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Register a new userOn low for speed oracles
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Added by
Laurent Bienvenu
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender introduced the concept of low for speed oracles. An oracle A is low for speed if relativizing to A has essentially no effect on computational complexity, meaning that if a decidable language can be decided in time $f(n)$ with access to oracle A, then it can be decided in time poly(f(n)) without any oracle. The existence of non-computable such As was later proven by Bayer and Slaman, who even constructed a computably enumerable one, and exhibited a number of properties of these oracles as well as interesting connections with computability theory. In this paper, we pursue this line of research, answering the questions left by Bayer and Slaman and give further evidence that the structure of the class of low for speed oracles is a very rich one.
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A classic result in algorithmic information theory is that every infinite binary sequence is computable from a Martin-Loef random infinite binary sequence. Proved independently by Kucera and Gacs, this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n + g(n) bits of the random oracle, then g is the redundancy of the computation. Kucera implicitly achieved redundancy n log n while Gacs used a more elaborate block-coding procedure which achieved redundancy sqrt(n) log n. Different approaches to coding such as the one by Merkle and Mihailovic have not improved this redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. Our redundancy bound is exponentially smaller than the previously best known bound and is known to be the best possible. It follows that redundancy r log n in computation from a random oracle is possible for every stream, if and only if r > 1.
The aim of this short note is mainly pedagogical. It summarizes some knowledge about Boolean satisfiability (SAT) and the P=NP? problem in an elementary mathematical language. A convenient scheme to visualize and manipulate CNF formulae is introduced. Also some results like the formulae for the number of unsatisfied clauses and the number of solutions might be unknown.
We prove that every key agreement protocol in the random oracle model in which the honest users make at most $n$ queries to the oracle can be broken by an adversary who makes $O(n^2)$ queries to the oracle. This improves on the previous $widetilde{Omega}(n^6)$ query attack given by Impagliazzo and Rudich (STOC 89) and resolves an open question posed by them. Our bound is optimal up to a constant factor since Merkle proposed a key agreement protocol in 1974 that can be easily implemented with $n$ queries to a random oracle and cannot be broken by any adversary who asks $o(n^2)$ queries.
For random CNF formulae with m clauses, n variables and an unrestricted number of literals per clause the transition from high to low satisfiability can be determined exactly for large n. The critical density m/n turns out to be strongly n-dependent, ccr = ln(2)/(1-p)^^n, where pn is the mean number of positive literals per clause.This is in contrast to restricted random SAT problems (random K-SAT), where the critical ratio m/n is a constant. All transition lines are calculated by the second moment method applied to the number of solutions N of a formula. In contrast to random K-SAT, the method does not fail for the unrestricted model, because long range interactions between solutions are not cut off by disorder.
A heuristic model procedure for determining satisfiability of CNF-formulae is set up and described by nonlinear recursion relations for m (number of clauses), n (number of variables) and clause filling k. The system mimicked by the recursion undergoes a sharp transition from bounded running times (easy) to uncontrolled runaway behaviour (hard). Thus the parameter space turns out to be separated into regions with qualitatively different efficiency of the model procedure. The transition results from a competition of exponential blow up by branching versus growing number of orthogonal clauses.
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