scholarly journals Quantum algorithmic randomness

2021 ◽  
Vol 62 (2) ◽  
pp. 022202
Author(s):  
Tejas Bhojraj
Keyword(s):  
Algorithmic Randomness

DEEP CLASSES

2016 ◽  
Vol 22 (2) ◽  
pp. 249-286 ◽  
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Mutual Information ◽  
Initial Segment ◽  
Computability Theory ◽  
Binary Sequences ◽  
Algorithmic Randomness ◽  
Positive Probability ◽  
Probabilistic Algorithm ◽  
Image Position ◽  
The Relationship ◽  
Mass Problems

AbstractA set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.

Algorithmic randomness

2019 ◽  
Vol 62 (5) ◽  
pp. 70-80
Author(s):  
Rod Downey ◽  
Denis R. Hirschfeldt
Keyword(s):  
Algorithmic Randomness

RELATIVIZING CHAITIN'S HALTING PROBABILITY

2005 ◽  
Vol 05 (02) ◽  
pp. 167-192 ◽  
Author(s):  
ROD DOWNEY ◽  
DENIS R. HIRSCHFELDT ◽  
JOSEPH S. MILLER ◽  
ANDRÉ NIES
Keyword(s):  
Computability Theory ◽  
Algorithmic Randomness ◽  
Jump Operator ◽  
Random Real ◽  
Primary Object ◽  
Object Of Study ◽  
Free Machine ◽  
Computable Functions ◽  
Omega Operator

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

scholarly journals Randomness, relativization and Turing degrees

2005 ◽  
Vol 70 (2) ◽  
pp. 515-535 ◽  
Author(s):  
André Nies ◽  
Frank Stephan ◽  
Sebastiaan A. Terwijn
Keyword(s):  
Random Sets ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Schnorr Randomness ◽  
Minimal Pairs ◽  
High Degree

AbstractWe compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅(n − 1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ ∣x∣ − c. The ‘only if’ direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity.Next we prove some results on lowness. Among other things, we characterize the 2-random sets as those l-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω. sets coincide with the r.e. K-trivial ones.Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees.

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