Electro Thermal Dynamic Stripping Process - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10067)

2014-03-29T16:59:23Z

WP:CHECKWIKI error fixes + other fixes using AWB (10067)

← Older revision		Revision as of 18:59, 29 March 2014
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en>Chris the speller: /* Extraction System */replaced: setup → set up; comma using AWB

2014-03-02T22:15:21Z

Extraction System <ref name="test"/>: replaced: setup → set up; comma using AWB

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en>Chowbok: clean up, removed: ™ (2), ^TM (33), added orphan tag using AWB

2010-10-18T04:12:38Z

clean up, removed: ™ (2), TM (33), added orphan tag using AWB

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The '''Ehrenfeucht–Mycielski sequence''' is a recursively defined sequence of [[binary digits]] with [[pseudorandom]] properties, defined by {{harvs|txt | last1 = Ehrenfeucht | first1 = Andrzej | author1-link = Andrzej Ehrenfeucht | last2 = Mycielski | first2 = Jan | author2-link = Jan Mycielski|year=1992}}.

==Definition==
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. (The empty string is a suffix and prefix of every string.)
For example, the first few steps of this construction process are:
#0: initial bit
#01: the suffix "" of "0" occurs earlier, most-recently followed by a 0, so add 1
#010: the suffix "" of "01" occurs earlier, most-recently followed by a 1, so add 0
#0100: the suffix "0" of "010" occurs earlier, most-recently followed by a 1, so add 0
#01001: the suffix "0" of "0100" occurs earlier, most-recently followed by a 0, so add 1
#010011: the suffix "01" of "01001" occurs earlier, most-recently followed by a 0, so add 1
#0100110: the suffix "1" of "010011" occurs earlier, most-recently followed by a 1, so add 0

The first few digits of the sequence are:

:010011010111000100001111... {{OEIS|A038219}}.

==Algorithms==
A naive algorithm that generates each bit in the sequence by comparing each suffix with the entire previous sequence could take as much as O(''n''3) time to generate the first ''n'' bits of the sequence; however, as {{harvtxt|Herman|Soltys|2009}} show using a data structure related to a [[suffix tree]], the sequence can be generated much more efficiently, in [[constant time]] per generated digit.

==Universality==
The sequence is [[disjunctive sequence|disjunctive]]; i.e., every finite subsequence of bits occurs contiguously, infinitely often within the sequence {{harv|Ehrenfeucht|Mycielski|1992}}. More explicitly, the position by which every subsequence of length ''i'' can be guaranteed to have occurred at least ''j'' times is at most ''A''(4''i'',''j''), where ''A'' is the [[Ackermann function]] {{harv|Herman|Soltys|2009}}. Experimentally, however, each subsequence appears much earlier in this sequence than this upper bound would suggest: the position by which all length-''i'' sequences occur, up to the limit of experimental testing, is close to the minimum possible value it could be, 2''i'' + ''i'', the position by which a [[de Bruijn sequence]] contains all length-''i'' substrings {{harv|Sutner|2003}}.

==Normality==
{{harvtxt|Ehrenfeucht|Mycielski|1992}} conjecture that the numbers of 0 and 1 bits each converge to a limiting density of 1/2. That is, if ''f''(''i'') denotes the number of 0 bits in the first ''i'' positions of the Ehrenfeucht–Mycielski sequence, then it should be the case that
:<math>\lim_{i\rightarrow\infty} \frac{f(i)}{i}=\frac{1}{2}.</math>
More strongly, [[I. J. Good]] suggests that the [[convergence rate]] of this limit should be significantly faster than that of a [[random sequence|random binary sequence]], for which (by the [[law of the iterated logarithm]]) <math>\scriptstyle \frac{f(i)}{i}=\frac{1}{2}+O(\sqrt{\frac{\log\log i}{i}})</math> {{harv|Ehrenfeucht|Mycielski|1992}}.
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 {{harv|Herman|Soltys|2009}}; however, it is known that every limit point of the sequence of values ''f''(''i'')/''i'' lies within the interval [1/4,3/4] {{harv|Kieffer|Szpankowski|2007}}.

==References==
*{{citation
| last1 = Ehrenfeucht | first1 = Andrzej | author1-link = Andrzej Ehrenfeucht
| last2 = Mycielski | first2 = Jan | author2-link = Jan Mycielski
| contribution = A pseudorandom sequence: how random is it?
| editor-last = Guy | editor-first = Richard K. | editor-link = Richard K. Guy
| journal = [[American Mathematical Monthly]]
| pages = 373–375
| title = Unsolved problems
| jstor = 2324917
| volume = 99
| year = 1992}}.
*{{citation
| last1 = Herman | first1 = Grzegorz
| last2 = Soltys | first2 = Michael
| doi = 10.1016/j.jda.2009.01.002
| issue = 4
| journal = Journal of Discrete Algorithms
| pages = 500–508
| title = On the Ehrenfeucht–Mycielski sequence
| url = http://www.cas.mcmaster.ca/~soltys/papers/herman-soltys-jda.pdf
| volume = 7
| year = 2009}}.
*{{citation
| last1 = Kieffer | first1 = John C.
| last2 = Szpankowski | first2 = Wojciech
| contribution = On the Ehrenfeucht–Mycielski balance conjecture
| pages = 19–28
| series = Discrete Mathematics and Theoretical Computer Science
| title = Proc. Conf. Analysis of Algorithms (AofA 2007)
| url = http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAH0102
| year = 2007}}.
*{{citation
| last = Sutner | first = Klaus
| contribution = The Ehrenfeucht–Mycielski sequence
| doi = 10.1007/3-540-45089-0_26
| editor1-last = Ibarra | editor1-first = O. H.
| editor2-last = Dang | editor2-first = Z.
| pages = 73–82
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = [[Conference on Implementation and Application of Automata|Proc. Conf. Implementation and Application of Automata (CIAA 2003)]]
| url = http://www.cs.cmu.edu/afs/cs/Web/People/sutner/papers/em-sequence.pdf
| volume = 2759
| year = 2003}}.

==External links==
*[http://www.cs.cmu.edu/~cdm/Projects/em-sequence.pdf One million terms of the sequence] calculated by Terry McConnell at Syracuse University.

{{DEFAULTSORT:Ehrenfeucht-Mycielski sequence}}
[[Category:Binary sequences]]
[[Category:Pseudorandomness]]

Electro Thermal Dynamic Stripping Process - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10067)

en>Chris the speller: /* Extraction System */replaced: setup → set up; comma using AWB

en>Chowbok: clean up, removed: ™ (2), TM (33), added orphan tag using AWB

en>Chowbok: clean up, removed: ™ (2), ^TM (33), added orphan tag using AWB