Modelling practical certainty and its link with classical propositional logic

We model practical certainty in the language of accept & reject statement-based uncertainty models. We present three different ways, each time using a different nature of assessment: we study coherent models following from (i) favourability assessments, (ii) acceptability assessments, and (iii) indifference assessments. We argue that a statement of favourability, when used with an appropriate background model, essentially boils down to stating a belief of practical certainty using acceptability assessments. We show that the corresponding models do not form an intersection structure, in contradistinction with the coherent models following from an indifferenc assessment. We construct embeddings of classical propositional logic into each of our models for practical certainty

Algebraic Aspects of Conditional Independence and Graphical Models

This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. This is accomplished by computing primary decompositions of conditional independence ideals. As examples the chapter presents in detail the graphical model of a four cycle and the intersection axiom, a certain implication of conditional independence statements. Another important problem in the area is to determine all constraints on a graphical model, for example, equations determined by trek separation. The full set of equality constraints can be determined by computing the model's vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.Comment: 20 pages, 1 figur

Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology

Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing

How scaling of the disturbance set affects robust positively invariant sets for linear systems

This paper presents new results on robust positively invariant (RPI) sets for linear discrete-time systems with additive disturbances. In particular, we study how RPI sets change with scaling of the disturbance set. More precisely, we show that many properties of RPI sets crucially depend on a unique scaling factor which determines the transition from nonempty to empty RPI sets. We characterize this critical scaling factor, present an efficient algorithm for its computation, and analyze it for a number of examples from the literature

A Discrete Choquet Integral for Ordered Systems

A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lov\'asz' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities

Characterisation of Strongly Normalising lambda-mu-Terms

We provide a characterisation of strongly normalising terms of the lambda-mu-calculus by means of a type system that uses intersection and product types. The presence of the latter and a restricted use of the type omega enable us to represent the particular notion of continuation used in the literature for the definition of semantics for the lambda-mu-calculus. This makes it possible to lift the well-known characterisation property for strongly-normalising lambda-terms - that uses intersection types - to the lambda-mu-calculus. From this result an alternative proof of strong normalisation for terms typeable in Parigot's propositional logical system follows, by means of an interpretation of that system into ours.Comment: In Proceedings ITRS 2012, arXiv:1307.784