161 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    An aperiodic monotile

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    A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Computability and Tiling Problems

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    In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles SS has total planar tilings, which we denote TILETILE, or whether it has infinite connected but not necessarily total tilings, WTILEWTILE (short for `weakly tile'). We show that both TILE≡mILL≡mWTILETILE \equiv_m ILL \equiv_m WTILE, and thereby both TILETILE and WTILEWTILE are Σ11\Sigma^1_1-complete. We also show that the opposite problems, ¬TILE\neg TILE and SNTSNT (short for `Strongly Not Tile') are such that ¬TILE≡mWELL≡mSNT\neg TILE \equiv_m WELL \equiv_m SNT and so both ¬TILE\neg TILE and SNTSNT are both Π11\Pi^1_1-complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets PTilePTile of periodic tilings, and ATileATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form (Σ11∧Π11)(\Sigma^1_1 \wedge \Pi^1_1). We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle CTCT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, CωωC_{\omega^\omega}. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to CωωC_{\omega^\omega}. Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    On constructing topology from algebra

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    In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on the set N. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the following semigroups: the monoid of all injective functions on N, the monoid of continuous transformations of the Hilbert cube [0, 1]N, the monoid of continuous transformations of the Cantor space 2N, and the monoid of endomorphisms of the countably infinite atomless boolean algebra. With the exception of the continuous transformation monoid of the Hilbert cube, we also show that all of the above semigroups admit a second countable semigroup topology such that every semigroup homomorphism from the semigroup to a second countable topological semigroup is continuous. In a recent paper, Bleak, Cameron, Maissel, Navas, and Olukoya use a theorem of Rubin to describe the automorphism groups of the Higman-Thompson groups Gâ‚™,áµ£ via their canonical Rubin action on the Cantor space. In particular they embed these automorphism groups into the rational group R of transducers introduced by Grigorchuk, Nekrashevich, and Sushchanskii. We generalise these transducers to be more suitable to higher dimensional Cantor spaces and give a similar description of the automorphism groups of the Brin-Thompson groups Vâ‚™ (although we do not give an embedding into R). Using our description, we show that the outer automorphism group Out(Vâ‚‚) of Vâ‚‚ is isomorphic to the wreath product of Out(1Vâ‚‚) with the symmetric group on points

    Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs

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    This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs. In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest. In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures

    Applications of tensor networks to open problems in many-body quantum physics

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    Non-equilibrium dynamics of bulk-deterministic cellular automata

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    In this thesis we study simple one-dimensional nonequilibirum many-body systems, namely, reversible cellular automata (RCA). These are discrete time lattice models exhibiting emergent collective excitations---solitons---that move with fixed velocities and that interact via pairwise scattering. In particular, we study the attractively interacting Rule 201 RCA and noninteracting Rule 150 RCA which, together with the extensively studied repulsively interacting Rule 54 RCA constitute arguably the simplest one-dimensional microscopic physical models of strongly interacting and asymptotically freely propagating particles, to investigate interacting nonequilibrium many-body dynamics. After a brief literature review of the field, we present the first publication-style chapter which considers the Rule 201 RCA. Here, we study the stationary or steady state properties of systems with periodic, deterministic, and stochastic boundary conditions. We demonstrate that, despite the complexities of the model, specifically, a reducible state space and nontrivial topological vacuum, the model exhibits a simple and intuitive quasiparticle interpretation, reminiscent of the simpler Rule 54 RCA. This enables us to obtain exact expressions for the steady states in terms of a highly versatile matrix product state (MPS) representation that takes an instructive generalized Gibbs ensemble form. In the second publication-style chapter, we study the Rule 150 RCA. Due to its simplicity, originating from the noninteracting dynamics, we are able to obtain many exact results relating to its dynamics. To start, we generalize the MPS ansatz used to study the Rule 201 RCA, and find its exact steady state distribution for identical boundary conditions. We proceed to extend the MPS ansatz further and obtain a class of eigenvectors that form the dominant decay modes of the Markov propagator. Following this, we postulate a conjecture for the complete spectrum, which is in perfect agreement with numerics obtained via exact diagonalization of computationally tractable system sizes, providing access to the full relaxation dynamics. From here, we further utilise the ansatz to investigate the large deviation statistics and obtain exact expressions for its scaled cumulant generating function and rate function, which demonstrate the existence of a dynamical first order phase transition. The third and final publication-style chapter focuses on the exact dynamical large deviations statistics of the Rule 201 RCA. Specifically, we employ the methods introduced to study the large deviations of the Rule 54 RCA and show that they fail here to provide any insight into the atypical dynamical behaviour of the Rule 201 RCA. We suggest that this is due to the restrictions imposed by the local dynamical rules, which limits the support of the local observables. In spite of this, we explicitly derived an exact analytic expression for the dominant eigenvalue of the tilted Markov propagator, from which several large deviation statistics can be obtained
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