On dice and coins: models of computation for random generation

David Feldman, Russell Impagliazzo, Moni Naor, Noam Nisan, Steven Rudich, Adi Shamir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


To distinguish between random generation in bounded, as opposed to expected, polynomial time, a model of Probabalisic Turing Machine (PTM) with the ability to make random choices with any (small) rational bias is necessary. This ability is equivalent to that of being able to simulate rolling any k-sided die (where |k| is polynomial in the length of the input). We would like to minimize the amount of hardware required for a machine with this capability. This leads to the problem of efficiently simulating a family of dice with as few different types of biased coins as possible. In the special case of simulaing one n-sided die, we prove that only two types of biased coins are necessary, which can be reduced to one if we allow irrationally biased coins. This simulation is efficient, taking O(log n) coin flips. For the general case we get a tight time vs. number of biases tradeoff; for example, with O(log n) different biases, we can simulate, for any i<n, an i-sided die in O(log n) coin flips.

Original languageAmerican English
Title of host publicationAutomata, Languages and Programming - 16th International Colloquium, Proceedings
EditorsMariangiola Dezani-Ciancaglini, Simonetta Ronchi Della Rocca, Giorgio Ausiello
PublisherSpringer Verlag
Number of pages22
ISBN (Print)9783540513711
StatePublished - 1989
Externally publishedYes
Event16th International Colloquium on Automata, Languages and Programming, 1989 - Stresa, Italy
Duration: 11 Jul 198915 Jul 1989

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume372 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference16th International Colloquium on Automata, Languages and Programming, 1989

Bibliographical note

Publisher Copyright:
© 1989, Springer-Verlag.


Dive into the research topics of 'On dice and coins: models of computation for random generation'. Together they form a unique fingerprint.

Cite this