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The Domino Problem for Self-similar Structures [chapter]

Sebastián Barbieri, Mathieu Sablik
2016 Lecture Notes in Computer Science  
In this setting we exhibit non-trivial families of subsets with decidable and undecidable domino problem.  ...  We define the domino problem for tilings over self-similar structures of Z d given by forbidden patterns.  ...  Connectivity Isthmus Weak grid Strong grid DP decidable Unknown DP undecidable DP undecidable Concluding remarks In this article we introduced a version of the domino problem on self-similar structures  ... 
doi:10.1007/978-3-319-40189-8_21 fatcat:uuiessd3tbca7ekqjroygyzana

On the solvability of domino snake problems

Yael Etzion-Petruschka, David Harel, Dale Myers
1994 Theoretical Computer Science  
Myers, On the solvability of domino snake problems, Theoretical Computer Science 131 (1994) 243-269.  ...  In this paper we present an extensive treatment of tile connecrability problems, sometimes called domino snake problems.  ...  Less known is the family of tile connectability problems, or domino snake problems.  ... 
doi:10.1016/0304-3975(94)90174-0 fatcat:mrdbkye4zfdexgty4psngbojye

The domino problem on groups of polynomial growth [article]

Alexis Ballier, Maya Stein
2018 arXiv   pre-print
We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually  ...  free groups, i.e. finite groups, and those having as a subgroup of finite index.  ...  DECIDABILITY OF THE DOMINO PROBLEM In this section we reduce the undecidability of the domino problem in groups with the 'half-grid' structure found in Section 3 to the undecidability of the domino problem  ... 
arXiv:1311.4222v4 fatcat:af5ubc5vuzbz5iqmfchgts4d7m

Monadic second-order logic and the domino problem on self-similar graphs [article]

Laurent Bartholdi
2020 arXiv   pre-print
On the other hand, we already prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and some examples including one of linear growth  ...  We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic.  ...  Undecidability results We now prove that the domino problem is undecidable on some examples of self-similar graphs.  ... 
arXiv:2011.02735v1 fatcat:vun3sghucve7zglgusdaq65a54

Undecidability of PDL with L = {a2i |; i ⩾ 0}

D. Harel, M.S. Paterson
1984 Journal of computer and system sciences (Print)  
It is shown that the validity problem for propositional dynamic logic (PDL), which is decidable and actually DEXPTIME-complete for the usual class of regular programs, becomes highly undecidable, viz.  ...  This answers a question of Harel, Pnueli, and Stavi. 0 1984 Academic Press, Inc.  ...  UNDECIDABILITY OF PDL WITH I!, = (u" 1 i > o}  ... 
doi:10.1016/0022-0000(84)90005-9 fatcat:f7udt723qvapddi7nllcfp4npe

The Domino Problem is Undecidable on Surface Groups

Nathalie Aubrun, Sebastián Barbieri, Etienne Moutot, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2.  ...  We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property.  ...  An important property of the domino problem is that groups which contain subgroups with undecidable domino problem have themselves undecidable domino problem [6, Proposition 9.3.30] .  ... 
doi:10.4230/lipics.mfcs.2019.46 dblp:conf/mfcs/AubrunBM19 fatcat:pzzwocsetnbv7pqnuo3ir2344e

The domino problem on groups of polynomial growth

Alexis Ballier, Maya Stein
2018 Groups, Geometry, and Dynamics  
Our proof uses a reduction of the undecidability of the domino problem on groups with infinite tree-width to the undecidability of the domino problem on Z 2 .  ...  We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually  ...  This means that v i w i 1 . . . w i ki = v i ′ w i ′ 1 . . . w i k i ′ for some i ̸ = i ′ , DECIDABILITY OF THE DOMINO PROBLEM In this section we reduce the undecidability of the domino problem in  ... 
doi:10.4171/ggd/439 fatcat:3yrldil3bveutoxjtsgquoa7fe

Translation-like Actions and Aperiodic Subshifts on Groups [article]

Emmanuel Jeandel
2015 arXiv   pre-print
weakly aperiodic SFT (and actually a undecidable domino problem).  ...  It is well known that if G admits a f.g. subgroup H with a weaklyaperiodic SFT (resp. an undecidable domino problem), then Gitself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove  ...  has undecidable domino problem.  ... 
arXiv:1508.06419v1 fatcat:dnrdibpfuvbkfgcml2e7kfoaee

Undecidability of the transitive graded modal logic with converse

Evgeny Zolin
2016 Journal of Logic and Computation  
The method of proof deserves a few words. We prove the undecidability by reduction from the undecidable domino problem for Z×Z.  ...  As a consequence, for the "unrestricted version" of the description logic SIQ, the problem of concept satisfiability (even with respect to the empty terminology) is undecidable.  ...  Domino problem Our undecidability proofs are given by reduction from the undecidable "domino problem".  ... 
doi:10.1093/logcom/exw026 fatcat:kqxfawxah5f5djgsv2dnugwseu

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Jakub Michaliszyn, Jan Otop, Piotr Witkowski
2012 Electronic Proceedings in Theoretical Computer Science  
finite satisfiability problem, but undecidable general satisfiability problem.  ...  with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable  ...  Is there a universal first-order formula Φ such that the global satisfiability problem of modal logic over K Φ is NEXPTIME-hard and decidable?  ... 
doi:10.4204/eptcs.96.11 fatcat:mo6aflstjndjtn6qkikytufo7e

Consistency of multidimensional combinatorial substitutions

Timo Jolivet, Jarkko Kari
2012 Theoretical Computer Science  
We prove that it is undecidable whether a two-dimensional substitution is consistent or overlapping, and we provide practical algorithms to decide these properties in some particular cases.  ...  Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the patterns in an image by the substitution might overlap.  ...  This research is supported by the Academy of Finland Grant 131558.  ... 
doi:10.1016/j.tcs.2012.03.050 fatcat:qsa7zlfv3fgs3du6vt6b3zheui

Consistency of Multidimensional Combinatorial Substitutions [chapter]

Timo Jolivet, Jarkko Kari
2012 Lecture Notes in Computer Science  
We prove that it is undecidable whether a two-dimensional substitution is consistent or overlapping, and we provide practical algorithms to decide these properties in some particular cases.  ...  Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the patterns in an image by the substitution might overlap.  ...  This research is supported by the Academy of Finland Grant 131558.  ... 
doi:10.1007/978-3-642-30642-6_20 fatcat:b2rwzp7b4neznoi6fbbafu345a

The domino problem is undecidable on surface groups [article]

Nathalie Aubrun, Sebastián Barbieri, Etienne Moutot
2018 arXiv   pre-print
As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2.  ...  We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property.  ...  We would also grateful to Yann Ollivier for his pictures of the Cayley graph of the surface group [16] , the one in the front page being one of them.  ... 
arXiv:1811.08420v1 fatcat:qjkw6u2pz5hulb4lyuozx2o4ly

Preface

Nataša Jonoska, Jarkko Kari
2009 Theoretical Computer Science  
The domino problem is the decision problem for determining whether a given set of Wang tiles admits a tiling of the infinite plane. Proved undecidable by R.  ...  Berger in the 60's, the domino problem has become an important tool for establishing undecidability results in various systems, including cellular automata. The paper by V.  ...  Jarkko Kari Department of Mathematics, University of Turku, FI-20014, Finland  ... 
doi:10.1016/j.tcs.2008.12.002 fatcat:ghm2oc3k35hrzncrp7flzothhq

Consensus Game Acceptors [chapter]

Dietmar Berwanger, Marie van den Bogaard
2015 Lecture Notes in Computer Science  
The players have a joint objective to avoid an inadmissible decision, in spite of the uncertainty induced by the input.  ...  Uniform encoding of domino problems in games Game formulations of domino tiling problems are standard in complexity theory, going back to the early work of Chlebus [7] .  ...  As the emptiness problem for contextsensitive languages is undecidable [12] , it follows that the solvability problem is undecidable for consensus game acceptors.  ... 
doi:10.1007/978-3-319-21500-6_8 fatcat:6jzwrwgegbdi5a7rblyf7buhbe
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