The present work studies clustering from an abstract point of view
and investigates its properties in the framework of inductive inference.
Any class $S$ considered is given by a numbering
$A_0,A_1,...$ of nonempty subsets of the natural numbers
or the rational k-dimensional vector space as a hypothesis space.
A clustering ... more >>>

{\bf Abstract}

Isometries on formal power series over the finite field $\ff_2$
or on $2$--adic integers can be
computed by invertible transducers on inputs from $\{0,1\}^\infty$.
We consider the structural complexity of an isometry $f$,
measured as {\it tree complexity} $T(f,h)$, $h$ the tree height
[H.~Niederreiter, M.~Vielhaber, {\it J.~Cpx.}, ... more >>>

In this short note we show that for any integer k, there are
languages in the complexity class PP that do not have Boolean
circuits of size $n^k$.

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