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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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TL;DR: In this article, the authors studied low levels of the arithmetical hierarchy of non-uniformly computable functions on Baire space and showed that error-bounded-learnability is equivalent to finite piecewise computability.
Abstract: Every computable function has to be continuous To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability For instance, we show that mind-change-bounded-learnability is equivalent to finite $(\Pi^0_1)_2$-piecewise computability (where $(\Pi^0_1)_2$ denotes the difference of two $\Pi^0_1$ sets), error-bounded-learnability is equivalent to finite $\Delta^0_2$-piecewise computability, and learnability is equivalent to countable $\Pi^0_1$-piecewise computability (equivalently, countable $\Sigma^0_2$-piecewise computability) Second, we introduce disjunction-like operations such as the coproduct based on BHK-like interpretations, and then, we see that these operations induce Galois connections between the Medvedev degree structure and associated Medvedev/Muchnik-like degree structures Finally, we interpret these results in the context of the Weihrauch degrees and Wadge-like games
9 citations
TL;DR: This paper shows that (a) the problem of determining the number of attractors in a given compact set is in general undecidable, even for analytic systems and (b) the attractors are semi-computable for stable systems.
Abstract: In this paper we explore the problem of computing attractors and their respective basins of attraction for continuous-time planar dynamical systems. We consider C1 systems and show that stability is in general necessary (but may not be sufficient) to attain computability. In particular, we show that (a) the problem of determining the number of attractors in a given compact set is in general undecidable, even for analytic systems and (b) the attractors are semi-computable for stable systems. We also show that the basins of attraction are semi-computable if and only if the system is stable.
9 citations
TL;DR: It seems that a robust and well defined notion of time complexity exists for the GPAC, or equivalently for computations by polynomial ordinary differential equations, as well as a rather nice and robust notion of ODE programming.
9 citations
01 Jan 2006
9 citations
TL;DR: In this article, it was shown that the Julia set of the Feigenbaum map is computable in polynomial time with a recurrent critical point, which is the first poly-time computable Julia set with a critical point.
Abstract: We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.
9 citations