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A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Infinite computations with random oracles
We consider the following problem for various infinite time machines. If a
real is computable relative to large set of oracles such as a set of full
measure or just of positive measure, a comeager set, or a nonmeager Borel set,
is it already computable? We show that the answer is independent from ZFC for
ordinal time machines (OTMs) with and without ordinal parameters and give a
positive answer for most other machines. For instance, we consider, infinite
time Turing machines (ITTMs), unresetting and resetting infinite time register
machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible
ordinals \alpha
Turing machines can be efficiently simulated by the General Purpose Analog Computer
The Church-Turing thesis states that any sufficiently powerful computational
model which captures the notion of algorithm is computationally equivalent to
the Turing machine. This equivalence usually holds both at a computability
level and at a computational complexity level modulo polynomial reductions.
However, the situation is less clear in what concerns models of computation
using real numbers, and no analog of the Church-Turing thesis exists for this
case. Recently it was shown that some models of computation with real numbers
were equivalent from a computability perspective. In particular it was shown
that Shannon's General Purpose Analog Computer (GPAC) is equivalent to
Computable Analysis. However, little is known about what happens at a
computational complexity level. In this paper we shed some light on the
connections between this two models, from a computational complexity level, by
showing that, modulo polynomial reductions, computations of Turing machines can
be simulated by GPACs, without the need of using more (space) resources than
those used in the original Turing computation, as long as we are talking about
bounded computations. In other words, computations done by the GPAC are as
space-efficient as computations done in the context of Computable Analysis
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