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The Boolean Hierarchy over Level 1/2 of the Straubing-Therien Hierarchy [article]

Heinz Schmitz, Klaus W. Wagner
1998 arXiv   pre-print
We give a membership criterion for the single classes of the boolean hierarchy over L(1/2).  ...  For some fixed alphabet A, a language L of A* is in the class L(1/2) of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A*aA*bA*...A*cA*, where a,b,...  ...  In the last section we state a result of this type relating the boolean hierarchy over level 1/2 of the Straubing-Thérien hierarchy to the boolean hierarchy over NP.  ... 
arXiv:cs/9809118v1 fatcat:gez5cgx7efgjhevgwqthywniq4

SUBHIERARCHIES OF THE SECOND LEVEL IN THE STRAUBING–THÉRIEN HIERARCHY

ONDŘEJ KLÍMA, LIBOR POLÁK
2011 International journal of algebra and computation  
Here we apply our constructions for particular examples of varieties of languages obtaining four hierarchies of (positive) varieties which have the 3/2 level and the second level of the Straubing-Thérien  ...  In a recent paper we assigned to each positive variety V and a fixed natural number k the class of all (positive) boolean combinations of the restricted polynomials, i.e. the languages of the form L0a1L1a2  ...  Notice that PPol(S + ) is the 3/2 level of the Straubing-Thérien hierarchy and that BPol(S + ) is the second level of this hierarchy (see Theorem 8.8 of [8] ).  ... 
doi:10.1142/s021819671100690x fatcat:hhu4fq4lzjeqholbzp7ziezuc4

New decidable upper bound of the second level in the Straubing-Therien concatenation hierarchy of star-free languages

Jorge Almeida, Ondrej Klima
2010 Discrete Mathematics & Theoretical Computer Science  
concerning the characterization of regular languages of level 2 in the Straubing-Therien concatenation hierarchy of star-free languages.  ...  special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience In a recent paper we gave a counterexample to a longstanding conjecture  ...  The work of the second author was supported, in part, by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and by the Grant no. 201/09/1313 of the Grant Agency of the Czech  ... 
doi:10.46298/dmtcs.490 fatcat:7cfdzapnendsvbwe27ug4wbe6a

Generic Results for Concatenation Hierarchies [article]

Thomas Place, Marc Zeitoun
2017 arXiv   pre-print
In the theory of formal languages, the understanding of concatenation hierarchies of regular languages is one of the most fundamental and challenging topic.  ...  In this paper, we survey progress made in the comprehension of this problem since 1971, and we establish new generic statements regarding this problem.  ...  separation for levels 1 2 , 1, 3 2 , 2 and 5 2 in the Straubing-Thérien hierarchy, and decidability of membership for level 7 2 .  ... 
arXiv:1710.04313v1 fatcat:xcssvb427fhwfhiscfcnsnb5si

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 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  ...  Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy.  ...  Straubing-Thérien hierarchy.  ... 
arXiv:2110.01279v1 fatcat:snxam2lzsnclrkzsyv3b3gphme

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

A counterexample to a conjecture concerning concatenation hierarchies

Jorge Almeida, Ondřej Klíma
2009 Information Processing Letters  
We give a counterexample to the conjecture which was originally formulated by Straubing in 1986 concerning a certain algebraic characterization of regular languages of level 2 in the Straubing-Thérien  ...  concatenation hierarchy of star-free languages.  ...  The work of the second author was supported, in part, by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and by the Grant no. 201/09/1313 of the Grant Agency of the Czech  ... 
doi:10.1016/j.ipl.2009.09.011 fatcat:zt3i56v3fveb3idvasjszlaqs4

New decidable upper bound of the second level in the Straubing-ThérienTh´Thérien concatenation hierarchy of star-free languages

Jorge Almeida, Ondřej Klíma
2010 Discrete Mathematics and Theoretical Computer Science DMTCS vol   unpublished
In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Thérien concatenation hierarchy of star-free languages  ...  In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound.  ...  )", benefiting in particular from an exchange grant for the visit of the second author to the University of Porto.  ... 
fatcat:i2jod2fd35h63mlgmc66yan6am

The Complexity of Separation for Levels in Concatenation Hierarchies

Thomas Place, Marc Zeitoun, Michael Wagner
2018 Foundations of Software Technology and Theoretical Computer Science  
It was recently proved that the problem is always decidable for levels 1/2 and 1 of any such hierarchy (with inefficient algorithms).  ...  been investigated so far (except for the class of piecewise testable languages [3, 11, 5] , which is level 1 in the Straubing-Thérien hierarchy).  ...  The full level n + 1 is the Boolean closure of the previous half level, i.e., of level n + 1/2. 0 1/2 1 3/2 2 5/2 P ol Bool P ol Bool P ol We write 1 2 N = {0, 1/2, 1, 2, 3/2, 3, . . . } for the set of  ... 
doi:10.4230/lipics.fsttcs.2018.47 dblp:conf/fsttcs/PlaceZ18 fatcat:wbhnpdtok5f37pwghrju7v4l2m

Bridges for concatenation hierarchies [chapter]

Jean-éric Pin
1998 Lecture Notes in Computer Science  
We establish a simple algebraic connection between the Straubing-Thérien hierarchy, whose basis is the trivial variety, and the group hierarchy, whose basis is the variety of group languages.  ...  Thanks to a recent result of Almeida and Steinberg, this reduces the decidability problem for the group hierarchy to a property stronger than decidability for the Straubing-Thérien hierarchy.  ...  The languages of level n+1 are the boolean combinations 1 of languages of level n+ 1 2 . Thus a concatenation hierarchy H is fully determined by its level zero H 0 .  ... 
doi:10.1007/bfb0055073 fatcat:xedoargdmzc75mln3y2quvxv7m

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 study (i) regular languages that are polylog-time reducible to languages of dot-depth 1/2 and (ii) regular languages that are polylog-time decidable.  ...  Hemaspaandra, Pascal Tesson, and Stephen Travers for drawing my attention to characterizations of AC 0 ∩ REG, for discussions about possible interpretations of Corollary 27, and for other helpful hints  ... 
doi:10.1016/j.jcss.2006.09.004 fatcat:2pwxmb63yrbcrh75ixflfjm7ea

On Decidability of Intermediate Levels of Concatenation Hierarchies [chapter]

Jorge Almeida, Jana Bartoňová, Ondřej Klíma, Michal Kunc
2015 Lecture Notes in Computer Science  
It is proved that in every concatenation hierarchy of regular languages, decidability of one of its half levels, obtained by polynomial closure, implies decidability of the intersection of the following  ...  In terms of the quantifier-alternation hierarchy of sentences in the first-order logic of finite words, this means that decidability of (definability in) the Σn fragment implies that of ∆n+1.  ...  from the case of the Straubing-Thérien hierarchy to an arbitrary concatenation hierarchy.  ... 
doi:10.1007/978-3-319-21500-6_4 fatcat:tyhp3ovqsbafzez37kkxjvskme

Efficient algorithms for membership in boolean hierarchies of regular languages

Christian Glaßer, Heinz Schmitz, Victor Selivanov
2016 Theoretical Computer Science  
restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ2 of the Straubing-Thérien hierarchy are decidable in NL.  ...  We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: • The classes of the Boolean hierarchy over level Σ1 of the dot-depth hierarchy  ...  We would like to thank the anonymous referees for their valuable comments.  ... 
doi:10.1016/j.tcs.2016.07.017 fatcat:tqu5qwscmbbbvebstf2lkexy2m

Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages [article]

Christian Glasser, Heinz Schmitz, Victor Selivanov
2008 arXiv   pre-print
restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ_2 of the Straubing-Thérien hierarchy are decidable in NL.  ...  We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level Σ_1 of the dot-depth hierarchy  ...  We would like to thank the anonymous referees for their valuable comments.  ... 
arXiv:0802.2868v1 fatcat:euqbn7hprrgdbkqnuocyqwhxyu

THE DOT-DEPTH AND THE POLYNOMIAL HIERARCHIES CORRESPOND ON THE DELTA LEVELS

BERND BORCHERT, KLAUS-JÖRN LANGE, FRANK STEPHAN, PASCAL TESSON, DENIS THÉRIEN
2005 International Journal of Foundations of Computer Science  
It is well-known that the Σ k -and Π k -levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages.  ...  The same methods are used to give evidence against an earlier conjecture of Straubing and Thérien about a leaf-language upper bound for BPP.  ...  Furthermore, the k-th full level DD k of the dot-depth hierarchy (the Boolean closure of Σ L k ) and the Boolean closure of Σ p k (for k = 1 called the Boolean hierarchy over NP) correspond via leaf languages  ... 
doi:10.1142/s0129054105003200 fatcat:ewx73a5ibnhblfrkd776sbq4du
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