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Register a new userLayerwise computability and image randomness
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Added by
Alexander Shen
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only if it has a random preimage. This result (for computable distributions and mappings, and Martin-Lof randomness) was known for a long time (folklore); in this paper we prove its natural generalization for layerwise computable mappings, and discuss the related quantitative results.
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Many constructions in computability theory rely on time tricks. In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which $Psi(A) e X$ for any higher functional~$Psi$ which is consistent on all oracles. We construct an oracle~$A$ relative to which there is no universal higher ML-test. On the other hand, we show that badness has its limits: there are no higher self-PA oracles, and for no~$A$ can we construct a higher $A$-c.e. set which is also higher $A$-ML-random. We study various classes of bad oracles and differentiate between them using other familiar classes. For example, bad oracles for consistent reductions can be higher ML-random, whereas bad oracles for universal tests cannot.
The paper considers quantitati
Randomness plays a central rol in the quantum mechanical description of our interactions. We review the relationship between the violation of Bell inequalities, non signaling and randomness. We discuss the challenge in defining a random string, and show that algorithmic information theory provides a necessary condition for randomness using Borel normality. We close with a view on incomputablity and its implications in physics.
Steganographic protocols enable one to embed covert messages into inconspicuous data over a public communication channel in such a way that no one, aside from the sender and the intended receiver, can even detect the presence of the secret message. In this paper, we provide a new provably-secure, private-key steganographic encryption protocol secure in the framework of Hopper et al. We first present a one-time stegosystem that allows two parties to transmit messages of length at most that of the shared key with information-theoretic security guarantees. The employment of a pseudorandom generator (PRG) permits secure transmission of longer messages in the same way that such a generator allows the use of one-time pad encryption for messages longer than the key in symmetric encryption. The advantage of our construction, compared to all previous work is randomness efficiency: in the information theoretic setting our protocol embeds a message of length n bits using a shared secret key of length (1+o(1))n bits while achieving security 2^{-n/log^{O(1)}n}; simply put this gives a rate of key over message that is 1 as n tends to infinity (the previous best result achieved a constant rate greater than 1 regardless of the security offered). In this sense, our protocol is the first truly randomness efficient steganographic system. Furthermore, in our protocol, we can permit a portion of the shared secret key to be public while retaining precisely n private key bits. In this setting, by separating the public and the private randomness of the shared key, we achieve security of 2^{-n}. Our result comes as an effect of the application of randomness extractors to stegosystem design. To the best of our knowledge this is the first time extractors have been applied in steganography.
We investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several characterizations. We show that some natural effectivizations of quasi-metric spaces are strictly stronger.
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