## Algorithmic randomness, quantum physics, and incompleteness (2004)

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Venue: | PROCEEDINGS OF THE CONFERENCE “MACHINES, COMPUTATIONS AND UNIVERSALITY” (MCU’2004), LECTURES NOTES IN COMPUT. SCI. 3354 |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Calude04algorithmicrandomness,,

author = {Cristian S. Calude},

title = {Algorithmic randomness, quantum physics, and incompleteness},

booktitle = {PROCEEDINGS OF THE CONFERENCE “MACHINES, COMPUTATIONS AND UNIVERSALITY” (MCU’2004), LECTURES NOTES IN COMPUT. SCI. 3354},

year = {2004},

pages = {1--17},

publisher = {Springer}

}

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### Abstract

### Citations

696 |
Quantum Theory: Concepts and Methods
- Peres
- 1993
(Show Context)
Citation Context ... (v1,1, v2,1, . . . , vn,1) T is measured it yields outcome i with probability |vi,1| 2 . In this sense, according to Milburn (see [37], p. 1), the “physical reality is irreducibly random”. For Peres =-=[42]-=-, “in a strict sense quantum theory is a set of rules allowing the computation of probabilities for the outcomes of tests which follow specific preparations”. In the standard model (Copenhagen interpr... |

540 | 2002 A new kind of science
- Wolfram
(Show Context)
Citation Context ...be quantum phenomena. Being pragmatic, perhaps we can accept the randomness because of the immense success of the applications of quantum mechanics. But even here there is room for doubt. As Wolfram (=-=[54]-=-, p. 1064) has 2 The conclusion of [4] is : “We find no evidence for short- or long-term correlations in the intervals of the quantum jumps or in the decay of the quantum states, in agreement with qua... |

412 | Simulating physics with computers
- Feynman
- 1982
(Show Context)
Citation Context ...hine, not even on a probabilistic Turing machine (in which the available transitions are chosen randomly with equal probability at each step). The reason is Bell’s Theorem, which, in Feynman’s words (=-=[24]-=-, p. 476), reads: “It is impossible to represent the results of quantum mechanics with a classical universal device.” A recently proposed complexity-theoretic analysis [9] of Heisenberg’s uncertainty ... |

339 | A theory of program size formally identical to information theory
- Chaitin
- 1975
(Show Context)
Citation Context ...with computability theory is given by the following theorem: A set is computably enumerable (shortly, c.e.) iff can be generated by some self-delimiting Turing machine. The Omega number introduced in =-=[11]-=- ΩU = � U(x) stops 2 −|x| = 0.ω1ω2 . . . ωn . . . (1) is the halting probability of U; |x| denotes the length of the (binary) string x. Omega is one of the most important concepts in algorithmic infor... |

229 |
Statistical mechanics of cellular automata
- Wolfram
- 1983
(Show Context)
Citation Context ...e systems in which unobservably small causes can produce large effects) or pseudo-random numbers (generated by software functions; an elegant solution is the so-called “rule 30” discovered by Wolfram =-=[53]-=-). The strong uncomputability of algorithmic randomness is expressed by the theorem: The set of algorithmic random strings is immune. That is, no infinite set of algorithmic random strings is c.e. (se... |

221 |
Various techniques used in connection with random digits
- Neumann
- 1951
(Show Context)
Citation Context ...ect at first sight. Postprocessing algorithms can be used to remove bias from a sequence of quantum random numbers affected by bias. The simplest unbiasing procedure was first proposed by von Neumann =-=[38]-=-. 8 The bits of a sequence are grouped in strings of two bits. The strings 00 and 11 are discarded; the string 01 is replaced by 0 and the string 10 is replaced by 1. After this procedure, the bias is... |

216 |
Riemann's Hypothesis and Tests for Primality
- Miller
- 1976
(Show Context)
Citation Context ...n discovered (see [1] 6 ), in practice they are quite slow and do not pose any immediate risk to the security of electronic communication. Probabilistic algorithms, first discovered in the mid 1970s, =-=[41, 44]-=-, can help speed things up, but such probabilistic tests—which essentially use a coinflipping source of pseudo-random bits to search for a number’s factors—are only “probably” correct. 7 If you run th... |

163 |
Algorithmic Randomness and Complexity
- Downey, Hirshfeldt
- 2009
(Show Context)
Citation Context ...ions, complex finite set, (substantially) smaller program, etc. A convenient way is to code all objects as binary strings and use Turing machines as a model of computation. For technical reasons (see =-=[5, 21]-=-), our model is a self-delimiting Turing machine, that is a Turing machine C which processes binary strings into binary strings and has a prefix-free domain: if C(x) is defined and y is either a prope... |

147 |
Hilbert’s Tenth Problem
- Matiyasevich
- 1993
(Show Context)
Citation Context ...mulated by Goldbach 262 years ago in a letter to Euler (see [35]). Many other problems can be solved in a similar manner, including the famous Riemann Hypothesis (see [17]), as Matiyasevich proved in =-=[36]-=-, p. 121–122. 12 All theoretical proposals for transcending the Turing barrier (see, for example, [23, 8, 32]) have been challenged on grounds of physical in-feasibility (see [19]): they require infin... |

140 |
Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik
- Heisenberg
- 1927
(Show Context)
Citation Context ...s: “It is impossible to represent the results of quantum mechanics with a classical universal device.” A recently proposed complexity-theoretic analysis [9] of Heisenberg’s uncertainty principle (see =-=[27]-=-) reveals more facts. The uncertainty principle states that the more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. In its exact form (... |

124 |
Information and Randomness: An Algorithmic Perspective
- CALUDE
- 2002
(Show Context)
Citation Context ...dom, but it’s not truly random. Is quantum randomness “truly random”? Our working model of “truly random” is “algorithmic randomness” in the sense of Algorithmic Information Theory (see, for example, =-=[5]-=-). In this paper we compare quantum randomness with algorithmic randomness in an attempt to obtain partial answers to the following questions: Is randomness in quantum mechanics “algorithmically rando... |

100 |
Probabilistic algorithms
- Rabin
- 1976
(Show Context)
Citation Context ...n discovered (see [1] 6 ), in practice they are quite slow and do not pose any immediate risk to the security of electronic communication. Probabilistic algorithms, first discovered in the mid 1970s, =-=[41, 44]-=-, can help speed things up, but such probabilistic tests—which essentially use a coinflipping source of pseudo-random bits to search for a number’s factors—are only “probably” correct. 7 If you run th... |

66 | Non-Turing computations via Malament-Hogarth spacetimes
- Etesi, Nemeti
- 2002
(Show Context)
Citation Context ...n a similar manner, including the famous Riemann Hypothesis (see [17]), as Matiyasevich proved in [36], p. 121–122. 12 All theoretical proposals for transcending the Turing barrier (see, for example, =-=[23, 8, 32]-=-) have been challenged on grounds of physical in-feasibility (see [19]): they require infinite time, infinite memory resources (or both), infinite precision measurements, etc. It’s ironic that we now ... |

49 |
Relative to a random oracle P A = NP A = coNP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ...t in both of them as well as in the fast Monte-Carlo simulation, primes play a central role. 10 The insight provided by the refutation (see [33, 16]) of Bennett and Gill’s “Random Oracle Hypothesis”, =-=[3]-=-—which basically states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unrelativized case—suggests that random oracles are extreme... |

46 |
The Universal Computer, The Road from Leibniz to Turing. W.W
- Davis
- 2000
(Show Context)
Citation Context ..., would probably not surprise Leibniz, who designed a succession of mechanical calculators, wrote on the binary notation (in 1679) and proposed the famous “let us calculate” dictum; see more in Davis =-=[18]-=-, chapter one.sHC(x) > |x| − k, then C cannot compress more than k − 1 bits of x. A string x is algorithmically k–random with respect to C if the complexity HC(x) is maximal up to k among the complexi... |

34 |
Iterating von Neumann’s procedure for extracting random bits
- Peres
- 1992
(Show Context)
Citation Context ... shortened; in the case of von Neumann’s procedure, the length of the unbiased sequence will be at most 25% of the length of the raw sequence. Other, more efficient, unbiasing procedures exist. Peres =-=[43]-=- proved that the number of bits produced by iterating von Neumann’s procedure is arbitrarily close to the entropy bound. Thirdly, another open question is: Exactly how much more powerful a Turing mach... |

31 | Computing the noncomputable
- Kieu
- 2003
(Show Context)
Citation Context ...n a similar manner, including the famous Riemann Hypothesis (see [17]), as Matiyasevich proved in [36], p. 121–122. 12 All theoretical proposals for transcending the Turing barrier (see, for example, =-=[23, 8, 32]-=-) have been challenged on grounds of physical in-feasibility (see [19]): they require infinite time, infinite memory resources (or both), infinite precision measurements, etc. It’s ironic that we now ... |

28 |
The myth of hypercomputation
- Davis
- 2004
(Show Context)
Citation Context ...r not algorithmically random. Is this interesting? For some authors, the analysis of this type of ‘oracle’ machine is pointless and “one can only pity those engaged in this misguided enterprise” (cf. =-=[19]-=-, p. 207). As the reader arriving at this point can expect, I do not share this view. First, it seems that the computing device “PC plus a quantum generator of random bits”, whose existence can be har... |

24 | The random oracle hypothesis is false
- Chang, Chor, et al.
- 1994
(Show Context)
Citation Context ... [46] and [7], chapter 11). Maybe it’s not a random fact that in both of them as well as in the fast Monte-Carlo simulation, primes play a central role. 10 The insight provided by the refutation (see =-=[33, 16]-=-) of Bennett and Gill’s “Random Oracle Hypothesis”, [3]—which basically states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unre... |

22 | The quantum coin toss—testing microphysical undecidability
- Svozil
- 1990
(Show Context)
Citation Context ...t is being measured, or from the actual process of measurement.” So, what is the relation between algorithmic randomness and quantum randomness? A detailed discussion appears in Svozil [51] (see also =-=[50]-=-). Yurtsever [55] argued that a string of quantum random bits is, almost certainly, algorithmically random. Here we take a different approach. First and foremost, there is a strong computational simil... |

21 |
On the random oracle hypothesis
- Kurtz
- 1983
(Show Context)
Citation Context ... [46] and [7], chapter 11). Maybe it’s not a random fact that in both of them as well as in the fast Monte-Carlo simulation, primes play a central role. 10 The insight provided by the refutation (see =-=[33, 16]-=-) of Bennett and Gill’s “Random Oracle Hypothesis”, [3]—which basically states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unre... |

20 |
Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44
- Kennard
- 1927
(Show Context)
Citation Context ...ncertainty principle states that the more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. In its exact form (first published by Kennard =-=[31]-=-), for all normalized state vectors |Ψ〉, ∆p · ∆q ≥ ¯h/2, where ∆p and ∆q are standard deviations of momentum and position, i.e. ∆ 2 p = 〈Ψ|p 2 |Ψ〉 − 〈Ψ|p|Ψ〉 2 ; ∆ 2 q = 〈Ψ|q 2 |Ψ〉 − 〈Ψ|q|Ψ〉 2 . 3 It i... |

19 | What can be efficiently reduced to the Kolmogorov-random strings
- Allender, Buhrman, et al.
(Show Context)
Citation Context ...at random oracles are extremely powerful; contrast this scenario with the behaviour of probabilistic primality tests run with algorithmically random bits, cf. [15]. 11 Related results can be found in =-=[2]-=-. 12 A rough estimation shows that solving the Goldbach Conjecture is equivalent to deciding the halting status of a RAM program of less than 2,000 bits; for Riemann Hypothesis the program will have a... |

18 | Is independence an exception
- Calude, Jürgensen, et al.
- 1994
(Show Context)
Citation Context ...lexity higher than N is independent of the theory. On this base we can show (see [7]) that, probabilistically, incompleteness is widespread, thus complementing the result of Calude, Jürgensen, Zimand =-=[9]-=- stating that the set of unprovable statements is topologically large: Consider a consistent, sound, finitely-specified theory strong enough to formalise arithmetic. The probability that a statement o... |

18 |
A version of Ω for which ZFC can not predict a single bit
- Solovay
- 2000
(Show Context)
Citation Context ... 1’s in the expansion of ΩU =0.11 ...10 .... As soon as the first 0 appears, the theory becomes useless. If ΩV < 1/2, then the binary expansion of ΩV starts with 0, and so we obtain Solovay’s theorem =-=[51]-=-: Consider a consistent, sound, finitely-specified theory strong enough to formalise arithmetic. There effectively exists a universal machine V such that the theory can determine no digit of ΩV . We f... |

14 | Chaitin Ω numbers, Solovay machines and incompleteness, Theoret
- Calude
(Show Context)
Citation Context ...pleteness is not artificial—it’s ubiquitous, pervasive. We can ask ourselves: How large is the constant N in the above theorem? The answer depends on the chosen universal machine U. Indeed, in Calude =-=[6]-=- one proves the following result: Consider a consistent, sound, finitely-specified theory strong enough to formalise arithmetic. Then, for each universal machine U we can effectively construct a unive... |

14 | A fast and compact quantum random number generator
- Jennewein, Achleitner, et al.
- 2000
(Show Context)
Citation Context ...se of radioactive materials may cause health problems. Fig. 1. Optical system for generating quantum random bitssFortunately, a beam of light offers an excellent alternative source of randomness (see =-=[30]-=-). Light consists of elementary particles called photons; they exhibit in certain situations a random behaviour. The transmission upon a semitransparent mirror is an example. A photon generated by a s... |

13 |
quantum measurements
- Coins
(Show Context)
Citation Context ...n a similar manner, including the famous Riemann Hypothesis (see [17]), as Matiyasevich proved in [36], p. 121–122. 12 All theoretical proposals for transcending the Turing barrier (see, for example, =-=[23, 8, 32]-=-) have been challenged on grounds of physical in-feasibility (see [19]): they require infinite time, infinite memory resources (or both), infinite precision measurements, etc. It’s ironic that we now ... |

10 |
The Music of the Primes
- Sautoy, Marcus
- 2003
(Show Context)
Citation Context ...ustify mathematically because it rests on the assumption that quantum processes are genuinely random. The relation between the Riemann Hypothesis and quantum randomness seems to be more profound (see =-=[46]-=- and [7], chapter 11). Maybe it’s not a random fact that in both of them as well as in the fast Monte-Carlo simulation, primes play a central role. 10 The insight provided by the refutation (see [33, ... |

9 |
A version of Ω for which ZF C can not predict a single bit
- Solovay
- 2000
(Show Context)
Citation Context ...n the expansion of ΩU = 0.11 . . . 10 . . .. As soon as the first 0 appears, the theory becomes useless. If ΩV < 1/2, then the binary expansion of ΩV starts with 0, and so we obtain Solovay’s theorem =-=[49]-=-: Consider a consistent, sound, finitely-specified theory strong enough to formalise arithmetic. There effectively exists a universal machine V such that the theory can determine no digit of ΩV . We f... |

9 | Is Complexity a Source of Incompleteness
- Calude, Jürgensen
- 2004
(Show Context)
Citation Context ...exity. Of course, a statement with a large δ–complexity has also a large H-complexity, but the converse is not true. We can now state the complexity-theoretic theorem obtained in Calude and Jürgensen =-=[7]-=-: Consider a consistent, sound, finitely-specified theory strong enough to formalise arithmetic and denote by T its set of theorems. Then, there exists a constant N, which depends upon U and T , such ... |

7 |
Dynamics based computation
- Sinha, Ditto
- 1998
(Show Context)
Citation Context ...n using traditional pseudo-random number generation techniques and [20] for Nescape error. 8 Unbiasing is a compression procedure. 9 The idea of computing with deterministic chaos was investigated in =-=[47, 48]-=-.ssequence, but (theoretically) to an unbounded finite set of quantum random bit strings. 10 Can this immense power be exploited? 11 A superficial attack suggests that it is unlikely that a Turing mac... |

6 |
Discours de métaphysique
- Leibniz
- 1995
(Show Context)
Citation Context ...ntative and raises more questions than offers answers. 2 Algorithmic Randomness The main idea of Algorithmic Information Theory (shortly, AIT) was traced back in time (see [13, 14]) to Leibniz, 1686 (=-=[34]-=-, 40–41). If we have a finitesset of points (e.g. say, observations of an experiment), then one can find many mathematical formulae each of which produces a curve passing through them all, in the orde... |

6 |
Computing with distributed chaos
- Sinha, Ditto
- 1999
(Show Context)
Citation Context ...n using traditional pseudo-random number generation techniques and [20] for Nescape error. 8 Unbiasing is a compression procedure. 9 The idea of computing with deterministic chaos was investigated in =-=[47, 48]-=-.ssequence, but (theoretically) to an unbounded finite set of quantum random bit strings. 10 Can this immense power be exploited? 11 A superficial attack suggests that it is unlikely that a Turing mac... |

4 | Upper bound by Kolmogorov complexity for the probability in computable POVM measurement, Los Alamos preprint archive
- Tadaki
- 2002
(Show Context)
Citation Context ...s described by R: R(ω1 . . . ωs) is the measurement outcome, and tr(ρR(ω1 . . . ωs)) is the probability of getting that outcome when we measure ρ. Under these hypotheses, Tadaki’s inequality (1) (see =-=[52]-=-, p. 2), and the relation (2) imply the existence of a constant τ (depending upon R) such that for all ρ and s we have: ∆s . 1 ≥ τ. tr(ρR(ω1 . . . ωs)) In other words, there is no algorithm that, for ... |

3 |
The Feynman Processor. An Introduction to Quantum
- Milburn
- 1998
(Show Context)
Citation Context ...urement postulate: When a closed quantum physical system in state V = (v1,1, v2,1, . . . , vn,1) T is measured it yields outcome i with probability |vi,1| 2 . In this sense, according to Milburn (see =-=[37]-=-, p. 1), the “physical reality is irreducibly random”. For Peres [42], “in a strict sense quantum theory is a set of rules allowing the computation of probabilities for the outcomes of tests which fol... |

2 | Do the zeros of Riemann's zeta-function form a random sequence
- Calude, Hertling, et al.
- 1997
(Show Context)
Citation Context ...thematically because it rests on the assumption that quantum processes are genuinely random. The relation between the Riemann Hypothesis and quantum randomness seems to be more profound (see [46] and =-=[7]-=-, chapter 11). Maybe it’s not a random fact that in both of them as well as in the fast Monte-Carlo simulation, primes play a central role. 10 The insight provided by the refutation (see [33, 16]) of ... |

2 | A relation between correctness and randomness in the computation of probabilistic algorithms - Calude, Zimand - 1984 |

2 |
The Quest for Omega, Pantheon
- MATH
- 2005
(Show Context)
Citation Context ...leteness? Our analysis is tentative and raises more questions than offers answers. 2 Algorithmic Randomness The main idea of Algorithmic Information Theory (shortly, AIT) was traced back in time (see =-=[13, 14]-=-) to Leibniz, 1686 ([34], 40–41). If we have a finitesset of points (e.g. say, observations of an experiment), then one can find many mathematical formulae each of which produces a curve passing throu... |

1 |
Tests for non-randomness in quantum jumps, Los Alamos preprint archive, http://arxiv.org/abs/physics
- Berkeland, Raymondson, et al.
- 2004
(Show Context)
Citation Context ...perhaps we can accept the randomness because of the immense success of the applications of quantum mechanics. But even here there is room for doubt. As Wolfram ([54], p. 1064) has 2 The conclusion of =-=[4]-=- is : “We find no evidence for short- or long-term correlations in the intervals of the quantum jumps or in the decay of the quantum states, in agreement with quantum theory”.spointed out, “a priori, ... |

1 |
From Heinsenberg to Gödel via Chaitin
- Calude, Stay
- 2004
(Show Context)
Citation Context ... which, in Feynman’s words ([24], p. 476), reads: “It is impossible to represent the results of quantum mechanics with a classical universal device.” A recently proposed complexity-theoretic analysis =-=[9]-=- of Heisenberg’s uncertainty principle (see [27]) reveals more facts. The uncertainty principle states that the more precisely the position is determined, the less precisely the momentum is known in t... |

1 |
A note on Monte-Carlo primality tests and algorithmic information theory
- Chaitin, Schwartz
(Show Context)
Citation Context ...are only “probably” correct. 7 If you run the probabilistic algorithm using a source of algorithmically random bits, however, it would not only be fast, it would also be correct every single time (cf =-=[15]-=-). One of the principal tools used in computer simulation, known as fast Monte-Carlo algorithms, can derive a similar benefit from the use of algorithmically random numbers (cf. [10]; see more in [5])... |

1 |
L’Intelligence and le Calcul
- Delahaye
- 2002
(Show Context)
Citation Context ...asymptotic time complexity of the algorithm is bounded by (log n) q(log log n), where q is some polynomial. 7 See [22] for pitfalls in using traditional pseudo-random number generation techniques and =-=[20]-=- for Nescape error. 8 Unbiasing is a compression procedure. 9 The idea of computing with deterministic chaos was investigated in [47, 48].ssequence, but (theoretically) to an unbounded finite set of q... |

1 |
Generation of uniformly distributed random variables on electronic computers
- Golenko
- 1966
(Show Context)
Citation Context .... The basic table was produced during May and June 1947; exhaustive tests found small but statistically significant biases and adjustments were made. Some of the early methods can be found in Golenko =-=[26]-=- who describes noise generators based on a germanium triode, on a gas-discharge tube with magnet, on an electronic trigger circuit with a switch in its anode supply (photograph in Fig. 44), on a gasot... |

1 |
Email to C
- Oliver
- 2004
(Show Context)
Citation Context ...t dense matter”. Collisions destroy the purity of otherwise coherent states, so quantum randomness (as well as deterministic chaos) may be a manifestation of the incompleteness of dynamical laws, cf. =-=[39]-=-.sThe expectation value for energy, therefore, is exactly the same as that of Y , but with units of energy, i.e. ∆C(ω1ω2 . . . ωs)[J] · ∆s ≥ ε[J], where [J] indicates Joules of energy. Now define σQ ∆... |

1 | Quantum mechanics and algorithmic randomness
- Yurtsever
(Show Context)
Citation Context ...ed, or from the actual process of measurement.” So, what is the relation between algorithmic randomness and quantum randomness? A detailed discussion appears in Svozil [51] (see also [50]). Yurtsever =-=[55]-=- argued that a string of quantum random bits is, almost certainly, algorithmically random. Here we take a different approach. First and foremost, there is a strong computational similarity: both algor... |