Disjunctive Deductive Databases Research Articles

Quasi-classical reasoning in paraconsistent databases

The well-founded model for any general deductive database computed using the paraconsistent relational model, based on four-valued logic, does not support inference rules such as disjunctive syllogism. In order to support disjunctive syllogism, we utilize the generalization of the relational model to quasi-classic (QC) logic, whose inference power is much closer to classical logic. As the paraconsistent relational model is capable of representing incomplete and inconsistent data, we propose an algorithm to find QC model for inconsistent positive extended disjunctive deductive databases. We also provide the proof for the algorithm.

Computing Only Minimal Answers in Disjunctive Deductive Databases

A method is presented for computing minimal answers of the form \(\bigvee {\cal A}\) in disjunctive deductive databases under the disjunctive stable model semantics. Such answers are constructed by repeatedly extending partial answers. Our method is complete (in that every minimal answer can be computed) and does not admit redundancy (in the sense that every partial answer generated can be extended to a minimal answer), thus no non-minimal answer is generated. The method does not (necessarily) require the computation of models of the database in their entirety. A partitioning of the database into extensional and intensional components is employed in order to overcome the problems caused by the possible non-existence of disjunctive stable models, and a form of compilation is presented as a means of simplifying and improving the efficiency of the run-time computation, which then reduces to relatively trivial processing within the extensional database. In addition, the output from this compilation process has the significant advantage of being immune to updates to the extensional database. Other forms of database pre-processing are also considered, and three transformations are presented mapping a database onto an equivalent positive database, non-disjunctive database, and set of conditional facts.

Optimization of bound disjunctive queries with constraints

This paper presents a technique for the optimization of bound queries over disjunctive deductive databases with constraints. The proposed approach is an extension of the well-known Magic-Set technique and is well-suited for being integrated in current bottom-up (stable) model inference engines. More specifically, it is based on the exploitation of binding propagation techniques which reduce the size of the data relevant to answer the query and, consequently, reduces both the complexity of computing a single model and the number of models to be considered. The motivation of this work stems from the observation that traditional binding propagation optimization techniques for bottom-up model generator systems, simulating the goal driven evaluation of top-down engines, are only suitable for positive (disjunctive) queries, while hard problems are expressed using unstratified negation. The main contribution of the paper consists in the extension of a previous technique, defined for positive disjunctive queries, to queries containing both disjunctive heads and constraints (a simple and expressive form of unstratified negation). As the usual way of expressing declaratively hard problems is based on the guess-and-check technique, where the guess part is expressed by means of disjunctive rules and the check part is expressed by means of constraints, the technique proposed here is highly relevant for the optimization of queries expressing hard problems. The value of the technique has been proved by several experiments.

Deductive databases for computing certain and consistent answers from mediated data integration systems

We address the problem of retrieving certain and consistent answers to queries posed to a mediated data integration system under the local-as-view paradigm with open sources and conjunctive and disjunctive view definitions. For obtaining certain answers a query program is run under the cautious stable model semantics on top of a normal deductive database with choice operator that specifies the class of minimal legal instances of the integration system. This methodology works for all monotone Datalog queries. To compute answers to queries that are consistent with respect to given global integrity constraints, the specification of minimal legal instances is combined with another disjunctive deductive database that specifies the repairs of those legal instances. This allows to retrieve the answers to any Datalog ¬ query that are consistent with respect to global universal and referential integrity constraints.

Binding propagation techniques for the optimization of bound disjunctive queries

This paper presents a technique for the optimization of bound queries on disjunctive deductive databases. The optimization is based on the rewriting of the source program into an equivalent program which can be evaluated more efficiently. The proposed optimization reduces the amount of data needed to answer the query and, consequently, 1) reduces the complexity of computing a single model and, more importantly, 2) greatly reduces the number of models to be considered. Although, in this paper, we consider the application of the magic-set method, other rewriting techniques defined for special classes of queries can also be applied. To show the relevance of our technique, we have implemented a prototype of an optimizer. Several experiments have confirmed the value of the technique.

On the complexity of data disjunctions

We study the complexity of data disjunctions in disjunctive deductive databases (DDDBs). A data disjunction is a disjunctive ground clause R( c ̄ 1)⋯R( c ̄ k) , k⩾2, which is derived from the database such that all atoms in the clause involve the same predicate R. We consider the complexity of deciding existence and uniqueness of a minimal data disjunction, as well as actually computing one, both for propositional (data) and nonground (program) complexity of the database. Our results extend and complement previous results on the complexity of disjunctive databases, and provide newly developed tools for the analysis of the complexity of function computation.

Duality for Goal-Driven Query Processing in Disjunctive Deductive Databases

Bottom-up query-answering procedures tend to explore a much larger search space than what is strictly needed. Top-down processing methods use the query to perform a more focused search that can result in more efficient query answering. Given a disjunctive deductive database, i>DB, and a query, i>Q, we establish a strong connection between model generation and clause derivability in two different representations of i>DB and i>Q. This allows us to use a bottom-up procedure for evaluating i>Q against i>DB in a top-down fashion. The approach requires no extensive rewriting of the input theory and introduces no new predicates. Rather, it is based on a certain i>duality principle for interpreting logical connectives. The duality transformation is achieved by reversing the direction of implication arrows in the clauses representing both the theory and the negation of the query. The application of a generic bottom-up procedure to the transformed clause set results in top-down query answering. Under favorable conditions efficiency gains are substantial, as shown by our preliminary testing. We give the logical meaning of the duality transformation and point to the conditions and sources of improved efficiency. We show how the duality approach can be used for i>refined query answering by specifying the i>minimal conditions (weakest updates) to i>DB under which i>Q becomes derivable. This is shown to be useful for view updates in disjunctive deductive databases as well as for other interesting applications.

Closed world assumption for disjunctive reasoning

In this paper, the relationship between argumentation and closed world reasoning for disjunctive information is studied. In particular, the authors propose a simple and intuitive generalization of the closed world assumption (CWA) for general disjunctive deductive databases (with default negation). This semantics called DCWA, allows a natural argumentation-based interpretation and can beused to represent reasoning for disjunctive information. We compare DCWA with GCWA and prove that DCWA extends Minker’s GCWA to the class of disjunctive databases with default negation. Also we compare our semantics with some related approaches. In addition, the computational complexity of DCWA is investigated.

On the equivalence of the static and disjunctive well-founded semantics and its computation

In recent years, much work was devoted to the study of theoretical foundations of Disjunctive Logic Programming and Disjunctive Deductive Databases. While the semantics of non-disjunctive programs is fairly well understood, the declarative and computational foundations of disjunctive logic programming proved to be much more elusive and difficult. Recently, two new and promising semantics have been proposed for the class of disjunctive logic programs. The first one is the static semantics STATIC, proposed by Przymusinski, and, the other is the disjunctive well-founded semantics D-WFS, proposed by Brass and Dix. Although the two semantics are based on very different ideas, both of them have been shown to share a number of natural and intuitive properties. In particular, both of them extend the well-founded semantics of normal logic programs. Nevertheless, since the static semantics employs a much richer underlying language than the D-WFS semantics, in general, the two semantics are different. The main result of this paper shows, however, that, when restricted to a common language, the two semantics in fact coincide. This important result provides additional and powerful argument in favor of the two semantics. It also allows us to use a recently developed minimal model theorem prover for an efficient implementation of the two semantics.

An Extension to GCWA and Query Evaluation for Disjunctive Deductive Databases

We present a simple and intuitive extension GCWAG of the generalized closed world assumption (GCWA) from positive disjunctive deductive databases to general disjunctive deductive databases (with default negation). This semantics is defined in terms of unfounded sets and possesses an argumentation-theoretic characterization. We also provide a top-down procedure for GCWAG, which is sound and complete with respect to GCWAG. We investigate two query evaluation methods for GCWAG: database partition, and database splitting. The basic idea of these methods is to divide the original deductive database into several smaller sub-databases and the query evaluation in the original database is transformed into the problem of query evaluation in smaller or simplified components. We prove that these two methods of query evaluation are all sound with respect to GCWAG.

Using clausal deductive databases for defining semantics in disjunctive deductive databases

This paper investigates the novel concept of i>clausal deductive databases (cd-databases), which are special normal deductive databases – i.e., deductive databases which may contain default negation in rule bodies – over a typed meta-language i>Lcd with a fixed set of predicate symbols, namely i>dis, i>con, and some i>built-in predicate symbols. The arguments of the literals in i>Lcd are given by disjunctive and conjunctive clauses of a basic first-order language l. i>Disjunctive deductive databases (dd-databases) generalize normal deductive databases by allowing for disjunctions of atoms or literals in rule heads. We present a i>transformation which maps a dd-database i>D into a cd-database i>Dcd that talks about the clauses of i>D. cd-databases provide a i>flexible framework for declaratively specifying the semantics of dd-databases: we can fix a standard i>control strategy, e.g., stable model or well-founded semantics, and vary the i>logical description i>Dcd for specifying different semantics. The i>transformed database i>Dcd usually consists of a part i>D⊗ which naturally expresses the rules of i>D, and two generic parts which are independent of i>D: i>Dlogic specifies i>logical inference rules like resolution and subsumption, and i>Dcwa specifies non-monotonic inference rules like i>closed-world-assumptions. Another i>program transformation, which uses the idea of bringing sets of clauses to the argument level is given for i>hypothetical logic programs: a hypothetical logic program i>D is mapped to a classical normal logic program i>Do, such that well-known semantics and inference methods for normal logic programs can be used for hypothetical reasoning as well.

Minimal model generation for refined answering of generalized queries in disjunctive deductive databases

Generalized queries are defined as sets of clauses in implication form. They cover several tasks of practical importance for database maintenance such as answering positive queries, computing database completions and integrity constraints checking. We address the issue of answering generalized queries under the minimal model semantics for the class of disjunctive deductive databases (DDDBs). The advanced approach is based on having the query induce an order on the models returned by a sound and complete minimal model generating procedure. We consider answers that are true in all and those that are true in some minimal models of the theory. We address the issue of answering positive queries through the construction of the minimal model state of the DDDB, using a minimal model generating procedure. The refinements allowed by the procedure include isolating a minimal component of a disjunctive answer, the specification of possible updates to the theory to enable the derivability of certain queries and deciding the monotonicity properties of answers to different classes of queries.

Argumentation-based abduction in disjunctive logic programming

In this paper, we propose an argumentation-based semantic framework, called DAS, for disjunctive logic programming. The basic idea is to translate a disjunctive logic program into an argumentation-theoretic framework. One unique feature of our proposed framework is to consider the disjunctions of negative literals as possible assumptions so as to represent incomplete information. In our framework, three semantics preferred disjunctive hypothesis (PDH), complete disjunctive hypothesis (CDH) and well-founded disjunctive hypothesis (WFDH) are defined by three kinds of acceptable hypotheses to represent credulous, moderate and skeptical reasoning in artificial intelligence (AI), respectively. Furthermore, our semantic framework can be extended to a wider class than that of disjunctive programs (called bi-disjunctive logic programs). In addition to being a first serious attempt in establishing an argumentation-theoretic framework for disjunctive logic programming, DAS integrates and naturally extends many key semantics, such as the minimal models, extended generalized closed world assumption (EGCWA), the well-founded model, and the disjunctive stable models. In particular, novel and interesting argumentation-theoretic characterizations of the EGCWA and the disjunctive stable semantics are shown. Thus the framework presented in this paper does not only provide a new way of performing argumentation (abduction) in disjunctive deductive databases, but also is a simple, intuitive and unifying semantic framework for disjunctive logic programming.

An architecture for a parallel disjunctive deductive database system

Parallel database systems are designed to handle data-intensive and logic-intensive queries. Disjunctive deductive databases (DDDB) — logic databases which allow for indefinite, or partial information - must handle such queries at the extreme. Therefore, a parallel architecture would be ideal for implementing a DDDB. This paper presents a brief description of a non-parallel DDDB, a prototype of which has been implemented at the University of Maryland, along with its query processing algorithm. The main design issues in parallel relational database architectures that are applicable to disjunctive databases are then reviewed. Finally, we investigate the sources of parallelism in DDDB and present a potential architecture for a parallel DDDB system.

Expressive power and complexity of partial models for disjunctive deductive databases

This paper investigates the expressive power and complexity of partial model semantics for disjunctive deductive databases. In particular, partial stable, regular model, maximal stable (M-stable), and least undefined stable (L-stable) semantics for function-free disjunctive logic programs are considered, for which the expressiveness of queries based on possibility and certainty inference is determined. On the complexity side, we determine the data and expression complexity of query evaluation. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of disjunction and negation. It appears that the considered semantics capture complexity classes at the lower end of the polynomial hierarchy. In fact, each class ∑ i P , Π i P , 1 ⩽ i⩽ 3 is captured by some semantics using appropriate syntactical restrictions. Partial stable models have exactly the same expressive power and complexity as total stable models (∑ 2 P resp. Π 2 P ), while a higher degree of expressiveness is obtained by the semantics which minimize undefinedness (M-stable, regular, and L-stable semantics). In particular, L-stable semantics has the highest expressive power (∑ 3 P resp. Π 3 P ). An interesting result in this course is that, in contrast with total stable models, negation is for partial stable models more expressive than disjunction. For the data complexity of queries, we obtain completeness results for the classes ∑ i P , Π i P , i⩽3, and, for the expression complexity, completeness results for the analogous classes at the lower end of the weak exponential hierarchy. The results of this paper complement and extend previous results, and contribute to a more complete picture of the computational aspects of disjunctive logic programming and databases, which supports in choosing an appropriate setting that fits the needs in practice.

Credulous argumentation with the disjunctive stable semantics

The relationship between the disjunctive stable semantics and argumentation is rarely explored. In particular, the problem of how to perform argumentation with disjunctive logic programs by the disjunctive stable semantics is still open. This paper attempts to address this problem and a satisfied solution is provided, in which a disjunctive logic program is naturally transformed into an argument framework and the credulous argumentation is characterized as the maximal members of all acceptable hypotheses. In this semantic framework, some interesting results are obtained. In particular, it is shown that the formalism of credulous argumentation can be implemented through the disjunctive stable models. As a result, the work provides not only a new way of performing argumentation (abduction) in disjunctive deductive databases, but also a natural and complete extension for the disjunctive stable semantics.

Reasoning with minimal models: efficient algorithms and applications

Reasoning with minimal models is at the heart of many knowledge-representation systems. Yet it turns out that this task is formidable, even when very simple theories are considered. In this paper, we introduce the elimination algorithm, which performs, in linear time, minimal model finding and minimal model checking for a significant subclass of positive CNF theories which we call positive head-cycle-free (HCF) theories. We also prove that the task of minimal entailment is easier for positive HCF theories than it is for the class of all positive CNF theories. Finally, we show how variations of the elimination algorithm can be applied to allow queries posed on disjunctive deductive databases and disjunctive default theories to be answered in an efficient way.

On the partial semantics for disjunctive deductive databases

Partial stable models for deductive databases, i.e., normal function-free logic programs (also called datalog programs), have two equivalent definitions: one based on 3-valued logics and another based on the notion of unfounded set. The notion of partial stable model has been extended to disjunctive deductive databases using 3-valued logics. In this paper, a characterization of partial stable models for disjunctive datalog programs is given using a suitable extension of the notion of unfounded set. Two interesting sub-classes of partial stable models, M-stable (Maximal-stable) (also called regular models, preferred extension,and maximal stable classes) and L-stable (Least undefined-stable) models, are then extended from normal to disjunctive datalog programs. On the one hand, L-stable models are shown to be the natural relaxation of the notion of total stable model; on the other hand the less strict M-stable models, endowed with a nice modularity property, may be appealing from the programming and computational point of view. M-stable and L-stable models are also compared with the regular models for disjunctive datalog programs recently proposed in the literature.

Model theoretic approach to view updates in deductive databases

The view update problem for deductive databases has been defined as the problem of accomplishing the update of an intensional predicate by modifying appropriately the extensional database. A previous paper by Grant, Horty, Lobo, and Minker developed algorithms for the insertion and the deletion of an intensional predicate in certain important classes of stratified disjunctive deductive databases. This paper introduces a model theoretic approach which encompasses a wide class of Herbrand semantics, including the perfect model and stable model semantics, for disjunctive databases including negation. This generalizes the earlier results: now the intensional database may contain disjunctive and denial rules, and the database may be required to satisfy integrity constraints. As in the previous paper, the algorithms are proved to be correct and best according to the criterion of causing minimal change to the database, where the first priority is to minimize deletions.

Implementing deductive databases by mixed integer programming

Existing and past generations of Prolog compilers have left deduction to run-time and this may account for the poor run-time performance of existing Prolog systems. Our work tries to minimize run-time deduction by shifting the deductive process to compile-time. In addition, we offer an alternative inferencing procedure based on translating logic to mixed integer programming. This makes available for research and implementation in deductive databases, all the theorems, algorithms, and software packages developed by the operations research community over the past 50 years. The method keeps the same query language as for disjunctive deductive databases, only the inferencing procedure changes. The language is purely declarative, independent of the order of rules in the program, and independent of the order in which literals occur in clause bodies. The technique avoids Prolog's problem of infinite looping. It saves run-time by doing primary inferencing at compile-time. Furthermore, it is incremental in nature. The first half of this article translates disjunctive clauses, integrity constraints, and database facts into Boolean equations, and develops procedures to use mixed integer programming methods to compute equations, and develops procedures to use mixed integer programming methods to compute equations, and develops procedures to use mixed integer programming methods to compute equations, and develops procedures to use mixed integer programming methods to compute —least models of definite deductive databases, and —minimal models and the Generalized Closed World Assumption of disjunctive databases.