Formal methods for the description of spatial relations can be based on mathematical theories of order. Subdivisions of land are represented as partially ordered sets (posets), a model that is general enough to answer spatial queries about inclusion and containment of spatial areas. After a brief introduction to the basic concepts of posets and lattices, their applications to modeling spatial relations and operations for spatial regions in terms of containment and overlay are presented. An interpretation is given for new geographic elements that are created by the completion from a poset to a lattice. It is shown that a novel approach to characterize certain topological relations based on a lattice of a simplicial complex is a model for spatial regions that combines both topological and order relations and allows spatial queries to be answered in a unified way.

: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are well-known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 06--01, 06A06 Secondary: 54-01, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...

. To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures. 1 Introduction Most spaces associated to mathematical structures: spectrum of a ring, space of valuations of a field, space of bounded linear functionals, . . . can be represented as distributive lattices. The key to have a natural definition in these cases is to use the notion of entailment relation due to Dana Scott. This note explains the connection between entailment relations and distributive lattices. An entailment relation may be seen as a logical description of a distributive lattice. Furthermore, most operations on distributive lattices are simpler when formulated as operations on entailment relations. A special kind of distribu...

The behavioral-inheritance relations of [7, 8] can be used to compare the life cycles of objects defined in terms of Petri nets. They yield partial orders on object life cycles (OLCs). Based on these orders, we define concepts such as the greatest common divisor and the least common multiple of a set of OLCs. These concepts have practical relevance: In component-based design, workflow management, ERP reference models, and electronic-trade procedures, there is a constant need for identifying commonalities and di erences in OLCs. Our results provide the theoretical basis for comparing, customizing, and unifying OLCs.

Access control has been an important issue in military systems for many years and is becoming in-creasingly important in commercial systems. There are three important access control paradigms: the Bell-LaPadula model, the protection matrix model and the role-based access control model. Each of these models has its advantages and disadvantages. Partial orders play a significant part in the role-based access control model and are also important in defining the security lattice in the Bell-LaPadula model. The main goal of this thesis is to improve the understanding and specification of access control models through a rigorous mathematical approach. We examine the mathematical foundations of the role-based access control model and conclude that antichains are a fundamental concept in the model. The analytical approach we adopt enables us to identify where improvements in the administration of role-based access control could be made. We then develop a new administrative model for role-based access control based on a novel, mathematical interpretation of encapsulated ranges. We show that this model supports discretionary access control features which have hitherto been difficult to incorporate into role-based access control frameworks.

Java source code is strongly typed, but the translation from Java source to bytecode omits much of the type information originally contained within methods. Type elaboration is a technique for reconstructing strongly typed programs from incompletely typed bytecode by inferring types for local variables. There are situations where, technically, there are not enough types in the original type hierarchy to type a bytecode program. Subtype completion is a technique for adding necessary types to an arbitrary type hierarchy to make type elaboration possible for all verifiable Java bytecode. Type elaboration with subtype completion has been implemented as part of the Marmot Java compiler.

This thesis is concerned with the transformation of the Large Deviation properties of a stochastic process under a random time-change. A random time-change is the reparametrisation of a process by a lower adjoint process; a simple example is the sequence of random variables f S Tn g n2N constructed by stopping a process f S t g at a sequence of random times f T n g n2N . Such a sequence of random times is lower adjoint to its associated point process f N t g N t := sup f n 2 N : T n 6 t g : In order to study the Large Deviations of random time-changes, we need to understand how the Large Deviation behaviour of adjoint processes are related. We prove that, under simple and very general hypotheses, an upper adjoint process satisfies a one-dimensional Large Deviation principle with rate-function U if and only if its lower adjoint does with rate-function V , where mixing condition. A similar one-dimensional approach to the case of random time-changes would require even stronger mixing conditions and would be somewhat awkward. Since a random time-change is a transformation of sample-paths, and the mixing conditions required to establish the relationship between the Large Deviations of a process and a random timechange of it are most conveniently stated in terms of sample-paths, it is most natural to consider the sample-path Large Deviations of adjoint processes and random time-changes. Adjoint processes are probability measures on the space of Galois connections between two partially ordered sets. We present the properties of Galois connections in general, and show how to construct the space of all adjunctions between I and J , where I and J , the dual of J , are continuous Heyting algebras. We study the order-theoretic properties of the random time-change transf...

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this paper, we explore one possible effective version of this theorem, that uses topological models in a point-free setting, following Sambin [11]. The truth-values, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this more abstract notion of model. The first is that, by using formal topology, we get a remarkably simple completeness proof; it seems indeed simpler than the usual classical completeness proof. The second is that this completeness proof is now constructive and elementary. In particular, it does not use any impredicativity and can be formalized in intuitionistic type theory; this is of importance for us, since we want to develop model theory in a computer system for type theory. Formal topology has been developed in the type theory implementation ALF [1] by Cederquist [2] and the completeness proof we use has been checked in ALF by Persson [9]. In view of the extreme simplicity of this proof, it might be feared that it has no interesting applications. We show that this is not the case by analysing a conservativity theorem due to Dragalin [4] concerning a non-standard extension of Heyting arithmetic. We can transpose directly the usual model theoretic conservativity argument, that we sketched above, in this framework. It seems likely that a direct syntactical proof of this result would have to be more involved. The first part of this paper presents a definition of topological models, Sambin's completeness proof, and an alternative completeness proof; we also discuss how Beth models relate to our approach. The second part shows how to use this in order to give a proof of Dragalin's conservativity result; our proof is different from his and, we believe, simpler. In [8] a stronger ...

. Lattice structures are often used in knowledge processing, but starting from a partial order, completion into a lattice poses efficiency problems. We use some recent results on lattice theory to propose an online algorithm for efficiently computing of the smallest lattice containing a given partial order, called the Dedekind-MacNeille completion. By online we mean that the elements are added if necessary when we compute the greatest lower bound or the least upper bound of two elements that do not already exist. This result can be used in knowledge processing for maintaining the hierarchy of types in an on-line fashion. Keywords: algorithm, partial order, lattice, Dedekind-MacNeille completion, encoding. 1 Introduction and Motivations Lattices are an important class of partial orders because of their structural properties and their interest in many areas as knowledges processing with the hierarchy of types [4,6,1,13,7]. The problem of minimally extending a partial order to the small...