The Ehrenfeucht-Mycielski Sequence

We show that the Ehrenfeucht-Mycielski sequence U is strongly balanced in the following sense: for any finite word w of length k, the limiting frequency of w in U is 2. 1. The Ehrenfeucht-Mycielski Sequence In [2] Ehrenfeucht and Mycielski introduced an infinite binary word based on avoiding repetitions. More precisely, to construct the Ehrenfeucht-Mycielski (EM) sequence U , start with a single bit 0. Suppose the first n bits Un = u1u2 . . . un have already been chosen. Find the longest suffix v of Un that appears already in Un−1. Find the last occurrence of v in Un−1, and let b be the first bit following that occurrence of v. Lastly, set un+1 = b, the complement of b. It is understood that if there is no prior occurrence of any non-empty suffix the last bit in the sequence is flipped. The resulting sequence starts like so: 01001101011100010000111101100101001001110 see also sequence A038219 in [7]. Since the Ehrenfeucht-Mycielski sequence is defined to avoid repetitions, one might suspect that it contains all finite words as factors; in the reference the authors show that this is indeed the case. The language pref(U) of all prefixes of U fails to be regular. Hence it follows from the gap theorem in [1] that pref(U) cannot be context-free. On the other hand, it is clear that a linear bounded automaton can recognize pref(U), so this language is contextsensitive. Indeed, it follows from the results in section 2 that one can recognize prefixes of the Ehrenfeucht-Mycielski word in logarithmic space and quadratic time using KMP. Much better results can be achieved with a hash-based algorithm, see [6, 3]. The second reference shows that under the assumption of near-monotonicity, see 1.2, one can generate a bit of the sequence in amortized constant time. Moreover, only linear space is required to construct an initial segment of the sequence, so that a simple laptop computer suffices to generate the first billion bits of the sequence in less than an hour, see [3]. Storing the first billion bits in the obvious bit-packed format requires 125 million bytes, and there is little hope to decrease this amount of space using data compression: the very definition of the EM sequence foils standard algorithms. For example, the Lemple-Ziv-Welch based gzip algorithm produces a “compressed” file of size 159,410 bytes from the first million bits of the EM sequence. The Burrows-Wheeler type bzip2 algorithm even produces a file of size 165,362 bytes. Date: March 28, 2007.