Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits

Ebert, Todd; Merkle, Wolfgang

doi:10.1007/3-540-45687-2_18

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…The hat problem and its variations have many applications and connections to different areas of science (for a survey on this topic, see [22]), for example: information technology [5], linear programming [16], genetic programming [9], economics [1,18], biology [17], approximating Boolean functions [2], and autoreducibility of random sequences [3,[12][13][14][15]. Therefore, it is hoped that the hat problem on a graph considered in this paper is worth exploring as a natural generalization, and may also have many applications.…”

Section: Introductionmentioning

confidence: 99%

On the hat problem on a graph

Krzywkowski¹

2012

OpMath

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The topic of this paper is the hat problem in which each of \(n\) players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities

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Section: Introductionmentioning

confidence: 99%

On the hat problem on a graph

Krzywkowski¹

2012

OpMath

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“…Related work. The current article is an extended joint version of conference articles by Ebert and Vollmer [21] and Ebert and Merkle [20]. The hat problem was originally introduced in the literature by Ebert [19] in order to illustrate the problem of autoreducing random sequences.…”

Section: Introductionmentioning

confidence: 99%

On the Autoreducibility of Random Sequences

Ebert¹,

Merkle²,

Vollmer³

2003

SIAM J. Comput.

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A binary sequence A = A(0)A(1). .. is called infinitely often (i.o.) Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don't-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table-autoreducible. We obtain the somewhat counterintuitive result that every Martin-Löf random sequence, in fact even every rec-random or p-random sequence, is i.o. truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o. autoreducing a random sequence. We show that rec-random sequences are never i.o. truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for Martin-Löf random sequences and i.o. Turing autoreducibility. On the other hand, we show that for any rational-valued computable function r that goes nonascendingly to zero, any rec-random sequence is i.o. truth-table-autoreducible such that on any prefix of length m at least a fraction of r(m) of the m bits in the prefix are guessed. We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o. autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o. autoreducibility.

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On the Construction of Effective Random Sets

Merkle

Mihailovic

2002

Lecture Notes in Computer Science

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Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits

Cited by 3 publications

References 14 publications

On the hat problem on a graph

On the hat problem on a graph

On the Autoreducibility of Random Sequences

On the Construction of Effective Random Sets

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