Effective randomness and computability
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- Franklin Parks
- 6 years ago
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1 University of Wisconsin October 2009
2 What is this about? Let s begin by examining the title: Effective randomness (from algorithmic point of view) Computability (study of the limits of algorithms)
3 Algorithms Etymology: Al-Khwā-rizmī, Persian astronomer and mathematician. He wrote a treatise in 825 AD, On Calculation with Hindu Numerals" The Latin translation is Algoritmi de numero Indorum" There is no generally accepted formal definition of "algorithm" What we intuitively mean is there is a mechanical procedure (devoid of intelligence), and gives the desired result after a finite number of steps. Notice the word finite". I will try and give an overview of the subject, and talk about some of my own work.
4 Algorithms Etymology: Al-Khwā-rizmī, Persian astronomer and mathematician. He wrote a treatise in 825 AD, On Calculation with Hindu Numerals" The Latin translation is Algoritmi de numero Indorum" There is no generally accepted formal definition of "algorithm" What we intuitively mean is there is a mechanical procedure (devoid of intelligence), and gives the desired result after a finite number of steps. Notice the word finite". I will try and give an overview of the subject, and talk about some of my own work.
5 Algorithms Etymology: Al-Khwā-rizmī, Persian astronomer and mathematician. He wrote a treatise in 825 AD, On Calculation with Hindu Numerals" The Latin translation is Algoritmi de numero Indorum" There is no generally accepted formal definition of "algorithm" What we intuitively mean is there is a mechanical procedure (devoid of intelligence), and gives the desired result after a finite number of steps. Notice the word finite". I will try and give an overview of the subject, and talk about some of my own work.
6 Algorithms
7 Algorithms
8 Algorithms In these cases you specify an input, or set of ingredients. The algorithm applies a mechanical method to get the desired result. Euclid s algorithm for finding greatest common divisor: Input: a pair of numbers (1001,357) = = = = 7 10 Output: the gcd(1001,357)= 7
9 Algorithms In these cases you specify an input, or set of ingredients. The algorithm applies a mechanical method to get the desired result. Euclid s algorithm for finding greatest common divisor: Input: a pair of numbers (1001,357) = = = = 7 10 Output: the gcd(1001,357)= 7
10 Into the 20 th century David Hilbert had a grand plan to finitely mechanize" all of mathematics. Based on the idea that in mathematics there should be no "ignorabimus" (statement that the truth can never be known), A machine, which you can feed Input: a statement about mathematics Process: the machine uses a reasonable formal system to generate proofs" Output: True or False. Equivalently, can you have a mechanical procedure that enumerates" all the truths in a system (e.g. number theory)?
11 Into the 20 th century David Hilbert had a grand plan to finitely mechanize" all of mathematics. Based on the idea that in mathematics there should be no "ignorabimus" (statement that the truth can never be known), A machine, which you can feed Input: a statement about mathematics Process: the machine uses a reasonable formal system to generate proofs" Output: True or False. Equivalently, can you have a mechanical procedure that enumerates" all the truths in a system (e.g. number theory)?
12 Into the 20 th century
13 Into the 20 th century Gödel proved his two famous Incompleteness Theorems. First Incompleteness Theorem: Any sufficiently strong formal system of axioms has a statement P for which neither P nor P can be proven. Furthermore if you add P to the system, there will still be another statement P independent from the augmented system. Second Incompleteness Theorem: Any such system of axioms cannot prove the statement I am consistent", unless it is itself inconsistent. The collective intuition of generations of mathematicians were wrong.
14 Into the 20 th century Gödel proved his two famous Incompleteness Theorems. First Incompleteness Theorem: Any sufficiently strong formal system of axioms has a statement P for which neither P nor P can be proven. Furthermore if you add P to the system, there will still be another statement P independent from the augmented system. Second Incompleteness Theorem: Any such system of axioms cannot prove the statement I am consistent", unless it is itself inconsistent. The collective intuition of generations of mathematicians were wrong.
15 Into the 20 th century
16 Independence Some systems are decidable. For example, real closed fields, Euclidean geometry. Alfred Tarski and quantifier elimination. Gödel s example was artificial. Are there statements which matter in working mathematics? Yes... from Peano s arithmetic (PA).
17 Independence Classically Kruskal s Tree Theorem states that the set of finite trees under homeomorphic embedding is a well-quasi-ordering (i.e. no infinite anti-chain). Friedman noted a special case of this is independent of PA. For all n there is a k so large such that if {T i : i < k} are finite trees such that T i = n + i, then there is a pair where T i embeds into T j. We can state this in PA but you need very strong induction (beyond PA) to prove it.
18 Independence Another example is the Goodstein sequence" shown by Kirby and Paris to be independent of PA. This was the analogy given by Kirby and Paris: The "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. The theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time.
19 Independence Another example is the Goodstein sequence" shown by Kirby and Paris to be independent of PA. Start with a number say 4 and write it in base 2: 2 2 = 4 Replace the base 2 with 3 and subtract 1: = = 26 Replace base 3 with 4 and subtract 1: = 41 Amazingly, every such sequence converges to 0!
20 Independence Another example is the Goodstein sequence" shown by Kirby and Paris to be independent of PA. Start with a number say 4 and write it in base 2: 2 2 = 4 Replace the base 2 with 3 and subtract 1: = = 26 Replace base 3 with 4 and subtract 1: = 41 Amazingly, every such sequence converges to 0!
21 Independence Another example is the Goodstein sequence" shown by Kirby and Paris to be independent of PA. Start with a number say 4 and write it in base 2: 2 2 = 4 Replace the base 2 with 3 and subtract 1: = = 26 Replace base 3 with 4 and subtract 1: = 41 Amazingly, every such sequence converges to 0!
22 Independence Another example is the Goodstein sequence" shown by Kirby and Paris to be independent of PA. Start with a number say 4 and write it in base 2: 2 2 = 4 Replace the base 2 with 3 and subtract 1: = = 26 Replace base 3 with 4 and subtract 1: = 41 Amazingly, every such sequence converges to 0!
23 Independence Kirby-Paris showed this cannot be proved nor refuted from PA. Need transfinite induction. During this time, various people were working on formalizing what we mean by algorithms" and mechanical method"... There were several notable models..
24 Formalizing computations Stephen C. Kleene µ-recursive functions Alonzo Church λ calculus
25 Formalizing computations Alan Turing Turing machines A Turing machine
26 Formalizing computations
27 Formalizing computations The fact that these models were all the same, lent support to the Church-Turing thesis: All mechanical and intuitively computable processes can be simulated on a Turing machine." Turing worked on code breaking during WW2 (Enigma machine) His fundamental paper was part of the inspiration for the first computers.
28 Formalizing computations Enigma A Turing machine made of Lego
29 Formalizing computations Small-Scale Experimental Machine, known as "Baby". University of Manchester, June 21st It was the first machine that could store data electronically.
30 Formalizing computations Not every set of natural numbers was computable, the most notable example is the Halting problem. Input: A pair of numbers e, x. Output: To tell whether the e th TM on input x halts. This is not computable. Code this set into others, to get other non-computable sets.
31 Formalizing computations Hilbert s tenth problem. Input: A polynomial p(x 1,, x n ) with integer coefficients. Output: To tell if p(x 1,, x n ) = 0 has integer solutions. There is no computable process to decide such problems (Matiyasevich, after Julia Robinson). Recently Braverman and Yampolsky showed that Julia sets can be non-computable, by coding the Halting problem. Many mathematical objects can be coded and used to simulate computations.
32 Formalizing computations Hilbert s tenth problem. Input: A polynomial p(x 1,, x n ) with integer coefficients. Output: To tell if p(x 1,, x n ) = 0 has integer solutions. There is no computable process to decide such problems (Matiyasevich, after Julia Robinson). Recently Braverman and Yampolsky showed that Julia sets can be non-computable, by coding the Halting problem. Many mathematical objects can be coded and used to simulate computations.
33 Measuring unsolvability We want to measure how impossible it is to compute a set, relative to other sets. Give two sets A, B N, we say A T B if whenever given a way to solve B, we have a way of solving A. The equivalence classes are called Turing degrees. A degree is computably enumerable (c.e.) if it contains the Halting set of some machine.
34 Measuring unsolvability Since there are only countably many ways of measuring relative unsolvability, there are continuum many Turing degrees. Structurally the Turing degrees form an upper-semilattice with minimal elements (Spector, Sacks). The c.e. Turing degrees were at some point very well studied. A motivation for looking at c.e. degrees are related to decidability of theories and formal systems.
35 C.e. degrees (Post s problem) Are there any c.e. Turing degree a which is not computable, yet does not compute the Halting problem? Friedberg, Muchnik (1956) developed a new important technique, the priority method. The structure of c.e. degrees is dense (Sacks 1962), and in fact much more complicated than originally thought. Recently Downey, Hirschfeldt, Nies, Stephan proved a priority-free, natural solution to Post s problem. From effective randomness
36 Feebleness If you relativize the construction of the Halting problem to a set X, you get the jump operator taking X X. In an influential paper of Kleene and Post, they asked about the range of this operator. Work of Friedberg, Shoenfield and Sacks collectively showed the range is the largest possible (even on restricted domains). Fundamental operator.
37 Feebleness The jump operator gives a way of measuring the computational feebleness of a set. A set A is low if A T - useless as an oracle. Recent work of Downey, Hirschfeldt, Nies have shown nice relationships of low sets with effective randomness.
38 What makes a string random? A real is a member of Cantor space 2 N with topology with basic clopen sets [σ] = {σα : α 2 ω }. Its measure is µ([σ]) = 2 σ. Strings = members of 2 <N = {0, 1}. We want to try and see which infinite binary strings are random.
39 What makes a string random? Which of the following sequences seem random? Not random: Sequence of zeroes A B C D E F
40 What makes a string random? Which of the following sequences seem random? A B C D E F
41 What makes a string random? Which of the following sequences seem random? Not random: 001 and 101 repeated A B C D E F
42 What makes a string random? Which of the following sequences seem random? A B C D E F
43 What makes a string random? Which of the following sequences seem random? Random: Sequence from random coin tosses A B C D E F
44 What makes a string random? Which of the following sequences seem random? A B C D E F
45 What makes a string random? Which of the following sequences seem random? Not random: 0,1,2,3,4,5 in binary A B C D E F
46 What makes a string random? Which of the following sequences seem random? A B C D E F
47 What makes a string random? Which of the following sequences seem random? Semi-random: Odd digits 0, Even digits coin tosses A B C D E F
48 What makes a string random? Which of the following sequences seem random? A B C D E F
49 What makes a string random? Which of the following sequences seem random? Not-random: Binary expansion of every other digit of π A B C D E F
50 What makes a string random? In many of these cases, the string is non-random because we can easily describe it / predict the next digit. In terms of probability and measure theory, these are all equally likely. No single element of the sample space can be random.. but how do we separate them? The first attempt was made by statistician von Mises 1919: to have an acceptable selection rule that generalizes the weak law of large numbers. If α = 0.a 0 a 1 a 2, then whenever we select a subsequence via the selection rule, the number of n where a f (n) = 1 should be asymptotically 1 2.
51 What makes a string random? In many of these cases, the string is non-random because we can easily describe it / predict the next digit. In terms of probability and measure theory, these are all equally likely. No single element of the sample space can be random.. but how do we separate them? The first attempt was made by statistician von Mises 1919: to have an acceptable selection rule that generalizes the weak law of large numbers. If α = 0.a 0 a 1 a 2, then whenever we select a subsequence via the selection rule, the number of n where a f (n) = 1 should be asymptotically 1 2.
52 What makes a string random? What would these acceptable selection rules be...? With the development of computability, Church linked these rules to computable functions. Take all computable stochastic properties. As pointed out by Ville, this was not good enough. He showed there were reals which passed all such selection rules, yet look inuitively non-random. Eventually Martin-Löf hit upon the idea of using effectively presented sets of Lebesgue measure 0, called Martin-Löf tests. A real is ML-random if it does not belong to any of these effectively presented statistical tests.
53 What makes a string random? What would these acceptable selection rules be...? With the development of computability, Church linked these rules to computable functions. Take all computable stochastic properties. As pointed out by Ville, this was not good enough. He showed there were reals which passed all such selection rules, yet look inuitively non-random. Eventually Martin-Löf hit upon the idea of using effectively presented sets of Lebesgue measure 0, called Martin-Löf tests. A real is ML-random if it does not belong to any of these effectively presented statistical tests.
54 What makes a string random? There are three main approaches to defining algorithmic randomness. We just gave the first: (I) Statistician s approach: A ML-random real possesses no algorithmically distinguishable trait (e.g ). Deals with rare patterns using measure theory. (II) Coder s approach: Since algorithmically distinguishable traits can be used to compress information, a random string should be incompressible. Eg. Text file can be zipped up to 50%, but a Jpeg file can hardly be compressed.
55 What makes a string random? There are three main approaches to defining algorithmic randomness. We just gave the first: (I) Statistician s approach: A ML-random real possesses no algorithmically distinguishable trait (e.g ). Deals with rare patterns using measure theory. (II) Coder s approach: Since algorithmically distinguishable traits can be used to compress information, a random string should be incompressible. Eg. Text file can be zipped up to 50%, but a Jpeg file can hardly be compressed.
56 What makes a string random? There are three main approaches to defining algorithmic randomness. We just gave the first: (I) Statistician s approach: A ML-random real possesses no algorithmically distinguishable trait (e.g ). Deals with rare patterns using measure theory. (II) Coder s approach: Since algorithmically distinguishable traits can be used to compress information, a random string should be incompressible. Eg. Text file can be zipped up to 50%, but a Jpeg file can hardly be compressed.
57 (II): The coder s approach Another example: We need to know (i) the pattern 01 (ii) the length 20 (log 20 bits) altogether 2 + log 20 bits 20. To output (sequence of coin tosses), we need to hardwire it into our system.
58 (II): The coder s approach Another example: We need to know (i) the pattern 01 (ii) the length 20 (log 20 bits) altogether 2 + log 20 bits 20. To output (sequence of coin tosses), we need to hardwire it into our system.
59 (II): The coder s approach Mandelbrot set fractal Storing the colours in each pixel requires 1.62 million bits Storing its generating program requires much less resources
60 (II): The coder s approach To formalize this, take a fixed machine M, and define: The plain complexity C(σ) of a string σ 2 <N is the length of the shortest τ where M(τ) converges and gives σ (due to Kolmogorov). Kolmogorov showed that universal machines exist, i.e. a machine U such that for every other machine M, C U (σ) C M (σ) + O(1). You might be tempted to say that a real X is random if for every n, C(X n) n + O(1), where X n denotes the first n bits of X.
61 (II): The coder s approach To formalize this, take a fixed machine M, and define: The plain complexity C(σ) of a string σ 2 <N is the length of the shortest τ where M(τ) converges and gives σ (due to Kolmogorov). Kolmogorov showed that universal machines exist, i.e. a machine U such that for every other machine M, C U (σ) C M (σ) + O(1). You might be tempted to say that a real X is random if for every n, C(X n) n + O(1), where X n denotes the first n bits of X.
62 (II): The coder s approach Unfortunately, Theorem (Martin-Löf ) There is no real X such that C(X n) n + O(1) for every n. The trick is to observe that any finite binary string is associated with a number - number off 2 <N from left to right. Eg 0 1, 1 2, 00 3, 01 4, 10 5, etc
63 (II): The coder s approach Unfortunately, Theorem (Martin-Löf ) There is no real X such that C(X n) n + O(1) for every n. The trick is to observe that any finite binary string is associated with a number - number off 2 <N from left to right. Eg 0 1, 1 2, 00 3, 01 4, 10 5, etc
64 (II): The coder s approach Consider the machine M that does the following. M takes the string σ and computes the string τ where τ is the σ th string to be named. It then outputs τσ. For any n, we can look at the number k where α n is the k th string to be named. Then let τ be the next k bits of X. I.e. τ = X(n + 1)X(n + 2) X(n + m). Then M(τ) = (X n)τ = X n + m. Hence C(X n + m) m + O(1). Infinitely many segments of X can be compressed.
65 (II): The coder s approach Consider the machine M that does the following. M takes the string σ and computes the string τ where τ is the σ th string to be named. It then outputs τσ. For any n, we can look at the number k where α n is the k th string to be named. Then let τ be the next k bits of X. I.e. τ = X(n + 1)X(n + 2) X(n + m). Then M(τ) = (X n)τ = X n + m. Hence C(X n + m) m + O(1). Infinitely many segments of X can be compressed.
66 (II): The coder s approach Consider the machine M that does the following. M takes the string σ and computes the string τ where τ is the σ th string to be named. It then outputs τσ. For any n, we can look at the number k where α n is the k th string to be named. Then let τ be the next k bits of X. I.e. τ = X(n + 1)X(n + 2) X(n + m). Then M(τ) = (X n)τ = X n + m. Hence C(X n + m) m + O(1). Infinitely many segments of X can be compressed.
67 (II): The coder s approach Under this system, the length of a string is used to give extra non-trivial information. To avoid this, (Chaitin, Levin, Schnorr) looked at specific machines where the domain is an anti-chain under string extension. The prefixfree complexity K (σ) is the length of the shortest string τ where M(τ) = σ and M is a universal prefix free machine. Then there are reals X such that K (X n) n + O(1), formalizing the notion of random in terms of incompressible. Schnorr showed that approaches I and II were the same!
68 (II): The coder s approach Under this system, the length of a string is used to give extra non-trivial information. To avoid this, (Chaitin, Levin, Schnorr) looked at specific machines where the domain is an anti-chain under string extension. The prefixfree complexity K (σ) is the length of the shortest string τ where M(τ) = σ and M is a universal prefix free machine. Then there are reals X such that K (X n) n + O(1), formalizing the notion of random in terms of incompressible. Schnorr showed that approaches I and II were the same!
69 (III): The gambler s approach The third approach to calibrating randomness is through the intuition that you should not be able to make arbitrarily much money when trying to predict the digits of a random string. Suppose you walk into a casino, with a certain amount of money (say $10). The manager has in his pocket the digits of a real X (which you don t know, naturally) At the n th round, you are given X n. You have to try and guess X(n), the next digit. You decide a weight p 2. Assign p to X(n) = 0 and 2 p to X(n) = 1
70 (III): The gambler s approach The third approach to calibrating randomness is through the intuition that you should not be able to make arbitrarily much money when trying to predict the digits of a random string. Suppose you walk into a casino, with a certain amount of money (say $10). The manager has in his pocket the digits of a real X (which you don t know, naturally) At the n th round, you are given X n. You have to try and guess X(n), the next digit. You decide a weight p 2. Assign p to X(n) = 0 and 2 p to X(n) = 1
71 (III): The gambler s approach The third approach to calibrating randomness is through the intuition that you should not be able to make arbitrarily much money when trying to predict the digits of a random string. Suppose you walk into a casino, with a certain amount of money (say $10). The manager has in his pocket the digits of a real X (which you don t know, naturally) At the n th round, you are given X n. You have to try and guess X(n), the next digit. You decide a weight p 2. Assign p to X(n) = 0 and 2 p to X(n) = 1
72 (III): The gambler s approach The manager then reveals the next digit X(n) to you. Your capital C n is pc n 1 if X(n) = 0 and (2 p)c n 1 if X(n) = 1, where C n is your stage n capital. You win if in the limit, if your capital. A real X is random if you cannot win against X using only certain effective betting strategies. For example it is easy to win against the sequence (Schnorr) The gambler s approach III give exactly the same class as I and II.
73 (III): The gambler s approach The manager then reveals the next digit X(n) to you. Your capital C n is pc n 1 if X(n) = 0 and (2 p)c n 1 if X(n) = 1, where C n is your stage n capital. You win if in the limit, if your capital. A real X is random if you cannot win against X using only certain effective betting strategies. For example it is easy to win against the sequence (Schnorr) The gambler s approach III give exactly the same class as I and II.
74 (III): The gambler s approach The manager then reveals the next digit X(n) to you. Your capital C n is pc n 1 if X(n) = 0 and (2 p)c n 1 if X(n) = 1, where C n is your stage n capital. You win if in the limit, if your capital. A real X is random if you cannot win against X using only certain effective betting strategies. For example it is easy to win against the sequence (Schnorr) The gambler s approach III give exactly the same class as I and II.
75 An example of a ML-random real The most famous example is Ω = µ(domu) = U(σ) 2 σ, where U is the universal prefixfree machine. Ω is a left-c.e. real, i.e. there is a computable increasing sequence of rationals q 0 < q 1 < q 2 Ω In fact any left-c.e. random real is µ(domm) for some prefixfree machine M. Analogues of c.e. sets.
76 An example of a ML-random real Theorem (Chaitin) Ω is ML-random. Sketch of proof. We can look at the stage s approximation of Ω s. We build a prefixfree machine M. Whenever we see K s (Ω s n) drops below n O(1), then we make M(τ) converge on some τ of length roughly n. This has to be reflected by Ω = µ(dom(u)) and U is universal. So, Ω s n Ω n has to increase.
77 Class of random reals There are lots of ML-random reals, and has measure 1. They are all contained in a Σ 0 2 class {X : c n K (X n) > n c} Hence there are randoms with low Turing degree, and hyperimmune-free degree. On the other hand they combine nicely with the jump operator: (Kučera) The class of ML-random have all possible jumps (Downey, Miller) The class of ML-random computable from the Halting problem,, also have all possible jumps
78 Class of random reals There are lots of ML-random reals, and has measure 1. They are all contained in a Σ 0 2 class {X : c n K (X n) > n c} Hence there are randoms with low Turing degree, and hyperimmune-free degree. On the other hand they combine nicely with the jump operator: (Kučera) The class of ML-random have all possible jumps (Downey, Miller) The class of ML-random computable from the Halting problem,, also have all possible jumps
79 My work What other ways are there of measuring randomness? What level of randomness is needed for different applications? How does one measure relative randomness? What are the ways to extend the concept of feebleness to randomness notions? What do they have to do with feebleness in computability theory? How do randomness and computability interact? Must random reals be computationally powerful?
80 The computational strength of random reals We expect not to be able to effectively extract a lot of coherent data from a random real. E.g. How do we effectively (mechanically) extract useful data out of random coin tosses? So, random reals should not be computationally powerful, in terms of classical computability theoretic notions. Unfortunately Kučera and Gács proved that any real X 2 N can be computed from a ML-random real R (i.e. X T R). ML-randoms can contain as much non-trivial information as we want!
81 The computational strength of random reals We expect not to be able to effectively extract a lot of coherent data from a random real. E.g. How do we effectively (mechanically) extract useful data out of random coin tosses? So, random reals should not be computationally powerful, in terms of classical computability theoretic notions. Unfortunately Kučera and Gács proved that any real X 2 N can be computed from a ML-random real R (i.e. X T R). ML-randoms can contain as much non-trivial information as we want!
82 The computational strength of random reals We expect not to be able to effectively extract a lot of coherent data from a random real. E.g. How do we effectively (mechanically) extract useful data out of random coin tosses? So, random reals should not be computationally powerful, in terms of classical computability theoretic notions. Unfortunately Kučera and Gács proved that any real X 2 N can be computed from a ML-random real R (i.e. X T R). ML-randoms can contain as much non-trivial information as we want!
83 The computational strength of random reals A very closely related notion... what s the easiest way of constructing a function f which is computationally non-trivial? List out all the Turing machines M 0, M 1, M 2 and let ϕ 0, ϕ 1, ϕ 2, be the functions simulated by the TMs. Define f to be different from each ϕ e, i.e. f (x e ) ϕ e (x e ) for some x e. A function f : N N is diagonally non-computable (d.n.c.) if for every e, f (e) ϕ e (e).
84 The computational strength of random reals A very closely related notion... what s the easiest way of constructing a function f which is computationally non-trivial? List out all the Turing machines M 0, M 1, M 2 and let ϕ 0, ϕ 1, ϕ 2, be the functions simulated by the TMs. Define f to be different from each ϕ e, i.e. f (x e ) ϕ e (x e ) for some x e. A function f : N N is diagonally non-computable (d.n.c.) if for every e, f (e) ϕ e (e).
85 The computational strength of random reals A d.n.c. function is non-computable, and is frequently much more than that. E.g. if a d.n.c. function f is of c.e. degree, then it computes the Halting problem f T. On the other hand, there are d.n.c. functions of low Turing degree. Longstanding question: Must a d.n.c. function always be strictly more than being non-computable? (Kumabe) There is a d.n.c. function f such that f computes nothing else other than itself and the computable sets.
86 The computational strength of random reals A d.n.c. function is non-computable, and is frequently much more than that. E.g. if a d.n.c. function f is of c.e. degree, then it computes the Halting problem f T. On the other hand, there are d.n.c. functions of low Turing degree. Longstanding question: Must a d.n.c. function always be strictly more than being non-computable? (Kumabe) There is a d.n.c. function f such that f computes nothing else other than itself and the computable sets.
87 The computational strength of random reals A d.n.c. function is non-computable, and is frequently much more than that. E.g. if a d.n.c. function f is of c.e. degree, then it computes the Halting problem f T. On the other hand, there are d.n.c. functions of low Turing degree. Longstanding question: Must a d.n.c. function always be strictly more than being non-computable? (Kumabe) There is a d.n.c. function f such that f computes nothing else other than itself and the computable sets.
88 The computational strength of random reals Another example that d.n.c. = computationally strong: take the class of d.n.c. functions whose range {0, 1}. (Jockusch, Soare) A binary valued function f is d.n.c. iff f computes a total extension of PA. Every f of PA degree (i.e. binary valued d.n.c.) computes a ML-random. Not true of every d.n.c. function (e.g. Kumabe). There have been increasing amounts of evidence that the bound of the range of a d.n.c. f is strongly related to prefixfree complexity.
89 The computational strength of random reals Another example that d.n.c. = computationally strong: take the class of d.n.c. functions whose range {0, 1}. (Jockusch, Soare) A binary valued function f is d.n.c. iff f computes a total extension of PA. Every f of PA degree (i.e. binary valued d.n.c.) computes a ML-random. Not true of every d.n.c. function (e.g. Kumabe). There have been increasing amounts of evidence that the bound of the range of a d.n.c. f is strongly related to prefixfree complexity.
90 The computational strength of random reals Another example that d.n.c. = computationally strong: take the class of d.n.c. functions whose range {0, 1}. (Jockusch, Soare) A binary valued function f is d.n.c. iff f computes a total extension of PA. Every f of PA degree (i.e. binary valued d.n.c.) computes a ML-random. Not true of every d.n.c. function (e.g. Kumabe). There have been increasing amounts of evidence that the bound of the range of a d.n.c. f is strongly related to prefixfree complexity.
91 The computational strength of random reals Theorem (Stephan) A ML-random real A computes the Halting problem iff it computes a binary valued d.n.c. function This result says there are only two kinds of ML-random reals: 1 The first kind resemble Ω, and are so smart that they know how to be stupid. 2 The second really are stupid (fail to compute a binary valued d.n.c.) Recent work (Franklin, Ng) have made the second class ( true" ML-randoms) more well-understood.
92 The computational strength of random reals Theorem (Stephan) A ML-random real A computes the Halting problem iff it computes a binary valued d.n.c. function This result says there are only two kinds of ML-random reals: 1 The first kind resemble Ω, and are so smart that they know how to be stupid. 2 The second really are stupid (fail to compute a binary valued d.n.c.) Recent work (Franklin, Ng) have made the second class ( true" ML-randoms) more well-understood.
93 The computational strength of random reals Theorem (Stephan) A ML-random real A computes the Halting problem iff it computes a binary valued d.n.c. function This result says there are only two kinds of ML-random reals: 1 The first kind resemble Ω, and are so smart that they know how to be stupid. 2 The second really are stupid (fail to compute a binary valued d.n.c.) Recent work (Franklin, Ng) have made the second class ( true" ML-randoms) more well-understood.
94 Variations on ML-randomness Two stronger forms of ML-randomness have been studied. The first, weak 2-randomness have to avoid every Π 0 2 -null class. Theorem (Downey, Nies, Weber, Yu) A is weakly 2-random iff A is random and contains no common information with the Halting problem. The second, 2-randomness also exhibit properties demonstrating weakness. Theorem Every 2-random real A is generalized low, i.e. A T A.
95 Variations on ML-randomness Two stronger forms of ML-randomness have been studied. The first, weak 2-randomness have to avoid every Π 0 2 -null class. Theorem (Downey, Nies, Weber, Yu) A is weakly 2-random iff A is random and contains no common information with the Halting problem. The second, 2-randomness also exhibit properties demonstrating weakness. Theorem Every 2-random real A is generalized low, i.e. A T A.
96 Variations on ML-randomness The class of weakly 2-randoms remains poorly understood. For example, is there a definition in terms of initial segment complexity? (Barmpalias, Downey, Ng) Unlike the ML-randoms, the weakly 2-randoms do not have all possible jumps. The jumps of weakly 2-randoms are very closely related to -domination and the functions d.n.c. relative to. Suppose you take two ML-random strings A and B. Then A B = {2n : n A} {2n + 1 : n B} is generally not ML-random.
97 Variations on ML-randomness The class of weakly 2-randoms remains poorly understood. For example, is there a definition in terms of initial segment complexity? (Barmpalias, Downey, Ng) Unlike the ML-randoms, the weakly 2-randoms do not have all possible jumps. The jumps of weakly 2-randoms are very closely related to -domination and the functions d.n.c. relative to. Suppose you take two ML-random strings A and B. Then A B = {2n : n A} {2n + 1 : n B} is generally not ML-random.
98 Variations on ML-randomness The class of weakly 2-randoms remains poorly understood. For example, is there a definition in terms of initial segment complexity? (Barmpalias, Downey, Ng) Unlike the ML-randoms, the weakly 2-randoms do not have all possible jumps. The jumps of weakly 2-randoms are very closely related to -domination and the functions d.n.c. relative to. Suppose you take two ML-random strings A and B. Then A B = {2n : n A} {2n + 1 : n B} is generally not ML-random.
99 Variations on ML-randomness (van Lambalgen) A B is ML-random iff A is random and B is random relative to A. (Barmpalias, Downey, Ng) The corresponding fact for weakly 2-randoms: ( ) fails but ( ) holds. Theorem (Barmpalias, Downey, Ng) Given any function f : N N there is a weakly 2-random A and some g T A where g is not dominated by f. Weakly 2-random reals are computationally weak (having no common information with but still strong enough to escape domination.
100 Variations on ML-randomness (van Lambalgen) A B is ML-random iff A is random and B is random relative to A. (Barmpalias, Downey, Ng) The corresponding fact for weakly 2-randoms: ( ) fails but ( ) holds. Theorem (Barmpalias, Downey, Ng) Given any function f : N N there is a weakly 2-random A and some g T A where g is not dominated by f. Weakly 2-random reals are computationally weak (having no common information with but still strong enough to escape domination.
101 Variations on ML-randomness A special case of this theorem gives that weakly 2-random reals can be array non-computable. This notion arises in the study of c.e. degrees. It has found use in an impressive array of areas within computability theory: lattice embeddings, computable analysis, c.e.a. operators, genericity. (Brodhead, Downey, Ng) Recently it was linked to randomness. A certain weakening of ML-randomness, called computably bounded randomness was characterized in terms of array computability amongst the c.e. degrees.
102 Hausdorff dimension Another way of measuring semi-randomness is looking at effective Hausdorff and packing dimensions. Classically, the Hausdorff dimension gives a way of refine sets of measure 0. The effective version of Hausdorff dimension has been studied in the work of Lutz, Mayordomo. For 0 < s 1, an s-gale is a function F : 2 <N R such that F(σ0) + F(σ1) F(σ) = 2 s These are betting strategies. When s = 1 it s what we talked about for ML-randomness.
103 Hausdorff dimension Another way of measuring semi-randomness is looking at effective Hausdorff and packing dimensions. Classically, the Hausdorff dimension gives a way of refine sets of measure 0. The effective version of Hausdorff dimension has been studied in the work of Lutz, Mayordomo. For 0 < s 1, an s-gale is a function F : 2 <N R such that F(σ0) + F(σ1) F(σ) = 2 s These are betting strategies. When s = 1 it s what we talked about for ML-randomness.
104 Hausdorff dimension in terms of s-gales Suppose now you return to the casino with $1 and plays against the real X in the manager s pocket. The casino has a new rule: for every round you stay in, it takes a fraction (depending on s) of your current capital. Previously (if s = 1) we could refrain from favouring one side simply be betting an equal amount on 0 and 1. Now if we do this we only get back 2 s 1 F(σ) < F(σ), if s < 1. Now it s much harder for you to win, because you need to have a lot more knowledge about X in order to win.
105 Hausdorff dimension in terms of s-gales Suppose now you return to the casino with $1 and plays against the real X in the manager s pocket. The casino has a new rule: for every round you stay in, it takes a fraction (depending on s) of your current capital. Previously (if s = 1) we could refrain from favouring one side simply be betting an equal amount on 0 and 1. Now if we do this we only get back 2 s 1 F(σ) < F(σ), if s < 1. Now it s much harder for you to win, because you need to have a lot more knowledge about X in order to win.
106 Hausdorff dimension in terms of s-gales Suppose now you return to the casino with $1 and plays against the real X in the manager s pocket. The casino has a new rule: for every round you stay in, it takes a fraction (depending on s) of your current capital. Previously (if s = 1) we could refrain from favouring one side simply be betting an equal amount on 0 and 1. Now if we do this we only get back 2 s 1 F(σ) < F(σ), if s < 1. Now it s much harder for you to win, because you need to have a lot more knowledge about X in order to win.
107 Hausdorff dimension in terms of s-gales Lutz showed that effective Hausdorff dimension can be characterized in terms of s-gales: Theorem (Lutz,Mayordomo) For a class X of reals the following are equivalent: dim(x)= s s = inf{s Q : for some s-gale F, X S[F ]}
108 Effective hausdorff dimension C.e. martingales give effective version of Hausdorff dimension. Remarkably, Theorem (Mayordomo) The effective Hausdorff dimension of a real A is lim inf n K (A n) n = (lim inf n C(A n) ) n For instance if Ω = 0.a 1 a 2 a 3 then 0.a 1 0a 2 0a 3 0 has effective Hausdorff dimension 1 2.
109 Effective hausdorff dimension C.e. martingales give effective version of Hausdorff dimension. Remarkably, Theorem (Mayordomo) The effective Hausdorff dimension of a real A is lim inf n K (A n) n = (lim inf n C(A n) ) n For instance if Ω = 0.a 1 a 2 a 3 then 0.a 1 0a 2 0a 3 0 has effective Hausdorff dimension 1 2.
110 Hausdorff dimension extraction Is this the only way to construct a real of effective Hausdorff dimension 1 2? A lot of work done on Hausdorff dimension extraction, and many lovely results (Greenberg, Miller, Reimann). Having effective Hausdorff dimension 1 is closely related to the d.n.c. functions. Some questions still remain, e.g. when can a degree compute another of effective Hausdorff dimension 1?
111 Extracting packing dimension Idea is to replace outer measure by inner measure. Look for a dense packing. Classically this is known as packing dimension. Athreya, Hitchcock, Lutz, Mayordomo also characterized packing dimension in terms of martingales. Can define the effective packing dimension. This is characterized as lim sup n K (α n) (= lim sup n n C(α n) ). n
112 Extracting packing dimension Fundamental question: what Turing degrees contain reals of high packing dimension? Work of Greenberg, Downey, Ng: Amongst the c.e. degrees, packing dimension 1 = array noncomputability, but it doesn t extend beyond the c.e. degrees. Theorem (Fortnow, Hitchcock, Aduri, Vinochandran, Wang) If α has packing dimension > 0, then for any ε > 0, β wtt α of packing dimension 1 ε. Hence for degrees a 0-1 Law for effective packing dimension (no broken dimension). Open Question: is there a real of effective packing dimension 1 inside each degree of packing dimension 1?
113 Extracting packing dimension Fundamental question: what Turing degrees contain reals of high packing dimension? Work of Greenberg, Downey, Ng: Amongst the c.e. degrees, packing dimension 1 = array noncomputability, but it doesn t extend beyond the c.e. degrees. Theorem (Fortnow, Hitchcock, Aduri, Vinochandran, Wang) If α has packing dimension > 0, then for any ε > 0, β wtt α of packing dimension 1 ε. Hence for degrees a 0-1 Law for effective packing dimension (no broken dimension). Open Question: is there a real of effective packing dimension 1 inside each degree of packing dimension 1?
114 The proof This proof is due to Bienvenu. Have n(c(α n) tn) for some t. Break α into pieces of exponential length α m 0, α m 1, α m 2, [m k, m k+1 ) a large number. If m k 1 n < m k, then C(α n) C(α m k ) + O(log n). Choose t small, C(α m k ) t m k for infinitely many k. We re going to swap these pieces α m k with more complex pieces τ 0, τ 1,.
115 The proof This proof is due to Bienvenu. Have n(c(α n) tn) for some t. Break α into pieces of exponential length α m 0, α m 1, α m 2, [m k, m k+1 ) a large number. If m k 1 n < m k, then C(α n) C(α m k ) + O(log n). Choose t small, C(α m k ) t m k for infinitely many k. We re going to swap these pieces α m k with more complex pieces τ 0, τ 1,.
116 The proof This proof is due to Bienvenu. Have n(c(α n) tn) for some t. Break α into pieces of exponential length α m 0, α m 1, α m 2, [m k, m k+1 ) a large number. If m k 1 n < m k, then C(α n) C(α m k ) + O(log n). Choose t small, C(α m k ) t m k for infinitely many k. We re going to swap these pieces α m k with more complex pieces τ 0, τ 1,.
117 The proof Now let s = lim sup k C(α m k ) m k t > 0. Swap α m k with τ k, where U(τ k ) = α m k, and demand that τ k s m k. What is C(τ k )? C(τ k ) C(α m k ) s.m k infinitely often. Hence C(τ k )/ τ k is infinitely often close to 1, by adjusting m and the tolerance in. The original proof was a bit different, gave polynomial time reductions using complex multisource extractors of Impagliazzo and Widgerson.
118 The proof Now let s = lim sup k C(α m k ) m k t > 0. Swap α m k with τ k, where U(τ k ) = α m k, and demand that τ k s m k. What is C(τ k )? C(τ k ) C(α m k ) s.m k infinitely often. Hence C(τ k )/ τ k is infinitely often close to 1, by adjusting m and the tolerance in. The original proof was a bit different, gave polynomial time reductions using complex multisource extractors of Impagliazzo and Widgerson.
119 The proof Now let s = lim sup k C(α m k ) m k t > 0. Swap α m k with τ k, where U(τ k ) = α m k, and demand that τ k s m k. What is C(τ k )? C(τ k ) C(α m k ) s.m k infinitely often. Hence C(τ k )/ τ k is infinitely often close to 1, by adjusting m and the tolerance in. The original proof was a bit different, gave polynomial time reductions using complex multisource extractors of Impagliazzo and Widgerson.
120 The proof Now let s = lim sup k C(α m k ) m k t > 0. Swap α m k with τ k, where U(τ k ) = α m k, and demand that τ k s m k. What is C(τ k )? C(τ k ) C(α m k ) s.m k infinitely often. Hence C(τ k )/ τ k is infinitely often close to 1, by adjusting m and the tolerance in. The original proof was a bit different, gave polynomial time reductions using complex multisource extractors of Impagliazzo and Widgerson.
121 Lots of other work Lowness in terms of randomness notions. Different combinatorial notions relating to these lowness notions, such as traceability and the diamond classes. Work on other randomness notions such as Schnorr and computable randomness. Many intermediate randomness notions defined using different kinds of betting strategies.
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