On Buffon machines and numbers
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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.With an appendix by Christopher Hillar‡, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA, chillar@math.berkeley.edu