Randomness? What Randomness?

Landsman, Klaas

doi:10.1007/s10701-020-00318-8

Cited by 21 publications

(54 citation statements)

References 102 publications

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“…25 Every last scrap of our external experience is of virtual reality (Deutsch, 2011). 26 For discussion see Svozil (2018) and Landsman (2019). programmable device that can carry out any algorithm it is given (Moore and Mertens, 2011).…”

Section: Cortical Operators Include Unitary Operatorsmentioning

confidence: 99%

“…In the cortex these distributions may not be static -as Koch said, a machine changing its instruction set as a function of its input (Koch, 1999;Ebert and Wegner, 2011). This model's kind of randomness (freedom in algebraic basis) captures both senses of a lack of predictability and a measure of (un)certainty (Landsman, 2019). Humans are a proven source of true "random" numbers for some computational states -insofar as any sequence may be considered "random".…”

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confidence: 99%

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The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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A model of computation describes how units of information and an organizational architecture transform input to output. In the cerebral cortex the unit of information is a normalized collection of neuronal membrane potentials representing a vector of complex amplitudes. The organizational architecture of the cortex allows for programming operator matrices in dendritic arborizations on vectors of complex amplitudes. Three well-documented facts are sufficient to define the cortical model of computation: first, cortical dendritic inputs to compute membrane potentials are, in general, complex numbers (amplitudes) that may be configured as state vectors in complex Hilbert spaces; second, normalization is a canonical neural computation, specifically, for vectors of cortical membrane amplitudes representing orthogonal states; and, third, dendritic arborizations are programmable devices in a universal sense for mathematical and logical functions. These three facts, well studied and accepted for years, are all that is needed for a simple, well-defined, and well-known model of computation. A great deal of research is presently devoted to understanding particular neural algorithms, circuits, and programming. Understanding the algorithm that a particular circuit implements is not to be confused with identifying the much more basic underlying model of computation. The analogy for classical computing is that the classical model of computing stores and manipulates information fundamentally in the form of bits – essentially a finite set of possible states, typically, {0,1}. The primitive classical container of information is analogous to a two-state switch. Nevertheless, sophisticated computational circuitry and algorithms are developed on that simple classical realization. In contrast to the finite set of states found in the fundamental unit of classical information, the fundamental units of information in the cerebral cortex form a model of computation over the complex field in normalized sets of mutually inhibitory amplitudes. This means that the cortex represents information in a fundamental container of information (a vector of amplitudes in membrane potentials) that can encode discrete states over a continuum of values (no doubt down to some constant.) A well-known result from probability theory and standard neuroimaging analysis argue that these mutually inhibitory fundamental units of information may normalize complex amplitudes under the 2-norm. Sufficiently sophisticated computational means are available in the cortex to program unitary operators, state reduction and expansion operators and tensor products that operate on vectors of normalized membrane amplitudes. This model of computation is typically viewed as a probabilistic or predictive model of computation. Again, the model of computation is not a particular circuit or implementation of any one algorithm. The model of computation is the arena in which cortical circuits and algorithms are implemented. The fundamental arena then for cortical computations is a field of complex amplitudes normalized for probabilistic computation.

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Section: Cortical Operators Include Unitary Operatorsmentioning

confidence: 99%

mentioning

confidence: 99%

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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“…This shows that experimental verification might be a hard task (if possible at all). Another aspect which matters here is the fact that infinite binary sequences which represent infinite process of fair coin tossing is impossible classically and its realisation require nondeterministic theory like QM 4 . Thus any verification of randomness as above should presumably be designed as a quantum process.…”

Section: Introduction -Randomness Formalization and Quantum Probabilitymentioning

confidence: 99%

“…The method adopted in this work relies on formal power of models of set theory and on showing that they are necessary in some essential respect in QM. The fundamental role of randomness of binary sequences in QM has been already analysed from different perspectives in a variety of works [2][3][4][6][7][8][9] . Our concern here is to uncover even more fundamental role of models of set theory underlying both QM and randomness of the sequences.…”

Section: Introduction -Randomness Formalization and Quantum Probabilitymentioning

confidence: 99%

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Random World and Quantum Mechanics

Król¹,

Bielas²,

Asselmeyer-Maluga³

2021

Preprint

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Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

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Indeterminism and Undecidability

Landsman

2021

Undecidability, Uncomputability, and Unpredictability

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The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Gödel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e. almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories. Contents 1 Introduction: Gödel and Bell 2 Randomness and its unprovability 3 Rethinking Bell's theorem 4 Are deterministic hidden variable theories deterministic? 5 Conclusion and discussion A The Born rule B 1-Randomness C Bell's theorem and free will theorem References

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Randomness? What Randomness?

Cited by 21 publications

References 102 publications

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Random World and Quantum Mechanics

Indeterminism and Undecidability

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