Optimum basis of finite convex geometry

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in [6] for UC- systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satis es the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L. Hammer and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be ope

Optimum basis of finite convex geometry

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in [6] for UC- systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satis es the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L. Hammer and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be ope

On implicational bases of closure systems with unique critical sets

We show that every optimum basis of a finite closure system, in D.Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce a K-basis of a general closure system, which is a refinement of the canonical basis of Duquenne and Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with the unique criticals and some of its subclasses, where the K-basis is unique. A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be effectively recognized. While E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into plenary talk on conference Universal Algebra and Lattice Theory, June 2012, Szeged, Hungary 29 pages and 2 figure

Quasi-closed elements in fuzzy posets

We generalize the notion of quasi-closed element to fuzzy posets in two stages: First, in the crisp style in which each element in a given universe either is quasi-closed or not. Second, in the graded style by defining degrees to which an element is quasi-closed. We discuss the different possible definitions and comparing them with each other. Finally, we show that the most general one has good properties to be used when we have a complete fuzzy lattice as a frame

Ordered direct implication basis of a finite closure system

Closure system on a nite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of nite lattices, and the tools of economic description of a nite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis in time polynomial in the size s( ), and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direc

Horn axiomatizations for sequential data

AbstractWe propose a notion of deterministic association rules for ordered data. We prove that our proposed rules can be formally justified by a purely logical characterization, namely, a natural notion of empirical Horn approximation for ordered data which involves background Horn conditions; these ensure the consistency of the propositional theory obtained with the ordered context. The whole framework resorts to concept lattice models from Formal Concept Analysis, but adapted to ordered contexts. We also discuss a general method to mine these rules that can be easily incorporated into any algorithm for mining closed sequences, of which there are already some in the literature

Ordered direct implicational basis of a finite closure system

Closure system on a finite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of finite lattices, and the tools of economic description of a finite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis S in time polynomial in the size of S, and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direct.Comment: 25 pages, 10 figures; presented at AMS conference, TACL-2011,ISAIM-2012 and at RUTCOR semina

On implicational bases of closure system with unique critical sets

We show that every optimum basis of a nite closure system, in D. Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce the K-basis of a general closure system, which is a re nement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further re nement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be e ectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete proble

A Semantics-Based Approach to Design of Query Languages for Partial Information

Most of work on partial information in databases asks which operations of standard languages, like relational algebra, can still be performed correctly in the presence of nulls. In this paper a different point of view is advocated. We believe that the semantics of partiality must be clearly understood and it should give us new design principles for languages for databases with partial information. There are different sources of partial information, such as missing information and conflicts that occur when different databases are merged. In this paper, we develop a common semantic framework for them which can be applied in a context more general than the flat relational model. This ordered semantics, which is based on ideas used in the semantics of programming languages, cleanly intergrates all kinds of partial information and serves as a tool to establish connections between them. Analyzing properties of semantic domains of types suitable for representing partial information, we come up with operations that are naturally associated with those types, and we organize programming syntax around these operations. We show how the languages that we obtain can be used to ask typical queries about incomplete information in relational databases, and how they can express some previously proposed languages. Finally, we discuss a few related topics such as mixing traditional constraints with partial information and extending semantics and languages to accommodate bags and recursive types

On implicational bases of closure system with unique critical sets

We show that every optimum basis of a nite closure system, in D. Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce the K-basis of a general closure system, which is a re nement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further re nement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be e ectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete proble