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Some Perspectives of Infinite-State Verification
[chapter]
2005
Lecture Notes in Computer Science
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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]
2014
Lecture Notes in Computer Science
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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
2010
Foundations of Software Technology and Theoretical Computer Science
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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]
2003
Lecture Notes in Computer Science
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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]
2022
arXiv
pre-print
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2021 pre-print arXiv:2111.10881v1 fatcat:ednjglc3g5dundxpejuyrrqwpi
, 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]
2004
Lecture Notes in Computer Science
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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
2012
2012 27th Annual IEEE Symposium on Logic in Computer Science
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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]
2006
Lecture Notes in Computer Science
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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]
2020
arXiv
pre-print
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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]
2009
Lecture Notes in Computer Science
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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
2006
Theoretical Computer Science
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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]
2006
arXiv
pre-print
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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
2004
Proceedings. 11th International Symposium on Temporal Representation and Reasoning, 2004. TIME 2004.
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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]
2016
arXiv
pre-print
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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
2015
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
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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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