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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}}.
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==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''<sup>3</sup>) 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<sup>''i''</sup>&nbsp;+&nbsp;''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&nbsp;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]]

Latest revision as of 18:59, 29 March 2014



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The repair blade variety and the manage are two various parts held collectively by a nut. This is what makes it weak and someday the two pieces ultimately independent. Pick a knife in which the blade goes all the way via the manage. Fastened sort is the most basic variety and it has a solitary piece of steel related to a handle and is created to tackle challenging work. Knives are between the most vital requirements for a kitchen. A correct knife can make your function a fantastic deal less difficult, also as faster. It's as a result crucial which you have a established of wonderful knives inside of your kitchen. study a lot more
Even though these pocket knives or multi-instruments are easily accessible a lot of gentlemen nevertheless opt to carry a straightforward variation like the jack knife which has a straightforward hinged blade and nothing more or a pen knife. The pen knife was at first created to reduce or sharpen pen quills for producing. The pen knife is good if you want much more than a single sort of blade. Excellent Knife! I purchased it and it shaved four inches of my leg hair. Razor sharp, durable serrations. I used it to pores and skin a squirrel and it still isn't really dull! Great knife. Fantastic job Gerber!
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For any individual that is looking at diverse designs for knives are the ability to merge usefulness, security and some of the most current designs in knives via folding blade knives The several types and appears with these various knives are known for delivering some of the least difficult resources to use for your chopping wants, while supplying you a mixture of functions that makes slicing and reducing even less difficult. I've got this knife recently in Malaysia. It's a sharp knife. I punctured a hole in my bolster when I accidently dropped this knife from only a ten centimetre drop. Hard knife for a folding blade. You will find not regret in receiving this knife.
This is without a question a single of the greater kniveshave owned. It is durable and trustworthy and it has yet to are unsuccessful me! Exceptional acquire if you will not want to get the entire sized sheath knife. An superb everyday have knife. Little and easily concealed however sturdy and razor sharp. Can be used for just about any work you need to have a knife for and is just as durable as any set blade knife. I certainly reccomend this to anybody hunting for a very good folding knife. I really like this knife.I set it on the stop of a adhere and killed a cat that by some means got under my residence.excellent knife
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MercWorx is an American business that prides alone in producing excellent, substantial-quality knives that are reliable and incredibly successful. Proof of their need to regularly produce quality knives, MercWorx knives are produced by one craftsmen, not equipment. Every knife is manufactured and assemble by a solitary individual, who will constantly examination and change it to make certain that the knife is in key condition. Amongst the very best companies producing conventional pocket knives are Situation Knives (or Scenario XX Knives ), Buck Knives , Queen Cutlery, Mooremaker, and(Wonderful Jap Cutlery). Some first rate, but cheaper and lesser quality historically-styled knives are currently being made by Tough Rider, Steel Warrior, and Remington.