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Monadic Second-Order Logic for Graphs: Algorithmic and Language Theoretical Applications
[chapter]

2009
*
Lecture Notes in Computer Science
*

This tutorial will present an overview

doi:10.1007/978-3-642-00982-2_2
fatcat:ryjvoaa7dnefrcruygfnpqso2i
*of**the*use*of**Monadic*Second-Order Logic to describe sets*of*finite graphs and graph transformations, in relation with*the*notions*of**tree*-width and clique-width. ... It will review applications to*the*construction*of*algorithms, to Graph*Theory*and to*the*extension to graphs*of*Formal Language*Theory*concepts. ... However,*the*two variants*of**monadic*secondorder logic have*the*same expressive power on words, on*trees*and on certain classes*of*graphs*like*those*of*planar graphs or, for each k,*of*graphs*of*degree ...##
###
Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

2012
*
Logical Methods in Computer Science
*

We present axiomatizations

doi:10.2168/lmcs-8(4:12)2012
fatcat:qzkujokj4bczthib4b67hduaj4
*of**the**monadic*second-order logic (MSO),*monadic*transitive closure logic (FO(TC1)) and*monadic*least fixed-point logic (FO(LFP1))*theories**of*this class*of**structures*. ... We consider a specific class*of**tree**structures*that can represent basic*structures*in linguistics and computer science such as XML documents, parse*trees*, and treebanks, namely, finite node-labeled sibling-ordered ... More precisely, by combining winning strategies in these games, we show that for every n ∈ N,*the*MSO n-*theory**of**the*fusion*structure*reduces to*the*MSO n-*theories**of**the*components*structures*(by MSO ...##
###
Semantical Evaluations as Monadic Second-Order Compatible Structure Transformations
[chapter]

2002
*
Lecture Notes in Computer Science
*

*The*unfolding

*of*a directed graph into a

*tree*is an MS-compatible transformation that is not an MS-transduction.

*The*MS-compatibility

*of*various transformations

*of*semantical interest follows. ... , for every

*structure*S, if T is

*the*transformed

*structure*τ (S), then P (T ) holds iff Q(S) holds. ... In such a case, if

*the*MS

*theory*

*of*an infinite

*structure*S is decidable (which means that there exists an algorithm that decides whether a

*monadic*secondorder formula is true or not in S), then so is ...

##
###
Types are Internal ∞-Groupoids
[article]

2021
*
arXiv
*
pre-print

That is, every type admits

arXiv:2105.00024v1
fatcat:lrv2nlt5cna7dfrh2aeaisuu3u
*the**structure**of*an ∞-groupoid internally, and this*structure*is unique. ... In particular, our approach leads to a definition*of*∞-groupoid internal to type*theory*and we prove that*the*type*of*such ∞-groupoids is equivalent to*the*universe*of*types. ... Category*theory*suggests that one way*of*representing a*structure*is by*the**monad*on U which it defines, so we might think*of*S as a universe*of**monads*. ...##
###
A descriptive characterization of tree-adjoining languages

1998
*
Proceedings of the 36th annual meeting on Association for Computational Linguistics -
*

Since

doi:10.3115/980691.980752
dblp:conf/acl/Rogers98
fatcat:3q77wdm7yjemhj5enpzx6cgkem
*the*early Sixties and Seventies it has been known that*the*regular and context-free languages are characterized by definability in*the**monadic*second-order*theory**of*certain*structures*. ... In this paper, we exploit these automatatheoretic results to obtain a characterization*of**the**tree*-adjoining languages by definability in*the**monadic*second-order*theory**of*these three-dimensional*tree*...*the*yield*of*sets*of*finite*trees*definable in*the*weak*monadic*second-order*theory**of*multiple successors (wSnS). ...##
###
General Operads and Multicategories
[article]

1998
*
arXiv
*
pre-print

Explicitly: given a

arXiv:math/9810053v1
fatcat:lbjcyxwtjrf6pdk6lmzgdtwfhe
*monad** on a category S, we define*the*term '(S,*)-multicategory', subject to certain conditions on S and *. ... Different choices*of*S and * give some*of**the*existing notions*of*operad and multicategory. ... My thanks go to Martin Hyland for encouraging me to pursue this line*of*enquiry. I am also grateful to Peter Johnstone, Craig Snydal and Robin Cockett, for help and useful discussions. ...##
###
Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects
[chapter]

2008
*
Lecture Notes in Computer Science
*

Hierarchical decompositions yielding

doi:10.1007/978-3-540-70575-8_1
fatcat:ftpcqszcb5fs5nswbqnbnmy5ja
*the*notions*of**tree*-width and clique-width, expressed by terms written with appropriate graph operations and associated with*Monadic*Second-order Logic are important ... tools for*the*construction*of*Fixed-Parameter Tractable algorithms and also for*the*extension*of*methods and results*of*Formal Language*Theory*to*the*description*of*sets*of*finite graphs. ... Walukiewicz for many useful comments on a first draft*of*this overview.*The*first 11 references are books and survey articles. ...##
###
Page 7064 of Mathematical Reviews Vol. , Issue 99j
[page]

1999
*
Mathematical Reviews
*

A collection-

*like*type former which does not seem to fit into*the*framework is given by labelled binary*trees*over some set*of*labels X. ... By seeing them as collection*monads*one can inherit*the*programming constructs and general*theory*developed for collection*monads*. ...##
###
Terms and infinite trees as monads over a signature
[chapter]

1989
*
Lecture Notes in Computer Science
*

In this way, we obtain a family

doi:10.1007/3-540-50939-9_126
fatcat:o56sqlpl2rgddismzmygdzpegu
*of*semantical domains having a*tree*-*like**structure*and appearing as*the*completions*of**the*corresponding finite*structures*. ... In this paper, we prove that*the*usual construction*of*terms and infinite*trees*over a signature is a particular case*of*a more general construction*of**monads*over*the*category*of*sets. ... Acknowledgements : I would*like*to thank Philippe Darondeau for his comments on earlier drafts*of*this paper. ...##
###
Logic and Graphs

1995
*
Electronical Notes in Theoretical Computer Science
*

This chapter is entitled: \

doi:10.1016/s1571-0661(05)80182-3
fatcat:kbsczdodn5fazjim4pssn3u44a
*The*expression*of*graph properties and graph transformations in*monadic*second-order logic". ...*The*lecture is based on a chapter*of*a forthcoming handbook*of*graph grammars, edited by G. Rozenberg. ...*The*lecture will survey*the*following notions: representations*of*graphs by logical*structures*, expressive p o wer*of**monadic*second-order logic from*the*point*of*view*of*graph*theory*, representation*of*...##
###
Self-referential noise as a fundamental aspect of reality

2000
*
AIP Conference Proceedings
*

Noise is often used in

doi:10.1063/1.60018
fatcat:vwkbyprorzahjfak5mwgxa4mba
*the*study*of*open systems, such as in classical Brownian motion and in Quantum Dynamics, to model*the*influence*of**the*environment. ... We show how a three-dimensional process-space may arise, as a Prigogine dissipative*structure*, from a non-geometric order-disorder model driven by, what is termed, self-referential noise. ... These results imply that*the*most*likely**tree*-graph*structure*to which a*monad*can belong has a distance distribution {D k } which indicates that*the**tree*-graph is embeddable in a 3-dimensional hypersphere ...##
###
Page 9540 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

Summary: “We study

*the*succinctness*of**monadic*second-order logic and a variety*of**monadic*fixed point logics on*trees*. ... ideal,*the**theory*Th,,*of*atomic Boolean algebras with an ideal such that*the*supremum*of**the*ideal exists, and*the**theory*Th,,*of*atomless Boolean algebras with an ideal such that*the*supremum*of**the*...##
###
Polynomial Functors and Trees

2010
*
International mathematics research notices
*

In

doi:10.1093/imrn/rnq068
fatcat:vm6daubnavegbe3ok67gohv3o4
*the*second part we describe polynomial endofunctors and*monads*as*structures*built from*trees*, characterising*the*images*of*several nerve functors from polynomial endofunctors and*monads*into presheaves ... Included in Part 1 is also an explicit construction*of**the*free*monad*on a polynomial endofunctor, given in terms*of**trees*. ... Let PolyMnd denote*the*category whose objects are polynomial*monads*and whose morphisms are diagrams*like**the*previous one, required to respect*the**monad**structure*. ...##
###
Two modes of recognition: algebra, coalgebra, and languages
[article]

2019
*
arXiv
*
pre-print

*The*aim

*of*

*the*paper is to build a connection between two approaches towards categorical language

*theory*:

*the*coalgebraic and algebraic language

*theory*for

*monads*. ... For a pair

*of*

*monads*modelling

*the*branching and

*the*linear type we defined regular maps that generalize regular languages known in classical non-deterministic automata

*theory*. ... p to q forms a subtheory

*of*

*the*

*theory*T T S .As a direct corollary

*of*

*the*above theorem we get a Kleene-

*like*theorem characterisation

*of*

*the*regular map

*theory*. ...

##
###
Don't Try This at Home: No-Go Theorems for Distributive Laws
[article]

2018
*
arXiv
*
pre-print

It negatively resolves

arXiv:1811.06460v2
fatcat:ifdjnpi4wzftxi3jt7amwkk7te
*the*open question*of*whether*the*list*monad*distributes over itself. ... This includes a detailed analysis*of*distributive laws for members*of*an extension*of**the*Boom type hierarchy, well known to functional programmers. ...*The*lack*of*this equation in*the**theory*for*the*multiset*monad*therefore means that Theorem 4.24 does not apply to multiset either. ...
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