178 research outputs found
Demonic Kleene Algebra
Nous rappelons dâabord le concept dâalgĂšbre de Kleene avec domaine (AKD). Puis, nous expliquons comment utiliser les opĂ©rateurs des AKD pour dĂ©finir un ordre partiel appelĂ© raffinement dĂ©moniaque ainsi que dâautres opĂ©rateurs dĂ©moniaques (plusieurs de ces dĂ©finitions proviennent de la littĂ©rature). Nous cherchons Ă comprendre comment se comportent les AKD munies des opĂ©rateurs dĂ©moniaques quand on exclut les opĂ©rateurs angĂ©liques usuels. Câest ainsi que les propriĂ©tĂ©s de ces opĂ©rateurs dĂ©moniaques nous servent de base pour axiomatiser une algĂšbre que nous appelons AlgĂšbre dĂ©moniaque avec domaine et opĂ©rateur t-conditionnel (ADD-[opĂ©rateur t-conditionnel]). Les lois des ADD-[opĂ©rateur t-conditionnel] qui ne concernent pas lâopĂ©rateur de domaine correspondent Ă celles prĂ©sentĂ©es dans lâarticle Laws of programming par Hoare et al. publiĂ© dans la revue Communications of the ACM en 1987. Ensuite, nous Ă©tudions les liens entre les ADD-[opĂ©rateur t-conditionnel] et les AKD munies des opĂ©rateurs dĂ©moniaques. La question est de savoir si ces structures sont isomorphes. Nous dĂ©montrons que ce nâest pas le cas en gĂ©nĂ©ral et nous caractĂ©risons celles qui le sont. En effet, nous montrons quâune AKD peut ĂȘtre transformĂ©e en une ADD-[opĂ©rateur t-conditionnel] qui peut ĂȘtre transformĂ©e Ă son tour en lâAKD de dĂ©part. Puis, nous prĂ©sentons les conditions nĂ©cessaires et suffisantes pour quâune ADD-[opĂ©rateur t-conditionnel] puisse ĂȘtre transformĂ©e en une AKD qui peut ĂȘtre transformĂ©e Ă nouveau en lâADD-[opĂ©rateur t-conditionnel] de dĂ©part. Les conditions nĂ©cessaires et suffisantes mentionnĂ©es prĂ©cĂ©demment font intervenir un nouveau concept, celui de dĂ©composition. Dans un contexte dĂ©moniaque, il est difficile de distinguer des transitions qui, Ă partir dâun mĂȘme Ă©tat, mĂšnent Ă des Ă©tats diffĂ©rents. Le concept de dĂ©composition permet dây arriver simplement. Nous prĂ©sentons sa dĂ©finition ainsi que plusieurs de ses propriĂ©tĂ©s.We first recall the concept of Kleene algebra with domain (KAD). Then we explain how to use the operators of KAD to define a demonic refinement ordering and demonic operators (many of these definitions come from the literature). We want to know how do KADs with the demonic operators but without the usual angelic ones behave. Then, taking the properties of the KAD-based demonic operators as a guideline, we axiomatise an algebra that we call Demonic algebra with domain and t-conditional (DAD-[opĂ©rateur t-conditionnel]). The laws of DAD-[opĂ©rateur t-conditionnel] not concerning the domain operator agree with those given in the 1987 Communications of the ACM paper Laws of programming by Hoare et al. Then, we investigate the relationship between DAD-[opĂ©rateur t-conditionnel] and KAD-based demonic algebras. The question is whether every DAD-[opĂ©rateur t-conditionnel] is isomorphic to a KAD-based demonic algebra. We show that it is not the case in general. However, we characterise those that are. Indeed, we demonstrate that a KAD can be transformed into a DAD-[opĂ©rateur t-conditionnel] which can be transformed back into the initial KAD. We also establish necessary and sufficient conditions for which a DAD-[opĂ©rateur t-conditionnel] can be transformed into a KAD which can be transformed back into the initial DAD-[opĂ©rateur t-conditionnel]. Finally, we define the concept of decomposition. This notion is involved in the necessary and sufficient conditions previously mentioned. In a demonic context, it is difficult to distinguish between transitions that, from a given state, go to different states. The concept of decomposition enables to do it easily. We present its definition together with some of its properties
Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts
We seek to augment a geometric network in the Euclidean plane with shortcuts
to minimize its continuous diameter, i.e., the largest network distance between
any two points on the augmented network. Unlike in the discrete setting where a
shortcut connects two vertices and the diameter is measured between vertices,
we take all points along the edges of the network into account when placing a
shortcut and when measuring distances in the augmented network.
We study this network augmentation problem for paths and cycles. For paths,
we determine an optimal shortcut in linear time. For cycles, we show that a
single shortcut never decreases the continuous diameter and that two shortcuts
always suffice to reduce the continuous diameter. Furthermore, we characterize
optimal pairs of shortcuts for convex and non-convex cycles. Finally, we
develop a linear time algorithm that produces an optimal pair of shortcuts for
convex cycles. Apart from the algorithms, our results extend to rectifiable
curves.
Our work reveals some of the underlying challenges that must be overcome when
addressing the discrete version of this network augmentation problem, where we
minimize the discrete diameter of a network with shortcuts that connect only
vertices
Probing Convex Polygons with a Wedge
Minimizing the number of probes is one of the main challenges in
reconstructing geometric objects with probing devices. In this paper, we
investigate the problem of using an -wedge probing tool to determine
the exact shape and orientation of a convex polygon. An -wedge consists
of two rays emanating from a point called the apex of the wedge and the two
rays forming an angle . To probe with an -wedge, we set the
direction that the apex of the probe has to follow, the line , and the initial orientation of the two rays. A valid -probe of a
convex polygon contains within the -wedge and its outcome
consists of the coordinates of the apex, the orientation of both rays and the
coordinates of the closest (to the apex) points of contact between and each
of the rays.
We present algorithms minimizing the number of probes and prove their
optimality. In particular, we show how to reconstruct a convex -gon (with
all internal angles of size larger than ) using -probes;
if , the reconstruction uses -probes. We show
that both results are optimal. Let be the number of vertices of whose
internal angle is at most , (we show that ). We
determine the shape and orientation of a general convex -gon with
(respectively , ) using (respectively , )
-probes. We prove optimality for the first case. Assuming the algorithm
knows the value of in advance, the reconstruction of with or
can be achieved with probes,- which is optimal.Comment: 31 pages, 27 figure
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