, Volume 25, Issue 6, pp 707724
First online:
New Coins From Old: Computing With Unknown Bias
 Elchanan Mossel*Affiliated withUniversity of California, Berkeley Email author
 , Yuval Peres†Affiliated withUniversity of California, Berkeley
 , With an appendix by Christopher Hillar‡, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 947203840, USA, chillar@math.berkeley.edu
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Get AccessSuppose that we are given a function f : (0, 1)→(0,1) and, for some unknown p∈(0, 1), a sequence of independent tosses of a pcoin (i.e., a coin with probability p of “heads”). For which functions f is it possible to simulate an f(p)coin? This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function f(p)≡1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in general, an f(p)coin can be simulated by a finite automaton for all p ∈ (0, 1), if and only if f is a rational function over ℚ. We also show that if an f(p)coin can be simulated by a pushdown automaton, then f is an algebraic function over ℚ; however, pushdown automata can simulate f(p)coins for certain nonrational functions such as \( f{\left( p \right)} = {\sqrt p } \). These results complement the work of Keane and O’Brien (1994), who determined the functions f for which an f(p)coin can be simulated when there are no computational restrictions on the simulation scheme.
Mathematics Subject Classification (2000):
68Q70 14P10 65C50 Title
 New Coins From Old: Computing With Unknown Bias
 Journal

Combinatorica
Volume 25, Issue 6 , pp 707724
 Cover Date
 200512
 DOI
 10.1007/s0049300500431
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 68Q70
 14P10
 65C50
 Industry Sectors
 Authors
 Author Affiliations

 1. University of California, Berkeley, 367 Evans Hall, Berkeley, CA 947203860, USA
 2. University of California, Berkeley, 367 Evans Hall, Berkeley, CA 947203860, USA