11 research outputs found

    SEARCHING FOR AN ANALOGUE OF ATR0 IN THE WEIHRAUCH LATTICE

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    Connected Choice and the Brouwer Fixed Point Theorem

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    We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.Comment: 36 page

    Le direzioni della logica in Italia: la reverse mathematics e l'analisi computazionale

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    Nelle conversazioni tra matematici non \ue8 infrequente sentire affermazioni del tipo \u201ci teoremi \u3a6 e \u3a8 sono equivalenti\u201d, oppure \u201cil teorema \u3a6 \ue8 pi\uf9 forte del teorema \u3a8\u201d. Dato che \u3a6 e \u3a8 (essendo teoremi) sono entrambi dimostrabili, prendendo alla lettera le due affermazioni abbiamo che la prima \ue8 banalmente vera e la seconda banalmente falsa. Sappiamo tutti per\uf2 che queste affermazioni hanno un altro significato, molto meno banale, e c\u2019\ue8 quindi una ragione per cui vengono fatte. Negli ultimi decenni la logica matematica ha sviluppato alcuni strumenti in grado di rendere precise, e suscettibili di dimostrazione o refutazione, affermazioni come le precedenti. In particolare ci riferiamo alla reverse mathematics e all\u2019analisi computazionale. Questi sono due programmi di ricerca di origine diverse che nell\u2019ultimo decennio, anche grazie al contributo di alcuni ricercatori italiani, hanno trovato significativi punti di contatto. In questo lavoro presenteremo i due programmi, con particolare riferimento alle loro aree di contatto. Evidenzieremo in particolare i contributi dei ricercatori italiani attivi in queste aree, e concluderemo indicando alcune prospettive di sviluppo su cui anche in Italia si sta cercando di lavorare

    Monte Carlo Computability

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    We introduce Monte Carlo computability as a probabilistic concept of computability on infinite objects and prove that Monte Carlo computable functions are closed under composition. We then mutually separate the following classes of functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable functions, and that of Monte Carlo computable functions. We give natural examples of computational problems witnessing these separations. As a specific problem which is Monte Carlo computable but neither Las Vegas computable nor non-deterministically computable, we study the problem of sorting infinite sequences that was recently introduced by Neumann and Pauly. Their results allow us to draw conclusions about the relation between algebraic models and Monte Carlo computability

    Computability and Complexity

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    We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility. We show that a more detailed picture emerges than what a related study by Giusto, Brown, and Simpson has revealed in the setting of reverse mathematics. In particular, different formulations of the Vitali Covering Theorem turn out to have different uniform computational content. These versions are either computable or closely related to uniform variants of Weak Weak K\H{o}nig's Lemma.Comment: 13 page

    Probabilistic Computability and Choice

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    We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a probabilistic Turing machine that receives a random binary sequence as auxiliary input. The machine can take advantage of this random sequence, but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful. With positive probability the random advice has to be successful. We characterize the class of Las Vegas computable functions in the Weihrauch lattice with the help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among other things we prove an Independent Choice Theorem that implies that Las Vegas computable functions are closed under composition. In a case study we show that Nash equilibria are Las Vegas computable, while zeros of continuous functions with sign changes cannot be computed on Las Vegas machines. However, we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms. The last mentioned results can be interpreted such that the Intermediate Value Theorem is reducible to the jump of Weak Weak K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These examples also demonstrate that Las Vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non-deterministically computable functions. We also study the impact of specific lower bounds on the success probabilities, which leads to a strict hierarchy of classes. In particular, the classical technique of probability amplification fails for computations on infinite objects. We also investigate the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
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