Graphs are not universal for online computability

Downey, Rodney G.; Harrison-Trainor, Matthew; Kalimullin, I. Sh.; Melnikov, Alexander G.; Turetsky, Daniel

doi:10.1016/j.jcss.2020.02.004

Cited by 15 publications

(16 citation statements)

References 26 publications

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“…This is fine to delay one point from the diagram of A. Just consider the next point u in The sketch above describes the worst case scenario, but there will be various cases which also had to be incorporated into the formal argument in [19]. To maintain the inequality |N A (pφ(x))| = |N B (φ(x))| at every stage we will have to consider the cases when either N A (pφ(x)) or N B (φ(x)) starts growing faster than anticipated.…”

Section: Punctual Versions Of Known Resultsmentioning

confidence: 99%

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Foundations of Online Structure Theory

Bazhenov¹,

Downey²,

Kalimullin³

et al. 2019

Bull. symb. log

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Section: Punctual Versions Of Known Resultsmentioning

confidence: 99%

“…Theorem 8.1 ( [19]). The class of structures with only one binary functional symbol is punctually universal.…”

Section: Problem 66 Investigate Into the Punctual Degrees Of Some Ementioning

confidence: 99%

Foundations of Online Structure Theory

Bazhenov¹,

Downey²,

Kalimullin³

et al. 2019

Bull. symb. log

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“…So, for instance, the transformations witnessing punctual universality preserve not only punctual categoricity but also computable categoricity and its generalization ∆ 0 α -categoricity (see [1]). A natural example of a punctually universal class is the class of structures in the language of one binary functional symbol [9]. It is well-known that graphs are Turing universal.…”

Section: Categoricity and Universalitymentioning

confidence: 99%

“…It is well-known that graphs are Turing universal. It has recently been discovered that every punctually categorical graph becomes automorphically trivial after fixing finitely many constants [9]. In particular, every punctually categorical graph must be (relatively) computably categorical.…”

Section: Categoricity and Universalitymentioning

confidence: 99%

Punctual Categoricity and Universality

Downey¹,

Greenberg²,

Melnikov³

et al. 2020

J. symb. log.

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We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

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“…We note that except the reductions ≤ c and ≤ tc , there are many other approaches to comparing computability-theoretic complexity of classes of structures. These approaches include: transferring degree spectra and other algorithmic properties [13], Σ-reducibility [8,17], computable functors [11,16], Borel functors [12], primitive recursive functors [1,7], etc.…”

Section: For Any a B ∈ K 0 We Have A ∼ = B If And Only If φ(A) ∼ =mentioning

confidence: 99%

A Note on Computable Embeddings for Ordinals and Their Reverses

Bazhenov

Vatev

2020

Lecture Notes in Computer Science

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We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that although {ω • 2, ω • 2} is computably embeddable in {ω 2 , (ω 2) }, the class {ω • k, ω • k} is not computably embeddable in {ω 2 , (ω 2) } for any natural number k ≥ 3.

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Graphs are not universal for online computability

Cited by 15 publications

References 26 publications

Foundations of Online Structure Theory

Foundations of Online Structure Theory

Punctual Categoricity and Universality

A Note on Computable Embeddings for Ordinals and Their Reverses

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