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Some Perspectives of Infinite-State Verification
[chapter]

2005
*
Lecture Notes in Computer Science
*

We report on recent progress in the study of

doi:10.1007/11562948_3
fatcat:z6i5hzxbvndrvj3rq52gqym5ju
*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*. ...##
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On the Decidability of MSO+U on Infinite Trees
[chapter]

2014
*
Lecture Notes in Computer Science
*

We conjecture that the

doi:10.1007/978-3-662-43951-7_5
fatcat:og5wdmnapnb6zfcm4qo22qp3cm
*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*...##
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First-Order Logic with Reachability Predicates on Infinite Systems

2010
*
Foundations of Software Technology and Theoretical Computer Science
*

It allows to transfer the

doi:10.4230/lipics.fsttcs.2010.493
dblp:conf/fsttcs/Schulz10
fatcat:26webfty2rfhxmja2wuj3kcoue
*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. ...##
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The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata
[chapter]

2003
*
Lecture Notes in Computer Science
*

In this paper we give two equivalent characterizations of the Caucal hierarchy,

doi:10.1007/978-3-540-24597-1_10
fatcat:hfabghdqxvan5ne5axcsk65exu
*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. ...##
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Solving Infinite Games in the Baire Space
[article]

2022
*
arXiv
*
pre-print

, the winner can be computed, and an ℕ-

arXiv:2111.10881v2
fatcat:hwukltkbvrca5jled4huu2jyaq
*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] )). ...##
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Decidability of MSO Theories of Tree Structures
[chapter]

2004
*
Lecture Notes in Computer Science
*

equivalence to reduce (

doi:10.1007/978-3-540-30538-5_36
fatcat:lcfh5pmbzfd65l4kzdnhswvjbm
*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. ...##
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Regular Transformations of Infinite Strings

2012
*
2012 27th Annual IEEE Symposium on Logic in Computer Science
*

The

doi:10.1109/lics.2012.18
dblp:conf/lics/AlurFT12
fatcat:c4uu62eylfhhha2vb4qjsskfsm
*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 ...##
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Tree Automata Make Ordinal Theory Easy
[chapter]

2006
*
Lecture Notes in Computer Science
*

We give

doi:10.1007/11944836_27
fatcat:lhvcjbdb5ndyroma2moiollgfa
*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. ...##
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MSO Undecidability for some Hereditary Classes of Unbounded Clique-Width
[article]

2020
*
arXiv
*
pre-print

Seese's conjecture for finite

arXiv:2011.02894v1
fatcat:birb72uxozcwxhafoo7dijq77a
*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*. ...##
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Facets of Synthesis: Revisiting Church's Problem
[chapter]

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

doi:10.1007/978-3-642-00596-1_1
fatcat:uhny7wacofafzl3s4fcd6sc37e
*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. ...##
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Trees, grids, and MSO decidability: From graphs to matroids

2006
*
Theoretical Computer Science
*

More exactly we prove, for every finite field F, that any class of F-representable matroids

doi:10.1016/j.tcs.2005.10.006
fatcat:dtsh4w7hs5dbtkz6nyqjef3cp4
*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. ...##
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Tree Automata Make Ordinal Theory Easy
[article]

2006
*
arXiv
*
pre-print

We give

arXiv:cs/0610166v1
fatcat:imb6eujq25ba5agu4hypao4lhm
*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 ...##
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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.
*

fixed vertex-colored tree; then, we exploit

doi:10.1109/time.2004.1314434
dblp:conf/time/MontanariP04
fatcat:ein7ak22qbbphizngvnisyvl7e
*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. ...##
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Marking Shortest Paths On Pushdown Graphs Does Not Preserve MSO Decidability
[article]

2016
*
arXiv
*
pre-print

Our main result is that the edge-marked version of

arXiv:1510.04000v2
fatcat:adzwyif2qfbwbk6jr2yjrmejoy
*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. ...##
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Higher-Order Model Checking: An Overview

2015
*
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
*

The past fifteen years or so have seen considerable progress in both

doi:10.1109/lics.2015.9
dblp:conf/lics/Ong15
fatcat:55bfxqlkuzhbtgmzc32m46ooku
*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*. ...
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