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Ramsey's theorem and products in the Weihrauch degrees [article]

Damir D. Dzhafarov, Jun Le Goh, Denis R. Hirschfeldt, Ludovic Patey, Arno Pauly
2019 arXiv   pre-print
We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem.  ...  Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for pairs (RT^2_2) is strictly Weihrauch below the parallel product of the stable Ramsey's theorem for  ...  In this paper, we turn the lens of Weihrauch reducibility on various results concerning Ramsey's theorem and its products with other mathematical principles.  ... 
arXiv:1804.10968v2 fatcat:445m77aomvfx5h6mv46nw6vism

ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY'S THEOREM

VASCO BRATTKA, TAHINA RAKOTONIAINA
2017 Journal of Symbolic Logic (JSL)  
We study the uniform computational content of Ramsey's theorem in the Weihrauch lattice.  ...  Our central results provide information on how Ramsey's theorem behaves under product, parallelization, and jumps.  ...  to the classification of the exact Borel degree of Ramsey's theorem in Corollary 4.19 .  ... 
doi:10.1017/jsl.2017.43 fatcat:dwflarkmszeupeqkq354jxt43y

An inside/outside Ramsey theorem and recursion theory [article]

Marta Fiori-Carones, Paul Shafer, Giovanni Soldà
2021 arXiv   pre-print
We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees.  ...  We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle  ...  We thank the  ... 
arXiv:2006.16969v2 fatcat:65zioadfzrfyvhxanelyqbs47e

Weihrauch Complexity in Computable Analysis [article]

Vasco Brattka, Guido Gherardi, Arno Pauly
2018 arXiv   pre-print
We provide a self-contained introduction into Weihrauch complexity and its applications to computable analysis.  ...  This includes a survey on some classification results and a discussion of the relation to other approaches.  ...  That CC [0,1] is not a cylinder was proved in [27, Theorem 9.5]. ⊓ ⊔  ... 
arXiv:1707.03202v4 fatcat:kifs2jmtfzgtnmylg6wj3oud2y

Measuring the Complexity of Computational Content (Dagstuhl Seminar 15392)

Vasco Brattka, Akitoshi Kawamura, Alberto Marcone, Arno Pauly, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 15392 "Measuring the Complexity of Computational Content: Weihrauch Reducibility and Reverse Analysis."  ...  It includes abstracts on most talks presented during the seminar, a list of open problems that were discussed and partially solved during the meeting as well as a bibliography on the seminar topic that  ...  Problem (Fernandes, Ferreira and Ferreira . How do different combinations of the stable version of Ramsey's Theorem for pairs and the cohesiveness problem compare to Ramsey's Theorem for pairs?  ... 
doi:10.4230/dagrep.5.9.77 dblp:journals/dagstuhl-reports/BrattkaKMP15 fatcat:vhiz2jlmsbbq7d53mwazzg53ku

On uniform relationships between combinatorial problems

François G. Dorais, Damir D. Dzhafarov, Jeffry L. Hirst, Joseph R. Mileti, Paul Shafer
2015 Transactions of the American Mathematical Society  
We also study Weak König's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature.  ...  We show, for instance, that for all n, j, k ≥ 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors  ...  The alternative product is called the co-product in the Weihrauch lattice, and was originally introduced in this context by Pauly [28] .) for all x < y < z.  ... 
doi:10.1090/tran/6465 fatcat:onf4yqdrw5f3nmvdlr57eqsz3a

On uniform relationships between combinatorial problems [article]

François G. Dorais, Damir D. Dzhafarov, Jeffry L. Hirst, Joseph R. Mileti, Paul Shafer
2015 arXiv   pre-print
We also study Weak König's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature.  ...  We show, for instance, that for all n,j,k ≥ 1, if j < k then Ramsey's theorem for n-tuples and k many colors does not uniformly reduce to Ramsey's theorem for j many colors.  ...  The alternative product is called the co-product in the Weihrauch lattice, and was originally introduced in this context by Pauly [28] .) for all x < y < z.  ... 
arXiv:1212.0157v5 fatcat:i6bax73y2zhale2za6tzayxoyy

The open and clopen Ramsey theorems in the Weihrauch lattice [article]

Alberto Marcone, Manlio Valenti
2021 arXiv   pre-print
We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice.  ...  We identify 8 different multivalued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong  ...  In Section 3 we will recall the precise statement for the open and clopen Ramsey theorems and prove some lemmas that will be useful in proving the results on the Weihrauch degrees.  ... 
arXiv:2003.04245v3 fatcat:yshhfkctlbfzxbtyzaaxam3xqu

On the algebraic structure of Weihrauch degrees [article]

Vasco Brattka, Arno Pauly
2018 arXiv   pre-print
We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure.  ...  Introducing the notion of an ideal with respect to the compositional product, we can consider suitable quotients of the Weihrauch degrees.  ...  Alternatively, we could match ≥ W to , and would obtain the exchange law as in Definition 78 (4) (cf. Proposition 38 (6) ).  ... 
arXiv:1604.08348v7 fatcat:n6hcxihfnrc43dejplr2eury7i

Finding descending sequences through ill-founded linear orders [article]

Jun Le Goh and Arno Pauly and Manlio Valenti
2021 arXiv   pre-print
To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree.  ...  In this work we investigate the Weihrauch degree of the problem 𝖣𝖲 of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem 𝖡𝖲 of finding  ...  This will come into play in Theorem 4.31 and in Theorem 4.33. Corollary 3.3. Det X (·) is an interior degree-theoretic operator on Weihrauch degrees, i.e.  ... 
arXiv:2010.03840v2 fatcat:ec5ajagq7jhy7kgwh6rkp7rarq

Degrees of incomputability, realizability and constructive reverse mathematics [article]

Takayuki Kihara
2020 arXiv   pre-print
In our setting, we make use of Weihrauch degrees (degrees of incomputability/discontinuity of partial multi-valued functions) to obtain Lifschitz-like relative realizability predicates.  ...  In this note, we present sample examples on how to lift some separation results on Weihrauch degrees to those over intuitionistic Zermelo-Fraenkel set theory IZF.  ...  The author's research was partially supported by JSPS KAKENHI Grant 19K03602, 15H03634, and the JSPS Coreto-Core Program (A. Advanced Research Networks).  ... 
arXiv:2002.10712v1 fatcat:2l7xoxzak5hl3feqflx5ynguve

Many-one reductions and the category of multivalued functions

ARNO PAULY
2015 Mathematical Structures in Computer Science  
However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).  ...  In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless  ...  Acknowledgements The author is tremendously grateful for the very helpful referee reports.  ... 
doi:10.1017/s0960129515000262 fatcat:m24jkc7si5eczam5lzcxksbmbq

SRT22 does not imply RT22 in omega-models [article]

Benoit Monin, Ludovic Patey
2019 arXiv   pre-print
The meta-mathematical study of Ramsey has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's theorem in 2012 about Ramsey's theorem  ...  We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014.  ...  Among the theorems studied in reverse mathematics, Ramsey's theorem received a special attention from the community, since Ramsey's theorem for pairs historically was the first theorem known to escape  ... 
arXiv:1905.08427v3 fatcat:tmmqon4djbaflbman6g4rd7uvi

Milliken's tree theorem and its applications: a computability-theoretic perspective [article]

Paul-Elliot Anglès d'Auriac, Peter A. Cholak, Damir D. Dzhafarov, Benoît Monin, Ludovic Patey
2021 arXiv   pre-print
Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences.  ...  in the case of Devlin's theorem.  ...  Weihrauch reducibility was introduced by Weihrauch [50] in the 1990s, and has since been widely deployed in computable analysis and other fields; for a recent survey, see Brattka, Gherardi, and Pauly  ... 
arXiv:2007.09739v2 fatcat:kpp3nalf75b5tlnqeeqqpwcs4i

Reduction games, provability, and compactness [article]

Damir D. Dzhafarov and Denis R. Hirschfeldt and Sarah C. Reitzes
2021 arXiv   pre-print
Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between Π^1_2 principles over  ...  exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game.  ...  As discussed in [7] , and in more detail in [8] , this top degree is usually added to the lattice of Weihrauch degrees as a formal object.  ... 
arXiv:2008.00907v2 fatcat:xteewlogc5gfhg6nacr75k6e2m
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