The Ehrenfeucht–Mycielski sequence is a recursively defined sequence of binary digits with pseudorandom properties, defined by Andrzej Ehrenfeucht and Jan Mycielski (1992).
The sequence starts with the single bit 0; each successive digit is formed by finding the longest suffix of the sequence that also occurs earlier within the sequence, and complementing the bit following the most recent earlier occurrence of that suffix.
For example, the first few steps of this construction process are:[1] The first few digits of the sequence are:[1] A naive algorithm that generates each bit in the sequence by comparing each suffix with the entire previous sequence could take as much as
However, using a data structure related to a suffix tree, the sequence can be generated much more efficiently, in constant time per generated digit.
[2] The sequence is disjunctive, meaning that every finite subsequence of bits occurs contiguously, infinitely often within the sequence.
[3] More explicitly, the position by which every subsequence of length
[2] Experimentally, however, each subsequence appears much earlier in this sequence than this upper bound would suggest: the position by which all length-
sequences occur, up to the limit of experimental testing, is close to the minimum possible value it could be,
, the position by which a de Bruijn sequence contains all length-
[4] Ehrenfeucht & Mycielski (1992) conjecture that the numbers of 0 and 1 bits each converge to a limiting density of 1/2.
denotes the number of 0 bits in the first
positions of the Ehrenfeucht–Mycielski sequence, then it should be the case that
Good suggests that the convergence rate of this limit should be significantly faster than that of a random binary sequence, for which (by the law of the iterated logarithm)[3]
The Ehrenfeucht–Mycielski balance conjecture suggests that the binary number 0.01001101... (the number having the Ehrenfeucht–Mycielski sequence as its sequence of binary digits after the binary point) may be a normal number in base 2.
As of 2009 this conjecture remains unproven;[2] however, it is known that every limit point of the sequence of values