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doi:10.1016/j.tcs.2005.11.001 
Copyright © 2005 Elsevier B.V. All rights reserved.
Received 2 November 2005;
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Abstract
We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes and
of functions
which are pointwise maximum of partial or total computable sequences of functions where
is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for
, leading to a variant
of Kolmogorov complexity. We characterize the orders
such that the enumeration theorem (resp. the invariance theorem) also holds for
. It turns out that
may satisfy the invariance theorem but not the enumeration theorem. Also, when
satisfies the invariance theorem then the Kolmogorov complexities associated to
and
are equal (up to a constant).
Letting , where
is the reverse order, we prove that either
(=ct is equality up to a constant) or
are
ct incomparable and and
. We characterize the orders leading to each case. We also show that
cannot be both much smaller than KD at any point.
These results are proved in a more general setting with two orders on D, one extending the other.
Keywords: Kolmogorov complexity; Infinite computation
