9,434 research outputs found
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
Computational Soundness of Formal Encryption in Coq
We formalize Abadi and Rogaway's computational soundness result in the
Coq interactive theorem prover. This requires to model notions of provable
cryptography like indistinguishability between ensembles of
probability distributions, PPT reductions, and security notions for
encryption schemes.
Our formalization is the first computational soundness result to be
mechanized, and it shows the feasibility of rigorous reasoning of
computational cryptography inside a generic interactive theorem prover
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
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The universality of polynomial time Turing equivalence
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees
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