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Scattered deletion and commutativity

1994
*
Theoretical Computer Science
*

This paper deals with the

doi:10.1016/0304-3975(94)90259-3
fatcat:rajlysmdsrgj5ezvvnzybd56i4
*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*. ...##
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Limited Set quantifiers over Countable Linear Orderings
[chapter]

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

doi:10.1007/978-3-662-47666-6_12
fatcat:dv6ak7eqgrh2fhs6ncjafwffve
*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. ...##
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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 ...##
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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*. ...##
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Level Two of the Quantifier Alternation Hierarchy over Infinite Words
[chapter]

2016
*
Lecture Notes in Computer Science
*

This paper is

doi:10.1007/978-3-319-34171-2_16
fatcat:hztry7ymsjh75j6xgnjwukti24
*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. ...##
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Block Products and Nesting Negations in FO2
[chapter]

2014
*
Lecture Notes in Computer Science
*

Among other techniques, the proof relies

doi:10.1007/978-3-319-06686-8_14
fatcat:avobzglq4vatphl46l35gg3q7q
*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. ...##
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Level Two of the Quantifier Alternation Hierarchy over Infinite Words
[article]

2015
*
arXiv
*
pre-print

This paper is

arXiv:1509.06207v1
fatcat:agg5o2xkizg7hcs46ie2izpw5a
*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*. ...##
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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. ...##
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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 ...##
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Star-free languages are Church–Rosser congruential

2012
*
Theoretical Computer Science
*

Acknowledgements The first author's work

doi:10.1016/j.tcs.2012.01.028
fatcat:qojk42tvc5cado275mrhsryiri
*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. ...##
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The Half-Levels of the FO2 Alternation Hierarchy

2016
*
Theory of Computing Systems
*

., for every integer m

doi:10.1007/s00224-016-9712-2
fatcat:aymcfkllrjbm7cwwatuiej6adq
*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 . ...##
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Recognisable languages over monads
[article]

2015
*
arXiv
*
pre-print

To talk about compositionality,

arXiv:1502.04898v1
fatcat:2in7ckqcvje7ddulnd7nmfn5jm
*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". ...##
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Piecewise testable languages via combinatorics on words

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. ...

##
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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]

2020
*
arXiv
*
pre-print

These are lecture notes

arXiv:2008.11635v1
fatcat:r6jidclh2zdtznsfsfsyfs3eam
*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. ...##
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Conelikes and Ranker Comparisons
[article]

2021
*
arXiv
*
pre-print

Most instances of such problems were solved by the connection of regular languages and finite

arXiv:2105.09346v2
fatcat:b4djro4evbcodnnpyt2ntaubsi
*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. ...
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