1,913 research outputs found
On -maximality
AbstractThis paper investigates a connection between the semantic notion provided by the ordering ◁∗ among theories in model theory and the syntactic (N)SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article (Ann. Pure Appl. Logic 80 (1996) 229) it was shown that SOP3 implies ◁∗-maximality and we prove here that ◁∗-maximality in a model of GCH implies a property called SOP2″. It has been subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between ◁∗-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent.Together with the known results about the connection between the (N)SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property
Formal Design of Asynchronous Fault Detection and Identification Components using Temporal Epistemic Logic
Autonomous critical systems, such as satellites and space rovers, must be
able to detect the occurrence of faults in order to ensure correct operation.
This task is carried out by Fault Detection and Identification (FDI)
components, that are embedded in those systems and are in charge of detecting
faults in an automated and timely manner by reading data from sensors and
triggering predefined alarms. The design of effective FDI components is an
extremely hard problem, also due to the lack of a complete theoretical
foundation, and of precise specification and validation techniques. In this
paper, we present the first formal approach to the design of FDI components for
discrete event systems, both in a synchronous and asynchronous setting. We
propose a logical language for the specification of FDI requirements that
accounts for a wide class of practical cases, and includes novel aspects such
as maximality and trace-diagnosability. The language is equipped with a clear
semantics based on temporal epistemic logic, and is proved to enjoy suitable
properties. We discuss how to validate the requirements and how to verify that
a given FDI component satisfies them. We propose an algorithm for the synthesis
of correct-by-construction FDI components, and report on the applicability of
the design approach on an industrial case-study coming from aerospace.Comment: 33 pages, 20 figure
Compositional abstraction and safety synthesis using overlapping symbolic models
In this paper, we develop a compositional approach to abstraction and safety
synthesis for a general class of discrete time nonlinear systems. Our approach
makes it possible to define a symbolic abstraction by composing a set of
symbolic subsystems that are overlapping in the sense that they can share some
common state variables. We develop compositional safety synthesis techniques
using such overlapping symbolic subsystems. Comparisons, in terms of
conservativeness and of computational complexity, between abstractions and
controllers obtained from different system decompositions are provided.
Numerical experiments show that the proposed approach for symbolic control
synthesis enables a significant complexity reduction with respect to the
centralized approach, while reducing the conservatism with respect to
compositional approaches using non-overlapping subsystems
Maximality of hyperspecial compact subgroups avoiding Bruhat-Tits theory
Let be a complete non-archimedean field (non trivially valued). Given a
reductive -group , we prove that hyperspecial subgroups of (i.e.
those arising from reductive models of ) are maximal among bounded
subgroups. The originality resides in the argument: it is inspired by the case
of and avoids all considerations on the Bruhat-Tits building of
.Comment: To appear at "Annales de l'Institut Fourier". This version avoids
completely Berkovich geometr
Tracking chains revisited
The structure , introduced and first
analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary
recursive. Here, denotes the proof-theoretic ordinal of the fragment
- of second order number theory, or equivalently the
set theory , which axiomatizes limits of models of
Kripke-Platek set theory with infinity. The partial orderings and
denote the relations of - and -elementary
substructure, respectively. In a subsequent article we will show that the
structure comprises the core of the structure of pure
elementary patterns of resemblance of order . In Carlson and Wilken 2012
(APAL) the stage has been set by showing that the least ordinal containing a
cover of each pure pattern of order is . However, it is not
obvious from Carlson and Wilken 2012 (APAL) that is an elementary
recursive structure. This is shown here through a considerable disentanglement
in the description of connectivity components of and . The key
to and starting point of our analysis is the apparatus of ordinal arithmetic
developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012
(JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for
the analysis of .Comment: The text was edited and aligned with reference [10], Lemma 5.11 was
included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section
5.3 on greatest immediate -successors. Updated publication
information. arXiv admin note: text overlap with arXiv:1608.0842
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