New Coins from old: Computing with unknown bias

1. New Coins from old: Computing with unknown bias Elchanan Mossel, U.C. Berkeley mossel@stat.berkeley.edu, http://www.cs.berkeley.edu/~mossel/ Joint work with Yuval Peres, U.C. Berkeley peres@stat.berkeley.edu, http://www.stat.berkeley.edu/~peres/ Supported by Microsoft Research and the Miller institute

2. von Neumann extractor (1951) Given a sequence of i.i.d. coins (p,1-p) coins want to toss a fair coin. 0 < p < 1 is unknown. Want Efficient in randomness (preserves entropy) Computationally simple (finite automaton) Efficient in time (small expected running time)

3. von Neumann extractor (1951) P[01] = p(1-p) = P[10]. Map 01  0, 10  1 and delete00,11. Properties Linear time. Rate p(1-p) (compared to H(p)). Easy to generalize to Markov chains. Can implement via finite automata.

4. Since von Neumann In information theory: extracting (1 – ε) of entropy Elias (72) – block construction. Peres (92) – iterative construction. In computer science, named extractors. General distributions with bound on (min) entropy. Extra randomness needed. Nearly optimal constructions in recent years.

5. Our model Input is i.i.d. (p,1-p) coins, where 0 < p < 1 is unknown. Want to Output(f(p), 1 – f(p)) coin. Simulation power? Unbounded (infinite memory), Turing machines, Pushdown automata, Finite automata. Asked byS. Asmussen and J. Propp.

6. Our model Keane & O’Brien : If there are no computational restriction can simulate any continuous function f : (0,1)  (0,1). Ergodic theory techniques. Need min(f(x),1-f(x)) > min(x,1-x)n, for some n.

7. Coins via finite and pushdown automata Which functions can be simulated via finite automata? Which functions can be simulated via push-down automata? Examples: f(p) = p2 ? f(p) = p/2 ? f(p) = 2p for 0 < p < 1/4 ? f(p) = p ? f(p) = p2 / (p2 + (1 – p)2) ? f(p) = π/6 ?

8. Exact simulation, computability, etc. Theory of exact simulation: simulating complicated distributions from simples ones. Here both are simple. But, some examples: Simulating percolation configuration on the triangular lattice from a configuration on the square lattice with same distance 2 connectivity functions (p unknown!). Given a sequence {yi AND zi} where yi, zi are i.i.d. with unkown mean p, find {xi} i.i.d. with mean p. Theory of computability – the computation of real functions. Trivial : for every computable constant 0 < q < 1, the function f(p) = q can be simulated via a Turing machine.

9. Coins via finite automata

10. Coins via finite automata • f(p) = p2 V • f(p) = p/2 V • f(p) = 2p for 0 < p < 1/4 X • f(p) = p X • f(p) = p2 / (p2 + (1 – p)2) V • f(p) = π/6 X

11. Coins via pushdown automata • f(p) = 2p for 0 < p < 1/4 ?? • f(p) = p V • f(p) = π/6 X Open problem: Does a converse hold? If f :(0,1)  (0,1) is algebraic over Q, does there exist a push down automata simulating f?

12. Pushdown automaton simulating p • Take a random walk on the ladder with P[up/down] = (1 – p)/2, P[left/right] = p. • Let t be the probability starting at (0,1) that the 1sthitting of level 0 is (0,0). • t = (1 – p)/2 + p (1 – t) + (1 – p)(t2 + (1-t)2)/2 • Can simulate the random walk with a push down automaton and t = (1 - p)/(1-p) • With another coin toss can get p ½ - p/2 ½ - p/2 p ½ - p/2 ½ - p/2

13. Finite automaton implies rationality

14. Finite automaton implies rationality • Proof 1: • Let F(p; s) be the probability to stop at 1 given that the current state is s. • F(p; s) = p F(p; δ(s,1)) + (1-p) F(p; δ(s,0)), and f(p) = F(p; s0). • Equations determine F by maximum principle for harmonic functions on directed graphs. • By Cramer’s rule F(p) = g(p) / h(p).

15. Finite automaton implies rationality • Proof 2 (Chomsky-Schützenberger): • If L is a regular language, then is a rational function in the non-commutative variables x0 and x1. • Let L be the language where the automaton stops at 1. • Looking at the homomorphism 0  1-p, 1  p, we see that is a rational function.

16. Algebraic properties of Pushdown automata • Our results for push-down automata do not follow from Chomsky-Schützenberger theorem. • Instead, we prove that f(p) is determined by a set of polynomial equations. • The proof uses the fact that bounded harmonic functions on recurrent infinite graphs are determined by boundary values. • Then we invoke the following result due to Hillar (2002).

17. Hillar 2002:

18. Rationality implies simulation by finite automata Definition:block simulation of f, is given by • A0, A1 disjoint subsets of {0,1}k, A' = {0,1}k \ (A0  A1), and the following procedure. • Read a k bit string w. • For i=1,2, if w Ai, output i. • Otherwise, discard w and reads a new k bit string. • Block simulation  automata simulation, and

19. Rationality implies simulation by finite automata Already seen I  II  III

20. Rationality implies simulation by finite automata Need to show: If f(p) = g(p)/h(p), where g,h  Z[p], and 0 < f(p) < 1, then f(p) is block simulated. Easier to show: Remark: claim doesn’t cover (p2 – 2p(1-p) + 2(1-p)2)/2

21. Rationality implies simulation by finite automata Proof: 1…10…0y0…yr if 0  y0…yr  di 1 i k-i 1…10…0y0…yr if di y0…yr  ei 0 i k-i 1…10…0y0…yr if ei y0…yr X i k-i

22. The general case

23. The general case • Remark:The lemma reduces the general case to the easy case. • Proof of Lemma: • f(p) = D(p)/E(p), where D, E  Z[p] have positive values. • Write D and E as homogenous polynomials in p, 1 – p: D(p) = δ(p,1-p) and E(p) = ε(p,1-p). • δ(p,1-p), ε(p,1-p) and (ε – δ)(p,1-p) are positive for all 0 < p < 1,  for large enough n all the coefficients of d(p,q) = (p+q)n δ(p,q), e(p,q) = (p+q)n ε(p,q), and (e – d)(p,q) = (p+q)n (ε – δ)(p,q) are positive. - Write f(p) = δ(p,1-p)/ε(p,1-p).

24. Open problems • Can every algebraic function f :(0,1)  (0,1) be simulated via a pushdown automaton? • For rational functions f, what is the minimal size of automaton simulating f? • The best bound known in Polya’s Theorem for f is (Powers and Reznick 2002). • For a rational function f, does there exist a finite automaton which extracts almost all entropy?