# New Coins From Old: Computing With Unknown Bias

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- Revised:

DOI: 10.1007/s00493-005-0043-1

- Cite this article as:
- Mossel*, E., Peres†, Y. & With an appendix by Christopher Hillar‡, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA, chillar@math.berkeley.edu Combinatorica (2005) 25: 707. doi:10.1007/s00493-005-0043-1

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Suppose 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 *p*-coin (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.