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Some Perspectives of Infinite-State Verification [chapter]

Wolfgang Thomas
2005 Lecture Notes in Computer Science  
We report on recent progress in the study of infinite transition systems for which interesting properties (like reachability of designated states) can be checked algorithmically.  ...  Two methods for the generation of such models are discussed: the construction from simpler models via operations like unfolding and synchronized product, and the internal representation of state spaces  ...  We shall address two methods for constructing infinite transition graphs where model-checking (with respect to MSO or FO(R)) is decidable.  ... 
doi:10.1007/11562948_3 fatcat:z6i5hzxbvndrvj3rq52gqym5ju

On the Decidability of MSO+U on Infinite Trees [chapter]

Mikołaj Bojańczyk, Tomasz Gogacz, Henryk Michalewski, Michał Skrzypczak
2014 Lecture Notes in Computer Science  
We conjecture that the mso+u theory of the complete binary tree is undecidable. We prove a weaker statement: there is no algorithm which decides this theory and has a correctness proof in zfc.  ...  This is because the theory is undecidable, under a settheoretic assumption consistent with zfc, namely that there exists of projective well-ordering of 2 ω of type ω1.  ...  Apart from the obvious question about unconditional undecidability, a natural question is about the decidability of mso+u on infinite words: is the mso+u theory of the natural numbers with successor decidable  ... 
doi:10.1007/978-3-662-43951-7_5 fatcat:og5wdmnapnb6zfcm4qo22qp3cm

First-Order Logic with Reachability Predicates on Infinite Systems

Stefan Schulz, Marc Herbstritt
2010 Foundations of Software Technology and Theoretical Computer Science  
It allows to transfer the decidability of MSO to FO(Reg) onto the class of transformed structures. Finally we extend regular ground tree rewriting with a skeleton tree.  ...  Furthermore we introduce a transformation for infinite graphs called setbased unfolding which is based on an idea of Lohrey and Ondrusch.  ...  Acknowledgements With thanks to Arnaud Carayol and Christof Löding for introducing me to these topics and constantly sharing their valuable ideas.  ... 
doi:10.4230/lipics.fsttcs.2010.493 dblp:conf/fsttcs/Schulz10 fatcat:26webfty2rfhxmja2wuj3kcoue

The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata [chapter]

Arnaud Carayol, Stefan Wöhrle
2003 Lecture Notes in Computer Science  
In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic second-order (MSO) theory.  ...  We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the ε-closure of configuration graphs of higher-order pushdown automata.  ...  Colcombet for pointing to a solution for Lem. 1, T. Cachat for reading a first draft of Sect. 4, and the anonymous referees for many useful remarks.  ... 
doi:10.1007/978-3-540-24597-1_10 fatcat:hfabghdqxvan5ne5axcsk65exu

Solving Infinite Games in the Baire Space [article]

Benedikt Brütsch, Wolfgang Thomas
2022 arXiv   pre-print
, the winner can be computed, and an ℕ-MSO-transducer realizing a winning strategy for the winner can be constructed.  ...  Infinite games (in the form of Gale-Stewart games) are studied where a play is a sequence of natural numbers chosen by two players in alternation, the winning condition being a subset of the Baire space  ...  sequences in which some number occurs infinitely often (which gives a proper Σ 3 -set; cf. [30] )).  ... 
arXiv:2111.10881v2 fatcat:hwukltkbvrca5jled4huu2jyaq

Decidability of MSO Theories of Tree Structures [chapter]

Angelo Montanari, Gabriele Puppis
2004 Lecture Notes in Computer Science  
equivalence to reduce (a number of instances of) the latter problem to the decidable case of regular trees.  ...  We prove that such a reduction works for a large class of trees, that we call residually regular trees. We conclude the paper with a short discussion of related work.  ...  We are currently trying to determine whether or not any deterministic tree in the Caucal hierarchy can be embedded into a suitable residually regular tree.  ... 
doi:10.1007/978-3-540-30538-5_36 fatcat:lcfh5pmbzfd65l4kzdnhswvjbm

Regular Transformations of Infinite Strings

Rajeev Alur, Emmanuel Filiot, Ashutosh Trivedi
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
The theory of regular transformations of finite strings is quite mature with appealing properties.  ...  Finally, we show that type checking and functional equivalence are decidable for MSO-definable transformations of infinite strings.  ...  MSO transducers work by introducing a fixed number of copies of the vertices of the input graph; and the domain, the labels and the edges of the output graph are defined by MSO formulas with zero, one  ... 
doi:10.1109/lics.2012.18 dblp:conf/lics/AlurFT12 fatcat:c4uu62eylfhhha2vb4qjsskfsm

Tree Automata Make Ordinal Theory Easy [chapter]

Thierry Cachat
2006 Lecture Notes in Computer Science  
We give a new simple proof of the decidability of the First Order Theory of (ω ω i , +) and the Monadic Second Order Theory of (ω i , <), improving the complexity in both cases.  ...  Dagstuhl Seminar Proceedings 07441 Algorithmic-Logical Theory of Infinite Structures  ...  and to the participants of Dagstuhl-Seminar "Algorithmic-Logical Theory of Infinite Structures" for interesting discussions.  ... 
doi:10.1007/11944836_27 fatcat:lhvcjbdb5ndyroma2moiollgfa

MSO Undecidability for some Hereditary Classes of Unbounded Clique-Width [article]

Anuj Dawar, Abhisekh Sankaran
2020 arXiv   pre-print
Seese's conjecture for finite graphs states that monadic second-order logic (MSO) is undecidable on all graph classes of unbounded clique-width.  ...  We explore a number of known examples of the former category and establish that grids of unbounded size can indeed be interpreted in them.  ...  Thus, if the MSO theory of C is decidable, we can decide if a given MSO sentence ϕ is true in all graphs in C ↓, by deciding if ϕ ′ ∶= ∀Xϕ X (X) is in this theory.  ... 
arXiv:2011.02894v1 fatcat:birb72uxozcwxhafoo7dijq77a

Facets of Synthesis: Revisiting Church's Problem [chapter]

Wolfgang Thomas
2009 Lecture Notes in Computer Science  
We recall the fundamental questions raised more than 50 years ago in "Church's Synthesis Problem" that led to the foundation of the algorithmic theory of infinite games.  ...  Tree Automata A very strong decidability result of logic is Rabin's Tree Theorem, saying that the MSO-theory of the infinite binary tree is decidable [21] .  ...  For MSO-logic this is compatible with the definition given above: A strategy f is MSO-definable iff the graph of the associated causal function f ω is MSO-definable.  ... 
doi:10.1007/978-3-642-00596-1_1 fatcat:uhny7wacofafzl3s4fcd6sc37e

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

Petr Hliněný, Detlef Seese
2006 Theoretical Computer Science  
More exactly we prove, for every finite field F, that any class of F-representable matroids with a decidable MSO theory must have uniformly bounded branch-width.  ...  Moreover, we show that bounding the branch-width of all matroids in general is not sufficient to obtain a decidable MSO theory.  ...  Acknowledgements It is a pleasure for us to express our thanks to M. Fellows, M. Hallett, R. Niedermeier and N.  ... 
doi:10.1016/j.tcs.2005.10.006 fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4

Tree Automata Make Ordinal Theory Easy [article]

Thierry Cachat
2006 arXiv   pre-print
We give a new simple proof of the decidability of the First Order Theory of (omega^omega^i,+) and the Monadic Second Order Theory of (omega^i,<), improving the complexity in both cases.  ...  Our algorithm is based on tree automata and a new representation of (sets of) ordinals by (infinite) trees.  ...  Acknowledgments Many thanks to Wolfgang Thomas for always saying that we have a proof without theorem (the proof that tree automata are closed under intersection, complementation and projection), to him  ... 
arXiv:cs/0610166v1 fatcat:imb6eujq25ba5agu4hypao4lhm

Decidability of the theory of the totally unbounded /spl omega/-layered structure

A. Montanari, G. Puppis
2004 Proceedings. 11th International Symposium on Temporal Representation and Reasoning, 2004. TIME 2004.  
fixed vertex-colored tree; then, we exploit a suitable notion of tree equivalence to reduce the latter problem to the decidable case of regular trees.  ...  a structure totally unbounded).  ...  The upshot of such a result is that infinite complete residually regular trees enjoy a decidable MSO theory. 6.  ... 
doi:10.1109/time.2004.1314434 dblp:conf/time/MontanariP04 fatcat:ein7ak22qbbphizngvnisyvl7e

Marking Shortest Paths On Pushdown Graphs Does Not Preserve MSO Decidability [article]

Arnaud Carayol, Olivier Serre
2016 arXiv   pre-print
Our main result is that the edge-marked version of a pushdown graph may itself no longer be a pushdown graph, as we prove that this enrich graph may have an undecidable MSO theory.  ...  Our main result is that the edge-marked version of a pushdown graph may itself no longer be a pushdown graph, as we prove that the MSO theory of this enriched graph may be undecidable.  ...  1 In programming languages, reflection is the process by which a computer program can observe and dynamically modify its own structure and behaviour.  ... 
arXiv:1510.04000v2 fatcat:adzwyif2qfbwbk6jr2yjrmejoy

Higher-Order Model Checking: An Overview

Luke Ong
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
The past fifteen years or so have seen considerable progress in both theory and practice.  ...  Outline: In Section II, we introduce two families of generators of infinite structures: recursion schemes and higher-order  ...  , they do not have a decidable MSO theory.  ... 
doi:10.1109/lics.2015.9 dblp:conf/lics/Ong15 fatcat:55bfxqlkuzhbtgmzc32m46ooku
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