We aim to construct a formal theory of affordances seen as ternary relations. Beginning with a characterization of affordances proposed by James J. Gibson, and utilizing the tools provided by Zdzisław Pawlak's information systems and rough sets, we construct a mathematically precise definition of both crisp and rough affordances. Then, we analyze modal and approximation operators that enable reasoning about affordances in both scenarios.

Abstract We introduce and study a class of betweenness algebras —Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and of sufficiency .

We consider the two-pronged fork frame $F$ and the variety $\mathbf{Eq}(B_F)$ generated by its dual closure algebra $B_F$. We describe the finite projective algebras in $\mathbf{Eq}(B_F)$ and give a purely semantic proof that unification in $\mathbf{Eq}(B_F)$ is finitary and not unitary.

Abstract We investigate modal algebras that generalize the unary discriminator into two directions related to an ideal of the algebra. It turns out that some classes lead to well-known logics, while others have not yet been explored.

Confusion matrices and their associated statistics are a well established tool in machine learning to evaluate the accuracy of a classifier. In the present study, we define a rough confusion matrix based on a very general classifier, and derive various statistics from it which are related to common rough set estimators. In other words, we perform a rough set–like analysis on a confusion matrix, which is the converse of the usual procedure; in particular, we consider upper approximations. A suitable index for measuring the tightness of the upper bound uses a ratio of odds. Odds ratios offer a symmetric interpretation of lower and upper precision, and remove the bias in the upper approximation. We investigate rough odds ratios of the parameters obtained from the confusion matrix; to guard against undue random influences, we also approximate their standard errors.

Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of g eneral lattices.

In previous work we have presented a class of algebras enhanced with two contact relations representing rough set style approximations of a spatial contact relation. In this paper, we develop a class of relational systems which is mutually interpretable with that class of algebras, and we consider a relational logic whose semantics is determined by those relational systems. For this relational logic we construct a proof system in the spirit of Rasiowa-Sikorski, and we outline the proofs of its soundness and completeness.

In this paper we review Lattice Machine, a learning paradigm that “learns” by generalising data in a consistent, conservative and parsimonious way, and has the advantage of being able to provide additional reliability information for any classification. More specifically, we review the related concepts such as hyper tuple and hyper relation, the three generalising criteria (equilabelledness, maximality, and supportedness) as well as the modelling and classifying algorithms. In an attempt to find a better method for classification in Lattice Machine, we consider the contextual probability which was originally proposed as a measure for approximate reasoning when there is insufficient data. It was later found to be a probability function that has the same classification ability as the data generating probability called primary probability. It was also found to be an alternative way of estimating the primary probability without much model assumption. Consequently, a contextual probability based Bayes classifier can be designed. In this paper we present a new classifier that utilises the Lattice Machine model and generalises the contextual probability based Bayes classifier. We interpret the model as a dense set of data points in the data space and then apply the contextual probability based Bayes classifier. A theorem is presented that allows efficient estimation of the contextual probability based on this interpretation. The proposed classifier is illustrated by examples.

Rough relation algebras are a generalization of relation algebras such that the underlying lattice structure is a regular double Stone algebra. Standard models are algebras of rough relations. A discrete duality is a relationship between classes of algebras and classes of relational systems (frames). In this paper we prove a discrete duality for a class of rough relation algebras and a class of frames.

A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of "additive circuits". If they are confined to the usual Boolean operations together with the result of lifting addition and multiplication to the level of sets, we speak of "arithmetic circuits". In this paper, we investigate the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations.

In this paper we show that the problem of discrete duality can be extended beyond the clasical setting of duality between a class of algebras and a class of relational structures. Namely, for some classes of algebras, the relevant dual structures are the structures with multirelations. Several applications of multirelations will be described.

Timed contact algebras constitute an approach to a temporal version of a region based theory of space. The general theory does not provide a notion of an underlying static world, i.e. it does not explicitly contain a set of non moving regions. Furthermore, the model of time does not have any structure, i.e. time is neither ordered nor required to be discrete or continuous. In this paper we want to investigate two extensions of the basic theory. The first extension considers grounded timed contact algebras that make the underlying static world explicit. In this context we introduce the Axiom of Construction that relates the existence of certain regions and the time structure for the first time. The second addition is given by a betweenness relation on the set of time. In this context we introduce the Axiom of Continuity (CONT), ensuring "smooth'' movement of regions through time. Last but not least, we show that both axioms together do not allow finite models.

An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural numbers and make some observations regarding the complexity of various theories of such algebras.

Boolean contact algebras constitute a convenient approach to a region based theory of space. In this paper we want to extend this approach to regions moving in time - called timed contact structures. We study their canonical models using topological spaces. As the main contribution we prove a general representation theorem for this kind of algebras.

We explore the usage of the modal possibility operator (and its dual necessity operator) in qualitative data analysis, and show that it-quite literally-complements the derivation operator of formal concept analysis; we also propose a new generalization of the rough set approximation operators. As an example for the applicability of the concepts we investigate the Morse data set which has been frequently studied in multidimensional scaling procedures.

Representation theorems for systems of regions have been of interest for some time, and various contexts have been used for this purpose: Mormann [17] has demonstrated the fruitfulness of the methods of continuous lattices to obtain a topological representation theorem for his formalisation of Whiteheadian ontological theory of space; similar results have been obtained by Roeper [20]. In this note, we prove a topological representation theorem for a connection based class of systems, using methods and tools from the theory of proximity spaces. The key novelty is a new proximity semantics for connection relations.

This paper presents a novel method for classification, which makes use of models built by the lattice machine (LM). The LM approximates data resulting in, as a model of data, a set of hyper tuples that are equilabelled, supported and maximal. The method presented uses the LM model of data to classify new data with a view to maximising the density of the model. Experiments show that this method, when used with the LM, outperforms the C2 algorithm and is comparable to the C5.0 classification algorithm.

Rough set data analysis (RSDA), introduced by Pawlak, has become a much researched method of knowledge discovery with over 1200 publications to date. One feature which distinguishes RSDA from other data analysis methods is that, in its original form, it gathers all its information from the given data, and does not make external model assumptions as all statistical and most machine learning methods (including decision tree procedures) do. The price which needs to be paid for the parsimony of this approach, however, is that some statistical backup is required, for example, to deal with random influences to which the observed data may be subjected. In supplementing RSDA by such meta-procedures care has to be taken that the same non-invasive principles are applied. In a sequence of papers and conference contributions, we have developed the components of a non-invasive method of data analysis, which is based on the RSDA principle, but is not restricted to “classical” RSDA applications. In this article, we present for the first time in a unified way the foundation and tools of such rough information analysis. © 2001 John Wiley & Sons, Inc.

Aiming at a user-oriented approach in software evaluation on the basis of ISO 9241 Part 10, we present a questionnaire (IsoMetrics) which collects usability data for summative and formative evaluation, and document its construction. The summative version of IsoMetrics shows a high reliability of its subscales and gathers valid information about differences in the usability of different software systems. Moreover, we show that the formative version of IsoMetrics is a powerful tool for supporting the identification of software weaknesses. Finally, we propose a procedure to categorize and prioritize weak points, which subsequently can be used as basic input to usability reviews.

The formalization of the “part – of” relationship goes back to the mereology of S. Leśniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part-of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra.

We exhibit some new connections between structure of an information system and its corresponding semilattice of equivalence relations. In particular, we investigate dependency properties and introduce a partial ordering of information systems over a fixed object set U which reflects the sub-semilattice relation on the set of all equivalence relations on U.

Abstract We investigate in an algebraic setting the question of which logical languages can express the properties integral, permutational, and rigid for algebras of relations.

Rough relation algebras were introduced by S. Comer as a generalisation of algebras of Pawlak's rough sets and Tarski's relation algebras. In this paper, some algebraic and arithmetical properties of rough relation algebras are studied and the repres