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Scattered deletion and commutativity

Alexandru Mateescu
1994 Theoretical Computer Science  
This paper deals with the scattered deletion of a language by another language, i.e. with scattered residuals. We introduce the notion of the scattered syntactical monoid.  ...  We investigate some properties of these families of languages related to the cardinality of the scattered syntactical monoid.  ...  Moreover, b2 is a zero of the syntactic monoid and will be identified with 0 in the scattered syntactic monoid.  ... 
doi:10.1016/0304-3975(94)90259-3 fatcat:rajlysmdsrgj5ezvvnzybd56i4

Limited Set quantifiers over Countable Linear Orderings [chapter]

Thomas Colcombet, A. V. Sreejith
2015 Lecture Notes in Computer Science  
second-order logic with cuts, as well as fragments of monadic secondorder logic in which sets have to be well ordered or scattered.  ...  In this paper, we study several sublogics of monadic secondorder logic over countable linear orderings, such as first-order logic, firstorder logic on cuts, weak monadic second-order logic, weak monadic  ...  Establishing invariants The core result in the translation from logics to •-monoids is the following.  ... 
doi:10.1007/978-3-662-47666-6_12 fatcat:dv6ak7eqgrh2fhs6ncjafwffve

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

1998 Mathematical Reviews  
The proof of this result uses another, syntactic and algebraic, characterization of the same class of languages. Using the notion of ordered syntactic monoid which he introduced earlier [Izv. Vyssh.  ...  Mat. 1995, no. 1, 80-90; MR 97c:68111], the author shows that a language L is in the polynomial closure of group languages if and only if, in the syntactic ordered monoid of L, it holds that e < | for  ... 

Page 5985 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
{For the entire collection see MR 94i:68015.} 94j:68163 68Q45 68R15 Mateescu, Alexandru (SF-SAF; Helsinki) On scattered syntactic monoids.  ...  We show that in both cases not every w-language having a finite syntactic monoid is regular and we give a character- ization of those w-languages having finite syntactic monoids.  ... 

Level Two of the Quantifier Alternation Hierarchy over Infinite Words [chapter]

Manfred Kufleitner, Tobias Walter
2016 Lecture Notes in Computer Science  
This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment.  ...  Every pseudovariety is generated by its syntactic monoids [4] , i.e., every monoid in a given pseudovariety is a divisor of a direct product of syntactic monoids.  ...  The importance of the syntactic monoid of a language L ⊆ Γ ∞ is that it is the smallest monoid recognizing L: Lemma 1.  ... 
doi:10.1007/978-3-319-34171-2_16 fatcat:hztry7ymsjh75j6xgnjwukti24

Block Products and Nesting Negations in FO2 [chapter]

Lukas Fleischer, Manfred Kufleitner, Alexander Lauser
2014 Lecture Notes in Computer Science  
Among other techniques, the proof relies on an extension of block products to ordered monoids.  ...  More precisely, for every m we give ω-terms Um and Vm such that an FO 2 -definable language is in Σ 2 m if and only if its ordered syntactic monoid satisfies the identity Um Vm.  ...  Note that in condition 3. of Theorem 1 one cannot drop requiring that the syntactic monoid is in DA.  ... 
doi:10.1007/978-3-319-06686-8_14 fatcat:avobzglq4vatphl46l35gg3q7q

Level Two of the Quantifier Alternation Hierarchy over Infinite Words [article]

Manfred Kufleitner, Tobias Walter
2015 arXiv   pre-print
This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment.  ...  One can effectively compute the syntactic homomorphism of L.  ...  Every pseudovariety is generated by its syntactic monoids [5] , i.e., every monoid in a given pseudovariety is a divisor of a direct product of syntactic monoids.  ... 
arXiv:1509.06207v1 fatcat:agg5o2xkizg7hcs46ie2izpw5a

Page 3644 of Mathematical Reviews Vol. , Issue 80I [page]

1980 Mathematical Reviews  
Author’s summary: “A special type of context condition is con sidered that imposes restrictions on the set of combinations of syntactic constructions that simultaneously occur in the text ofa program.  ...  Ehrenfeucht, Andrzej; Rozenberg, Grzegorz An observation on scattered Inform. Process. Lett. 9 (1979), no. 2, 84-85. From the introduction: “Scattered context grammars were ‘intro- duced by S. A.  ... 

Page 430 of Mathematical Reviews Vol. , Issue 95a [page]

1995 Mathematical Reviews  
Summary: “This paper deals with the scattered deletion of a lan- guage by another language, i.e., with scattered residuals. We in- troduce the notion of the scattered syntactical monoid.  ...  We investigate some properties of these families of languages related to the cardinality of the scattered syntactical monoid.” 95a:68069 68Q55 03B60 03B70 68N17 68T27 Baral, Chitta R. (1-MD-CS; College  ... 

Star-free languages are Church–Rosser congruential

Volker Diekert, Manfred Kufleitner, Pascal Weil
2012 Theoretical Computer Science  
Acknowledgements The first author's work on this paper has been started during the program Automata Theory and Applications at the Institute for Mathematical Sciences, National University of Singapore  ...  ., for languages where the syntactic monoid is group-free.  ...  We put on the subsemigroup cM ∩ Mc a monoid structure by defining a new multiplication • as follows: xc • cy = xcy.  ... 
doi:10.1016/j.tcs.2012.01.028 fatcat:qojk42tvc5cado275mrhsryiri

The Half-Levels of the FO2 Alternation Hierarchy

Lukas Fleischer, Manfred Kufleitner, Alexander Lauser
2016 Theory of Computing Systems  
., for every integer m one can decide whether a given regular language is definable in Σ 2 m . Among other techniques, the proof relies on an extension of block products to ordered monoids.  ...  Note that in condition (c) of Theorem 4 one cannot drop the requirement of the syntactic monoid being in DA.  ...  (b) The ordered syntactic monoid of L is in W m . (c) The ordered syntactic monoid of L is in DA and satisfies U m V m . Lemma 8 . 8 Let m 1. If M ∈ W m , then M ∈ DA and M satisfies U m V m .  ... 
doi:10.1007/s00224-016-9712-2 fatcat:aymcfkllrjbm7cwwatuiej6adq

Recognisable languages over monads [article]

Mikołaj Bojańczyk
2015 arXiv   pre-print
To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad.  ...  The cases of countable well-founded and countable scattered chains are simple enough to warrant a self-contained proof, modulo the Hausdorff theorem on scattered chains.  ...  In this case, the left monoid is trivial, i.e. contains only the identity transformation, while the right monoid is the syntactic monoid of the language "words beginning with a".  ... 
arXiv:1502.04898v1 fatcat:2in7ckqcvje7ddulnd7nmfn5jm

Piecewise testable languages via combinatorics on words

Ondřej Klíma
2011 Discrete Mathematics  
monoid is J-trivial.  ...  Nowadays, there exist several proofs of this result based on various methods from algebraic theory of regular languages. Our contribution adds a new purely combinatorial proof.  ...  This relation is called the syntactic congruence of L and the corresponding quotient monoid A * / ∼ L is called the syntactic monoid of L.  ... 
doi:10.1016/j.disc.2011.06.013 fatcat:2p3bvdn74rajtl2viczxcrsdvy

Languages recognised by finite semigroups, and their generalisations to objects such as trees and graphs, with an emphasis on definability in monadic second-order logic [article]

Mikołaj Bojańczyk
2020 arXiv   pre-print
These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata.  ...  By Theorem 1.7 on syntactic monoids, there is a surjective homomorphism from the image of h, which is a sub-monoid of M, into the syntactic monoid.  ...  The syntactic monoid of a language.  ... 
arXiv:2008.11635v1 fatcat:r6jidclh2zdtznsfsfsyfs3eam

Conelikes and Ranker Comparisons [article]

Viktor Henriksson, Manfred Kufleitner
2021 arXiv   pre-print
Most instances of such problems were solved by the connection of regular languages and finite monoids. Both the membership problem and the separation problem were also extended to ordered monoids.  ...  In this paper, we consider the extension the computation of pointlikes to ordered monoids. This leads to the notion of conelikes for the corresponding algebraic framework.  ...  Let µ : A * → A * /≤ L be the canonical projection, then π is the syntactic morphism and A * /≤ L the syntactic monoid of L. A language is regular if and only if the syntactic monoid is finite.  ... 
arXiv:2105.09346v2 fatcat:b4djro4evbcodnnpyt2ntaubsi
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