Aspects of Chaitin’s Omega

Barmpalias, George

doi:10.1017/9781108781718.007

Cited by 6 publications

(3 citation statements)

References 45 publications

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“…In the same way, if we are interested in the rationality of finite agents-e.g., human beings and the machines that they build-then there is something unsatisfactory about any account of rationality that requires agents to be able to perform tasks that are impossible for finite agents-solving the Halting Problem, say. 4 We will see in Section 4 that each of our paradigm examples of theories of chance is unsatisfactory in this way. So there is a challenge here for those attracted to reductionist theories of chance and to the Principal Principle: to develop and to defend examples of reductionist theories of chance that are CCP-satisfiable by computable priors (or to explain how norms of rationality can require agents to perform tasks that are impossible for computable agents).…”

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confidence: 99%

“…For early results of this kind, see[54, 55]. For elaborations, see[36,41] 4. For a dissenting perspective, see[19, sec.…”

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confidence: 99%

“… 20 So here I am denying that numbers that can in a sense be defined but which cannot be computably approximated in a controlled way can play any role in science—maybe that is a mistake. For the most famous examples of such numbers, see, e.g., [4]. …”

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confidence: 99%

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Unprincipled

Belot

2023

The Review of Symbolic Logic

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It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped.

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“…For early results of this kind, see[54, 55]. For elaborations, see[36,41] 4. For a dissenting perspective, see[19, sec.…”

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confidence: 99%

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Unprincipled

Belot

2023

The Review of Symbolic Logic

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Chaitin’s Ω as a Continuous Function

Hölzl¹,

Merkle²,

Miller³

et al. 2019

J. symb. log.

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We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left( X \right)$ is rational.

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Some Questions of Uniformity in Algorithmic Randomness

Bienvenu¹,

Csima²,

Harrison-Trainor³

2021

J. symb. log.

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The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $ , a universal prefix-free machine U whose halting probability is $\alpha $ . We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $ , one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.

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Aspects of Chaitin’s Omega

Cited by 6 publications

References 45 publications

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Unprincipled

Chaitin’s Ω as a Continuous Function

Some Questions of Uniformity in Algorithmic Randomness

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