Editing CYK algorithm (section)

==Example==
[[File:CYK algorithm animation showing every step of a sentence parsing.gif|thumb|upright=2|Sentence parsing using the CYK algorithm]]
This is an example grammar:

:<math chem>\begin{align}
\ce{S} & \ \ce{-> NP\ VP}\\
\ce{VP} & \ \ce{-> VP\ PP}\\
\ce{VP} & \ \ce{-> V\ NP}\\
\ce{VP} & \ \ce{-> eats}\\
\ce{PP} & \ \ce{-> P\ NP}\\
\ce{NP} & \ \ce{-> Det\ N}\\
\ce{NP} & \ \ce{-> she}\\
\ce{V} & \ \ce{-> eats}\\
\ce{P} & \ \ce{-> with}\\
\ce{N} & \ \ce{-> fish}\\
\ce{N} & \ \ce{-> fork}\\
\ce{Det} & \ \ce{-> a}
\end{align}</math>

Now the sentence ''she eats a fish with a fork'' is analyzed using the CYK algorithm. In the following table, in <math>P[i,j,k]</math>, {{mvar|i}} is the number of the row  (starting at the bottom at 1), and {{mvar|j}} is the number of the column (starting at the left at 1).

{| class="wikitable" style="text-align:center"
|+CYK table
|-
| '''S'''
|-
|       || VP
|-
|       || &nbsp;||
|-
| '''S'''     ||       ||      ||
|-
|       || VP    ||      ||      || PP
|-
| '''S'''||       || NP   ||      ||       || NP
|-
| NP    || V, VP || Det. || N    || P     || Det || N
|- style="border-top:3px solid grey;"
| she   || eats  || a    || fish || with  || a    || fork
|}

For readability, the CYK table for ''P'' is represented here as a 2-dimensional matrix ''M'' containing a set of non-terminal symbols, such that {{mvar|R<sub>k</sub>}} is in {{tmath|M[i,j]}} if, and only if, {{tmath|P[i,j,k]}}.
In the above example, since a start symbol ''S'' is in {{tmath|M[7,1]}}, the sentence can be generated by the grammar.