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

Bruno Courcelle
2009 Lecture Notes in Computer Science
This tutorial will present an overview 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

Amélie GHEERBRANT, Balder Cate, Nicole Schweikardt
2012 Logical Methods in Computer Science
We present axiomatizations 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]

Bruno Courcelle
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]

Antoine Allioux, Eric Finster, Matthieu Sozeau
2021 arXiv   pre-print
That is, every type admits 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

James Rogers
1998 Proceedings of the 36th annual meeting on Association for Computational Linguistics -
Since 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]

Tom Leinster
1998 arXiv   pre-print
Explicitly: given a 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]

Bruno Courcelle
2008 Lecture Notes in Computer Science
Hierarchical decompositions yielding 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 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

Bruno Courcelle
1995 Electronical Notes in Theoretical Computer Science
This chapter is entitled: \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

Reginald T. Cahill
2000 AIP Conference Proceedings
Noise is often used in 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

J. Kock
2010 International mathematics research notices
In 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]

Tomasz Brengos, Marco Peressotti
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]

Maaike Zwart, Dan Marsden
2018 arXiv   pre-print
It negatively resolves 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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