Anti-Complex Sets and Reducibilities with Tiny Use

Franklin, Johanna N. Y.; Greenberg, Noam; Stephan, Frank; Wu, Guohua

doi:10.2178/jsl.7804170

Cited by 9 publications

(12 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similarly, it is not true that X is anti-complex if and only if for every computable order f, K(X↾n) ≤ f (n) for almost every n. This follows from the fact established by Bienvenu and Downey [26,Theorem 4.3] that (i) there is a computable order g such that K(X↾n) ≤ g(n) + O(1) if and only if X is K-trivial, that is, K(X↾n) ≤ K(n)+O(1), and that (ii) not every anti-complex sequence is K-trivial (Franklin et al [25] show that every high degree contains an anti-complex sequence, but every K-trivial sequence has low Turing degree).…”

Section: Notions Of Non-complexitymentioning

confidence: 99%

See 1 more Smart Citation

Randomness for computable measures and initial segment complexity

Hölzl

Porter

2017

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2 ω , the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure µ, some sequence in this family fails to be random with respect to µ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2 ω , namely diminutive measures and trivial measures.

show abstract

Section: Notions Of Non-complexitymentioning

confidence: 99%

“…We first discuss anti-complex, proper sequences. Franklin et al [25] showed that every high degree contains an anti-complex sequence. The converse fails, as an anti-complex sequence need not compute a fast-growing function.…”

Section: Anti-complexity and Randomnessmentioning

confidence: 99%

Randomness for computable measures and initial segment complexity

Hölzl

Porter

2017

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

show abstract

“…Complexity and autocomplexity were introduced by Kanovich [29] and the relation with DNC functions and traceability was studied in [31]. Franklin, Greenberg, Stephan and Wu [18] introduced anticomplexity and studied the relation with traceability. We will give some characterizations of variants of complexity via traceability in Sect.…”

Section: Traceability and Complexitymentioning

confidence: 99%

“…More directly, Hölzl and Merkle [28] gave characterizations of traceability notions by complexity concepts with respect to prefix-free machines and total machines, where a partial computable function M : 2 <ω → 2 <ω is said to be a total machine if dom(M) is total. Franklin et al [18] also introduced the notion of anticomplex.…”

Section: Traceability and Complexitymentioning

confidence: 99%

See 1 more Smart Citation

Unified characterizations of lowness properties via Kolmogorov complexity

Kihara

Miyabe

2014

Arch. Math. Logic

View full text Add to dashboard Cite

Consider a randomness notion C. A uniform test in the sense of C is a total computable procedure that each oracle X produces a test relative to X in the sense of C. We say that a binary sequence Y is C-random uniformly relative to X if Y passes all uniform C tests relative to X . Suppose now we have a pair of randomness notions C and D where C ⊆ D, for instance Martin-Löf randomness and Schnorr randomness. Several authors have characterized classes of the form Low(C, D) which consist of the oracles X that are so feeble that C ⊆ D X . Our goal is to do the same when the randomness notion D is relativized uniformly: denote by Low (C, D) the class of oracles X such that every C-random is uniformly D-random relative to X . (1) We show that X ∈ Low (MLR, SR) if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Löf packing measure zero with respect to all computable dimension functions.(2) We also show that X ∈ Low (SR, WR) if and only if X is computably i.o. tt-traceable if and only if X is not totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

show abstract