Redundancy in logic I: CNF propositional formulae

A knowledge base is redundant if it contains parts that can be inferred from the rest of it. We study some problems related to the redundancy of a CNF formula. In particular, any CNF formula can be made irredundant by deleting some of its clauses: what results is an irredundant equivalent subset. We study the complexity of problems related to irredundant equivalent subsets: verification, checking existence of an irredundant equivalent subset with a given size, checking necessary and possible presence of clauses in irredundant equivalent subsets, and uniqueness. We also consider the problem of redundancy with different definitions of equivalence.     

Generic dependencies and database design

The concept of functional dependencies in databases is generalized and called generic dependencies. Just as a Karnaugh map exhibits all the functional dependencies in a relation, an entropy map represents all the generic dependencies. A generalized normal form useful in database design is defined.     

Redundancy in logic II: 2CNF and Horn propositional formulae

We report results about the redundancy of formulae in 2CNF form. In particular, we give a slight improvement over the trivial redundancy algorithm and give some complexity results about some problems related to finding Irredundant Equivalent Subsets (i.e.s.) of 2CNF formulae. The problems of checking whether a 2CNF formula has a unique i.e.s. and checking whether a clause in is all its i.e.s.'s are polynomial. Checking whether a 2CNF formula has an i.e.s. of a given size and checking whether a clause is in some i.e.s.'s of a 2CNF formula are polynomial or NP-complete depending on whether the formula is cyclic. Some results about Horn formulae are also reported.     

Reasoning with propositional knowledge based on fuzzy neural logic

In this article, a new kind of reasoning for propositional knowledge, which is based on the fuzzy neural logic initialed by Teh, is introduced. A fundamental theorem is presented showing that any fuzzy neural logic network can be represented by operations: bounded sum, complement, and scalar product. Propositional calculus of fuzzy neural logic is also investigated. Linear programming problems risen from the propositional calculus of fuzzy neural logic show a great advantage in applying fuzzy neural logic to answer imprecise questions in knowledge-based systems. An example is reconsidered here to illustrate the theory. © 1996 John Wiley & Sons, Inc.     

Approximate dependencies in database systems

Functional dependencies are the most commonly used approach for capturing real-word integrity constraints which are to be reflected in a database. There are, however, many useful kinds of constraints, especially approximate ones, that cannot be represented correctly by functional dependencies and therefore are enforced via programs which update the database, if they are enforced at all. This tends to make such constraints invisible since they are not an explicit part of the database, increasing maintenance problems and the likelihood of inconsistencies. We propose a new approach, cluster dependencies, as a way to enforce approximate dependencies. By treating equality as a fuzzy concept and defining appropriate similarity measures, it is possible to represent a broad range of approximate constraints directly in the database by storing and accessing cluster definitions. We discuss different interpretations of cluster dependencies and describe the additional data structures needed to enforce them. We also contrast them with an existing approach, fuzzy functional dependencies, which are much more limited in the kind of approximate constraints they can represent.     

Inductive database languages: requirements and examples

Inductive databases (IDBs) represent a database perspective on Knowledge discovery in databases (KDD). In an IDB, the KDD application can express both queries capable of accessing and manipulating data, and queries capable of generating, manipulating, and applying patterns allowing to formalize the notion of mining process. The feature that makes them different from other data mining applications is exactly the idea of looking at the support for knowledge discovery as an extension of the query process. This paper draws a list of desirable properties to be taken into account in the definition of an IDB framework. They involve several dimensions, such as the expressiveness of the language in representing data and models, the closure principle, the capability to provide a support for an efficient algorithm programming. These requirements are a basis for a comparative study that highlights strengths and weaknesses of existing IDB approaches. The paper focuses on the SQL-based ATLaS language/system, on the logic-based LDL++{\mathcal{LDL}++} language/system, and on the XML-based KDDML language/system.     

Reasoning about equations and functional dependencies on complexobjects

Virtually all semantic or object-oriented data models assume that objects have an identity separate from any of their parts, and allow users to define complex object types in which part values may be any other objects. This often results in a choice of query language in which a user can express navigating from one object to another by following a property value path. We consider a constraint language in which one may express equations and functional dependencies over complex object types. The language is novel in the sense that component attributes of individual constraints may correspond to property paths. The kind of equations we consider are also important, because they are a natural abstraction of the class of conjunctive queries for query languages that support property value navigation. In our introductory comments, we give an example of such a query and outline two applications of the constraint theory to problems relating to a choice of access plan for the query. We present a sound and complete axiomatization of the constraint language for the case in which interpretations are permitted to be infinite, where interpretations themselves correspond to a form of directed labeled graph. Although the implication problem for our form of equational constraint alone over arbitrary schema is undecidable, we present decision procedures for the implication problem for both kinds of constraints when the problem schema satisfies a stratification condition, and when all input functional dependencies are keys     

On characterizing boundedness of database schemes with bounded dependencies

We investigate the effect of bounded dependencies on the boundedness of database schemes. The following results are proved. A database scheme with only bounded equality-generating dependencies is always bounded with respect to dependencies; a lossless database scheme with bounded full implicational dependencies is bounded w.r.t. dependencies if and only if the implicational dependencies are equivalent to a single join dependency and some equality-generating dependencies. By a known method, this condition can be tested effectively. These results are relevant in database theory in that they determine in a rather general case whether queries under the representative instance approach can be expressed in relational algebra.     

Independent database schemes under functional and inclusion dependencies

In a context considering in a unique framework all the relations in a database, by means of the notion of global consistency, independent database schemes allow enforcement of constraints to be performed locally, thus providing independent updatability of the various relations. Independent schemes have hitherto been studied in the presence of functional and join dependencies. In this paper the definition is extended and some characterizations are given for schemes whose sets of constraints contain functional and inclusion dependencies.An extended abstract of this paper appears in the Proceedings of the Thirteenth International Conference on Very Large Data Bases, Brighton, England, 1–4 September 1987, pp. 159–166. The work of the first author was supported byMinistero dell'Università e della Ricerca Scientifica e Tecnologica, within the Project Metodi formali e strumenti per basi di dati evolute and the work of the second author by the Natural Sciences and Engineering Research Council of Canada. Part of this work was performed when the first author was with IASI-CNR and then with Università di Napoli.     

Computational methods for database repair by signed formulae*

We introduce a simple and practical method for repairing inconsistent databases. Given a possibly inconsistent database, the idea is to properly represent the underlying problem, i.e., to describe the possible ways of restoring its consistency. We do so by what we call signed formulae, and show how the ‘signed theory’ that is obtained can be used by a variety of off-the-shelf computational models in order to compute the corresponding solutions, i.e., consistent repairs of the database. *This paper is a revised and extended version of [9].     

On the complementation rule for multivalued dependencies in database relations

Summary We study the interrelation between various versions of the complementation rule and other inference rules for multivalued dependencies in database relations. In particular we settle two open questions of [1] concerning the derivability of inference rules for Boolean operations on the right side of multivalued dependencies. Furthermore we prove that there is a trade-off between the complementation rule and the augmentation rule.     

Another view of functional and multivalued dependencies in the relational database model

In this paper, pseudo-functional and pseudo-multivalued dependencies are introduced. They are shown to be isomorphic with functional and multivalued dependencies, i.e., they behave in the same way with respect to implication. This formalism leads in a very natural way to a rather efficient algorithm for the inference of functional and multivalued dependencies. Some applications to acyclic join dependencies are discussed.     

A survey of some combinatorial results concerning functional dependencies in database relations

A databaseR has some obvious and less obvious parameters such as the number of attributes, the size |r|, the maximum size of a domain, the number of some special functional dependencies (e.g. the minimal keys), and so on. The main aim of this paper is to survey some of the results giving connections and inequalities among these parameters. The methods are of a combinatorial nature. A generalization of the numerical dependency is also considered.This work was supported by the Hungarian National Foundation for Scientific Research, Grant No. 2575.     

A new method for computing fuzzy functional dependencies in relational database systems

In this paper, we present a new method for computing fuzzy functional dependencies between attributes in fuzzy relational database systems. The method is based on the use of fuzzy implications. A literature analysis has shown that there is no algorithm that would enable the identification of attribute relationships in fuzzy relational schemas. This fact was the motive for development a new methodology in the analysis of fuzzy functional dependencies over a given set of attributes. Solving this, not so new problem, is not only research challenge having theoretical importance, but it also has practical significance. Possible applications of the proposed methodology include GIS, data mining, information retrieval, reducing data redundancy in fuzzy relations through implementation of logical database model, estimation of missing values etc.     

Uncertain lines and circles with dependencies

Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require the modeling of objects with geometric uncertainties. Existing geometric uncertainty models cannot handle dependencies among objects. This results in the overestimation of errors. We have developed the Linear Parametric Geometric Uncertainty Model, a general, computationally efficient, worst-case, linear approximation of geometric uncertainty that supports dependencies among uncertainties. In this paper, we present the properties of the uncertainty zones of a line and circle, defined using this model, and describe efficient algorithms to compute them. We show that the line’s envelope has linear space complexity and is computed in low polynomial time. The circle’s envelope has quadratic complexity and is also computed in low polynomial time.     

Process algebra with action dependencies

In this paper, we present a process algebra with a minimal form of semantics for actions given by dependencies. Action dependencies are interpreted in the Mazurkiewicz sense: independent actions should be able to commute, or (from a different perspective) should be unordered, whereas dependent actions are always ordered. In this approach, the process algebra operators are used to describe the conceptual behavioural structure of the system, and the action dependencies determine the minimal necessary orderings and thereby the additionally possible parallelism within this structure. In previous work on the semantics of specifications using Mazurkiewicz dependencies, the main interest has been on linear time. We present in this paper a branching time semantics, both operationally and denotationally. For this purpose, we introduce a process algebra that incorporates, besides some standard operators, also an operator for action refinement. For interpreting the operators in the presence of action dependencies, a new concept of partial termination has to be developed. We show consistency of the operational and denotational semantics; furthermore, we give a axiomatisation of bisimilarity, which is complete for finite terms. Some small examples demonstrate the flexibility of this process algebra in the design of distributed reactive systems. Received: 19 November 1998 / 18 July 2001