Noam Greenberg, Matthew Harrison-Trainor, Joseph S. Miller, Dan Turetsky
Slaman and Wehner independently built a family of sets with the property that every non-computable degree can compute an enumeration of the family, but there is no computable enumeration of the family. We call such a family a Slaman–Wehner family. The original Slaman–Wehner argument relies on all sets in the family constructed being finite, and in particular, it diagonalizes against computably enumerated families using only finite differences. In this paper we ask whether this is a necessary feature, that is, whether there is a Slaman–Wehner family closed under finite differences. This question remains open but we obtain a number of interesting partial results which can be interpreted as saying that the question is quite hard. First of all, no Slaman–Wehner family closed under finite differences can contain a finite set, and the enumeration of the family from a non-computable degree cannot be uniform (whereas, in the Slaman–Wehner construction, it is uniform). On the other hand, we build the following examples of families closed under finite differences which show the impossibility of several natural attempts to show that no Slaman–Wehner family exists: (1) a family that can be enumerated by every non-low degree, but not by any low degree; (2) a family that can be enumerated by any set in a given uniform list of c.e. sets, but which cannot be enumerated computably; and (3) a family that can be enumerated by a given Δ 2 0 set, but which cannot be computably enumerated.
{"title":"Enumerations of families closed under finite differences","authors":"Noam Greenberg, Matthew Harrison-Trainor, Joseph S. Miller, Dan Turetsky","doi":"10.3233/com-210349","DOIUrl":"https://doi.org/10.3233/com-210349","url":null,"abstract":"Slaman and Wehner independently built a family of sets with the property that every non-computable degree can compute an enumeration of the family, but there is no computable enumeration of the family. We call such a family a Slaman–Wehner family. The original Slaman–Wehner argument relies on all sets in the family constructed being finite, and in particular, it diagonalizes against computably enumerated families using only finite differences. In this paper we ask whether this is a necessary feature, that is, whether there is a Slaman–Wehner family closed under finite differences. This question remains open but we obtain a number of interesting partial results which can be interpreted as saying that the question is quite hard. First of all, no Slaman–Wehner family closed under finite differences can contain a finite set, and the enumeration of the family from a non-computable degree cannot be uniform (whereas, in the Slaman–Wehner construction, it is uniform). On the other hand, we build the following examples of families closed under finite differences which show the impossibility of several natural attempts to show that no Slaman–Wehner family exists: (1) a family that can be enumerated by every non-low degree, but not by any low degree; (2) a family that can be enumerated by any set in a given uniform list of c.e. sets, but which cannot be enumerated computably; and (3) a family that can be enumerated by a given Δ 2 0 set, but which cannot be computably enumerated.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135167787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.
如果对于每一个递增的可计算函数r: n→n，序列(d (ar (n + 1)， ar (n))) n可计算地收敛于0，我们称度量空间中元素的序列(a n) n可计算柯西。我们证明了存在一个几乎可计算的柯西无界有理数递增序列。如果存在一个收敛于α且柯西近似可计算的可计算有理数序列(an) n，则称实数α近似可计算。很明显，每个可计算的实数都是几乎可计算的，并且从Downey和LaForte(理论计算机科学284(2002)539-555)的结果可以得出，存在一个不可计算的几乎可计算和左可计算的数。我们观察到近可计算实数集合是一个实闭域，在具有开定义域的可计算实数函数下闭，而在任意可计算实数函数下不闭。除其他外，我们加强了Hoyrup(计算系统理论60(2017)396-420)和Stephan和Wu(新计算范式)的结果。第一次欧洲可计算性会议，CiE 2005, Proceedings (2005) 461-469 Springer)通过证明任何不可计算的近可计算实数是弱1-泛型(因此，超免疫而不是Martin-Löf随机)和强库尔茨随机(因此，不是K-trivial)，并且我们通过证明没有快速简单集可以图灵约简为近可计算实数来加强Downey和LaForte(理论计算机科学284(2002)539-555)的结果。
{"title":"Nearly computable real numbers","authors":"Peter Hertling, Philip Janicki","doi":"10.3233/com-230445","DOIUrl":"https://doi.org/10.3233/com-230445","url":null,"abstract":"We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Overt choice was recently introduced and thoroughly studied by de Brecht, Pauly and Schröder. They give estimates on the Weihrauch degree of overt choice on various spaces, and relate it to the topological properties of the space. In this article, we pursue this line of research, answering some of the questions that were left open. We show that overt choice on the rationals is not limit-computable. We identify the Weihrauch degree of overt choice on the space of natural numbers with the co-finite topology. We prove that the quasi-Polish spaces are the countably-based T 0 -spaces on which a variant of overt choice, called Π ~ 2 0 overt choice, is continuous. It extends a previous result that holds in the class of T 1 -spaces. We also prove an effective version of this equivalence.
{"title":"Notes on overt choice","authors":"Mathieu Hoyrup","doi":"10.3233/com-230458","DOIUrl":"https://doi.org/10.3233/com-230458","url":null,"abstract":"Overt choice was recently introduced and thoroughly studied by de Brecht, Pauly and Schröder. They give estimates on the Weihrauch degree of overt choice on various spaces, and relate it to the topological properties of the space. In this article, we pursue this line of research, answering some of the questions that were left open. We show that overt choice on the rationals is not limit-computable. We identify the Weihrauch degree of overt choice on the space of natural numbers with the co-finite topology. We prove that the quasi-Polish spaces are the countably-based T 0 -spaces on which a variant of overt choice, called Π ~ 2 0 overt choice, is continuous. It extends a previous result that holds in the class of T 1 -spaces. We also prove an effective version of this equivalence.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136356844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of eventually uniformly weak truth table array computable (e.u.wtt-a.c.) sets. As our main result, we show that a computably enumerable (c.e.) set has this property iff it is weak truth table ( wtt-) reducible to a maximal set. Moreover, in this equivalence we may replace maximal sets by quasi-maximal sets, hyperhypersimple sets or dense simple sets and we may replace wtt-reducibility by identity-bounded Turing reducibility (or any intermediate reducibility). Here, a set A is e.u.wtt-a.c. if there is an effective procedure which, for any given partial wtt-functional Φ ˆ, yields a computable approximation g ( x , s ) of the domain of Φ ˆ A together with a computable indicator function k ( x , s ) and a computable order h ( x ) such that, once the indicator becomes positive, i.e., k ( x , s ) = 1, the number of the mind changes of the approximation g on x after stage s is bounded by h ( x ) where, for total Φ ˆ A , the indicator eventually becomes positive on almost all arguments x of Φ ˆ A . In addition to our main result, we show several properties of the computably enumerable e.u.wtt-a.c. sets. For instance, the class of these sets is closed downwards under wtt-reductions and closed under join. Moreover, we relate this class to – and separate it from – well known classes in the literature. On the one hand, the class of the wtt-degrees of the c.e. e.u.wtt-a.c. sets is strictly contained in the class of the array computable c.e. wtt-degrees. On the other hand, every bounded low set is e.u.wtt-a.c. but there are e.u.wtt-a.c. c.e. sets which are not bounded low. Here a set A is bounded low if A † ⩽ wtt ∅ † , i.e., if A † is ω-c.a., where A † is the wtt-jump of A (Anderson, Csima and Lange (Archive for Mathematical Logic 56(5–6) (2017) 507–521)). Finally, we prove that there is a strict hierarchy within the class of the bounded low c.e. sets A depending on the order h that bounds the number of mind changes of a computable approximation of A † , and we show that there exists a Turing complete set A such that A † is h-c.a. for any computable order h with h ( 0 ) > 0.
我们引入了最终一致弱真值表数组可计算集的概念。作为我们的主要结果，我们证明了一个可计算枚举集(c.e)如果是弱真值表(wtt-)可约为极大集就具有这个性质。此外，在这个等价中，我们可以用拟极大集、超超简单集或密集简单集代替极大集，也可以用单位有界图灵可约性(或任何中间可约性)代替wtt可约性。这里，集合a是e.u wtt- ac。如果有一个有效的过程,对于任何给定的部分wtt-functionalΦˆ,产生一个可计算的近似g (x, s)域的Φˆ连同一个可计算的指标函数k (x,年代)和一个可计算的h (x),这样,一旦指标变得积极,也就是说,k (x) = 1,心灵的数量变化的x在舞台上近似g s以h (x)为界,总Φˆ,该指标最终对Φ * A的几乎所有参数x都变为正值。除了我们的主要结果之外，我们还展示了可计算枚举e.u.t -a.c的几个性质。集。例如，这些集合的类在wtt-约简下是向下闭的，在连接下是闭的。此外，我们将这类与文献中著名的类联系起来，并将其分开。一方面，cce的wtt度的类。set严格包含在可计算数组的类中，例如wtt-degrees。另一方面，每一个有界的低集都是e.u wtt-a.c。但也有e.u.wtt- ac。非低有界的集合。在这里，如果a†≥wtt∅†，即a†为ω- ca，则集合a有界为低。，其中A†为A (Anderson, Csima and Lange (Archive for Mathematical Logic 56(5-6)(2017) 507-521))的wtt-jump。最后，我们证明了在有界低c.e.集合a类中存在一个严格的层次结构，该层次结构依赖于约束a†的可计算近似的思想变化数的阶数h，并且我们证明了存在一个图灵完备集a，使得a†为h-c.a。对于h (0) >0.
{"title":"Lowness properties for strong reducibilities and the computational power of maximal sets","authors":"Klaus Ambos-Spies, Rod Downey, Martin Monath","doi":"10.3233/com-220432","DOIUrl":"https://doi.org/10.3233/com-220432","url":null,"abstract":"We introduce the notion of eventually uniformly weak truth table array computable (e.u.wtt-a.c.) sets. As our main result, we show that a computably enumerable (c.e.) set has this property iff it is weak truth table ( wtt-) reducible to a maximal set. Moreover, in this equivalence we may replace maximal sets by quasi-maximal sets, hyperhypersimple sets or dense simple sets and we may replace wtt-reducibility by identity-bounded Turing reducibility (or any intermediate reducibility). Here, a set A is e.u.wtt-a.c. if there is an effective procedure which, for any given partial wtt-functional Φ ˆ, yields a computable approximation g ( x , s ) of the domain of Φ ˆ A together with a computable indicator function k ( x , s ) and a computable order h ( x ) such that, once the indicator becomes positive, i.e., k ( x , s ) = 1, the number of the mind changes of the approximation g on x after stage s is bounded by h ( x ) where, for total Φ ˆ A , the indicator eventually becomes positive on almost all arguments x of Φ ˆ A . In addition to our main result, we show several properties of the computably enumerable e.u.wtt-a.c. sets. For instance, the class of these sets is closed downwards under wtt-reductions and closed under join. Moreover, we relate this class to – and separate it from – well known classes in the literature. On the one hand, the class of the wtt-degrees of the c.e. e.u.wtt-a.c. sets is strictly contained in the class of the array computable c.e. wtt-degrees. On the other hand, every bounded low set is e.u.wtt-a.c. but there are e.u.wtt-a.c. c.e. sets which are not bounded low. Here a set A is bounded low if A † ⩽ wtt ∅ † , i.e., if A † is ω-c.a., where A † is the wtt-jump of A (Anderson, Csima and Lange (Archive for Mathematical Logic 56(5–6) (2017) 507–521)). Finally, we prove that there is a strict hierarchy within the class of the bounded low c.e. sets A depending on the order h that bounds the number of mind changes of a computable approximation of A † , and we show that there exists a Turing complete set A such that A † is h-c.a. for any computable order h with h ( 0 ) > 0.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135898173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rademacher (Mathematische Annalen 87 (1922) 112–138), Steinhaus (Mathematische Zeitschrift 31 (1930) 408–416) and Paley and Zygmund (Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 337–257, Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 458–474, Mathematical Proceedings of the Cambridge Philosophical Society 28 (1932) 190–205) initiated the extensive study of random series. Using the theory of algorithmic randomness, which is a mix of computability theory and probability theory, we investigate the effective content of some classical theorems. We discuss how this is related to an old question of Kahane and Bollobás. We also discuss how considerations of such algorithmic questions about random series seem to lead to new notions of algorithmic randomness.
{"title":"Algorithmically random series","authors":"Rodney G. Downey, Noam Greenberg, Andrew Tanggara","doi":"10.3233/com-220433","DOIUrl":"https://doi.org/10.3233/com-220433","url":null,"abstract":"Rademacher (Mathematische Annalen 87 (1922) 112–138), Steinhaus (Mathematische Zeitschrift 31 (1930) 408–416) and Paley and Zygmund (Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 337–257, Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 458–474, Mathematical Proceedings of the Cambridge Philosophical Society 28 (1932) 190–205) initiated the extensive study of random series. Using the theory of algorithmic randomness, which is a mix of computability theory and probability theory, we investigate the effective content of some classical theorems. We discuss how this is related to an old question of Kahane and Bollobás. We also discuss how considerations of such algorithmic questions about random series seem to lead to new notions of algorithmic randomness.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the primitive recursive content of various closure results in algebra and model theory, including the algebraic, the real, and the differential closure theorems. In the case of ordered fields and their real closures, our result settles a question recently raised by Selivanova and Selivanov.
{"title":"Punctually presented structures I: Closure theorems","authors":"M. Dorzhieva, A. Melnikov","doi":"10.3233/com-230448","DOIUrl":"https://doi.org/10.3233/com-230448","url":null,"abstract":"We study the primitive recursive content of various closure results in algebra and model theory, including the algebraic, the real, and the differential closure theorems. In the case of ordered fields and their real closures, our result settles a question recently raised by Selivanova and Selivanov.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74558222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that W f ( x ) ≠ W x for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x ↦ W x with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.
Arslanov完备性判据指出一个c.e.集合a是图灵完备的，当且仅当存在一个没有不动点的a -可计算函数f，即对于每一个整数x，存在一个使得W f (x)≠W x的函数f。最近，Barendregt和Terwijn证明了当我们用一个任意的预完备可计算的编号代替Gödel编号x × W x时，完备性判据仍然成立。本文利用(预)完全可计算编号和不动点性质证明了c.e.集的不可计算性和高度性准则。我们还分别找到了相对于图灵完备和非图灵完备的任意族的预完备和弱预完备编号。
{"title":"Numberings, c.e. oracles, and fixed points","authors":"M. Faizrahmanov","doi":"10.3233/com-210387","DOIUrl":"https://doi.org/10.3233/com-210387","url":null,"abstract":"The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that W f ( x ) ≠ W x for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x ↦ W x with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82669109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove various sufficient conditions for the effective infinity of classes of computable numberings. Then we apply them to show that for every computable family of left-c.e. reals without the greatest element the class of its Friedberg computable numberings is effectively infinite. In particular, this result covers the families of all left-c.e. and all Martin-Löf random left-c.e. reals whose Friedberg computable numberings have been constructed by Broadhead and Kjos-Hanssen in their paper (In Mathematical Theory and Computational Practice, CiE 2009 (2009) 49–58 Springer). In addition, for every infinite computable family of left-c.e. reals we prove that the classes of all its computable, positive and minimal numberings are effectively infinite.
证明了一类可计算数的有效无穷的各种充分条件。然后应用它们证明了对于每一个可计算的左-c - e族。没有最大元的实数，它的弗里德伯格可计算数的类实际上是无限的。特别地，这个结果涵盖了所有左-c - e的科。和所有Martin-Löf随机左-c。在Broadhead和Kjos-Hanssen的论文(in Mathematical Theory and Computational Practice, CiE 2009 (2009) 49-58 Springer)中，他们构建了弗里德伯格可计算数。此外，对于每一个无限可计算的左-c族。实数证明了其所有可计算数、正数和极小数的类是有效无穷的。
{"title":"Effectively infinite classes of numberings and computable families of reals","authors":"M. Faizrahmanov, Zlata Shchedrikova","doi":"10.3233/com-230461","DOIUrl":"https://doi.org/10.3233/com-230461","url":null,"abstract":"We prove various sufficient conditions for the effective infinity of classes of computable numberings. Then we apply them to show that for every computable family of left-c.e. reals without the greatest element the class of its Friedberg computable numberings is effectively infinite. In particular, this result covers the families of all left-c.e. and all Martin-Löf random left-c.e. reals whose Friedberg computable numberings have been constructed by Broadhead and Kjos-Hanssen in their paper (In Mathematical Theory and Computational Practice, CiE 2009 (2009) 49–58 Springer). In addition, for every infinite computable family of left-c.e. reals we prove that the classes of all its computable, positive and minimal numberings are effectively infinite.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77010473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the computability degree of real numbers arising as L 2 -Betti numbers or L 2 -torsion of groups, parametrised over the Turing degree of the word problem.
我们研究了实数作为l2 -Betti数或l2 -群的可计算度，参数化在字问题的图灵度上。
{"title":"L 2 -Betti numbers and computability of reals","authors":"Clara Löh, Matthias Uschold","doi":"10.3233/com-220416","DOIUrl":"https://doi.org/10.3233/com-220416","url":null,"abstract":"We study the computability degree of real numbers arising as L 2 -Betti numbers or L 2 -torsion of groups, parametrised over the Turing degree of the word problem.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136296393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice.
{"title":"Banach’s theorem in higher-order reverse mathematics","authors":"J. Hirst, Carl Mummert","doi":"10.3233/com-230453","DOIUrl":"https://doi.org/10.3233/com-230453","url":null,"abstract":"In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77820325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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