On the Autoreducibility of Random Sequences

Abstract

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.

Derandomization of auctions

“…Such an objective might be for some of the players to guess their hat color correctly. Hat puzzles of various forms have been contemplated in the mathematics community, partially due to their connections to coding theory, discrepancy problems, and auto-reducibility of random sequences; and often simply because they make interesting brain-teasers [6,7,20,23]. Notice that the strategies in a hat puzzle have the same structure as the pricing function in the bid-independent representation of an incentive-compatible auction.…”

Hat Guessing Numbers of Degenerate Graphs

“…Many variations of this puzzle have been studied (see [9,14] for surveys), including one [5] where, like in this work, the sight graph is an arbitrary graph, rather than a clique. Hat guessing puzzles have found connections to coding theory [7], to auctions [1], to network coding [13,16], to finite dynamical systems [10], and possibly to understanding DNA [11].…”

On the hat problem on a graph

“…To prove that h(G)h(G) > α, it suffices to prove that (h(K n )) 2 > α, that is h(K n ) > √ α. The authors of [14] have proven that for the hat problem with n = 2 k −1 players there exists a strategy giving the chance of success (2 k −1)/2 k . Since lim k→∞ (2 k −1)/2 k = 1, for every α ∈ [1/4; 1) there exists a positive integer k such that for the hat problem with n = 2 k − 1 players there exists a strategy S such that p(S) ≥ 1 − 1/2 k = 1 − 1/(n + 1) > √ α.…”

“…The hat problem with 2 k − 1 players was solved in [14], and for 2 k players in [11]. The problem with n players was investigated in [7].…”

“…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.…”