Computable analysis of martingales and related measure-theoretic topics, with an emphasis on algorithmic randomness

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1 Computable analysis of martingales and related measure-theoretic topics, with an emphasis on algorithmic randomness Jason Rute Carnegie Mellon University PhD Defense August, Jason Rute (CMU) Randomness, martingales, and more PhD Defense 1 / 32

2 Martingales, computable analysis, and randomness What is a martingale? Informally: A martingale is a gambling strategy. Example: Bet on a fair coin. T 0 H T -1 1 H T H d(hh) M For logicians: If d(σ) is a martingale in the computability theory sense and x 2 N, then we are considering M n (x) = d(x n). Jason Rute (CMU) Randomness, martingales, and more PhD Defense 3 / 32

3 Martingales, computable analysis, and randomness What is a martingale in general? Formally: A martingale is a sequence of integrable functions M n : (Ω,P) R such that Slogan: E[M n+1 M 0,M 1,...,M n ] = M n. The expectation of the future is the present (conditioned on the past & present). Jason Rute (CMU) Randomness, martingales, and more PhD Defense 4 / 32

4 Martingales, computable analysis, and randomness An a.e. convergence theorem Example (Doob s martingale convergence theorem) Let (M n ) be a martingale. Assume sup n M n L 1 <. Then M n converges a.e. In particular, lim n M n < a.s. 4 Questions 1 (Computable analysis) Is the rate of convergence of (M n ) computable (from the martingale (M n ))? 2 (Computable analysis) If not, what other information is needed to compute a rate of convergence? 3 (Algorithmic randomness) For which points x does M n (x) converge for all computable L 1 -bounded martingales? 4 (Algorithmic randomness) Which assumptions are needed to characterize convergence on (insert favorite randomness notion)? Jason Rute (CMU) Randomness, martingales, and more PhD Defense 5 / 32

5 Martingales, computable analysis, and randomness Computable analysis Computable reals A real number a is computable if it can be effectively approximated by rationals. Example π is computable. We can compute a sequence of rationals q n such that q n π 2 n. (This sequence is called a name for x.) Computable points, functions This definition extends to any complete separable metric space with a nice countable set of simple points {q n } n. Examples Computable L 1 -functions, computable sequences of reals, computable sequences of L 1 -functions, computable Borel measures, etc. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 6 / 32

6 Martingales, computable analysis, and randomness Computable analysis Computable maps We say that y is computable from x if there is an algorithm which computes a name for y uniformly from a name for x, more formally takes in ε > 0 keeps reading the name for x: q 0,q 1,q 2,... when it has a close enough approximation q n, it returns r n such that r n y ε. Note Total computable maps are continuous. Most continuous maps in practice are computable. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 7 / 32

7 Question 1 Martingales, computable analysis, and randomness Question 1. Is the rate of a.e. convergence computable? A rate of a.e. convergence is some n(ε,δ) such that { } µ x sup f n (x) f (x) ε δ, n n(ε,δ) i.e. f n f with an ε-uniform rate of convergence outside a set of measure δ. Theorem The rate of a.e. convergence of a martingale M n is not necessarily computable. Proof sketch. Code in the halting problem. Enumerate the programs {e n } that halt and bet 3 e n dollars that the e n th program halts. Any rate of convergence would compute the halting problem. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 8 / 32

8 Martingales, computable analysis, and randomness Question 2 Question 2. What other information is needed to compute a rate of convergence? Theorem (R.) The rate of convergence of M n M is computable uniformly from (M n ) (as a sequence of L 1 functions), M (as an L 1 function), and sup n M n L 1. Note This is not just because we know the limit M. Without sup n M L 1 being computable, the limit could be 0 but the rate of convergence not computable. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 9 / 32

9 Martingales, computable analysis, and randomness Question 2 Question 2. What other information is needed to compute a rate of convergence? Theorem (R.) The rate of convergence of M n M is computable uniformly from (M n ) (as a sequence of L 1 functions), M (as an L 1 function), and sup n M n L 1. Theorem (R.) The rate of convergence (a.e. and L 2 ) of M n M is computable uniformly from (M n ) (as a sequence of L 2 functions) and sup n M n L 2 = M L 2. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 9 / 32

10 Martingales, computable analysis, and randomness Question 3 Question 3. For which points x does the following hold? M n (x) converges for all computable L 1 -bounded martingales (M n ) Theorem Every L 1 -bounded martingale converges on almost every point. Corollary For almost every point, every computable L 1 -bdd martingales converges. There are countably many computable martingales. Question What is this measure-one set of points? Jason Rute (CMU) Randomness, martingales, and more PhD Defense 10 / 32

11 Martingales, computable analysis, and randomness Algorithmic randomness π random.org Are either of these random? How can we check? 1 Are they normal? 2 Do they satisfy the law of the iterated logarithm? 3 Is the number not π? 4 It did not come from random.org? Wait! Is any bit sequence random? They all fail some statistical test! A bit sequence is algorithmically random if it passes all computable statistical tests. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 11 / 32

12 Martingales, computable analysis, and randomness The randomness zoo Randomness Zoo Antoine Taveneaux 1 1 R 1 1 MLR 1 1 R... 3MLR 2MLR SR 0 There are many randomness notions. Most start out on 2 N (coin flipping). Some of the more natural ones are: 2-randomness Weak 2-randomness Difference randomness Martin-Löf randomness Computable randomness Schnorr randomness Kurtz randomness The natural ones have connections with computable analysis. The natural ones can be extended to other computable probability spaces. FBoundR CBoundR InjR PCR CR SR PolyR WR P ermr PInjR MWCStoch dim 1 compr s<1 dim s compr s 0 <s dim s0 compr LimitR W 2R DemR W DemR BalancedR DiffR MLR KLR KLStoch PMWCStoch Cdim 1 R s<1 Cdim s R s 0 <s Cdim s0 R Randomness Zoo (Antoine Taveneaux) Jason Rute (CMU) Randomness, martingales, and more PhD Defense 12 / 32

13 Martingales, computable analysis, and randomness Martin-Löf and Schnorr randomness Definition A Martin-Löf test is a computable sequence (U n ) of effectively open sets (uniform sequence of Σ 0 1 sets) such that µ(u n) 2 n. A Schnorr test is a Martin-Löf test, where µ(u n ) is uniformly computable. x is Martin-Löf/Schnorr random (for the measure µ) if x n U n for each ML/Schnorr tests. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 13 / 32

14 Martingales, computable analysis, and randomness Question 3 Again Question 3. For which points x does the following hold? M n (x) converges for all computable L 1 -bounded martingales (M n ) Theorem (Takahashi; Merkle-Mihalovic-Slaman; Dean; R.) The answer is Martin-Löf randomness, even if the martingales are dyadic or nonnegative (but not both). Corollary Doob s martingale convergence theorem characterizes Martin-Löf randomness! Jason Rute (CMU) Randomness, martingales, and more PhD Defense 14 / 32

15 Question 4 Martingales, computable analysis, and randomness Question 4. Which assumptions characterize convergence on Schnorr randoms? Lemma (R.) If (f n ) and f are L 1 -computable, and f n f effectively a.e., then f n (x) f (x) on Schnorr randoms x. Here f (x) = limn p n (x) where (p n ) is a sequence of simple functions f p n L 1 < 2 n. (We need this since f is an equivalence class.) Jason Rute (CMU) Randomness, martingales, and more PhD Defense 15 / 32

16 Question 4 Martingales, computable analysis, and randomness Question 4. Which assumptions characterize convergence on Schnorr randoms? Lemma (R.) If (f n ) and f are L 1 -computable, and f n f effectively a.e., then f n (x) f (x) on Schnorr randoms x. Theorem (R.) Assume (M n ) is L 1 -computable, M is L 1 -computable, and sup n M n L 1 is computable. Then M n M effectively a.e. Therefore (for free!) M n (x) M (x) on Schnorr randoms x. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 15 / 32

17 Question 4 Martingales, computable analysis, and randomness Question 4. Which assumptions characterize convergence on Schnorr randoms? Lemma (R.) If (f n ) and f are L 1 -computable, and f n f effectively a.e., then f n (x) f (x) on Schnorr randoms x. Theorem (R.) Assume (M n ) is L 1 -computable, M is L 1 -computable, and sup n M n L 1 is computable. Then M n (x) M (x) on Schnorr randoms x. Theorem (R.) If x is not Schnorr random, there is some L 1 -computable martingale (M n ), with an L 1 -computable limit M, and sup n M n L 1 = 1 such that lim n M n (x) =. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 15 / 32

18 Martingales, computable analysis, and randomness Martingale convergence Theorem Assume (M n ) is L 1 -computable, M is L 1 -computable, and sup n M n L 1 is computable. Then M n M effectively a.e. Hence, M n (x) M (x) on Schnorr randoms x. Proof sketch. Decompose M n = N n + L n where N n converges in L 1 and L n converges to 0. Handle each case individually. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 16 / 32

19 Martingales, computable analysis, and randomness Martingale convergence in L 1 Theorem Assume (M n ) is L 1 -computable, M is L 1 -computable and M n M in L 1. Then Proof sketch. M n M effectively a.e. and in L 1. Fix k, and find n k such that M M nk L 1 2 2k. Facts: M n M nk L 1 is increasing and (M n M nk ) L1 (M M nk ). M n M effectively in L 1 since n n k M n M nk L 1 M M nk L 1 2 2k. M n M effectively a.e. since (by Kolmogorov s inequality) { } µ sup M n M nk 2 k sup n M n M nk L 1 n 2 k 2 2k 2 k 2 k. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 17 / 32

20 Other a.e. convergence theorems Similar results All these theorems can be used to characterize Schnorr randomness Differentiability of bounded variation functions Lebesgue differentiation theorem for signed measures Ergodic theorem (Avigad-Gerhardy-Towsner; Gács-Hoyrup-Rojas; Galatalo-Hoyrup-Rojas) Sub/supermartingale convergence theorem (nonnegative) Backwards martingale convergence theorem Monotone convergence theorem Strong law of large numbers De Finetti s theorem Jason Rute (CMU) Randomness, martingales, and more PhD Defense 19 / 32

21 Other a.e. convergence theorems An observation Observation In most common a.e. convergence theorems, the rate of convergence is computable from the sequence (f n ), the limit f, and the bounds sup n f n L 1 and inf n f n L 1. There are easy, but contrived, counterexamples. Can this observation be made into a theorem with a few more assumptions? Sub/supermartingales What about sub/supermartingales? This is one of the only cases I have not been able to work out. It is also one of the only cases where f n L 1 is not monotone (or nearly monotone). Jason Rute (CMU) Randomness, martingales, and more PhD Defense 20 / 32

22 Other a.e. convergence theorems Lebesgue differentiation theorem Theorem (R.; Pathak-Rojas-Simpson) Assume f is L 1 -computable on [0,1]. Then 1 2r x+r x r f (x)dx f (x) effectively a.e. r and 1 2r x+r x r f (x)dx f (x) on Schnorr random x. r Theorem (R.; Pathak-Rojas-Simpson) If x is not Schnorr random, there is some L 1 -computable f such that 1 2r x+r x r f (y)dy. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 21 / 32

23 Other a.e. convergence theorems Backwards martingales with an application Theorem (R.) Assume (M n ) is an L 1 -computable backwards martingale, M is L 1 -comp. Then M n M effectively a.e. and in L 1. Hence M n M on Schnorr random x. Corollary (Variation on Kučera s theorem, R.) Let C be a closed set of positive computable measure µ(c). Let x be Schnorr random. There is some y C such that y is the same as x but with finitely many bits permuted. Proof sketch. Let M n be the average of 1 C under all permutations of the first n bits. It turns out M n is a reverse martingale with limit µ(c). Then M n (x) µ(c) by the above theorem. Hence M n (x) > 0 for some n. Hence some y C where y is a permutation of first n bits of x. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 22 / 32

24 Computable randomness on other spaces The randomness zoo Randomness Zoo Antoine Taveneaux 1 1 R 1 1 MLR 1 1 R... 3MLR 2MLR SR 0 There are many randomness notions. Most start out on 2 N (coin flipping). Some of the more natural ones are: 2-randomness Weak 2-randomness Difference randomness Martin-Löf randomness Computable randomness Schnorr randomness Kurtz randomness The natural ones have connections with computable analysis. The natural ones can be extended to other computable probability spaces. FBoundR CBoundR InjR PCR CR SR PolyR WR P ermr PInjR MWCStoch dim 1 compr s<1 dim s compr s 0 <s dim s0 compr LimitR W 2R DemR W DemR BalancedR DiffR MLR KLR KLStoch PMWCStoch Cdim 1 R s<1 Cdim s R s 0 <s Cdim s0 R Randomness Zoo (Antoine Taveneaux) Jason Rute (CMU) Randomness, martingales, and more PhD Defense 24 / 32

25 Computable randomness on other spaces Computable randomness Definition A test for computable randomness is a nonnegative dyadic martingale M : 2 <ω R + such that µ(σ0)m(σ0) + µ(σ1)m(σ1) = µ(σ)m(σ) and M(σ) is computable from σ, provided that µ(σ) > 0. Definition We say that x (2 N,µ) is computably random if limsup n M(x n) < for all martingale tests M. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 25 / 32

26 Computable randomness on other spaces Computably random Brownian motion? B a computably random Brownian motion. Are these also computably random? B(1) (Gaussian distribution) Last hitting time before t = 1 (arcsin distribution) Maximum/minimum values for t [0,1] argmax/argmin set of zeros {t : B(t) = 0} We need a good definition of computable randomness for Brownian motion and for reals (with, say, Gaussian distribution). Which maps preserve computable randomness? For example B B(1)? Jason Rute (CMU) Randomness, martingales, and more PhD Defense 26 / 32

27 Computable randomness on other spaces Computable randomness on [0,1]. Base invariance Say that x is random on [0,1] (with Lebesgue measure) if its binary digits are random on 2 N. its decimal digits are random on 10 N. Are these the same? What about other bases? Easy 2-randomness, weak 2-randomness, difference randomness, Martin-Löf randomness, Schnorr randomness, and Kurtz randomness are base invarient! Brattka, Miller, Nies; Silveira Computable randomness is base invariant. The proofs for comp. randomness are not trivial. (The Brattka, Miller, Nies proof uses differentiability and does not even work in multiple dimensions.) Jason Rute (CMU) Randomness, martingales, and more PhD Defense 27 / 32

28 Computable randomness on other spaces Computable randomness on Polish space X. Let X be a computable Polish metric space. Let µ be a computable measure on X. Break up X into cells (Gács; Hoyrup-Rojas; Bosserhoff) Now (X,µ) looks like a measure (2 N,ν) on Cantor space Say x X is computably random if the corresponding point in (2 N,ν) is computably random. Theorem (R.) It does not matter how we break up X into cells. Example We then have computably random Brownian motion, Gaussian distributed reals, etc. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 28 / 32

29 Computable randomness on other spaces Computing randoms from randoms Almost-everwhere computable maps Let T : (2 N,µ 1 ) (2 N,µ 2 ) come from an algorithm which 1 Takes in a sequence of coin flips with distribution µ 1. 2 Outputs a sequence of coin flips with distribution µ 2. 3 Almost every input has an output. However, there is a problem with computable randomness! Theorem (Bienvenu-Porter; R.) There is an a.e. computable map T : (2 N,λ) (2 N,λ) where x is computably random, but T(x) is not computably random! Jason Rute (CMU) Randomness, martingales, and more PhD Defense 29 / 32

30 Computable randomness on other spaces Preservation of computable randomness Theorem (R.) Assume T : (2 N,µ 1 ) (2 N,µ 2 ) and T 1 : (2 N,µ 2 ) (2 N,µ 1 ) are a.e. computable such that T T 1 = id and T 1 T = id. Then T preserves computable randomness. Proof sketch. Take y (2 N,µ 2 ) not computably random. There is a martingale M which succeeds on y (lim n M 2 (y n) = ). Slow down the martingale by saving some of your money (savings trick). This gives an absolutely continuous measure ν 2 (σ) = M 2 (σ)µ 2 (σ). Since ν 2 µ 2, then T 1 is a.e. computable on ν 2. Let ν 1 = ν 2 T 1 (pushforward of ν 2 along T 1 ). Since T 1 is ν 2 -a.e. comp., then ν 1 is comp. Let M 1 (σ) = ν 1(σ) µ 1 (σ) and x = T 1 (y). It can be shown that M 1 (x n). So x is not computably random. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 30 / 32

31 Computable randomness on other spaces More recent work (not in the dissertation) Theorem (R.) Assume T : (X,µ 1 ) (Y,µ 2 ) is effectively measurable with a computable conditional probability µ 1 [ T]. Then T(x) preserves computable randomness. Theorem (R.) TFAE: TFAE: (X,µ) is computable. There exists a map T : (2 N,λ) (X,µ) as above. x is computably random on (X,µ). x = T(ω) for some computably random x (2 N,λ). Theorem (R.) If (M n,f n ) is an L 1 -comp. martingale, sup n M n L 1 is comp., and f E[ f F ] is (L 1 L 1 )-computable, then M n (x) converges on computable randoms. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 31 / 32

32 Thank You! Final remarks These slides will be available on my webpage: math.cmu.edu/~jrute Or just Google me, Jason Rute. Jason Rute (CMU) Randomness, martingales, and more PhD Defense 32 / 32