Generalization of One-Sided Concept Lattices

We provide a generalization of one-sided (crisp-fuzzy) concept lattices, based on Galois connections. Our approach allows analysis of object-attribute models with different structures for truth values of attributes. Moreover, we prove that this method of creating one-sided concept lattices is the most general one, i.e., with respect to the set of admissible formal contexts, it produces all Galois connections between power sets and the products of complete lattices. Some possible applications of this approach are also included

Interval-Dismantling for Lattices

Dismantling allows for the removal of elements of a set, or in our case lattice, without disturbing the remaining structure. In this paper we have extended the notion of dismantling by single elements to the dismantling by intervals in a lattice. We utilize theory from Formal Concept Analysis (FCA) to show that lattices dismantled by intervals correspond to closed subrelations in the respective formal context, and that there exists a unique kernel with respect to dismantling by intervals. Furthermore, we show that dismantling intervals can be identified directly in the formal context utilizing a characterization via arrow relations and provide an algorithm to compute all dismantling intervals.Comment: 12 pages, 5 figures, 1 algorith

Interpretation of Fuzzy Attribute Subsets in Generalized One-Sided Concept Lattices

In this paper we describe possible interpretation and reduction of fuzzy attributes in Generalized One-sided Concept Lattices (GOSCL). This type of concept lattices represent generalization of Formal Concept Analysis (FCA) suitable for analysis of datatables with different types of attributes. FCA as well as generalized one-sided concept lattices represent conceptual data miningmethods. With growing number of attributes the interpretation of fuzzy subsets may become unclear, hence another interpretation of this fuzzy attribute subsets can be valuable. The originality of the presented method is based on the usage of one-sided concept lattices derived from submodels of former object-attribute model by grouping attributes with the same truth value structure. This leads to new method for attribute reduction in GOSCL environment

Notes on integer partitions

Some observations concerning the lattices of integer partitions are presented. We determine the size of the standard contexts, discuss a recursive construction and show that the lattices have unbounded breadth

Dicomplemented Lattices: A Contextual Generalization of Boolean Algebras

Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen.The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a "negation" on formal concepts, "concept algebras" are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on "Contextual Logic", a Formal Concept Analysis approach, based on concepts as units of thought

Polynomial growth of concept lattices, canonical bases and generators:: extremal set theory in Formal Concept Analysis

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts

Selecting Attributes for Sport Forecasting using Formal Concept Analysis

In order to address complex systems, apply pattern recongnition on their evolution could play an key role to understand their dynamics. Global patterns are required to detect emergent concepts and trends, some of them with qualitative nature. Formal Concept Analysis (FCA) is a theory whose goal is to discover and to extract Knowledge from qualitative data. It provides tools for reasoning with implication basis (and association rules). Implications and association rules are usefull to reasoning on previously selected attributes, providing a formal foundation for logical reasoning. In this paper we analyse how to apply FCA reasoning to increase confidence in sports betting, by means of detecting temporal regularities from data. It is applied to build a Knowledge-Based system for confidence reasoning.Comment: Paper 3 for the Complex Systems in Sports Workshop 2011 (CS-Sports 2011

Complex concept lattices for simulating human prediction in sport

In order to address the study of complex systems, the detection of patterns in their dynamics could play a key role in understanding their evolution. In particular, global patterns are required to detect emergent concepts and trends, some of them of a qualitative nature. Formal concept analysis (FCA) is a theory whose goal is to discover and extract knowledge from qualitative data (organized in concept lattices). In complex environments, such as sport competitions, the large amount of information currently available turns concept lattices into complex networks. The authors analyze how to apply FCA reasoning in order to increase confidence in sports predictions by means of detecting regularities from data through the management of intuitive and natural attributes extracted from publicly available information. The complexity of concept lattices -considered as networks with complex topological structure- is analyzed. It is applied to building a knowledge based system for confidence-based reasoning, which simulates how humans tend to avoid the complexity of concept networks by means of bounded reasoning skills.Ministerio de Ciencia e Innovación TIN2009-09492Junta de Andalucía TIC-606