly linear” time. Much of the work along these lines DOI 10.17323/1609-4514-2014-14-4-711-744 (English, has been number-theoretic, connected to various notions with English and Russian summaries). MR3292047 [13] Greenberg N, Miller JS. Diagonally non-recursive func- of well-approximability of irrational reals, such as that of tions and effective Hausdorff dimension, Bull. Lond. Math. a Liouville number, which is an irrational 훼 such that for Soc., no. 4 (43):636–654, 2011. https://doi.org/10. every 푛 > 1, there are 푝, 푞 ∈ ℕ for which 1112/blms/bdr003. MR2820150 푝 −푛 |훼 − 푞 | < 푞 . For example, Becher, Heiber, and Sla- [14] Greenberg N, Miller JS, Shen A, Westrick LB. Dimen- man (2015) have constructed computable absolutely nor- sion 1 sequences are close to randoms, Theoret. Comput. mal Liouville numbers. This work has also produced Sci. (705):99–112, 2018, DOI 10.1016/j.tcs.2017.09.031. results in the classical theory of normal numbers, for in- MR3721461 stance by Becher, Bugeaud, and Slaman (2016). [15] Hochman M, Meyerovitch T. A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), no. 3 (171):2011–2038, 2010, DOI 10.4007/an- nals.2010.171.2011. MR2680402 [16] Hoyrup M, Rojas C. Computability of probability mea- sures and Martin-Löf randomness over metric spaces, In- 5See https://www-2.dc.uba.ar/staff/becher/publications.html for references to the papers cited here and below.

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1011 form. and Comput., no. 7 (207):830–847, 2009, DOI FEATURED TITLE FROM 10.1016/j.ic.2008.12.009. MR2519075 [17] Kritchman S, Raz R. The surprise examination paradox and the second incompleteness theorem, Notices Amer. Math. Soc., no. 11 (57):1454–1458, 2010. MR2761888 [18] Li M, Vit´anyi P. An introduction to Kolmogorov complex- ity and its applications, Third, Texts in Computer Science, Inequalities: Springer, New York, 2008. https://doi.org/10.1007/ An Approach 978-0-387-49820-1 MR2494387 through Problems [19] Lutz JH, Lutz N. Algorithmic information, plane Kakeya Second Edition sets, and conditional dimension, ACM Trans. Comput. The- B. J. Venkatachala, Homi Bhabha ory, no. 2 (10):Art. 7, 22, 2018, DOI 10.1145/3201783. Centre for Science Education, Mumbai, MR3811993 India [20] Lutz N. Fractal intersections and products via algorith- This book is an introduction to basic mic dimension (83):Art. No. 58, 12, 2017. MR3755351 topics in inequalities and their appli- cations. These include the arithmetic mean-geometric mean [21] Lutz N, Stull DM. Bounding the dimension of points inequality, Cauchy–Schwarz inequality, Chebyshev inequality, on a line. Theory and applications of models of computation: rearrangement inequality, convex and concave functions and Springer, Cham; 2017:425–439. MR3655512 Muirhead’s theorem. More than 400 problems are included in the book and their solutions are explained. A chapter on geometric [22] Messner J, Thierauf T. A Kolmogorov complexity proof inequalities is a special feature of this book. Most of these prob- of the Lov´aszlocal lemma for satisfiability, Theoret. Comput. lems are from International Mathematical Olympiads and from Sci. (461):55–64, 2012, DOI 10.1016/j.tcs.2012.06.005. many national Mathematical Olympiads. MR2988089 The book is intended to help students who are preparing for var- [23] Miller JS. Extracting information is hard: a Tur- ious mathematical competitions. It is also a good sourcebook for graduate students in consolidating their knowledge of inequalities ing degree of non-integral effective Hausdorff dimen- and their applications. sion, Adv. Math., no. 1 (226):373–384, 2011, DOI Hindustan Book Agency; 2018; 532 pages; Softcover; ISBN: 10.1016/j.aim.2010.06.024. MR2735764 978-93-86279-68-2; List US$72; AMS members US$57.60; Order [24] Nies A. Computability and randomness, Oxford Logic code HIN/75 Guides, vol. 51, Oxford University Press, Oxford, 2009. MR2548883 Titles published by the Hindustan Book Agency [25] Simpson SG. Symbolic dynamics: entropy = dimension (New Delhi, India) include studies in advanced = complexity, Theory Comput. Syst., no. 3 (56):527–543, mathematics, monographs, lecture notes, and/or 2015, DOI 10.1007/s00224-014-9546-8. MR3334259 conference proceedings on current topics of interest.

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Rod Downey Denis R. Hirschfeldt

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1012 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7