### Book

## Formal Concept Analysis. ICFCA 2019

This book constitutes the proceedings of the 15th International Conference on Formal Concept Analysis, ICFCA 2019, held in Frankfurt am Main, Germany, in June 2019.

The 15 full papers and 5 short papers presented in this volume were carefully reviewed and selected from 36 submissions. The book also contains four invited contributions in full paper length.

The field of Formal Concept Analysis (FCA) originated in the 1980s in Darmstadt as a subfield of mathematical order theory, with prior developments in other research groups. Its original motivation was to consider complete lattices as lattices of concepts, drawing motivation from philosophy and mathematics alike. FCA has since then developed into a wide research area with applications much beyond its original motivation, for example in logic, data mining, learning, and psychology.

In this paper, we consider computational problems related to finding implications in an explicitly given formal context or via queries to an oracle. We are concerned with two types of problems: enumerating implications (or association rules) and finding a single implication satisfying certain conditions. We present complexity results for some of these problems and leave others open. The paper is not meant as a comprehensive survey, but rather as a subjective selection of interesting problems.

A scalable method for mining graph patterns stable under subsampling is proposed. The existing subsample stability and robustness measures are not antimonotonic according to definitions known so far. We study a broader notion of antimonotonicity for graph patterns, so that measures of subsample stability become antimonotonic. Then we propose gSOFIA for mining the most subsample-stable graph patterns. The experiments on numerous graph datasets show that gSOFIA is very efficient for discovering subsample-stable graph patterns.

There is a lot of usefulness measures of patterns in data mining. This paper is focused on the measures used in Formal Concept Analysis (FCA). In particular, concept stability is a popular relevancy measure in FCA. Experimental results of this paper show that high stability of a pattern in a given dataset derived from the general population suggests that the stability of that pattern is high in another dataset derived from the same population. At the second part of the paper, a new estimate of stability is introduced and studied. It es performance is evaluated experimentally. And it is shown that it is more efficient.

Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classication, introduced and detailed in the book of Bernhard Ganter and Rudolf Wille, \Formal Concept Analysis", Springer 1999. The area came into being in the early 1980s and has since then spawned over 10000 scientic publications and a variety of practically deployed tools. FCA allows one to build from a data table with objects in rows and attributes in columns a taxonomic data structure called concept lattice, which can be used for many purposes, especially for Knowledge Discovery and Information Retrieval. The \Formal Concept Analysis Meets Information Retrieval" (FCAIR) workshop collocated with the 35th European Conference on Information Retrieval (ECIR 2013) was intended, on the one hand, to attract researchers from FCA community to a broad discussion of FCA-based research on information retrieval, and, on the other hand, to promote ideas, models, and methods of FCA in the community of Information Retrieval. This volume contains 11 contributions to FCAIR workshop (including 3 abstracts for invited talks and tutorial) held in Moscow, on March 24, 2013. All submissions were assessed by at least two reviewers from the program committee of the workshop to which we express our gratitude. We would also like to thank the co-organizers and sponsors of the FCAIR workshop: Russian Foundation for Basic Research, National Research University Higher School of Economics, and Yandex.

Relationships between proto-fuzzy concepts, crisply generated fuzzy concepts, and pattern structures are considered. It is shown that proto-fuzzy concepts are closely related to crisply generated fuzzy concepts in the sense that the mappings involved in the definitions coincide for crisp subsets of attributes. Moreover, a proto-fuzzy concept determines a crisp subset of attributes, which generates a (crisply generated) fuzzy concept. However, the reverse is true only in part: given a crisp subset of attributes, one can find a proto-fuzzy concept whose intent includes (but not necessarily coincides with) the given subset of attributes. Interval pattern concepts are shown to be related to crisply generated formal concepts. In particular, every crisply closed subset of objects is an extent of an interval pattern concept. Also, we establish some properties of the collection of formal concepts for a given fuzzy context.

Symbolic classifiers allow for solving classification task and provide the reason for the classifier decision. Such classifiers were studied by a large number of researchers and known under a number of names including tests, JSM-hypotheses, version spaces, emerging patterns, proper predictors of a target class, representative sets etc. Here we consider such classifiers with restriction on counter-examples and discuss them in terms of pattern structures. We show how such classifiers are related. In particular, we discuss the equivalence between good maximally redundant tests and minimal JSM-hyposethes and between minimal representations of version spaces and good irredundant tests.

This paper considers a data analysis system for collaborative platforms which was developed by the joint research team of the National Research University Higher School of Economics and the Witology company. Our focus is on describing the methodology and results of the first experiments. The developed system is based on several modern models and methods for analysing of object-attribute and unstructured data (texts) such as Formal Concept Analysis, multimodal clustering, association rule mining, and keyword and collocation extraction from texts.

During the last three decades, formal concept analysis (FCA) became a well-known formalism in data analysis and knowledge discovery because of its usefulness in important domains of knowledge discovery in databases (KDD) such as ontology engineering, association rule mining, machine learning, as well as relation to other established theories for representing knowledge processing, like description logics, conceptual graphs, and rough sets. In early days, FCA was sometimes misconceived as a static crisp hardly scalable formalism for binary data tables. In this paper, we will try to show that FCA actually provides support for processing large dynamical complex (may be uncertain) data augmented with additional knowledge.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

Existing approaches suggest that IT strategy should be a reflection of business strategy. However, actually organisations do not often follow business strategy even if it is formally declared. In these conditions, IT strategy can be viewed not as a plan, but as an organisational shared view on the role of information systems. This approach generally reflects only a top-down perspective of IT strategy. So, it can be supplemented by a strategic behaviour pattern (i.e., more or less standard response to a changes that is formed as result of previous experience) to implement bottom-up approach. Two components that can help to establish effective reaction regarding new initiatives in IT are proposed here: model of IT-related decision making, and efficiency measurement metric to estimate maturity of business processes and appropriate IT. Usage of proposed tools is demonstrated in practical cases.