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| {{Unreferenced|date=December 2009}}
| | Translator Philip Geddes from Cooksville, loves to spend some time horse riding, castle clash hack and ornithology. Last year just made a trip Tombs of Buganda Kings at Kasubi.<br><br>Here is my web page; [https://www.facebook.com/pages/Castle-Clash-Hack/262424220600747 castle clash hack no download] |
| The '''right quotient''' (or simply '''quotient''') of a [[formal language]] <math>L_1</math> with a formal language <math>L_2</math> is the language consisting of strings ''w'' such that ''wx'' is in <math>L_1</math> for some string ''x'' in <math>L_2</math>. In symbols, we write:
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| :<math>L_1 / L_2 = \{w \ | \ \exists x ((x \in L_2) \land (wx \in L_1)) \}</math>
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| In other words, each string in <math>L_1 / L_2</math> is the prefix of a string <math>wx</math> in <math>L_1</math>, with the remainder of the word being a string in <math>L_2</math>.
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| ==Examples==
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| Consider
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| <math>L_1 = \{ a^n b^n c^n \ \ |\ \ n\ge 0 \}</math>
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| <math>L_2 = \{ b^i c^j \ \ | \ \ i,j\ge 0 \}</math>.
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| Now, if we insert a divider into the middle of an element of <math>L_1</math>, the part on the right is in <math>L_2</math> only if the divider is placed adjacent to a ''b'' (in which case ''i'' ≤ ''n'' and ''j'' = ''n'') or adjacent to a ''c'' (in which case ''i'' = 0 and ''j'' ≤ ''n''). The part on the left, therefore, will be either <math>a^n b^{n-i}</math> or <math>a^n b^n c^{n-j}</math>; and <math>L_1 / L_2</math> can be written as
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| <center><math>\{ a^p b^q c^r \ \ | \ \ p = q \ge r \ \ \or \ \ p \ge q \and r = 0 \}</math>.</center>
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| ==Properties==
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| Some common closure properties of the right quotient include:
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| * The quotient of a [[regular language]] with any other language is regular.
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| * The quotient of a [[context free language]] with a regular language is context free.
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| * The quotient of two context free languages can be any [[recursively enumerable]] language.
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| * The quotient of two recursively enumerable languages is recursively enumerable.
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| ==Left and right quotients==
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| There is a related notion of [[left quotient]], which keeps the postfixes of <math>L_1</math> without the prefixes in <math>L_2</math>. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.
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| {{DEFAULTSORT:Right Quotient}}
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| [[Category:Formal languages]]
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Translator Philip Geddes from Cooksville, loves to spend some time horse riding, castle clash hack and ornithology. Last year just made a trip Tombs of Buganda Kings at Kasubi.
Here is my web page; castle clash hack no download