David Fernández-Duque

Universidad de Sevilla, Hispalis, Andalusia, Spain

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Publications (15)4.86 Total impact

  • David Fernández-Duque, Valentin Goranko
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    ABSTRACT: We consider the generic problem of Secure Aggregation of Distributed Information (SADI), where several agents acting as a team have information distributed among them, modeled by means of a publicly known deck of cards distributed among the agents, so that each of them knows only her cards. The agents have to exchange and aggregate the information about how the cards are distributed among them by means of public announcements over insecure communication channels, intercepted by an adversary "eavesdropper", in such a way that the adversary does not learn who holds any of the cards. We present a combinatorial construction of protocols that provides a direct solution of a class of SADI problems and develop a technique of iterated reduction of SADI problems to smaller ones which are eventually solvable directly. We show that our methods provide a solution to a large class of SADI problems, including all SADI problems with sufficiently large size and sufficiently balanced card distributions.
    07/2014;
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    David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: In this paper we consider transfinite provability logics where for each ordinal in some recursive well-order we have a corresponding modal provability operator. The modality [xi] will be interpreted as "provable in ACA_0 together with at most xi nested applications of the omega rule". We show how to formalize this in in second order number theory. Next we prove both soundness and completeness under this interpretation. We conclude by showing how one can lower the base theory ACA_0 to theories below RCA_0.
    02/2013;
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    ABSTRACT: In the generalized Russian cards problem, the three players Alice, Bob and Cath draw a,b and c cards, respectively, from a deck of a+b+c cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. The communication is said to be safe if Cath does not learn the ownership of any specific card; in this paper we consider a strengthened notion of safety introduced by Swanson and Stinson which we call k-safety. An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces rather than projective planes, and call it the `geometric protocol'. Given arbitrary c,k>0, this protocol gives an informative and k-safe solution to the generalized Russian cards problem for infinitely many values of (a,b,c) with b=O(ac). This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees $k$-safety when Cath has more than one card.
    Designs Codes and Cryptography 01/2013; · 0.78 Impact Factor
  • David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: This paper studies the transfinite propositional provability logics $\glp_\Lambda$ and their corresponding algebras. These logics have for each ordinal $\xi< \Lambda$ a modality $[\alpha]$. We have a particular focus on the closed fragment of $\glp_\Lambda$ and \emph{worms} therein. Worms are iterated consistency expressions of the form $\la \xi_n\ra \ldots \la \xi_1 \ra \top$. In [L.D. Beklemishev, Veblen hierarchy in the context of provability algebras, 2005] orderings $<_\xi$ on worms are defined and a calculus is presented to compute the respective order-types $o_\xi$. In the current paper we present a different calculus for $o_\xi$ which is based on so-called hyperations which are transfinite iterations of normal functions. The $o_\xi$ are defined only for worms whose modalities are all at least $\xi$. In this paper we drop this restriction and study the resulting order-types $\Omega_\xi$. We provide a reduction of $\Omega_\xi$ to $o_\zeta$ and give two different characterizations of the sequences $\la \Omega_\xi (A) \ra_{\xi \in \ord}$ that can occur for worms $A$. One of these characterizations is in terms of another kind of transfinite iteration which is called cohyperation.
    Logic Journal of IGPL 12/2012; · 1.14 Impact Factor
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    Lev D. Beklemishev, David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: We introduce the logics GLP(\Lambda), a generalization of Japaridze's polymodal provability logic GLP(\omega) where \Lambda is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP(\omega) yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP(\Lambda) and the decidability of GLP(\Lambda) for recursive orderings \Lambda. Further, we give a restricted axiomatization of the variable-free fragment of GLP(\Lambda).
    Studia Logica 10/2012; · 0.34 Impact Factor
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    David Fernández-Duque
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    ABSTRACT: Provability logics are modal or polymodal systems designed for modeling the behavior of G\"odel's provability predicate in arithmetical theories and its natural extensions. If \Lambda is any ordinal, the G\"odel-L\"ob calculus GLP(\Lambda) contains one modality [\lambda] for each \lambda<\Lambda, representing provability predicates of increasing strength. GLP(\Lambda) has no Kripke models, but Beklemishev and Gabelaia recently proved that GLP(\omega) is complete for its class of topological models. In this paper we generalize Beklemishev and Gabelaia's result to GLP(\Lambda) for arbitrary \Lambda. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP(\Lambda) for the class of provability ambiances based on Icard polytopologies.
    Archive for Mathematical Logic 07/2012; · 0.28 Impact Factor
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    ABSTRACT: In the Russian cards problem, Alice, Bob and Cath draw three, three and one cards, respectively, from a deck of seven. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card. Unlike many traditional problems in cryptography, however, they are not allowed to hide or codify the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. One elegant solution, due to Atkinson, considers the cards as points in a finite projective plane. In this paper we consider the generalized Russian cards problem, where the number of cards that each player draws is a,b and c, respectively, from a deck of $a+b+c$ cards. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces, and call it the "colouring protocol", as it involves colourings of affine subsets. Our main results show that the colouring protocol provides a solution to the generalized Russian cards problem in cases where $a$ is a power of a prime, c=O(a^2) and b=O(c^2). This improves substantially on the collection of parameters for which solutions are known. In particular, it is the first solution which allows the eavesdropper to have more cards than one of the players.
    Theoretical Computer Science 07/2012; · 0.49 Impact Factor
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    David Fernández-Duque
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    ABSTRACT: Dynamic topological logic (DTL) is a polymodal logic designed for reasoning about {\em dynamic topological systems. These are pairs (X,f), where X is a topological space and f:X->X is continuous. DTL uses a language L which combines the topological S4 modality [] with temporal operators from linear temporal logic. Recently, I gave a sound and complete axiomatization DTL* for an extension of the logic to the language L*, where <> is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known over L, although one proof system, which we shall call $\mathsf{KM}$, was conjectured to be complete by Kremer and Mints. In this paper we show that, given any language L' between L and L*, the set of valid formulas of L' is not finitely axiomatizable. It follows, in particular, that KM is incomplete.
    ACM Transactions on Computational Logic 07/2012; · 0.79 Impact Factor
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    David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation of a normal function f is a sequence of normal functions so that f^0= id, f^1 = f and for all ordinals \alpha, \beta we have that f^(\alpha + \beta) = f^\alpha f^\beta. These conditions do not determine f^\alpha uniquely; in addition, we require that the functions be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given \alpha, \beta, f^(\alpha + \beta)= f^\beta f^\alpha. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study cohyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.
    Annals of Pure and Applied Logic 05/2012; 164(s 7–8). · 0.50 Impact Factor
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    David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: For any ordinal \Lambda, we can define a polymodal logic GLP(\Lambda), with a modality [\xi] for each \xi<\Lambda. These represent provability predicates of increasing strength. Although GLP(\Lambda) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities. Later, Icard defined a topological model for the same fragment which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary \Lambda. More generally, for each \Theta,\Lambda we build a Kripke model I(\Theta,\Lambda) and a topological model T(\Theta,\Lambda), and show that the closed fragment of GLP(\Lambda) is sound for both of these structures, as well as complete, provided \Theta is large enough.
    Journal of Symbolic Logic 04/2012; 78(2). · 0.54 Impact Factor
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    David Fernández-Duque, Joost J. Joosten
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    ABSTRACT: The logic GLP is a polymodal logic that has for each ordinal \alpha an operator [\alpha ], whose intended interpretation is a provability predicate in a hierarchy of theories of increasing strength. Its corresponding algebra is called the (transfinite) Japaridze algebra. There are various natural orders in this algebra that are based on comparing consistency strength of its elements. In particular, for each \alpha we define A <_{\alpha} iff over GLP, B implies <\alpha> A. In this paper we shall consider worms, which are formulas of the form <\alpha_0>...<\alpha_n>T, and the partial orders <_\alpha on their images in the Japaridze algebra. Given a worm A and an ordinal \alpha, our goal is to show how one computes the order type that is naturally associated to \Omega_\alpha(A):={B:B<_\alpha A}. Our main results show how the sequences <\Omega_\alpha(A)> can be computed via hyperations and cohyperations, which are forms of transfinite iterations of ordinal functions closely related to Veblen hierarchies.
    04/2012;
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    ABSTRACT: Consider three players Alice, Bob and Cath who hold a, b and c cards, respectively, from a deck of d=a+b+c cards. The cards are all different and players only know their own cards. Suppose Alice and Bob wish to communicate their cards to each other without Cath learning whether Alice or Bob holds a specific card. Considering the cards as consecutive natural numbers 0,1,..., we investigate general conditions for when Alice or Bob can safely announce the sum of the cards they hold modulo an appropriately chosen integer. We demonstrate that this holds whenever a,b>2 and c=1. Because Cath holds a single card, this also implies that Alice and Bob will learn the card deal from the other player's announcement.
    11/2011;
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    David Fernández-Duque
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    ABSTRACT: Dynamic Topological Logic (DTL\mathcal{DTL}) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within DTL\mathcal{DTL} one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within DTL\mathcal{DTL} translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while DTL\mathcal{DTL}s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that DTL\mathcal{DTL} interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of DTL\mathcal{DTL} which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. KeywordsDynamic topological logic–Spatial logic–Temporal logic–Multimodal logic–Topological dynamics
    Journal of Philosophical Logic 01/2011; 40(6):767-804.
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    ABSTRACT: The articles of this volume will be reviewed individually. For the preceding M4M workshop see [Zbl 1281.03003].
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    David Fernández-Duque
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    ABSTRACT: Given a measure space µ we define its measure algebra Aµ as the quotient of the algebra of all measurable subsets of X modulo the relation X µ ∼ Y if µ(X) = 0. If further X is endowed with a topology T , we can define an interior operator on Aµ analogous to the interior operator on P(X). Formulas of S4u (the modal logic S4 with a universal modality ∀ added) can then be assigned elements of Aµ by interpreting 2 as the aforementioned interior operator. In this paper we prove a general completeness result which implies the following two facts: (i) the logic S4u is complete for interpretations on any subset of Euclidean space of positive Lebesgue measure; (ii) the logic S4u is complete for interpretations on the Cantor set equipped with its appropriate fractal measure. Further, our result implies in both cases that given ε > 0, a satisfiable formula can be satisfied every-where except in a region of measure at most ε.