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### Automata theory on trees and partial orders [chapter]

Wolfgang Thomas
1997 Lecture Notes in Computer Science
The paper reviews recent results which aim at generalizing nite automata theory from words and trees to labelled partial orders (presented as labelled directed acyclic graphs), with an emphasis on logical  ...  Graph acceptors and their specialization for pictures, \tiling systems", are presented, and their equivalence to existential monadic second-order logic is reviewed.  ...  Graph Acceptors and Existential Monadic Logic Over words or trees, there is a simple argument to show that monadic secondorder formulas and nite automata are expressively equivalent: Acceptance of a word  ...

### Uniform and nonuniform recognizability

Wolfgang Thomas
2003 Theoretical Computer Science
We show how this approach encompasses known models of nondeterministic automata over ÿnite words, trees, pictures, and graphs, and present some elementary metaresults connecting uniform recognizability  ...  , nonuniform recognizability, and monadic second-order logic.  ...  Starting from the expressive equivalence between ÿnite automata and monadic second-order logic over words (as shown in [2, 12] ) or over trees [9, 21] , one may compare the expressive power of recognizers  ...

### 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
However, the algebraic approach can be extended to structures beyond words, e.g.~infinite words, or trees or graphs.  ...  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.  ...  Show that for graphs without edges, mso 1 and mso 2 has the same expressive power as first-order logic. Exercise 172.  ...

### Languages, Automata, and Logic [chapter]

Wolfgang Thomas
1997 Handbook of Formal Languages
-words and their determinization, and a self-contained proof of the \Rabin Tree Theorem".  ...  Central points are the connection between nite automata and monadic second-order logic, the Ehrenfeucht-Fra ss e technique in the context of formal language theory, nite automata on !  ...  Zielonka on a pre-nal version contributed a lot to improve the text. Finally, I thank the members of the theory group in Kiel for e cient help and support, and G.  ...

### Page 7468 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews
Monadic second-order logic (an extension of first-order logic with quantified variables denoting sets of elements of the domain) is used to define sets of graphs.  ...  {For the entire collection see MR 94c:00024.} 94m:68107 68Q50 03B15 03C85 05C99 68R10 Courcelle, Bruno (F-BORD-LB; Talence) Graph grammars, monadic second-order logic and the theory of graph minors.  ...

### Page 5089 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews
- tifier alternation hierarchy over grids and pictures (441-460); Thomas Schwentick, Padding and the expressive power of existen- tial second-order logics (461—477); Ludwig Staiger, Rich w-words and monadic  ...  , Concurrent con- straint programming and non-commutative logic (406-423); Ina Schiering, A hierarchical approach to monadic second-order logic over graphs (424-440); Nicole Schweikardt, The monadic quan  ...

### Separator logic and star-free expressions for graphs [article]

Mikolaj Bojanczyk
2021 arXiv   pre-print
A corollary is that, given k and a graph language represented by an formula, one can decide if the language can be defined in either of two equivalent formalisms on graphs of pathwidth at most k.  ...  This is first-order logic on graphs which is allowed to use the edge relation, and for every n ∈{0,1,...} a relation of arity n+2 which says that "vertex s can be connected to vertex t by a path that avoids  ...  By saying that a graph is connected and has no cycles, we can define trees in separator logic. In this paper, we are most interested in the expressive power of separator logic.  ...

### Trees, grids, and MSO decidability: From graphs to matroids

Petr Hliněný, Detlef Seese
2006 Theoretical Computer Science
Monadic second order (MSO) logic has proved to be a useful tool in many areas of application, reaching from decidability and complexity to picture processing, correctness of programs and parallel processes  ...  Logic 53 (1991) ] that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees.  ...  Courcelle, the organizer of the workshop on "Logic and Graph Transformations" at the ICGT 2004 conference, for inviting us to present our research results there.  ...

### Two monads for graphs [article]

Mikolaj Bojanczyk
2018 arXiv   pre-print
An introduction to algebras for graphs, based on Courcelle's algebras of hyperedge replacement and vertex replacement. The paper uses monad notation.  ...  the expressive power of mso without counting.  ...  Monadic second-order logic A classical theme in language theory is to use logic to describe properties of objects such as words, trees, graphs, etc.  ...

### A Gentle Introduction to Applications of Algorithmic Metatheorems for Space and Circuit Classes

Till Tantau
2016 Algorithms
As an example, by Courcelle's Theorem, all monadic second-order ("in a certain logic") properties of graphs of bounded tree width ("structured in a certain way") can be solved in linear time ("with a certain  ...  The underlying algorithmic approach is quite 93 ingenious: On input of a graph, if the graph is tree-like, we apply an algorithmic metatheorem; and if 94 the graph is not tree-like, it must be "highly  ...  This takes the first step out of the picture and we get an algorithmic metatheorem where the algorithmic complexity hinges (only) on the complexity of evaluating a tree automaton on an input tree, and  ...

Wolfgang Thomas
2010 Lecture Notes in Computer Science
Pioneers of logic, among them J.R. Büchi, M.O. Rabin, S. Shelah, and Y. Gurevich, have shown that monadic second-order logic offers a rich landscape of interesting decidable theories.  ...  Prominent examples are the monadic theory of the successor structure S1 = (N, +1) of the natural numbers and the monadic theory of the binary tree, i.e., of the two-successor structure S2 = ({0, 1} * ,  ...  Acknowledgment Many thanks are due to Nachum Dershowitz for his patience and help and to Christof Löding and Alex Rabinovich for their comments.  ...

### On Decidability of MSO Theories of Representable Matroids [chapter]

Petr Hliněný, Detlef Seese
2004 Lecture Notes in Computer Science
We show that, for every finite field , the class of allrepresentable matroids of branch-width at most a constant t has a decidable MSO theory.  ...  In the other direction, we prove that every class of -representable matroids with a decidable MSO theory must have uniformly bounded branch-width.  ...  Acknowledgement It is a pleasure for us to express our thanks to M. Fellows, M. Hallett, R. Niedermeier and N.  ...

### Subject index volumes 1–200

1999 Theoretical Computer Science
3593 logic, 2574, 3580 first-order linear-time temporal logic, 3049 first-order logic, 823, 1057,2155,2156,2222,2407, 2479,2785,2791,3054,3242,3270 completeness-and -, 2187 expressive power of  ...  theory, 1483 weak logical reductions, 2269 weak model, 252 1 weak monadic formulas, 1570 logic, 1732, 1802 weak monadic theory of the tree. 1802 trees, 1802 weak monotonicity in laguage learning  ...

### Second-Order Logic over Strings: Regular and Non-regular Fragments [chapter]

Thomas Eiter, Georg Gottlob, Thomas Schwentick
2002 Lecture Notes in Computer Science
We also give a brief survey of recent results on the complexity of evaluating existential second-order logic over graphs, and a list of interesting open problems.  ...  By a well-known result due to Büchi and Trakhtenbrot, all monadic second-order sentences over words describe regular languages.  ...  of graphs of bounded tree-width.  ...

### Algebraic recognizability of languages [article]

Pascal Weil
2006 arXiv   pre-print
Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a cornerstone for applications and for theory.  ...  We would like to briefly explore why that is, and how this word-related notion extends to more complex models, such as those developed for modeling distributed or timed behaviors.  ...  Recognizability Based on the examples of words and trees, the natural language for logical definability of recognizable sets would seem to be MSO, monadic second order logic.  ...
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