78 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models

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    Chaotic dependence on temperature refers to the phenomenon of divergence of Gibbs measures as the temperature approaches a certain value. Models with chaotic behaviour near zero temperature have multiple ground states, none of which are stable. We study the class of uniformly chaotic models, that is, those in which, as the temperature goes to zero, every choice of Gibbs measures accumulates on the entire set of ground states. We characterise the possible sets of ground states of uniformly chaotic finite-range models up to computable homeomorphisms. Namely, we show that the set of ground states of every model with finite-range and rational-valued interactions is topologically closed and connected, and belongs to the class Π2\Pi_2 of the arithmetical hierarchy. Conversely, every Π2\Pi_2-computable, topologically closed and connected set of probability measures can be encoded (via a computable homeomorphism) as the set of ground states of a uniformly chaotic two-dimensional model with finite-range rational-valued interactions.Comment: 46 pages, 12 figure

    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 TILEmILLmWTILETILE \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 ¬TILEmWELLmSNT\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 (Σ11Π11)(\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

    Computable classifications of continuous, transducer, and regular functions

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    We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the Banach space C[0,1] of real-valued continuous functions on the unit interval. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipshitz functions is Σ20\Sigma^0_2-complete. We show that a function f ⁣:[0,1]Rf\colon [0,1] \rightarrow \mathbb{R} is (binary) transducer if and only if it is continuous regular; interestingly, this peculiar and nontrivial fact was overlooked by experts in automata theory. As one of many consequences, our Σ20\Sigma^0_2-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space C[0,1]C[0,1] of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.Comment: Revised argument in Section 5; results unchange

    Collected Papers (on Neutrosophic Theory and Applications), Volume VI

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    This sixth volume of Collected Papers includes 74 papers comprising 974 pages on (theoretic and applied) neutrosophics, written between 2015-2021 by the author alone or in collaboration with the following 121 co-authors from 19 countries: Mohamed Abdel-Basset, Abdel Nasser H. Zaied, Abduallah Gamal, Amir Abdullah, Firoz Ahmad, Nadeem Ahmad, Ahmad Yusuf Adhami, Ahmed Aboelfetouh, Ahmed Mostafa Khalil, Shariful Alam, W. Alharbi, Ali Hassan, Mumtaz Ali, Amira S. Ashour, Asmaa Atef, Assia Bakali, Ayoub Bahnasse, A. A. Azzam, Willem K.M. Brauers, Bui Cong Cuong, Fausto Cavallaro, Ahmet Çevik, Robby I. Chandra, Kalaivani Chandran, Victor Chang, Chang Su Kim, Jyotir Moy Chatterjee, Victor Christianto, Chunxin Bo, Mihaela Colhon, Shyamal Dalapati, Arindam Dey, Dunqian Cao, Fahad Alsharari, Faruk Karaaslan, Aleksandra Fedajev, Daniela Gîfu, Hina Gulzar, Haitham A. El-Ghareeb, Masooma Raza Hashmi, Hewayda El-Ghawalby, Hoang Viet Long, Le Hoang Son, F. Nirmala Irudayam, Branislav Ivanov, S. Jafari, Jeong Gon Lee, Milena Jevtić, Sudan Jha, Junhui Kim, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Darjan Karabašević, Songül Karabatak, Abdullah Kargın, M. Karthika, Ieva Meidute-Kavaliauskiene, Madad Khan, Majid Khan, Manju Khari, Kifayat Ullah, K. Kishore, Kul Hur, Santanu Kumar Patro, Prem Kumar Singh, Raghvendra Kumar, Tapan Kumar Roy, Malayalan Lathamaheswari, Luu Quoc Dat, T. Madhumathi, Tahir Mahmood, Mladjan Maksimovic, Gunasekaran Manogaran, Nivetha Martin, M. Kasi Mayan, Mai Mohamed, Mohamed Talea, Muhammad Akram, Muhammad Gulistan, Raja Muhammad Hashim, Muhammad Riaz, Muhammad Saeed, Rana Muhammad Zulqarnain, Nada A. Nabeeh, Deivanayagampillai Nagarajan, Xenia Negrea, Nguyen Xuan Thao, Jagan M. Obbineni, Angelo de Oliveira, M. Parimala, Gabrijela Popovic, Ishaani Priyadarshini, Yaser Saber, Mehmet Șahin, Said Broumi, A. A. Salama, M. Saleh, Ganeshsree Selvachandran, Dönüș Șengür, Shio Gai Quek, Songtao Shao, Dragiša Stanujkić, Surapati Pramanik, Swathi Sundari Sundaramoorthy, Mirela Teodorescu, Selçuk Topal, Muhammed Turhan, Alptekin Ulutaș, Luige Vlădăreanu, Victor Vlădăreanu, Ştefan Vlăduţescu, Dan Valeriu Voinea, Volkan Duran, Navneet Yadav, Yanhui Guo, Naveed Yaqoob, Yongquan Zhou, Young Bae Jun, Xiaohong Zhang, Xiao Long Xin, Edmundas Kazimieras Zavadskas

    Computability Theory (hybrid meeting)

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    Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

    A journey through computability, topology and analysis

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    This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility, as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of the open and clopen Ramsey theorems. In particular, since there is not a canonical way to phrase these theorems as multi-valued functions, we identify 8 different multi-valued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. These notions turn out to be extremely relevant when exploring the Weihrauch degree of the problem DS of computing descending sequences in ill-founded linear orders. They allow us to show that DS, and the Weihrauch equivalent problem BS of finding bad sequences through non-well quasi-orders, while being very "hard" to solve, are rather weak in terms of uniform computational strength. We then generalize DS and BS by considering Gamma-presented orders, where Gamma is a Borel pointclass or Delta11, Sigma11, Pi11. We study the obtained DS-hierarchy and BS-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and of Salem sets. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Pi03-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1]^d). We also generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Pi03-complete when we endow K(R^d) with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by showing how these results can be used to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions

    On cohesive powers of linear orders

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    Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let ω\omega, ζ\zeta, and η\eta denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of ω\omega. If L\mathcal{L} is a computable copy of ω\omega that is computably isomorphic to the standard presentation of ω\omega, then every cohesive power of L\mathcal{L} has order-type ω+ζη\omega + \zeta\eta. However, there are computable copies of ω\omega, necessarily not computably isomorphic to the standard presentation, having cohesive powers not elementarily equivalent to ω+ζη\omega + \zeta\eta. For example, we show that there is a computable copy of ω\omega with a cohesive power of order-type ω+η\omega + \eta. Our most general result is that if XN{0}X \subseteq \mathbb{N} \setminus \{0\} is either a Σ2\Sigma_2 set or a Π2\Pi_2 set, thought of as a set of finite order-types, then there is a computable copy of ω\omega with a cohesive power of order-type ω+σ(X{ω+ζη+ω})\omega + \sigma(X \cup \{\omega + \zeta\eta + \omega^*\}), where σ(X{ω+ζη+ω})\sigma(X \cup \{\omega + \zeta\eta + \omega^*\}) denotes the shuffle of the order-types in XX and the order-type ω+ζη+ω\omega + \zeta\eta + \omega^*. Furthermore, if XX is finite and non-empty, then there is a computable copy of ω\omega with a cohesive power of order-type ω+σ(X)\omega + \sigma(X)
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