arXiv subjects

David Fernández-Duque

Publications and source records attributed to David Fernández-Duque.

65 records · Page 4Linked to original sources

arXiv2014

A colouring protocol for the generalized Russian cards problem

In the generalized Russian cards problem, Alice, Bob and Cath draw $a$, $b$ and $c$ cards, respectively, from a deck of size $a+b+c$. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the "colouring protocol", as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve 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 set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players.

cs.IT↗

arXiv2014

Well-orders in the transfinite Japaridze algebra

This paper studies the transfinite propositional provability logics $\glp_Λ$ and their corresponding algebras. These logics have for each ordinal $ξ< Λ$ a modality $\la α\ra$. We will focus on the closed fragment of $\glp_Λ$ (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form $\la ξ_n\ra \ldots \la ξ_1 \ra \top$. Beklemishev has defined well-orderings $<_ξ$ on worms whose modalities are all at least $ξ$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_ξ$ orderings and provide a calculus for the corresponding generalized order-types $o_ξ$. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form $\la {\formerOmega}_ξ(A) \ra_{ξ\in \ord}$ for some worm $A$. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}

math.LO↗

arXiv2013

Well-orders in the transfinite Japaridze algebra II: Turing progressions and their well-orders

We study transfinite extensions of Japaridze's provability logic GLP and the well-founded relations that naturally occur within them. Every ordinal induces a partial order over the class of "words," which are iterated consistency statements expressible within GLP. Well-ordered restrictions of these partial orders have been studied previously; in this paper we consider the unrestricted partial orders, which are no longer linear but remain well-founded. These unrestricted partial orders bear important repercussions on modal semantics for GLP and on Turing progressions.

math.LO↗

arXiv2013

The polytopologies of transfinite provability logic

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel's provability predicate in arithmetical theories and its natural extensions. If Λis any ordinal, the Gödel-Löb calculus GLP(Λ) contains one modality [λ] for each λ<Λ, representing provability predicates of increasing strength. GLP(Λ) has no Kripke models, but Beklemishev and Gabelaia recently proved that GLP(ω) is complete for its class of topological models. In this paper we generalize Beklemishev and Gabelaia's result to GLP(Λ) for arbitrary Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP(Λ) for the class of provability ambiances based on Icard polytopologies.

math.LO↗

arXiv2013

Evidence and plausibility in neighborhood structures

The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an {\em evidence logic} for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a neighborhood semantics, where a neighborhood $N$ indicates that the agent has reason to believe that the true state of the world lies in $N$. Further notions of relative plausibility between worlds and beliefs based on the latter ordering are then defined in terms of this evidence structure, yielding our intended models for evidence-based beliefs. In addition, we also consider a second more general flavor, where belief and plausibility are modeled using additional primitive relations, and we prove a representation theorem showing that each such general model is a $p$-morphic image of an intended one. This semantics invites a number of natural special cases, depending on how uniform we make the evidence sets, and how coherent their total structure. We give a structural study of the resulting `uniform' and `flat' models. Our main result are sound and complete axiomatizations for the logics of all four major model classes with respect to the modal language of evidence, belief and safe belief. We conclude with an outlook toward logics for the dynamics of changing evidence, and the resulting language extensions and connections with logics of plausibility change.

math.LO↗

arXiv2013

The omega-rule interpretation of transfinite provability logic

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.

math.LO↗

arXiv2013

A geometric protocol for cryptography with cards

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.

cs.CR↗

arXiv2012

On provability logics with linearly ordered modalities

We introduce the logics GLP(Λ), a generalization of Japaridze's polymodal provability logic GLP(ω) where Λis any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP(ω) yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP(Λ) and the decidability of GLP(Λ) for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP(Λ).

math.LO↗

arXiv2012

Non-finite axiomatizability of Dynamic Topological Logic

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.

math.LO↗

arXiv2012

Hyperations, Veblen progressions and transfinite iterations of ordinal functions

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 α, βwe have that f^(α+ β) = f^αf^β. These conditions do not determine f^αuniquely; 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 α, β, f^(α+ β)= f^βf^α. 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.

math.LO↗

arXiv2012

Models of transfinite provability logic

For any ordinal Λ, we can define a polymodal logic GLP(Λ), with a modality [ξ] for each ξ<Λ. These represent provability predicates of increasing strength. Although GLP(Λ) 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 Λ. More generally, for each Θ,Λwe build a Kripke model I(Θ,Λ) and a topological model T(Θ,Λ), and show that the closed fragment of GLP(Λ) is sound for both of these structures, as well as complete, provided Θis large enough.

math.LO↗