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Computable structures of rank omega_1^CK [article]

Julia Knight, Jessica Millar
2005 arXiv   pre-print
We obtain a 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.  ... 
arXiv:math/0508507v1 fatcat:bu3gyfj6s5he7enhji37hkknry

Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

2014 Mathematical Structures in Computer Science  
A natural proper subclass 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.  ... 
doi:10.1017/s096012951300025x fatcat:awgtb4wbinhepdxiwbqng6xs7a

The determined property of Baire in reverse math [article]

Eric P. Astor, Damir Dzhafarov, Antonio Montalbán, Reed Solomon, Linda Brown Westrick
2018 arXiv   pre-print
We define the notion 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.  ... 
arXiv:1809.03940v2 fatcat:gcdbwftcwvffxcxrj6haz3xypa

Scott ranks of models of a theory

Matthew Harrison-Trainor
2018 Advances in Mathematics  
We also answer a question 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.  ... 
doi:10.1016/j.aim.2018.03.012 fatcat:3wwd5rdtprecbid5scvbfwpbyi

Wadge hardness in Scott spaces and its effectivization

2014 Mathematical Structures in Computer Science  
The length 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.  ... 
doi:10.1017/s0960129513000248 fatcat:alsjcmqztjbqhitovg7vfotko4

On bi-embeddable categoricity of algebraic structures [article]

Nikolay Bazhenov, Dino Rossegger, Maxim Zubkov
2020 arXiv   pre-print
We furthermore show that this is the best one can do: Let L be a 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.  ... 
arXiv:2005.07829v1 fatcat:rozqmdb77bflfoxdzaktxuz53a

Turing degrees in Polish spaces and decomposability of Borel functions [article]

Vassilios Gregoriades, Takayuki Kihara, Keng Meng Ng
2016 arXiv   pre-print
Additionally we prove results about the transfinite version as well as the 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.  ... 
arXiv:1410.1052v2 fatcat:rxyszurelbgg5l7p7awfojh5km

Model Theory in Computer Science: My Own Recurrent Themes Forty Years Ago

Johann Makowsky
I review my own experiences in research 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.  ... 

Analysis of spatially correlated functional data objects

Salihah Safar Alghamdi
First, we had to generalise the methodologies 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.  ... 
doi:10.5525/gla.thesis.71942 fatcat:rpcec34djneoplfkftva3y2dmq

Locality and complexity in simulations of complex quantum systems [article]

Martin Kliesch, Universitätsbibliothek Der FU Berlin, Universitätsbibliothek Der FU Berlin
Moreover, Markovian dynamics is quasi-local 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.  ... 
doi:10.17169/refubium-11923 fatcat:weifcouxhfdyhnlsox5kdx5yge