Papers by Kira Adaricheva
algebra universalis
The universal algebra and lattice theory community has suffered an immense loss. On January 29, 1... more The universal algebra and lattice theory community has suffered an immense loss. On January 29, 1999, the death occurred of Viktor Aleksandrovich Gorbunov, professor at the Novosibirsk State University and researcher in the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, a scientist of international renown, distinguished by his outstanding passion for Mathematics. After graduation from Novosibirsk State University in 1972, he started working as an instructor at the Department of Mechanics and Mathematics of the Novosibirsk State University and simultaneously entered the Ph.D. program in algebra and logic offered by the Institute of Mathematics in Novosibirsk. In 1978, he was awarded the Ph.D. degree on the basis of a dissertation "On the Theory of Quasivarieties of Algebraic Systems" of which Dr. Sc. D. M. Smirnov was his advisor. In 1996, he obtained the degree of Doctor of Sciences for his achievements in the algebraic theory of quasivarieties. He became a professor at the Novosibirsk State University, maintaining simultaneously the position of a leading researcher in the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. Developing the theory of quasivarieties and searching for its applications in other areas, mainly in lattice theory, algebra, graph theory, and formal linguistics, were two main goals Viktor undertook as a choice for his professional life while still early in his career. His publications, his Ph.D. students, and his textbook "Algebraic Theory of Quasivarieties" published in the Siberian School of Algebra and Logic series are luminous evidence of his success in achieving these goals. Viktor's name was and will always be affiliated with the Siberian School of Algebra and Logic, which was founded by Anatoliȋ Ivanovich Mal tsev. Viktor was proud to belong to the School and to continue its traditions. He created around himself a team of bright young mathematicians whom he inspired and with whom he shared his ideas. In the forum of universal algebra and lattice theory, his group was well respected, within the School as well as internationally. Mainly through him and his collaborators the theory of quasivarieties, the precursors of which were Garrett Birkhoff and Anatoliȋ I. Mal tsev, has been gaining both in recognition and status as a well-founded theory. The problem to which Viktor dedicated himself the most was the so-called Birkhoff-Mal tsev problem which asks for a description of all lattices that can be represented as lattices of quasivarieties. Although the problem remains open, all partial solutions found hitherto have been as result of his influence or of his own significant contributions.
Representing Distributive Dually Algebraic Lattices
CMS/CAIMS Books in Mathematics, 2022
Lattices 1 + L as $$\operatorname {L}_{\text{q}}(\mathcal K)$$
CMS/CAIMS Books in Mathematics, 2022
Finite Lattices as $$\operatorname {Sub}(\mathbf S,\wedge ,1,H)$$ : The Case $$\operatorname {J}(\mathbf L) \subseteq \tau (\mathbf L)$$
CMS/CAIMS Books in Mathematics, 2022
This report documents the program and the outcomes of Dagstuhl Seminar 14201 “Horn formulas, dire... more This report documents the program and the outcomes of Dagstuhl Seminar 14201 “Horn formulas, directed hypergraphs, lattices and closure systems: related formalisms and applications”. The seminar brought together researchers working in various areas of mathematics and computer science, mostly in algebra, logic, date base theory, artificial intelligence and data mining. A key objective of the seminar has been to bring together a critical mass of researchers and to provide a platform for personal contacts and scientific interchange between the different disciplines in an atmosphere that will stimulate collaboration and lead to new partnerships. The goal was to crystallize the main research directions and to disseminate challenging open problems across the different research areas.
Computer Science & Information Technology (CS & IT), 2018
We develop a new approach for distributed computing of the association rules of high confidence i... more We develop a new approach for distributed computing of the association rules of high confidence in a binary table. It is derived from the D-basis algorithm [1], which is performed on multiple sub-tables of a table given by removing several rows at a time. The set of rules is then aggregated using the same approach as the D-basis is retrieved from a larger set of implications. This allows to obtain a basis of association rules of high confidence, which can be used for ranking all attributes of the table with respect to a given fixed attribute using the relevance parameter introduced in [2]. This paper focuses on the technical implementation of the new algorithm. Some testing results are performed on transaction data and medical data.
We develop a new approach for distributed computing of the association rules of high confidence o... more We develop a new approach for distributed computing of the association rules of high confidence on the attributes/columns of a binary table. It is derived from the D-basis algorithm developed by K.Adaricheva and J.B.Nation (Theoretical Computer Science, 2017), which runs multiple times on sub-tables of a given binary table, obtained by removing one or more rows. The sets of rules retrieved at these runs are then aggregated. This allows us to obtain a basis of association rules of high confidence, which can be used for ranking all attributes of the table with respect to a given fixed attribute. This paper focuses on some algorithmic details and the technical implementation of the new algorithm. Results are given for tests performed on random, synthetic and real data.
Closure system is a fundamental concept appearing in several areas such as databases, formal conc... more Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties. In the literature, there is a common interest in the search of the mini- mality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The eciency of computation with the direct basis is achieved due to the fact that the closure is computed in one traver...
Combining Algorithms to Find Signatures That Predict Risk in Early-Stage Stomach Cancer
Journal of Computational Biology, 2021
This study applied two mathematical algorithms, lattice up-stream targeting (LUST) and D-basis, t... more This study applied two mathematical algorithms, lattice up-stream targeting (LUST) and D-basis, to the identification of prognostic signatures from cancer gene expression data. The LUST algorithm looks for metagenes, which are sets of genes that are either overexpressed or underexpressed in the same patients. Whereas LUST runs unsupervised by clinical data, the D-basis algorithm uses implications and association rules to relate gene expression to clinical outcomes. The D-basis selects a small subset of the metagene (a signature) to predict survival. The two algorithms, LUST and D-basis, were combined and applied to mRNA expression and clinical data from The Cancer Genome Atlas (TCGA) for 203 stage 1 and 2 stomach cancer patients. Two small (four-gene) signatures effectively predict survival in early-stage stomach cancer patients. These signatures could be used as a guide for treatment. The first signature (DU4) consists of genes that are underexpressed on the long-survival/low-risk group: FLRT2, KCNB1, MYOC, and TNXB. The second signature consists of genes that are overexpressed on the short-survival/high-risk group: ASB5, SFRP1, SMYD1, and TACR2. Another nine-gene signature (REC9) predicts recurrence: BNC2, CCDC8, DPYSL3, MOXD1, MXRA8, PRELP, SCARF2, TAGLN, and ZNF423. Each patient is assigned a score that is a linear combination of the expression levels for the genes in the signature. Scores below a selected threshold predict low-risk/long survival, whereas high scores indicate a high risk of short survival. The metagenes associate with TCGA cluster C1. Both our signatures and cluster C1 identify tumors that are genomically silent, and have a low mutation load or mutation count. Furthermore, our signatures identify tumors that are predominantly in the WHO classification of poorly cohesive and the Lauren class of diffuse samples, which have a poor prognosis.
The Electronic Journal of Combinatorics, 2016
Various characterizations of finite convex geometries are well known. This note provides similar ... more Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.
Algebraic atomistic lattices of quasivarieties
Algebra and Logic, 1997
The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for... more The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as Lq(K) for some algebraic quasivariety K; (2) L is represented as SΛ (A) for some algebraic lattice A which satisfies the minimality condition and
Fundamenta Mathematicae, 1993
Finite atomistic lattices that can be represented as lattices of quasivarieties by K. V. A d a r ... more Finite atomistic lattices that can be represented as lattices of quasivarieties by K. V. A d a r i c h e v a (Novosibirsk), W. D z i o b i a k (Toruń) and V. A. G o r b u n o v (Novosibirsk)
International Journal of General Systems, 2017
Due to its solid mathematical foundations, Formal Concept Analysis (FCA) has become an emergent t... more Due to its solid mathematical foundations, Formal Concept Analysis (FCA) has become an emergent topic in the area of data analysis and knowledge discovering. Information is represented in a binary table defining a relation between a set of objects and a set of attributes-the formal context. The knowledge extracted from the formal context allows to identify useful patterns in data in different forms. One very useful knowledge representation in FCA are implications among attributes which are validated over the objects. The most outstanding feature of implications is that they can be managed by means of inference systems. Equivalent sets of implications can be obtained using different logic-based transformations. The aim of these transformations is to turn the original set of implications into an equivalent one fulfilling some desired properties. Among them, the directness and optimality are very popular targets because getting a direct-optimal basis ensures that the closure of a set of attributes may be computed with lower cost (time and resources). In this work, we introduce a new method to compute the direct-optimal basis which improves the existing ones. The new method reduces the input in a first stage and is guided by the idea of limiting the growth of the intermediate sets of implications as a way to improve the performance. We illustrate the good features of the new method with both a detailed example and by experimental evaluation.
Interval Dismantlable Lattices
Order, 2017
A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, ea... more A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1-element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many minimal interval non-dismantlable lattices.
Bases of Closure Systems
Lattice Theory: Special Topics and Applications, 2016
In Section 4-2.2 we discussed the fundamental connection between closure systems and sets of impl... more In Section 4-2.2 we discussed the fundamental connection between closure systems and sets of implications. In this chapter, we will look into the canonical forms of representations of a closure system by implications. Most of the results are inspired by the structure of the closure lattice and its properties. In particular, we will be concerned with effective representations of closure systems whose closure lattices are join-semidistributive, lower bounded or locally distributive.
Order, 2016
A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of alg... more A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.
Theoretical Computer Science, 2017
Discovery of (strong) association rules, or implications, is an important task in data management... more Discovery of (strong) association rules, or implications, is an important task in data management, and it finds application in artificial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a specific set of implications, called the D-basis, that provides a representation for a reduced binary table, based on the structure of its Galois lattice. At the core of the method are the D-relation defined in the lattice theory framework, and the hypergraph dualization algorithm that allows us to effectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, first developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, artificial intelligence and game theory. One application of the method is for analysis of gene expression data related to a particular phenotypic variable, and some initial testing is done for the data provided by the University of Hawaii Cancer Center.
Lecture Notes in Computer Science, 2015
We introduce the parameter of relevance of an attribute of a binary table to another attribute of... more We introduce the parameter of relevance of an attribute of a binary table to another attribute of the same table, computed with respect to an implicational basis of a closure system associated with the table. This enables a ranking of all attributes, by relevance parameter to the same fixed attribute, and, as a consequence, reveals the implications of the basis most relevant to this attribute. As an application of this new metric, we test the algorithm for D-basis extraction presented in Adaricheva and Nation [1] on biomedical data related to the survival groups of patients with particular types of cancer. Each test case requires a specialized approach in converting the real-valued data into binary data and careful analysis of the transformed data in a multidisciplinary environment of cross-field collaboration.
A recent direction within belief revision theory is to develop a theory of belief change for the ... more A recent direction within belief revision theory is to develop a theory of belief change for the Horn knowledge representation framework. We consider questions related to the complexity aspects of previous work, leading to questions about Horn envelopes (or Horn LUB's), introduced earlier in the context of knowledge compilation. A characterization is obtained of the remainders of a Horn belief set with respect to a consequence to be contracted, as the Horn envelopes of the belief set and an elementary conjunction corresponding to a truth assignment satisfying a certain body building formula. This gives an efficient algorithm to generate all remainders, each represented by a truth assignment. On the negative side, examples are given of Horn belief sets and consequences where Horn formulas representing the result of most contraction operators, based either on remainders or on weak remainders, must have exponential size.
This report documents the programme and the outcomes of Dagstuhl Seminar 14202 on "Judgment Aggre... more This report documents the programme and the outcomes of Dagstuhl Seminar 14202 on "Judgment Aggregation for Artificial Intelligence". Judgment aggregation is a new group decisionmaking theory that lies in the intersection of logic and social choice; it studies how to reach group decisions on several logically interconnected issues by aggregation of individual judgments. Until recently research in judgment aggregation was dominated by its originating context of philosophy, political science and law. Presently, however we are witnessing increasing work in judgment aggregation from researchers in computer science. Since researchers from such diverse disciplinary backgrounds working on judgment aggregation each publish within their own discipline with virtually no cross-discipline cooperation on concrete projects, it is essential that they are given an opportunity to connect to each other and become aware of the workings of the other side. This seminar has provided such an opportunity.
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Papers by Kira Adaricheva