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The dot-depth hierarchy of star-free languages is infinite

J.A. Brzozowski, R. Knast
1978 Journal of computer and system sciences (Print)  
., &P),...) is called the dot-depth hierarchy. A language L is of (dot) depth 0 iff L E 99(s), and of depth k, K > 1, iff L E &7(k) -gck--l).  ...  It was conjectured in [3] that the dot-depth hierarchy is infinite if the alphabet has two or more letters, i.e., that for each K 2 0 there exists a language that is of depth R + 1 but not of depth k  ...  By Lemma 3+ Observe that the notion of k-mutativity defines an infinite hierarchy of finite semigroups. This follows from the example in Fig.  ... 
doi:10.1016/0022-0000(78)90049-1 fatcat:lm7pwd3adrdaldq3fyrqplriom

Hierarchies of aperiodic languages

Janusz A. Brzozowski
1976 Revue française d automatique informatique recherche opérationnelle Informatique théorique  
KNAST, The Dot-Dept h Hierarchy of Star-Free Languages is Infinité, Research Report CS-76-23, Computer Science Dept., University of Waterloo, Waterloo, Ont., Canada; April, 1976.  ...  THE DOT-DEPTH HIERARCHY [8] The séquence (^) of Boolean algebras, defmed below, is called the dot-depth hierarchy. Let This bound is met by L t above.  ... 
doi:10.1051/ita/197610r200331 fatcat:j4rnswgadvgd3m26iiteko3cpi

Games, equations and the dot-depth hierarchy

F. Blanchet-Sadri
1989 Computers and Mathematics with Applications  
This paper studies the fine structure of the Straubing hierarchy of star-free languages.  ...  The monoid varieties of some sublevels of level one of the hierarchy are shown to be characterized by certain natural equation systems.  ...  Many thanks to the referee for comments and suggestions.  ... 
doi:10.1016/0898-1221(89)90179-x fatcat:px6cx5qu25dzzfv2ay4mrzvrp4

Languages polylog-time reducible to dot-depth 1/2

Christian Glaßer
2007 Journal of computer and system sciences (Print)  
We construct star-free regular languages L n such that L n 's balanced leaf-language class is NP, but the unbalanced leaf-language class of L n contains level n of the unambiguous alternation hierarchy  ...  We construct languages of arbitrary dot-depth that are reducible to languages of dot-depth 1/2. (3) Unbalanced star-free leaf languages can be much stronger than balanced ones.  ...  Acknowledgments The author thanks Bernhard Schwarz, Victor Selivanov, and Klaus W. Wagner for exciting discussions about leaf languages and reducibility notions for star-free languages.  ... 
doi:10.1016/j.jcss.2006.09.004 fatcat:2pwxmb63yrbcrh75ixflfjm7ea

Languages vs. ω-Languages in Regular Infinite Games [chapter]

Namit Chaturvedi, Jörg Olschewski, Wolfgang Thomas
2011 Lecture Notes in Computer Science  
A game is specified by the ω-language which contains the plays won by Player 2.  ...  Winning strategies for infinite games can be represented again in terms of * -languages.  ...  Analogous to the dot-depth hierarchy, the Straubing-Thérien hierarchy is strict, infinite, and exhausts the class of all star-free languages.  ... 
doi:10.1007/978-3-642-22321-1_16 fatcat:ek47w5zwxbacveuoxcnxprrkku

Page 5688 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
In this paper, the authors study the second level of the dot- depth hierarchy for star-free regular languages.  ...  This is part of a continuing investigation of the dot-depth hierarchy of recognizable languages studied by Straubing [Theoret. Comput. Sci. 58 (1988), no. 1-3, 361-378; MR 90j:20139].  ... 

On the Complexity of Intersection Non-emptiness for Star-Free Language Classes [article]

Emmanuel Arrighi and Henning Fernau and Stefan Hoffmann and Markus Holzer and Ismaël Jecker and Mateus de Oliveira Oliveira and Petra Wolf
2021 arXiv   pre-print
We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels zero or one half and already PSPACE-hard when  ...  We analyze the complexity of the Intersection Non-Emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien  ...  We have investigated how the increase in complexity within the dot-depth and the Straubing-Thérien hierarchies is reflected in the complexity of the Intersection Non-emptiness problem.  ... 
arXiv:2110.01279v1 fatcat:snxam2lzsnclrkzsyv3b3gphme

Languages of Dot-depth One over Infinite Words [article]

Manfred Kufleitner, Alexander Lauser
2011 arXiv   pre-print
Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic.  ...  In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.  ...  Acknowledgments We thank the anonymous referees of the conference version of this paper for their suggestions which helped to improve the presentation.  ... 
arXiv:1101.4152v2 fatcat:so2ib3fu45e5le7fesl2qd7kcy

On dot-depth two

F. Blanchet-Sadri
1990 RAIRO - Theoretical Informatics and Applications  
We would like to thank the référées for their comments and suggestions,  ...  L<=;4* is star-free if and only if LeA*i r k for some A:^0. The dot-depth of L is the smallest such k.  ...  The language classes A + ^0, A + M x , ... form the so-called dot-depth hierarchy introduced by Cohen and Brzozowski in [4] , The union of the classes A + <% 0 , A + âS l9 . . . is the class of star-free  ... 
doi:10.1051/ita/1990240605211 fatcat:qfyo4wviyvflrbx3dz4wjz2mbm

Games, equations and dot-depth two monoids

F. Blanchet-Sadri
1992 Discrete Applied Mathematics  
,mJ to be of dotdepth exactly 2. Upper and lower bounds on the dot-depth of the A'/-(ml,. . .,mJ are discussed.  ...  ,mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A*/-(m"...,mL).  ...  Many thanks to the referee for comments and suggestions.  ... 
doi:10.1016/0166-218x(92)90161-3 fatcat:opqixodqf5eznc5ibrlgfozvny

Page 20 of Mathematical Reviews Vol. , Issue 85a [page]

1985 Mathematical Reviews  
Similarly, the Simon hierarchy (also called the dot-depth one hierarchy or (-hierarchy) is char- acterized by the quantifier rank of £,-sentences in prenex normal form.  ...  The au- thor refines this result by showing that a language has level n in the Brzozowski hierarchy (also called the dot-depth hierarchy) if and only if it can be described by a Boolean combination of  ... 

The Dot-Depth Hierarchy, 45 Years Later [chapter]

Jean-Éric Pin
2017 The Role of Theory in Computer Science  
Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy.  ...  The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971.  ...  The author was funded from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 670624).  ... 
doi:10.1142/9789813148208_0008 dblp:conf/birthday/Pin17a fatcat:u54iyat4cbgl7ov2eu64xhktma

Page 3874 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
This type 68 COMPUTER SCIENCE 3874 of construction is used in the definitions of the dot-depth and the Straubing hierarchies of star-free languages.  ...  A basis of open sets of the Hall topology on the free monoid A* consists precisely of the group languages, so that a language is open if and only if it is a union (finite or infinite) of group languages  ... 

Classifying regular languages by a split game

Qiqi Yan
2007 Theoretical Computer Science  
context of ω-languages.  ...  An extension of the split game to generalized ω-regular expressions is also established.  ...  Acknowledgments I thank Enshao Shen for pointing me to the star height 2 problem, as well as sharing his thoughts. I also thank Jean-Eric Pin and Wolfgang Thomas for their useful comments.  ... 
doi:10.1016/j.tcs.2006.12.041 fatcat:6jf4kttfrfgk7e4zhlq4cgr52q

Finite Semigroups and Recognizable Languages: An Introduction [chapter]

Jean-Eric Pin
1995 Semigroups, Formal Languages and Groups  
The various hierarchies of rational languages, based on star height, extended star height, dot-depth and concatenation level are introduced in section 5.  ...  This led to the natural notions of star height, extended star height, dot-depth and concatenation level.  ...  As product is often denoted by a dot, Brzozowski defined the "dot-depth" of languages of the free semigroup [5] .  ... 
doi:10.1007/978-94-011-0149-3_1 fatcat:6g24l3rfergcnaw2mumpz3lohu
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