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Changes plural to singular when describing a family of complexity cla…
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jfinkels committed Dec 6, 2016
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Expand Up @@ -110,7 +110,7 @@ \subsection{Definition of \texorpdfstring{$\para \NC$}{paraNC}}

If the depth of the circuit is instead bounded by $f(k) O(\log^d n)$, the class is denoted $\para \NC^{d \uparrow}$, a superclass of $\para \NC^d$.
If the circuits are of unbounded fan-in, the classes are $\para \AC^d$ and $\para \AC^{d \uparrow}$, respectively.
The classes $\para \AC^{d \uparrow}$ were first defined in \autocite{bst15}.
The class $\para \AC^{d \uparrow}$ was first defined in \autocite{bst15}.

A subtle point is that the value of the parameter $\kappa(x)$ must be non-constant but also independent of the size of the instance $x$ for the parameterized problem to be interesting.
First, if $\kappa(x)$ were bounded above by a constant for each $x$, then the parameter would be irrelevant and the problem would simply be in the standard complexity class $\NC^d$.
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