Functional dependencies among Boolean dependencies

In this paper, we consider functional dependencies among Boolean dependencies (BDs, for short). Armstrong relations are defined for BDs (called BD-Armstrong relations). For BDs, two necessary and sufficient conditions for the existence of BD-Armstrong relations are given. A necessary and sufficient condition for the existence of Armstrong relations for functional dependencies (FDs, for short) is given, which in some sense is more convenient than the condition given in [3]. We give an algorithm that solves the problem of deciding if two BDs imply the same set of functional dependencies. If the BDs are given in perfect disjunctive normal form, then the algorithm requires only polynomial time. Although Mannila and Räihä have shown that for some relations exponential time is needed for computing any cover of the set of FDs defined in this relation, as a consequence, we show that the problem of deciding if two relations satisfy the same set of FDs can be solved in polynomial time. Another consequence is a new correspondence of the families of functional dependencies to the families of Sperner systems. By this correspondence, the estimate of the number of databases given previously in [6] is improved. It is shown that there is a one-to-one correspondence between the closure of the FDs that hold in a BD and its so-calledbasic cover. As applications of basic covers, we obtain a representation of a key, the family of minimal keys and a representation of canonical covers.This research was supported by the Hungarian Foundation for Scientific Research, Grant Nos. OTKA 2575, 2149.     

Design by example for SQL table definitions with functional dependencies

A database is C-Armstrong for a given set of constraints in a class C if it satisfies every constraint of the set and violates every constraint in C not implied by the set. Therefore, Armstrong databases are test data that perfectly illustrate the current perceptions about the semantics of a schema. We extend the existing theory of Armstrong relations to a toolbox of Armstrong tables. That is, we investigate structural and computational properties of Armstrong tables for the class of functional dependencies (FDs) over SQL tables. Relations are special instances of SQL tables with no duplicate rows and no null value occurrences. While FDs do not enjoy Armstrong tables, the combined class of standard FDs and NOT NULL constraints does enjoy Armstrong tables. The problem of finding an Armstrong table is shown to be precisely exponential for this combined class. However, we establish an algorithm that computes Armstrong tables with a size at most quadratic in that of a minimum-sized Armstrong table. Our resulting toolbox of Armstrong tables can be applied by data engineers to concisely visualize constraints on SQL data. Such support can lead to designs that guarantee efficient data management in practice.     

About quotient and division of crisp and fuzzy relations

The role and properties of the division are very well known in the context of queries addressed to regular relational databases. However, Boolean queries whose result is expressed in terms of all or nothing may turn out to be too limited to answer certain user needs and it is desirable to envisage extended queries by introducing preferences in the conditions. In this paper, two lines of extension of the division operator are studied: (i) operand relations of the division are fuzzy relations (i.e., they are made of weighted tuples) and (ii) the universal quantifier underlying the division is weakened. Various approaches to these extensions can be considered and one of our goals is to point out those which ensure that the resulting relation is a quotient (in reference to the characterization of the quotient of two integers). So doing, a sound semantics for the extended division is guaranteed.     

Modelling and solving temporal reasoning as propositional satisfiability

Representing and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen's interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete.In this paper, we investigate how an interval-based representation, or IA network, can be encoded into a propositional formula of Boolean variables and/or predicates in decidable theories. Our task is to discover whether satisfying such a formula can be more efficient than finding a consistent scenario for the original problem. There are two basic approaches to modelling an IA network: one represents the relations between intervals as variables and the other represents the end-points of each interval as variables. By combining these two approaches with three different Boolean satisfiability (SAT) encoding schemes, we produced six encoding schemes for converting IA to SAT. In addition, we also showed how IA networks can be formulated into satisfiability modulo theories (SMT) formulae based on the quantifier-free integer difference logic (QF-IDL). These encodings were empirically studied using randomly generated IA problems of sizes ranging from 20 to 100 nodes. A general conclusion we draw from these experimental results is that encoding IA into SAT produces better results than existing approaches. More specifically, we show that the new point-based 1-D support SAT encoding of IA produces consistently better results than the other alternatives considered. In comparison with the six different SAT encodings, the SMT encoding came fourth after the point-based and interval-based 1-D support schemes and the point-based direct scheme. Further, we observe that the phase transition region maps directly from the IA encoding to each SAT or SMT encoding, but, surprisingly, the location of the hard region varies according to the encoding scheme. Our results also show a fixed performance ranking order over the various encoding schemes.     

Computing SSA Form with Matrices

Static Single-Assignment (SSA) form is an efficient intermediate representation used in virtual machines and modern compilers. It provides data flow information that simplifies the implementation of standard program optimisations such as constant propagation, dead code elimination, and partial redundancy elimination. Constructing SSA form involves the computation of graph relations such as dominance, and non-iterated and iterated dominance frontier. Although there exist efficient graph algorithms for these relations, the algorithms are elaborate to implement. In this paper we introduce a new approach to compute the dominance relation, the dominance frontiers, and the iterated dominance frontiers based on Boolean matrix calculus. We implemented our approach in an optimising backend for LCC bytecode and compared its performance with the state-of-the-art approaches. We use the Spec95 benchmark suite for our experimental evaluation.     

零化多项式与仿射空间关系研究

为把流密码的代数攻击问题转化为求解布尔函数的低次数零化多项式问题,讨论布尔函数的性质,介绍{0,1}上矩阵的特殊结构,研究两者间的关系,在此基础上探讨n元布尔函数f的零化多项式次数与f的支撑点集之间的关系,实验结果表明,寻找布尔函数零化多项式等价于在布尔函数的零点集合中寻找最大的仿射空间。     

The Use of Boolean Model for Texture Analysis of Grey Images

We generalize here the use of the 1D Boolean model for the analysis of grey level textures. Each grey image is first split into eight binary images using different criteria. Each of these binary images is separately analysed with the help of the 1D Boolean model and features are extracted from it. The final grey texture recognition is performed on the basis of these features using several classification criteria. Experiments have been carried out using an image database of 30 grey level textures, all of them with 512×512 pixels in size, obtaining correct classification rates between 95% and 100%, according to the classification criterion used.     

Empirical evidence for the usefulness of Armstrong relations in the acquisition of meaningful functional dependencies

Armstrong relations satisfy precisely those data dependencies that are implied by a given set of data dependencies. A common perception is that Armstrong relations are useful in the acquisition of data semantics, in particular since errors during the requirements elicitation have the most expensive consequences.     

Parity relations for linear uncertain dynamic systems

A new approach for the design of parity relations for linear dynamic systems with additive and multiplicative uncertainties is presented. Instead of cancelling uncertainties following the example of the so-called robust approaches, uncertain parity relations take uncertainties into account as bounded variables. The method is based on the analysis of zonotopes representing the uncertainties. It leads both to Boolean detection results and to an indicator representing the distance to the opposite decision.     

Quantified maximum satisfiability

In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies an optimization extension of QBF and considers the problem in a quantified MaxSAT setting. More precisely, the general QMaxSAT problem is defined for QBFs with a set of soft clausal constraints and consists in finding the largest subset of the soft constraints such that the remaining QBF is true. Two approaches are investigated. One is based on relaxing the soft clauses and performing an iterative search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the two concrete problems of computing smallest minimal unsatisfiable subformulas (SMUS) and smallest minimal equivalent subformulas (SMES), decision versions of which are well-known problems in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS and SMES problems outperforms the use of iterative search over the values of the cost function. More significantly, the core-guided approach to SMUS also outperforms the state-of-the-art SMUS extractor Digger.     

Peirce algebras

We present a two-sorted algebra, called aPeirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a set-forming operator on relations (the Peirce product of Boolean modules) and a relation-forming operator on sets (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-calledterminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.     

On the finite and general implication problems of independence atoms and keys

We investigate implication problems for keys and independence atoms in relational databases. For keys and unary independence atoms we show that finite implication is not finitely axiomatizable, and establish a finite axiomatization for general implication. The same axiomatization is also sound and complete for finite and general implication of unary keys and independence atoms, which coincide. We show that the general implication of keys and unary independence atoms and of unary keys and general independence atoms is decidable in polynomial time. For these two classes we also show how to construct Armstrong relations. Finally, we establish tractable conditions that are sufficient for certain classes of keys and independence atoms not to interact.     

Distributive mereotopology: extended distributive contact lattices

The notion of contact algebra is one of the main tools in the region based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. The elements of the Boolean algebra are considered as formal representations of spatial regions as analogues of physical bodies and Boolean operations are considered as operations for constructing new regions from given ones and also to define some mereological relations between regions as part-of, overlap and underlap. The contact relation is one of the basic mereotopological relations between regions expressing some topological nature. It is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact and others. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of the operation of Boolean complementation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement and replacing the Boolean part of the definition by that of a distributive lattice. First steps in this direction were made in (Düntsch et al. Lect. Notes Comput. Sci. 4136, 135–147, 2006, Düntsch et al. J. Log. Algebraic Program. 76, 18–34, 2008) presenting the notion of distributive contact lattice based on the contact relation as the only mereotopological relation. In this paper we consider as non-definable primitives the relations of contact, nontangential inclusion and dual contact, extending considerably the language of distributive contact lattices. Part I of the paper is devoted to a suitable axiomatization of the new language called extended distributive contact lattice (EDC-lattice) by means of universal first-order axioms true in all contact algebras. EDC-lattices may be considered also as an algebraic tool for a certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDC-lattice can be isomorphically embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDC-lattices, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space.     

Counterexample-guided abstraction refinement for linear programs with arrays

Predicate abstraction refinement is one of the leading approaches to software verification. The key idea is to abstract the input program into a Boolean Program (i.e. a program whose variables range over the Boolean values only and model the truth values of predicates corresponding to properties of the program state), and refinement searches for new predicates in order to build a new, more refined abstraction. Thus Boolean programs are commonly employed as a simple, yet useful abstraction. However, the effectiveness of predicate abstraction refinement on programs that involve a tight interplay between data-flow and control-flow is still to be ascertained. We present a novel counterexample guided abstraction refinement procedure for Linear Programs with arrays, a fragment of the C programming language where variables and array elements range over a numeric domain and expressions involve linear combinations of variables and array elements. In our procedure the input program is abstracted w.r.t. a family of sets of array indices, the abstraction is a Linear Program (without arrays), and refinement searches for new array indices. We use Linear Programs as the target of the abstraction (instead of Boolean programs) as they allow to express complex correlations between data and control. Thus, unlike the approaches based on predicate abstraction, our approach treats arrays precisely. This is an important feature as arrays are ubiquitous in programming. We provide a precise account of the abstraction, Model Checking, and refinement processes, discuss their implementation in the EUREKA tool, and present a detailed analysis of the experimental results confirming the effectiveness of our approach on a number of programs of interest.     

Structure identification of Boolean relations and plain bases for co-clones

We give a quadratic algorithm for the following structure identification problem: given a Boolean relation R and a finite set S of Boolean relations, can the relation R be expressed as a conjunctive query over the relations in the set S? Our algorithm is derived by first introducing the concept of a plain basis for a co-clone and then identifying natural plain bases for every co-clone in Post's lattice. In the process, we also give a quadratic algorithm for the problem of finding the smallest co-clone containing a Boolean relation.     

采用推理方法提高多面体Boolean运算的可靠性

提高实体Boolean运算的可靠性是几何造型中最基本也是最迫切的工作。通过对国内外几何造型系统在实体Boolean运算可靠性方面的测试,我们发现,实体Boolean运算不可靠是一个普遍现象,运算失败的根本原因在于数值计算存在误差。数值计算误差决定了我们不能精确地确定集合成员分类。集合成员分类的结果与选定的计算容差有关,具 有相对性。从而相关分类之间会发生冲突。一旦分类之间发生冲突,Boolean运算就不能得到正确的结果。我们提出了一个相当可靠的Boolean运算算法。该算法采用推理的方法在进行Boolean运算之前解决由数值计算误差所引起的相关分类之间冲突。这些算法已在Apollo和SUN工作站上实现,并取代了GEMS 2.0造型系统,构成了一个新的造型系统GEMS 2.1。经测试和比较,这个系统的可靠性比SDRC公司的Geomod 3.9和Intergraph公司的I/EMS高得多。     

EXTRACTING LAWS FROM DECISION TABLES: A ROUGH SET APPROACH

We present some methods, based on the rough set and Boolean reasoning approaches, for extracting laws from decision tables. First we discuss several procedures for decision rules synthesis from decision tables. Next we show how to apply some near-to-functional relations between data to data filtration. Two methods of searching for new classifiers (features) are described: searching for new classifiers in a given set of logical formulas, and searching for some functions approximating near-to-functional relations.     

Toward a dialectic relation between the results in CSCL: Three critical methodological aspects of content analysis schemes

The research field of Computer-Supported Collaborative Learning (CSCL) includes a large variety of approaches which present significant theoretical and methodological differences. This diversity complicates the articulation of the knowledge that is produced within this investigative framework. The paper addresses this problem from a dialectic view. We propose that the main reason for this problem is not the theoretical and methodological diversity itself, but rather the difficulty of situating one specific result within this diversity in a way that makes dialectic relations between results visible and mutual transformation of the approaches possible. In the present paper, we propose a set of indicators, applicable to content analysis approaches, aimed to facilitate this reciprocal positioning of the results in the field. These indicators come from what we term “critical methodological aspects”: those aspects of the methodological infrastructure that are directly related to theoretical positions. We consider three critical methodological aspects in content analysis schemes: the units of analysis, the relations to be established, and the dimensions of analysis. Indicators regarding these aspects are proposed and defined, and their use for facilitating dialectical relations between results is exemplified by means of the examination of five specific approaches.     

The use of data-derived label hierarchies in multi-label classification

Instead of traditional (multi-class) learning approaches that assume label independency, multi-label learning approaches must deal with the existing label dependencies and relations. Many approaches try to model these dependencies in the process of learning and integrate them in the final predictive model, without making a clear difference between the learning process and the process of modeling the label dependencies. Also, the label relations incorporated in the learned model are not directly visible and can not be (re)used in conjunction with other learning approaches. In this paper, we investigate the use of label hierarchies in multi-label classification, constructed in a data-driven manner. We first consider flat label sets and construct label hierarchies from the label sets that appear in the annotations of the training data by using a hierarchical clustering approach. The obtained hierarchies are then used in conjunction with hierarchical multi-label classification (HMC) approaches (two local model approaches for HMC, based on SVMs and PCTs, and two global model approaches, based on PCTs for HMC and ensembles thereof). The experimental results reveal that the use of the data-derived label hierarchy can significantly improve the performance of single predictive models in multi-label classification as compared to the use of a flat label set, while this is not preserved for the ensemble models.     

On lower and upper intension order relations by different cover concepts

In this paper, the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set. These types of ordering relations are distinct from the well-known ordering relation based on set inclusion. For these ordering relations, we consider the algebraic structures that arise in various types of covers. We show that in the case of a representative cover, the algebraic structure resulting from the lower intension inclusion is a double Stone algebra, while in the case of a reduced cover, it is a Boolean algebra. In addition, the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a Boolean algebra, and in the case of a reduced cover, the two Boolean algebraic structures from lower and upper intension inclusions are isomorphic.