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

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We exhibit a strong connection between the matchgate formalism introduced by Valiant and the ZW-calculus of Coecke and Kissinger. This connection provides a natural compositional framework for matchgate theory as well as a direct combinatorial interpretation of the diagrams of ZW-calculus through the perfect matchings of their underlying graphs.
We identify a precise fragment of ZW-calculus, the planar W-calculus, that we prove to be complete and universal for matchgates, that are linear maps satisfying the matchgate identities. Computing scalars of the planar W-calculus corresponds to counting perfect matchings of planar graphs, and so can be carried in polynomial time using the FKT algorithm, making the planar W-calculus an efficiently simulable fragment of the ZW-calculus, in a similar way that the Clifford fragment is for ZX-calculus. This work opens new directions for the investigation of the combinatorial properties of ZW-calculus as well as the study of perfect matching counting through compositional diagrammatical technics.

Titouan Carette, Etienne Moutot, Thomas Perez, and Renaud Vilmart. Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 120:1-120:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carette_et_al:LIPIcs.ICALP.2023.120, author = {Carette, Titouan and Moutot, Etienne and Perez, Thomas and Vilmart, Renaud}, title = {{Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {120:1--120:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.120}, URN = {urn:nbn:de:0030-drops-181726}, doi = {10.4230/LIPIcs.ICALP.2023.120}, annote = {Keywords: Perfect Matchings Counting, Quantum Computing, Matchgates, ZW-Calculus, String Diagrams, Completeness} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

In this paper we study low-complexity colorings (or tilings) of the two-dimensional grid ℤ². A coloring is said to be of low complexity with respect to a rectangle if there exists m,n∈ℕ such that there are no more than mn different rectangular m× n patterns in it. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. Suppose we are given at most nm rectangular patterns of size n× m. If Nivat’s conjecture is true, one can only build periodic colorings out of these patterns - meaning that if the m× n rectangular patterns of the coloring are among these mn patterns, it must be periodic. The main contribution of this paper proves that there exists at least one periodic coloring build from these patterns. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Finally, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. The results also extend to other convex shapes in place of the rectangle.

Jarkko Kari and Etienne Moutot. Decidability and Periodicity of Low Complexity Tilings. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 14:1-14:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kari_et_al:LIPIcs.STACS.2020.14, author = {Kari, Jarkko and Moutot, Etienne}, title = {{Decidability and Periodicity of Low Complexity Tilings}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {14:1--14:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.14}, URN = {urn:nbn:de:0030-drops-118752}, doi = {10.4230/LIPIcs.STACS.2020.14}, annote = {Keywords: Nivat’s conjecture, domino problem, decidability, low pattern complexity, 2D subshifts, symbolic dynamics} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property. 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.

Nathalie Aubrun, Sebastián Barbieri, and Etienne Moutot. The Domino Problem is Undecidable on Surface Groups. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{aubrun_et_al:LIPIcs.MFCS.2019.46, author = {Aubrun, Nathalie and Barbieri, Sebasti\'{a}n and Moutot, Etienne}, title = {{The Domino Problem is Undecidable on Surface Groups}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {46:1--46:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.46}, URN = {urn:nbn:de:0030-drops-109900}, doi = {10.4230/LIPIcs.MFCS.2019.46}, annote = {Keywords: tilings, substitutions, SFTs, decidability, domino problem} }

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