{"title":"Algebra for trees","authors":"M. Bojanczyk","doi":"10.4171/AUTOMATA-1/22","DOIUrl":"https://doi.org/10.4171/AUTOMATA-1/22","url":null,"abstract":"This chapter presents several algebraic approaches to tree languages. The idea is to design a notion for trees that resembles semigroups or monoids for words. The focus is on the connection between the structure of an algebra recognizing a tree language, and the kind of logic needed to define the tree language. Four algebraic approaches are described in this chapter: trees as terms of universal algebra, preclones, forest algebra, and seminearrings. Each approach is illustrated with an application to logic on trees.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124543393","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":"Profinite topologies","authors":"J. Almeida, A. Costa","doi":"10.4171/Automata-1/17","DOIUrl":"https://doi.org/10.4171/Automata-1/17","url":null,"abstract":"Profinite semigroups are a generalization of finite semigroups that come about naturally when one is interested in considering free structures with respect to classes of finite semigroups. They also appear naturally through dualization of Boolean algebras of regular languages. The additional structure is given by a compact zero-dimensional topology. Profinite topologies may also be considered on arbitrary abstract semigroups by taking the initial topology for homomorphisms into finite semigroups. This text is the proposed chapter of the Handdbook of Automata Theory dedicated to these topics. The general theory is formulated in the setting of universal algebra because it is mostly independent of specific properties of semigroups and more general algebras naturally appear in this context. In the case of semigroups, particular attention is devoted to solvability of systems of equations with respect to a pseudovariety, which is relevant for solving membership problems for pseudovarieties. Focus is also given to relatively free profinite semigroups per se, specially \"large\" ones, stressing connections with symbolic dynamics that bring light to their structure.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123515529","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":"Automata and Quantum Computing","authors":"A. Ambainis, A. Yakaryılmaz","doi":"10.4171/Automata-2/17","DOIUrl":"https://doi.org/10.4171/Automata-2/17","url":null,"abstract":"Quantum computing is a new model of computation, based on quantum physics. Quantum computers can be exponentially faster than conventional computers for problems such as factoring. Besides full-scale quantum computers, more restricted models such as quantum versions of finite automata have been studied. In this paper, we survey various models of quantum finite automata and their properties. We also provide some open questions and new directions for researchers. \u0000Keywords: quantum finite automata, probabilistic finite automata, nondeterminism, bounded error, unbounded error, state complexity, decidability and undecidability, computational complexity","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127348991","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":"Automata and rational expressions","authors":"J. Sakarovitch","doi":"10.4171/Automata-1/2","DOIUrl":"https://doi.org/10.4171/Automata-1/2","url":null,"abstract":"This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133273769","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 Cobham's theorem","authors":"F. Durand, M. Rigo","doi":"10.4171/Automata-2/4","DOIUrl":"https://doi.org/10.4171/Automata-2/4","url":null,"abstract":"In this chapter we essentially focus on the representation of non-negative integers in a given numeration system. The main role of such a system --- like the usual integer base $k$ numeration system --- is to replace numbers or more generally sets of numbers by their corresponding representations, {em i.e.}, by words or by languages. First we consider integer base numeration systems to present the main concepts but rapidly we will introduce non-standard systems and their relationships with substitutions.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115011207","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":"Rational subsets of groups","authors":"L. Bartholdi, Pedro V. Silva","doi":"10.4171/Automata-2/1","DOIUrl":"https://doi.org/10.4171/Automata-2/1","url":null,"abstract":"This text, Chapter 23 in the \"AutoMathA\" handbook, is devoted to the study of rational subsets of groups, with particular emphasis on the automata-theoretic approach to finitely generated subgroups of free groups. Indeed, Stallings' construction, associating a finite inverse automaton with every such subgroup, inaugurated a complete rewriting of free group algorithmics, with connections to other fields such as topology or dynamics. \u0000Another important vector in the chapter is the fundamental Benois' Theorem, characterizing rational subsets of free groups. The theorem and its consequences really explain why language theory can be successfully applied to the study of free groups. Rational subsets of (free) groups can play a major role in proving statements (a priori unrelated to the notion of rationality) by induction. The chapter also includes related results for more general classes of groups, such as virtually free groups or graph groups.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126506860","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":"Groups defined by automata","authors":"L. Bartholdi, Pedro V. Silva","doi":"10.4171/Automata-2/2","DOIUrl":"https://doi.org/10.4171/Automata-2/2","url":null,"abstract":"This is Chapter 24 in the \"AutoMathA\" handbook. Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups (including word-hyperbolic groups as a particular case) and automata groups (singled out among the more general self-similar groups). \u0000The first approach implements in the language of automata some tight constraints on the geometry of the group's Cayley graph, building strange, beautiful bridges between far-off domains. Automata are used to define a normal form for group elements, and to monitor the fundamental group operations. \u0000The second approach features groups acting in a finitely constrained manner on a regular rooted tree. Automata define sequential permutations of the tree, and represent the group elements themselves. The choice of particular classes of automata has often provided groups with exotic behaviour which have revolutioned our perception of infinite finitely generated groups.","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"30 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126965875","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":"Weighted automata","authors":"M. Droste, D. Kuske","doi":"10.4171/Automata-1/4","DOIUrl":"https://doi.org/10.4171/Automata-1/4","url":null,"abstract":"Weighted automata are classical finite automata in which the transitions carry weights. 7 These weights may model quantitative properties like the amount of resources needed for executing 8 a transition or the probability or reliability of its successful execution. Using weighted automata, 9 we may also count the number of successful paths labeled by a given word. 10 As an introduction into this field, we present selected classical and recent results concentrating 11 on the expressive power of weighted automata. 12 2 M. Droste, D. Kuske","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127044764","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":"Language equations","authors":"Michal Kunc, A. Okhotin","doi":"10.4171/Automata-1/21","DOIUrl":"https://doi.org/10.4171/Automata-1/21","url":null,"abstract":"","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116767442","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":"Automata on infinite trees","authors":"Christof Löding","doi":"10.4171/Automata-1/8","DOIUrl":"https://doi.org/10.4171/Automata-1/8","url":null,"abstract":"","PeriodicalId":267596,"journal":{"name":"Handbook of Automata Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122432168","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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