104 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Mathias and silver forcing parametrized by density

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    We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse 2ω to ω , while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property

    Asymmetric cut and choose games

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    We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness

    Constructivisation through Induction and Conservation

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    The topic of this thesis lies in the intersection between proof theory and alge- braic logic. The main object of discussion, constructive reasoning, was intro- duced at the beginning of the 20th century by Brouwer, who followed Kant’s explanation of human intuition of spacial forms and time points: these are constructed step by step in a finite process by certain rules, mimicking con- structions with straightedge and compass and the construction of natural numbers, respectively. The aim of the present thesis is to show how classical reasoning, which admits some forms of indirect reasoning, can be made more constructive. The central tool that we are using are induction principles, methods that cap- ture infinite collections of objects by considering their process of generation instead of the whole class. We start by studying the interplay between cer- tain structures that satisfy induction and the calculi for some non-classical logics. We then use inductive methods to prove a few conservation theorems, which contribute to answering the question of which parts of classical logic and mathematics can be made constructive.TĂ€mĂ€n opinnĂ€ytetyön aiheena on todistusteorian ja algebrallisen logiikan leikkauspiste. Keskustelun pÀÀaiheen, rakentavan pÀÀttelyn, esitteli 1900-luvun alussa Brouwer, joka seurasi Kantin selitystĂ€ ihmisen intuitiosta tilamuodoista ja aikapisteistĂ€: nĂ€mĂ€ rakennetaan askel askeleelta ÀÀrellisessĂ€ prosessissa tiettyjen sÀÀntöjen mukaan, jotka jĂ€ljittelevĂ€t suoran ja kompassin konstruktioita ja luonnollisten lukujen konstruktiota. TĂ€mĂ€n opinnĂ€ytetyön tavoitteena on osoittaa, kuinka klassista pÀÀttelyĂ€, joka mahdollistaa tietyt epĂ€suoran pÀÀttelyn muodot, voidaan tehdĂ€ rakentavammaksi. Keskeinen työkalu, jota kĂ€ytĂ€mme, ovat induktioperiaatteet, menetelmĂ€t, jotka kerÀÀvĂ€t ÀÀrettömiĂ€ objektikokoelmia ottamalla huomioon niiden luomisprosessin koko luokan sijaan. Aloitamme tutkimalla vuorovaikutusta tiettyjen induktiota tyydyttĂ€vien rakenteiden ja joidenkin ei-klassisten logiikan laskelmien vĂ€lillĂ€. Todistamme sitten induktiivisten menetelmien avulla muutamia sĂ€ilymislauseita, jotka auttavat vastaamaan kysymykseen siitĂ€, mitkĂ€ klassisen logiikan ja matematiikan osat voidaan tehdĂ€ rakentaviksi

    A survey on the model theory of tracial von Neumann algebras

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    We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II1_1 factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II1_1 factors and existentially closed II1_1 factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.Comment: 27 pages; first draft; comments welcome; to appear in the volume "Model theory of operator algebras" as part of DeGruyter's Logic and its Application Serie

    Trellis Decoding And Applications For Quantum Error Correction

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    Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost. Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products. Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.Ph.D

    Dynamical systems via domains:Toward a unified foundation of symbolic and non-symbolic computation

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    Non-symbolic computation (as, e.g., in biological and artificial neural networks) is astonishingly good at learning and processing noisy real-world data. However, it lacks the kind of understanding we have of symbolic computation (as, e.g., specified by programming languages). Just like symbolic computation, also non-symbolic computation needs a semantics—or behavior description—to achieve structural understanding. Domain theory has provided this for symbolic computation, and this thesis is about extending it to non-symbolic computation. Symbolic and non-symbolic computation can be described in a unified framework as state-discrete and state-continuous dynamical systems, respectively. So we need a semantics for dynamical systems: assigning to a dynamical system a domain—i.e., a certain mathematical structure—describing the system’s behavior. In part 1 of the thesis, we provide this domain-theoretic semantics for the ‘symbolic’ state-discrete systems (i.e., labeled transition systems). And in part 2, we do this for the ‘non-symbolic’ state-continuous systems (known from ergodic theory). This is a proper semantics in that the constructions form functors (in the sense of category theory) and, once appropriately formulated, even adjunctions and, stronger yet, equivalences. In part 3, we explore how this semantics relates the two types of computation. It suggests that non-symbolic computation is the limit of symbolic computation (in the ‘profinite’ sense). Conversely, if the system’s behavior is fairly stable, it may be described as realizing symbolic computation (here the concepts of ergodicity and algorithmic randomness are useful). However, the underlying concept of stability is limited by a no-go result due to a novel interpretation of Fitch’s paradox. This also has implications for AI-safety and, more generally, suggests fruitful applications of philosophical tools in the non-symbolic computation of modern AI

    Descriptive Set Theory and Applications

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    The systematic study of Polish spaces within the scope of Descriptive Set Theory furnishes the working mathematician with powerful techniques and illuminating insights. In this thesis, we start with a concise recapitulation of some classical aspects of Descriptive Set Theory which is followed by a succint review of topological groups, measures and some of their associated algebras.The main application of these techniques contained in this thesis is the study of two families of closed subsets of a locally compact Polish groupG, namely U(G) - closed sets of uniqueness - and U0(G) - closed sets of extended uniqueness. We locate the descriptive set theoretic complexityof these families, proving in particular that U(G) is \Pi_1^1-complete whenever G/\overline{[G,G]} is non-discrete, thereby extending the existing literature regarding the abelian case. En route, we establish some preservation results concerning sets of (extended) uniqueness and their operator theoretic counterparts. These results constitute a pivotal part in the arguments used and entail alternative proofs regarding the computation of the complexity of U(G) and U0(G) in some classes of the abelian case.We study U(G) as a calibrated \Pi_1^1 \sigma-ideal of F(G) - for G amenable - and prove some criteria concerning necessary conditions for the inexistence of a Borel basis for U(G). These criteria allow us to retrieve information about G after examination of its subgroups or quotients. Furthermore, a sufficient condition for the inexistence of a Borel basis for U(G) is proven for the case when G is a product of compact (abelian or not) Polish groupssatisfying certain conditions.\ua0Finally, we study objects associated with the point spectrum of linear bounded operators T\in L(X) acting on a separable Banach space X. We provide a characterization of reflexivity for Banach spaces with an unconditional basis : indeed such space X is reflexive if and only if for all closed subspaces Y\subset X;Z\subset X^{\ast} and T\in 2 L(Y); S\in 2 L(Z) it holds that the point spectra \sigma_p(T); \sigma_p(S) are Borel. We study the complexity of sets prescribed by eigenvalues and prove a stability criterion for Jamison sequences
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