{"title":"Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev","authors":"J. Shallit, S. Shan, Kai Hsiang Yang","doi":"10.48550/arXiv.2208.06025","DOIUrl":"https://doi.org/10.48550/arXiv.2208.06025","url":null,"abstract":"We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (—k) to express variables, thus permitting quantification over ℤ instead of ℕ. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shut involving bi-infinite binary words.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130118393","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}
{"title":"Properties of a Ternary Infinite Word","authors":"J. Currie, Pascal Ochem, N. Rampersad, J. Shallit","doi":"10.48550/arXiv.2206.01776","DOIUrl":"https://doi.org/10.48550/arXiv.2206.01776","url":null,"abstract":"We study the properties of the ternary infinite word\u0000p = 012102101021012101021012⋯,\u0000that is, the fixed point of the map h : 0 → 01, 1 → 21, 2 → 0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"197 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120976778","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}
{"title":"Critical factorisation in square-free words","authors":"T. Harju","doi":"10.1051/ita/2022003","DOIUrl":"https://doi.org/10.1051/ita/2022003","url":null,"abstract":"A position p in a word w is critical if the minimal local period at p is equal to the global period of w. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number η(w) of critical points of square-free ternary words w, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words w satisfy η(w) ≤|w|− 5 where |w| denotes the length of w. Moreover, the bound |w|− 5 is reached by infinitely many words. On the other hand, every square-free word w has at least |w|∕4 critical points, and there is a sequence of these words closing to this bound.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116092578","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}
{"title":"Almost all Classical Theorems are Intuitionistic","authors":"P. Lescanne","doi":"10.1051/ita/2022009","DOIUrl":"https://doi.org/10.1051/ita/2022009","url":null,"abstract":"Canonical expressions represent the implicative propositions (i.e., the propositions with only implications) up-to renaming of variables. Using a Monte-Carlo approach, we explore the model of canonical expressions in order to confirm the paradox that says that asymptotically almost all classical theorems are intuitionistic. Actually we found that more than 96.6% of classical theorems are intuitionistic among propositions of size 100.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"240 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120930594","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}
{"title":"The Simplest Binary Word with Only Three Squares","authors":"D. Gabric, J. Shallit","doi":"10.1051/ITA/2021001","DOIUrl":"https://doi.org/10.1051/ITA/2021001","url":null,"abstract":"We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the “simplest”, in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments to prove that the constructions have the desired property, we rely instead on theorem-proving software for their correctness.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131952597","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}
{"title":"On universal partial words for word-patterns and set partitions","authors":"Herman Z. Q. Chen, S. Kitaev","doi":"10.1051/ITA/2020004","DOIUrl":"https://doi.org/10.1051/ITA/2020004","url":null,"abstract":"Universal words are words containing exactly once each element from a given set of combinatorial structures admitting encoding by words. Universal partial words (u-p-words) contain, in addition to the letters from the alphabet in question, any number of occurrences of a special “joker” symbol. We initiate the study of u-p-words for word-patterns (essentially, surjective functions) and (2-)set partitions by proving a number of existence/non-existence results and thus extending the results in the literature on u-p-words and u-p-cycles for words and permutations. We apply methods of graph theory and combinatorics on words to obtain our results.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116341088","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}
{"title":"The Weak Circular Repetition Threshold Over Large Alphabets","authors":"Lucas Mol, N. Rampersad","doi":"10.1051/ita/2020006","DOIUrl":"https://doi.org/10.1051/ita/2020006","url":null,"abstract":"The repetition threshold for words on n letters, denoted RT(n), is the infimum of the set of all r such that there are arbitrarily long r-free words over n letters. A repetition threshold for circular words on n letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for n letters, denoted CRTW(n), CRTI(n), and CRTS(n), respectively. Currie and the present authors conjectured that CRTI(n) = CRTW(n) = RT(n) for all n ≥ 4. We prove that CRTW(n) = RT(n) for all n ≥ 45, which confirms a weak version of this conjecture for all but finitely many values of n.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"252 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122494834","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}
{"title":"Digital semigroups","authors":"H. Brunotte","doi":"10.1051/ita/2016005","DOIUrl":"https://doi.org/10.1051/ita/2016005","url":null,"abstract":"The well-known expansion of rational integers in an arbitrary integer base different from $0, 1, -1$ is exploited to study relations between numerical monoids and certain subsemigroups of the multiplicative semigroup of nonzero integers.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125770376","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}
{"title":"Injective envelopes of transition systems and Ferrers languages","authors":"M. Kabil, M. Pouzet","doi":"10.1051/ita/2020005","DOIUrl":"https://doi.org/10.1051/ita/2020005","url":null,"abstract":"We consider reflexive and involutive transition systems over an ordered alphabet A equipped with an involution. We give a description of the injective envelope of any two-element set in terms of Galois lattice, from which we derive a test of its finiteness. Our description leads to the notion of Ferrers language.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115203703","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}
{"title":"Edge-bipancyclicity in conditional edge-faulty k-ary n-cubes","authors":"Shiying Wang, Shurong Zhang","doi":"10.1051/ita/2019003","DOIUrl":"https://doi.org/10.1051/ita/2019003","url":null,"abstract":"The class of k-ary n-cubes represents the most commonly used interconnection topology for parallel and distributed computing systems. In this paper, we consider the faulty k-ary n-cube with even k ≥ 4 and n ≥ 2 such that each vertex of the k-ary n-cube is incident with at least two healthy edges. Based on this requirement, we investigate the fault-tolerant capabilities of the k-ary n-cube with respect to the edge-bipancyclicity. We prove that in the k-ary n-cube Qnk, every healthy edge is contained in fault-free cycles of even lengths from 6 to |V(Qnk)|, even if the Qnk has up to 4n − 5 edge faults and our result is optimal with respect to the number of edge faults tolerated.","PeriodicalId":438841,"journal":{"name":"RAIRO Theor. Informatics Appl.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116009516","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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