166 research outputs found
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 , the eavesdropper does not know with
certainty which agent holds . Here we present a perfectly safe protocol,
which does not alter the eavesdropper's perceived probability that any given
agent holds . 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 , the number of cards may be chosen so that each of the agents
holds more than cards and less than
The intuitionistic temporal logic of dynamical systems
A dynamical system is a pair , where is a topological space and
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 , and show
that it is decidable. We also show that minimality and Poincar\'e recurrence
are both expressible in the language of , thus providing a
decidable logic expressive enough to reason about non-trivial asymptotic
behavior in dynamical systems
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 -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 , 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
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
Well-orders in the transfinite Japaridze algebra
This paper studies the transfinite propositional provability logics
\glp_\Lambda and their corresponding algebras. These logics have for each
ordinal a modality \la \alpha \ra. We will focus on the closed
fragment of \glp_\Lambda (i.e., where no propositional variables occur) and
\emph{worms} therein. Worms are iterated consistency expressions of the form
\la \xi_n\ra \ldots \la \xi_1 \ra \top. Beklemishev has defined
well-orderings on worms whose modalities are all at least and
presented a calculus to compute the respective order-types.
In the current paper we present a generalization of the original
orderings and provide a calculus for the corresponding generalized order-types
. 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}_\xi (A) \ra_{\xi \in \ord}
for some worm . One of these characterizations is in terms of a second kind
of transfinite iteration called {\em cohyperation.}Comment: Corrected a minor but confusing omission in the relation between
Veblen progressions and hyperation
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 \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
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
, 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.Comment: arXiv admin note: text overlap with arXiv:1201.5162 by other author
Dynamic Topological Logic of Metric Spaces
Dynamic Topological Logic (DT L) is a modal framework for reasoning
about dynamical systems, that is, pairs hX; fi where X is a topological
space and f : X ! X a continuous function.
In this paper we consider the case where X is a metric space. We rst
show that any formula which can be satis ed on an arbitrary dynamic
topological system can be satis ed on one based on a metric space; in
fact, this space can be taken to be countable and have no isolated points.
Since any metric space with these properties is homeomorphic to the set of
rational numbers, it follows that any formula can be satis ed on a system
based on Q.
We then show that the situation changes when considering complete
metric spaces, by exhibiting a formula which is not valid in general but is
valid on the class of systems based on a complete metric space. While we
do not attempt to give a full characterization of the set of valid formulas
on this class we do give a relative completeness result; any formula which
is satis able on a dynamical system based on a complete metric space is
also satis ed on one based on the Cantor spac
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