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David Fernández-Duque

Publications and source records attributed to David Fernández-Duque.

At least 55 records · Page 3Linked to original sources
arXiv2019

Intuitionistic Linear Temporal Logics

We consider intuitionistic variants of linear temporal logic with `next', `until' and `release' based on expanding posets: partial orders equipped with an order-preserving transition function. This class of structures gives rise to a logic which we denote $\iltl$, and by imposing additional constraints we obtain the logics $\itlb$ of persistent posets and $\itlht$ of here-and-there temporal logic, both of which have been considered in the literature. We prove that $\iltl$ has the effective finite model property and hence is decidable, while $\itlb$ does not have the finite model property. We also introduce notions of bounded bisimulations for these logics and use them to show that the `until' and `release' operators are not definable in terms of each other, even over the class of persistent posets.

cs.LO↗
arXiv2019

Complete Intuitionistic Temporal Logics in Topological Dynamics

The language of linear temporal logic can be interpreted over the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${{\sf ITL}^{\sf c}}_{\Diamond,\forall}$, recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

cs.LO↗
arXiv2019

Predicatively unprovable termination of the Ackermannian Goodstein process

The classical Goodstein process gives rise to long but finite sequences of natural numbers whose termination is not provable in Peano arithmetic. In this manuscript we consider a variant based on the Ackermann function. We show that Ackermannian Goodstein sequences eventually terminate, but this fact is not provable using predicative means.

math.LO↗
arXiv2018

Frame-validity games and lower bounds on the complexity of modal axioms

We introduce frame-equivalence games tailored for reasoning about the size, modal depth, number of occurrences of symbols and number of different propositional variables of modal formulae defining a given frame-property. Using these games, we prove lower bounds on the above measures for a number of well-known modal axioms; what is more, for some of the axioms, we show that they are optimal among the formulae defining the respective class of frames.

cs.LO↗
arXiv2018

The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincaré recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems.

math.LO↗
arXiv2018

An intuitionistic axiomatization of `eventually'

Boudou and the authors have recently introduced the intuitionistic temporal logic $\sf ITL^e$ and shown it to be decidable. In this article we show that the `henceforth'-free fragment of this logic is complete for the class of non-deterministic quasimodels introduced by Fernández-Duque. From this and results of Boudou, Romero and the authors, we conclude that this fragment is also complete for the standard semantics of $\sf ITL^e$ over the class of expanding posets.

math.LO↗
arXiv2018

Bisimulations for intuitionistic temporal logics

We introduce bisimulations for the logic $ITL^e$ with `next', `until' and `release', an intuitionistic temporal logic based on structures equipped with a partial order used to interpret intuitionistic implication and a monotone function used to interpret the temporal modalities. Our main results are that `eventually', which is definable in terms of `until', cannot be defined in terms of `next' and `henceforth', and similarly that `henceforth', definable in terms of `release', cannot be defined in terms of `next' and `until', even over the smaller class of here-and-there models.

math.LO↗
arXiv2018

Axiomatic systems and topological semantics for intuitionistic temporal logic

We propose four axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the four logics obtained from the axiomatic systems are distinct. Finally, we show that when the language is restricted to the `henceforth'-free fragment, the set of valid formulas for the relational and topological semantics coincide.

math.LO↗
arXiv2017

Succinctness in subsystems of the spatial mu-calculus

In this paper we systematically explore questions of succinctness in modal logics employed in spatial reasoning. We show that the closure operator, despite being less expressive, is exponentially more succinct than the limit-point operator, and that the $μ$-calculus is exponentially more succinct than the equally-expressive tangled limit operator. These results hold for any class of spaces containing at least one crowded metric space or containing all spaces based on ordinals below $ω^ω$, with the usual limit operator. We also show that these results continue to hold even if we enrich the less succinct language with the universal modality.

math.LO↗
arXiv2017

A Decidable Intuitionistic Temporal Logic

We introduce the logic $\sf ITL^e$, an intuitionistic temporal logic based on structures $(W,\preccurlyeq,S)$, where $\preccurlyeq$ is used to interpret intuitionistic implication and $S$ is a $\preccurlyeq$-monotone function used to interpret temporal modalities. Our main result is that the satisfiability and validity problems for $\sf ITL^e$ are decidable. We prove this by showing that the logic enjoys the strong finite model property. In contrast, we also consider a `persistent' version of the logic, $\sf ITL^p$, whose models are similar to Cartesian products. We prove that, unlike $\sf ITL^e$, $\sf ITL^p$ does not have the finite model property.

math.LO↗
arXiv2017

Exploring the bidimensional space: A dynamic logic point of view

We present a family of logics for reasoning about agents' positions and motion in the plane which have several potential applications in the area of multi-agent systems (MAS), such as multi-agent planning and robotics. The most general logic includes (i) atomic formulas for representing the truth of a given fact or the presence of a given agent at a certain position of the plane, (ii) atomic programs corresponding to the four basic orientations in the plane (up, down, left, right) as well as the four program constructs of propositional dynamic logic (sequential composition, nondeterministic composition, iteration and test). As this logic is not computably enumerable, we study some interesting decidable and axiomatizable fragments of it. We also present a decidable extension of the iteration-free fragment of the logic by special programs representing motion of agents in the plane.

cs.AI↗
arXiv2016

Non-deterministic Semantics for Dynamic Topological Logic

Dynamic Topological Logic ($\mathcal{DTL}$) is a combination of $\mathcal{S}${\em 4}, under its topological interpretation, and the temporal logic $\mathcal{LTL}$ interpreted over the natural numbers. $\mathcal{DTL}$ is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples $\left $, where $\left $ is a topological space, $f$ a function on $X$ and $V$ a truth valuation assigning subsets of $X$ to propositional variables.

math.LO↗
arXiv2015

Strong Completeness of Provability Logic for Ordinal Spaces

Abashidze and Blass independently proved that the modal logic $\sf{GL}$ is complete for its topological interpretation over any ordinal greater than or equal to $ω^ω$ equipped with the interval topology. Icard later introduced a family of topologies $\mathcal I_λ$ for $λ< ω$, with the purpose of providing semantics for Japaridze's polymodal logic $\sf{GLP}$ $_ω$. Icard's construction was later extended by Joosten and the second author to arbitrary ordinals $λ\geq ω$. We further generalize Icard topologies in this article. Given a scattered space $\mathfrak X = (X, τ)$ and an ordinal $λ$, we define a topology $τ_{+λ}$ in such a way that $τ_{+0}$ is the original topology $τ$ and $τ_{+λ}$ coincides with $\mathcal I_λ$ when $\mathfrak X$ is an ordinal endowed with the left topology. We then prove that, given any scattered space $\mathfrak X$ and any ordinal $λ>0$ such that the rank of $(X, τ)$ is large enough, $\sf{GL}$ is strongly complete for $τ_{+λ}$. One obtains the original Abashidze-Blass theorem as a consequence of the special case where $\mathfrak X=ω^ω$ and $λ=1$.

math.LO↗
arXiv2015

Forgetting complex propositions

This paper uses possible-world semantics to model the changes that may occur in an agent's knowledge as she loses information. This builds on previous work in which the agent may forget the truth-value of an atomic proposition, to a more general case where she may forget the truth-value of a propositional formula. The generalization poses some challenges, since in order to forget whether a complex proposition $π$ is the case, the agent must also lose information about the propositional atoms that appear in it, and there is no unambiguous way to go about this. We resolve this situation by considering expressions of the form $[\boldsymbol{\ddagger} π]φ$, which quantify over all possible (but minimal) ways of forgetting whether $π$. Propositional atoms are modified non-deterministically, although uniformly, in all possible worlds. We then represent this within action model logic in order to give a sound and complete axiomatization for a logic with knowledge and forgetting. Finally, some variants are discussed, such as when an agent forgets $π$ (rather than forgets whether $π$) and when the modification of atomic facts is done non-uniformly throughout the model.

cs.LO↗
arXiv2015

Perfectly secure data aggregation via shifted projections

We study a general scenario where confidential information is distributed among a group of agents who wish to share it in such a way that the data becomes common knowledge among them but an eavesdropper intercepting their communications would be unable to obtain any of said data. The information is modelled as a deck of cards dealt among the agents, so that after the information is exchanged, all of the communicating agents must know the entire deal, but the eavesdropper must remain ignorant about who holds each card. Valentin Goranko and the author previously set up this scenario as the secure aggregation of distributed information problem and constructed weakly safe protocols, where given any card $c$, the eavesdropper does not know with certainty which agent holds $c$. Here we present a perfectly safe protocol, which does not alter the eavesdropper's perceived probability that any given agent holds $c$. In our protocol, one of the communicating agents holds a larger portion of the cards than the rest, but we show how for infinitely many values of $a$, the number of cards may be chosen so that each of the $m$ agents holds more than $a$ cards and less than $2m^2a$.

cs.CR↗
arXiv2015

A case study in almost-perfect security for unconditionally secure communication

In the Russian cards problem, Alice, Bob and Cath draw $a$, $b$ and $c$ cards, respectively, from a publicly known deck. Alice and Bob must then communicate their cards to each other without Cath learning who holds a single card. Solutions in the literature provide weak security, where Cath does not know with certainty who holds each card that is not hers, or perfect security, where Cath learns no probabilistic information about who holds any given card from Alice and Bob's exchange. We propose an intermediate notion, which we call $\varepsilon$-strong security, where the probabilities perceived by Cath may only change by a factor of $\varepsilon$. We then show that a mild variant of the so-called geometric strategy gives $\varepsilon$-strong safety for arbitrarily small $\varepsilon$ and appropriately chosen values of $a,b,c$.

cs.CR↗
arXiv2015

Secure aggregation of distributed information: How a team of agents can safely share secrets in front of a spy

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.

cs.CR↗