Feasible Analysis, Randomness, and Base Invariance

Figueira, Santiago; Nies, André

doi:10.1007/s00224-013-9507-7

Cited by 10 publications

(23 citation statements)

References 24 publications

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“…In §5.4 we show that the cumulative distribution function of µ M is polynomial time computable when restricted to β-adic inputs. Finally, in §5.5 we show the main result via an 'almost Lipschitz' property, as in [6].…”

Section: Polynomial Time Randomnessmentioning

confidence: 79%

“…Then Proof. The proof of [6,Lemma 6] basically works in this case. The only difference is that here N is real-valued instead of rational-valued.…”

Section: The Savings Propertymentioning

confidence: 96%

“…Proof. As in [6,Proposition], proof is by induction on the length of σ. For the inductive step, we only need notice that P (σb|σ) is well defined and positive.…”

Section: The Savings Propertymentioning

confidence: 99%

“…A result of Schnorr [16] states that if x is polynomial time random in base b then x is normal in base b. It was recently shown [6] that polynomial time randomness is base invariant, so that being polynomial time random in a single base implies being normal for all bases, i.e. being absolutely normal.…”

Section: Introductionmentioning

confidence: 99%

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Normality in non-integer bases and polynomial time randomness

Almarza

Figueira

2015

Journal of Computer and System Sciences

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It is known that if x ∈ [0, 1] is polynomial time random then x is normal in any integer base greater than one. We show that if x is polynomial time random and β > 1 is Pisot, then x is "normal in base β", in the sense that the sequence (xβ n ) n∈N is uniformly distributed modulo one. We work with the notion of P -martingale, a generalization of martingales to non-uniform distributions, and show that a sequence over a finite alphabet is distributed according to an irreducible, invariant Markov measure P if an only if no P -martingale whose betting factors are computed by a deterministic finite automaton succeeds on it. This is a generalization of Schnorr and Stimm's characterization of normal sequences in integer bases. Our results use tools and techniques from symbolic dynamics, together with automata theory and algorithmic randomness.

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Section: Polynomial Time Randomnessmentioning

confidence: 79%

“…Then Proof. The proof of [6,Lemma 6] basically works in this case. The only difference is that here N is real-valued instead of rational-valued.…”

Section: The Savings Propertymentioning

confidence: 96%

“…Proof. As in [6,Proposition], proof is by induction on the length of σ. For the inductive step, we only need notice that P (σb|σ) is well defined and positive.…”

Section: The Savings Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Normality in non-integer bases and polynomial time randomness

Almarza

Figueira

2015

Journal of Computer and System Sciences

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“…[21]). Given a program for the time function t, one can compute a t-random sequence in deterministic time O(t(n) · log(t(n)) · n 3 ).…”

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confidence: 99%

Nonsignaling Deterministic Models for Nonlocal Correlations have to be Uncomputable

et al. 2017

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Quantum mechanics postulates random outcomes. However, a model making the same output predictions but in a deterministic manner would be, in principle, experimentally indistinguishable from quantum theory. In this work we consider such models in the context of non-locality on a device independent scenario. That is, we study pairs of non-local boxes that produce their outputs deterministically. It is known that, for these boxes to be non-local, at least one of the boxes' output has to depend on the other party's input via some kind of hidden signaling. We prove that, if the deterministic mechanism is also algorithmic, there is a protocol which, with the sole knowledge of any upper bound on the time complexity of such algorithm, extracts that hidden signaling and uses it for the communication of information. Bell nonlocality [1] makes us choose between determinism and the non-signaling principle [2]. That is, if one wants to account in a deterministic manner for the non-local correlations that quantum mechanics predicts and which we are now almost certain [3][4][5] that Nature exhibits, one must allow for the existence of some kind of signaling mechanism that links distant measurement choices and outcomes. But, since quantum correlations are non-signaling, such signaling mechanism must be restricted to the so-called hidden variables, and not reach the phenomenological level. Known examples of deterministic non-local theories violating the non-signaling principle (also referred to as parameter independence [6]) at the hidden-variable level are: the hidden variable model with communication of Toner and Bacon [7] and, more prominently, Bohmian mechanics [8]. For those models that use classical communication to mimic nonlocality, one can in fact study the amount of communication needed (see, for example, [9-11]).A reasonable feature that one would expect of any physical model is that it is computable [12]. This means that, in principle, one should be able to write a computer program that given a description of an experiment (that is, the measurement choices and the state of the system) outputs the model's outcomes predictions (these being probabilities in the case of quantum mechanics).Our main result is that, on the contrary, deterministic models of non-local correlations need to be uncomputable if we want to prevent those correlations from being signaling. In other words, we show that if the deterministic model is computable, the hidden signaling mechanism used to exhibit non-locality can be extracted at the observation level and used for the communication of information. More specifically, we give a protocol to perform one-way communication between two observers holding computable non-local boxes.There are a few previous results on this direction. First, this result has a flavour similar to [13]. However, we obtain our result in a device-independent scenario, that is, without assuming quantum mechanics, and provide an explicit communication protocol. Second, in [14,15] it is shown that some non-local boxes fed with...

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Normal Numbers and Computer Science

Becher

Carton

2018

Trends in Mathematics

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Feasible Analysis, Randomness, and Base Invariance

Cited by 10 publications

References 24 publications

Normality in non-integer bases and polynomial time randomness

Normality in non-integer bases and polynomial time randomness

Nonsignaling Deterministic Models for Nonlocal Correlations have to be Uncomputable

Normal Numbers and Computer Science

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