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Computable structures of rank omega_1^CK
[article]

2005
*
arXiv
*
pre-print

We obtain a

arXiv:math/0508507v1
fatcat:bu3gyfj6s5he7enhji37hkknry
*computable*structure*of**Scott**rank*omega_1^*CK*(call this ock),*and*give a general coding procedure that transforms any hyperarithmetical structure A into a*computable*structure A' such that ... the*rank**of*A is ock, ock+1, or < ock iff the same is true*of*A'. ... Makkai [9] produced an example*of*an arithmetical structure*of**Scott**rank*ω*CK*1 . Here we obtain a*computable*structure*of**Scott**rank*ω*CK*1 in two different ways. ...##
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Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

2014
*
Mathematical Structures in Computer Science
*

A natural proper subclass

doi:10.1017/s096012951300025x
fatcat:awgtb4wbinhepdxiwbqng6xs7a
*of**approximation*spaces coincides with the class*of*quasi-Polish spaces. ... We study the Borel*and*Hausdorff difference hierarchies in*approximation*spaces, revisiting the work done for the other topological spaces. ... Moreover, in (iv-vi), one can suppose that, for each n ∈ N, the set*of*elements with*rank*exactly n*and*that with*ranks*in {ωα + n α < ω*CK*1 } are*computable*. Remark 5.10. ...##
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The determined property of Baire in reverse math
[article]

2018
*
arXiv
*
pre-print

We define the notion

arXiv:1809.03940v2
fatcat:gcdbwftcwvffxcxrj6haz3xypa
*of*a determined Borel code in reverse math,*and*consider the principle DPB, which states that every determined Borel set has the property*of*Baire. ... Any ω-model*of*DPB must be closed under hyperarithmetic reduction, but DPB is not a theory*of*hyperarithmetic analysis. ... There is a Z-*computable*function which, given the index*of*a truly well-founded Z-*computable**tree*, outputs an element*of*O Z which bounds the*rank**of*the*tree*. ...##
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Scott ranks of models of a theory

2018
*
Advances in Mathematics
*

We also answer a question

doi:10.1016/j.aim.2018.03.012
fatcat:3wwd5rdtprecbid5scvbfwpbyi
*of*Knight*and*Calvert by showing that there are*computable*models*of*high*Scott**rank*which are not*computably**approximable*by models*of*low*Scott**rank*. ... Finally, we answer a question*of*Sacks*and*Marker by showing that δ 1 2 is the least ordinal α such that if the models*of*a*computable*theory T have*Scott**rank*bounded below*ω1*, then their*Scott**ranks*... We show here that there are*computable*structures*of**Scott**rank*ω*CK*1*and*ω*CK*1 +1 which are not strongly*computably**approximable*. Theorem 4. ...##
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Wadge hardness in Scott spaces and its effectivization

2014
*
Mathematical Structures in Computer Science
*

The length

doi:10.1017/s0960129513000248
fatcat:alsjcmqztjbqhitovg7vfotko4
*of*these chains is the*rank**of*the considered class,*and*each element in one chain is incomparable with all the elements in the other chain. ... We prove some results on the Wadge order on the space*of*sets*of*natural numbers endowed with*Scott*topology,*and*more generally, on omega-continuous domains. ... Acknowledgements: The authors are indebted to an anonymous referee for a huge number*of*improvements in our original manuscript. ...##
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On bi-embeddable categoricity of algebraic structures
[article]

2020
*
arXiv
*
pre-print

We furthermore show that this is the best one can do: Let L be a

arXiv:2005.07829v1
fatcat:rozqmdb77bflfoxdzaktxuz53a
*computable*linear order*of*Hausdorff*rank*n≥ 1, then 0^(2n-2) does not*compute*embeddings between it*and*all its*computable*bi-embeddable ... We show that if L is a*computable*linear order*of*Hausdorff*rank*n, then for every bi-embeddable copy*of*it there is an embedding*computable*in 2n-1 jumps from the atomic diagrams. ... As L is indecomposable*of**rank*n + 1, there is an h-indecomposable linear order together with its signed*tree*T*of**rank*n + 1. ...##
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Turing degrees in Polish spaces and decomposability of Borel functions
[article]

2016
*
arXiv
*
pre-print

Additionally we prove results about the transfinite version as well as the

arXiv:1410.1052v2
fatcat:rxyszurelbgg5l7p7awfojh5km
*computable*version*of*the Decomposability Conjecture,*and*we explore the idea*of*applying the technique*of*turning Borel-measurable ... Our techniques employ deep results from effective descriptive set theory*and*recursion theory. ... Project no: 294962*COMPUTAL*,*and*the second named author was partially supported by a Grant-in-Aid for JSPS fellows. ...##
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Model Theory in Computer Science: My Own Recurrent Themes Forty Years Ago

unpublished

I review my own experiences in research

fatcat:mo7rciama5eonibpvtztfjfhry
*and*the management*of*science. ... Barwise showed that the admissible fragments*of*L*ω1*,ω satisfy the Craig Interpolation Theorem. D.*Scott**and*, independently before, E. ... Barvinok [4] If M has*rank*at most k, per(M ) can be*computed*in polynomial time, where the constants depend on k. ...##
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Analysis of spatially correlated functional data objects

2019

First, we had to generalise the methodologies

doi:10.5525/gla.thesis.71942
fatcat:rpcec34djneoplfkftva3y2dmq
*and*codes*of*both*of*these methods to analyse data with features they were not originally designed for. ... There are several approaches for analysis spatially correlated functional data, but most*of*them are designed for specific applications*and*there is no easy way*of*comparing these methods. ... Later, Wood et al. (2008) introduced a group*of*smoothers that consist*of*a low*rank*basis*and*a quadratic penalty. ...##
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Locality and complexity in simulations of complex quantum systems
[article]

2015

Moreover, Markovian dynamics is quasi-local

doi:10.17169/refubium-11923
fatcat:weifcouxhfdyhnlsox5kdx5yge
*and*can be locally simulated on classical*computers*with a cost scaling polynomially in the system size. ... However, also a major roadblock for making such simulations reliable is identified: Testing positivity*of*certain common*approximations*to mixed quantum states, called matrix product operators, is shown ... Campbell,*and*Rodrigo Gallego for insightful discussions*and**Scott*Aaronson*and*Alex Arkhipov for useful criticism. ...