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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete).

Pablo Barceló, Chih-Duo Hong, Xuan-Bach Le, Anthony W. Lin, and Reino Niskanen. Monadic Decomposability of Regular Relations (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 103:1-103:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barcelo_et_al:LIPIcs.ICALP.2019.103, author = {Barcel\'{o}, Pablo and Hong, Chih-Duo and Le, Xuan-Bach and Lin, Anthony W. and Niskanen, Reino}, title = {{Monadic Decomposability of Regular Relations}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {103:1--103:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.103}, URN = {urn:nbn:de:0030-drops-106790}, doi = {10.4230/LIPIcs.ICALP.2019.103}, annote = {Keywords: Transducers, Automata, Synchronized Rational Relations, Ramsey Theory, Variable Independence, Automatic Structures} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the special linear group following recent NP-completeness result for SL(2,Z) and the undecidability for SL(4,Z) generated by 48 matrices. First we show that there is no embedding from pairs of words into 3 x3 integer matrices with determinant one, i.e., into SL{(3,Z)} extending previously known result that there is no embedding into C^{2 x 2}. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in SL{(3,Z)} are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into SL{(3,Z)}, where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of SL(3,Z), the Heisenberg group H(3,Z). Furthermore, we extend the decidability result for H(n,Q) in any dimension n. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in SL{(4,Z)} substantially reducing the bound on the size of the generator set from 48 to 8 by developing a novel reduction technique.

Sang-Ki Ko, Reino Niskanen, and Igor Potapov. On the Identity Problem for the Special Linear Group and the Heisenberg Group. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 132:1-132:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ko_et_al:LIPIcs.ICALP.2018.132, author = {Ko, Sang-Ki and Niskanen, Reino and Potapov, Igor}, title = {{On the Identity Problem for the Special Linear Group and the Heisenberg Group}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {132:1--132:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.132}, URN = {urn:nbn:de:0030-drops-91367}, doi = {10.4230/LIPIcs.ICALP.2018.132}, annote = {Keywords: matrix semigroup, identity problem, special linear group, Heisenberg group, decidability} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Robot game is a two-player vector addition game played on the integer lattice Z^n. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases.

Reino Niskanen, Igor Potapov, and Julien Reichert. Undecidability of Two-dimensional Robot Games. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 73:1-73:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{niskanen_et_al:LIPIcs.MFCS.2016.73, author = {Niskanen, Reino and Potapov, Igor and Reichert, Julien}, title = {{Undecidability of Two-dimensional Robot Games}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {73:1--73:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.73}, URN = {urn:nbn:de:0030-drops-64839}, doi = {10.4230/LIPIcs.MFCS.2016.73}, annote = {Keywords: reachability games, vector addition game, decidability, winning strategy} }

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