248 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
Pseudodeterministic constructions in subexponential time
We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence {pn}n∈N of increasing primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often.
This result follows from a much more general theorem about pseudodeterministic constructions. A property Q ⊆ {0, 1}* is γ-dense if for large enough n, |Q ⋂ {0, 1}n| ≥ γ2n. We show that for each c > 0 at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family {Hn} of sets, Hn ⊆ {0, 1}n, such that for each (1=nc)-dense property Q ∈ DTIME(n^c) and every large enough n, Hn ⋂ Q ≠∅; or (2) There is a deterministic sub-exponential time construction of a family {H'n} of sets, H'n ⊆ {0, 1}n, such that for each (1/n^c)-dense property Q ∈ DTIME(n^c) and for infinitely many values of n, H'n ⋂ Q ≠∅.
We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.</p
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