We answer the following question by Arno Pauly: “Is there a square root operator on the Weihrauch degrees?”. In fact, we show that there are uncountably many pairwise incomparable Weihrauch degrees without any roots. We also prove that the omniscience principles of LPO and LLPO do not have roots.
{"title":"Weihrauch degrees without roots","authors":"Patrick Uftring","doi":"10.3233/com-230471","DOIUrl":"https://doi.org/10.3233/com-230471","url":null,"abstract":"We answer the following question by Arno Pauly: “Is there a square root operator on the Weihrauch degrees?”. In fact, we show that there are uncountably many pairwise incomparable Weihrauch degrees without any roots. We also prove that the omniscience principles of LPO and LLPO do not have roots.","PeriodicalId":515920,"journal":{"name":"Computability","volume":"83 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140709303","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 the context of constructive reverse mathematics, we characterize the difference between König’s lemma and weak König’s lemma by a particular fragment of the countable choice principle. Specifically, we show that König’s lemma can be decomposed into weak König’s lemma and the choice principle over a weak intuitionistic two-sorted arithmetic.
{"title":"Choice principles characterizing the difference between König’s lemma and weak König’s lemma in constructive reverse mathematics","authors":"Makoto Fujiwara, Takako Nemoto","doi":"10.3233/com-230478","DOIUrl":"https://doi.org/10.3233/com-230478","url":null,"abstract":"In the context of constructive reverse mathematics, we characterize the difference between König’s lemma and weak König’s lemma by a particular fragment of the countable choice principle. Specifically, we show that König’s lemma can be decomposed into weak König’s lemma and the choice principle over a weak intuitionistic two-sorted arithmetic.","PeriodicalId":515920,"journal":{"name":"Computability","volume":"40 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140752147","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}
Isometric words are those words whose occurrence as a factor in a transformation of a word u in a word v can be avoided, while preserving the minimal length of the transformation. Such minimal length refers to a distance between u and v. In the literature, isometric words have been considered with respect to the Hamming distance and the Lee distance; the former especially for binary words, while the latter for k-ary words, with k ⩾ 2. Ham- and Lee- isometric words have been characterized in terms of their overlaps with errors. In this paper, we give algorithms to decide whether a word f, of length n, is Ham- or Lee-isometric and provide evidence of the possible non-isometricity by returning a pair of words of minimal length whose transformation cannot avoid the factor f. Such a pair of words is called a pair of witnesses and the minimal length of the witnesses is called the index of f. The algorithms run in O ( n ) time with a preprocessing of O ( n ) time and space to construct a data structure that allows answering LCA queries on the suffix tree of f in constant time. The correctness of the algorithms lies on some theoretical results on the index and the witnesses of a word that are here presented. The investigation on the index is completed by the characterization of words with minimum/maximum index. All the results are shown referring to both Hamming and Lee distance.
等距词是指在词 v 中转换词 u 时，可以避免出现的词，同时保留转换的最小长度。这种最小长度指的是 u 和 v 之间的距离。在文献中，等距词是根据汉明距离（Hamming distance）和李距离（Lee distance）来考虑的；前者尤其适用于二元词，而后者适用于 k-ary 词（k ⩾ 2）。Ham 等距词和 Lee 等距词的特征在于它们的重叠误差。在本文中，我们给出了判断长度为 n 的词 f 是 Ham-isometric 还是 Lee-isometric 的算法，并通过返回一对最小长度的词来证明可能的非等距性，这对词的变换不能避开因子 f。这样的一对词称为一对见证词，见证词的最小长度称为 f 的索引。算法的运行时间为 O ( n )，预处理的时间和空间为 O ( n )，构建的数据结构可以在恒定时间内回答对 f 后缀树的 LCA 查询。算法的正确性取决于本文介绍的索引和单词见证的一些理论结果。通过对具有最小/最大索引的词的特征描述，完成了对索引的研究。所有结果均参考了汉明距离和李距离。
{"title":"Computing the index of non-isometric k-ary words with Hamming and Lee distance","authors":"M. Anselmo, Manuela Flores, Maria Madonia","doi":"10.3233/com-230441","DOIUrl":"https://doi.org/10.3233/com-230441","url":null,"abstract":"Isometric words are those words whose occurrence as a factor in a transformation of a word u in a word v can be avoided, while preserving the minimal length of the transformation. Such minimal length refers to a distance between u and v. In the literature, isometric words have been considered with respect to the Hamming distance and the Lee distance; the former especially for binary words, while the latter for k-ary words, with k ⩾ 2. Ham- and Lee- isometric words have been characterized in terms of their overlaps with errors. In this paper, we give algorithms to decide whether a word f, of length n, is Ham- or Lee-isometric and provide evidence of the possible non-isometricity by returning a pair of words of minimal length whose transformation cannot avoid the factor f. Such a pair of words is called a pair of witnesses and the minimal length of the witnesses is called the index of f. The algorithms run in O ( n ) time with a preprocessing of O ( n ) time and space to construct a data structure that allows answering LCA queries on the suffix tree of f in constant time. The correctness of the algorithms lies on some theoretical results on the index and the witnesses of a word that are here presented. The investigation on the index is completed by the characterization of words with minimum/maximum index. All the results are shown referring to both Hamming and Lee distance.","PeriodicalId":515920,"journal":{"name":"Computability","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140091350","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 two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill’s axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop’s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.
我们研究了在毕晓普式构造数学的达尼尔度量与积分方法的构造发展中引入的两个补集子代数。我们为这里所谓的分类互补子集和扩展互补子集介绍了它们的主要性质。我们通过建立互补子集（分类或外延）与布尔值部分函数（分类或外延）之间的双射，建构性地转换了子集与布尔值全函数之间的经典双射。我们讨论了迈希尔非选择公理在分类子集和外延子集之间等价性中的作用。我们将 (I) 和 (II) 型交换代数分别作为 (I) 和 (II) 型互补子集的毕肖普代数的抽象版本，并将交换环作为集合上布尔值偏函数结构的抽象版本。我们所举的交换代数和交换环的例子以及这里所包含的结果表明，它们的理论是对布尔代数和布尔环理论的某种概括。
{"title":"Complemented subsets and Boolean-valued, partial functions","authors":"Daniel Misselbeck-Wessel, I. Petrakis","doi":"10.3233/com-230462","DOIUrl":"https://doi.org/10.3233/com-230462","url":null,"abstract":"We study the two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill’s axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop’s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.","PeriodicalId":515920,"journal":{"name":"Computability","volume":"109 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139963789","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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