Approximation fixpoint theory (AFT) is a robust and popular mathematical framework that characterizes many nonmonotonic semantics, where the construction of stable fixpoints, called stable revision, play a central role. Nondeterministic AFT is a recent development that redefines AFT for a nondeterministic setting to capture disjunctive semantics. This theory departs from traditional AFT by introducing distinct definitions, thus raising the question of whether deterministic AFT can be adopted directly to define nondeterministic stable revision. This work proposes such an alternate theory and creates a new way to study disjunctive semantics in terms of normal (non-disjunctive) knowledge bases. To demonstrate the viability of our framework, we show how to capture stable and partial stable models for disjunctive logic programs. We then study the relationships between this alternative theory and the state-of-the-art nondeterministic AFT.

Boolean Satisfiability (SAT) is an NP-complete problem, indicating its inherent computational hardness. However, Conflict Driven Clause Learning (CDCL) SAT solvers efficiently tackle large instances in diverse domains. Swift conflict identification is crucial for effective problem-solving, as conflicts lead to the learning of search space pruning clauses, pinpointing the root causes of conflicts and preventing their recurrence. CDCL decision heuristics prioritize variables that participated in recent conflicts, anticipating rapid conflict generation and expediting additional clause learning. In practice, only a fraction of decisions lead to conflicts, yet some decisions may yield multiple conflicts. In this paper, we delve into a detailed study of conflict generating decisions in CDCL, distinguishing between single conflict (sc) decisions, generating only one conflict, and multi-conflict (mc) decisions, producing two or more conflicts. Our empirical analysis characterizes each decision type based on the quality of the learned clauses they produce. Furthermore, our theoretical analysis reveals a crucial distinction: consecutive clauses learned within the same mc decision form a chain of clauses, absent in learned clauses from sc decisions. This leads to the hypothesis that the reasons for conflicts in mc decisions are more closely related than the reasons for conflicts in sc decisions, empirically confirmed with our introduced notion of reason proximity. Finally, we propose score reduction (sr) as a novel decision strategy, reducing the selection priority of certain variables from learned clauses in mc decisions. With four sets of benchmarks, culminating in over 1200 benchmarks, empirical evaluation of sr implemented on top of the SAT competition 2023 winner solver reveals the merit of this new strategy.

Abstract Hybrid MKNF Knowledge Bases (HMKNF-KBs) constitute a formalism for tightly integrated reasoning over closed-world rules and open-world ontologies. This approach allows for accurate modeling of real-world systems, which often rely on both categorical and normative reasoning. Conflict-driven solving is the leading approach for computationally hard problems, such as satisfiability (SAT) and answer set programming (ASP), in which MKNF is rooted. This paper investigates the theoretical underpinnings required for a conflict-driven solver of HMKNF-KBs. The approach defines a set of completion and loop formulas, whose satisfaction characterizes MKNF models. This forms the basis for a set of nogoods, which in turn can be used as the backbone for a conflict-driven solver.

Combining the closed-world reasoning of answer set programming (ASP) with the open-world reasoning of ontologies broadens the space of applications of reasoners. Disjunctive hybrid MKNF knowledge bases succinctly extend ASP and in some cases without increasing the complexity of reasoning tasks. However, in many cases, solver development is lagging behind. As the result, the only known method of solving disjunctive hybrid MKNF knowledge bases is based on guess-and-verify, as formulated by Motik and Rosati in their original work. A main obstacle is understanding how constraint propagation may be performed by a solver, which, in the context of ASP, centers around the computation of \textit{unfounded atoms}, the atoms that are false given a partial interpretation. In this work, we build towards improving solvers for hybrid MKNF knowledge bases with disjunctive rules: We formalize a notion of unfounded sets for these knowledge bases, identify lower complexity bounds, and demonstrate how we might integrate these developments into a DPLL-based solver. We discuss challenges introduced by ontologies that are not present in the development of solvers for disjunctive logic programs, which warrant some deviations from traditional definitions of unfounded sets. We compare our work with prior definitions of unfounded sets.

Abstract The chase procedure for existential rules is an indispensable tool for several database applications, where its termination guarantees the decidability of these tasks. Most previous studies have focused on the skolem chase variant and its termination analysis. It is known that the restricted chase variant is a more powerful tool in termination analysis provided a database is given. But all-instance termination presents a challenge since the critical database and similar techniques do not work. In this paper, we develop a novel technique to characterize the activeness of all possible cycles of a certain length for the restricted chase, which leads to the formulation of a framework of parameterized classes of the finite restricted chase, called $k$-$\mathsf{safe}(\Phi)$ rule sets. This approach applies to any class of finite skolem chase identified with a condition of acyclicity. More generally, we show that the approach can be applied to the hierarchy of bounded rule sets previously only defined for the skolem chase. Experiments on a collection of ontologies from the web show the applicability of the proposed methods on real-world ontologies.

The efficiency of Conflict Driven Clause Learning (CDCL) SAT solving depends crucially on finding conflicts at a fast rate. State-of-the-art CDCL branching heuristics such as VSIDS, CHB and LRB conform to this goal. We take a closer look at the way in which conflicts are generated over the course of a CDCL SAT search. Our study of the VSIDS branching heuristic shows that conflicts are typically generated in short bursts, followed by what we call a conflict depression phase in which the search fails to generate any conflicts in a span of decisions. The lack of conflict indicates that the variables that are currently ranked highest by the branching heuristic fail to generate conflicts. Based on this analysis, we propose an exploration strategy, called expSAT, which randomly samples variable selection sequences in order to learn an updated heuristic from the generated conflicts. The goal is to escape from conflict depressions expeditiously. The branching heuristic deployed in expSAT combines these updates with the standard VSIDS activity scores. An extensive empirical evaluation with four state-of-the-art CDCL SAT solvers demonstrates good-to-strong performance gains with the expSAT approach.

Abstract We investigate the problem of cost-optimal planning in ASP. Current ASP planners can be trivially extended to a cost-optimal one by adding weak constraints, but only for a given makespan (number of steps). It is desirable to have a planner that guarantees global optimality. In this paper, we present two approaches to addressing this problem. First, we show how to engineer a cost-optimal planner composed of two ASP programs running in parallel. Using lessons learned from this, we then develop an entirely new approach to cost-optimal planning,stepless planning, which is completely free of makespan. Experiments to compare the two approaches with the only known cost-optimal planner in SAT reveal good potentials for stepless planning in ASP.

Knorr et al. (2011) formulated a three-valued formalism for the logic of Minimal Knowledge and Negation as Failure (MKNF) and proposed a well-founded semantics for hybrid MKNF knowledge bases (KBs). The main results state that if a hybrid MKNF KB has a three-valued MKNF model, its well-founded MKNF model exists, which is unique and can be computed by an alternating fixpoint construction. In this paper, we show that these claims are erroneous. We propose a classification of hybrid MKNF KBs into a hierarchy and show that its innermost subclass is what works for the well-founded semantics of Knorr et al. Furthermore, we provide a uniform characterization of well-founded, two-valued, and all three-valued MKNF models, in terms of stable partitions and the alternating fixpoint construction, which leads to updated complexity results as well as proof-theoretic tools for reasoning under these semantics.

Abstract Hybrid MKNF knowledge bases have been considered one of the dominant approaches to combining open world ontology languages with closed world rule-based languages. Currently, the only known inference methods are based on the approach of guess-and-verify, while most modern SAT/ASP solvers are built under the DPLL architecture. The central impediment here is that it is not clear what constitutes a constraint propagator, a key component employed in any DPLL-based solver. In this paper, we address this problem by formulating the notion of unfounded sets for non-disjunctive hybrid MKNF knowledge bases, based on which we propose and study two new well-founded operators. We show that by employing a well-founded operator as a constraint propagator, a sound and complete DPLL search engine can be readily defined. We compare our approach with the operator based on the alternating fixpoint construction by Knorr et al. (2011. Artificial Intelligence 175, 9, 1528–1554) and show that, when applied to arbitrary partial partitions, the new well-founded operators not only propagate more truth values but also circumvent the non-converging behavior of the latter. In addition, we study the possibility of simplifying a given hybrid MKNF knowledge base by employing a well-founded operator and show that, out of the two operators proposed in this paper, the weaker one can be applied for this purpose and the stronger one cannot. These observations are useful in implementing a grounder for hybrid MKNF knowledge bases, which can be applied before the computation of MKNF models.

Finite chase, or alternatively chase termination, is an important condition to ensure the decidability of existential rule languages. In the past few years, a number of rule languages with finite chase have been studied. In this work, we propose a novel approach for classifying the rule languages with finite chase. Using this approach, a family of decidable rule languages, which extend the existing languages with the finite chase property, are naturally defined. We then study the complexity of these languages. Although all of them are tractable for data complexity, we show that their combined complexity can be arbitrarily high. Furthermore, we prove that all the rule languages with finite chase that extend the weakly acyclic language are of the same expressiveness as the weakly acyclic one, while rule languages with higher combined complexity are in general more succinct than those with lower combined complexity.

We propose to demonstrate Rubato DB, a highly scalable NewSQL system, supporting various consistency levels from ACID to BASE for OLTP and big data applications. Rubato DB employs the staged grid architecture with a novel formula based protocol for distributed concurrency control. Our demonstration will present Rubato DB as one NewSQL database management system running on a collection of commodity servers against two of benchmark sets.

Big data applications demand and consequently lead to developments of large-scale data management systems, which provide high scalability by partitioning data across multiple servers. Since conventional transactional access is quite expensive, many real world large-scale distributed systems eschew transactional functionality and adopt semantics of atomic multi-partition operations. Accordingly, BASE, a consistency model weaker than ACID, is commonly used to guarantee availability. In this work, we identify a new consistency model-BASIC (Basic Availability, Scalability, Instant Consistency) that matches the requirements where extra efforts are not needed to manipulate inconsistent soft states. We present a timestamp-based formula protocol for BASIC that can enforce Instant Consistency while achieving linear scalability (via logical formula caching, dynamic timestamp ordering) and achieve Basic Availability in the presence of partial failure and network partition (via partition independence, genuine atomic commit). Our extensive experimental results verify the scalability of BASIC and demonstrate that the limited overhead induced by BASIC pays a reasonable price for keeping all soft states consistent.

This paper proposes a new formula protocol for distributed concurrency control, and specifies a staged grid architecture for highly scalable database management systems. The paper also describes novel implementation techniques of Rubato DB based on the proposed protocol and architecture. We have conducted extensive experiments which clearly show that Rubato DB is highly scalable with efficient performance under both TPC-C and YCSB benchmarks. Our paper verifies that the formula protocol and the staged grid architecture provide a satisfactory solution to one of the important challenges in the database systems: to develop a highly scalable database management system that supports various consistency levels from ACID to BASE.

Abstract Weight constraint and aggregate programs are among the most widely used logic programs with constraints. In this paper, we relate the semantics of these two classes of programs, namely, the stable model semantics for weight constraint programs and the answer set semantics based on conditional satisfaction for aggregate programs. Both classes of programs are instances of logic programs with constraints, and in particular, the answer set semantics for aggregate programs can be applied to weight constraint programs. We show that the two semantics are closely related. First, we show that for a broad class of weight constraint programs, called strongly satisfiable programs, the two semantics coincide. When they disagree, a stable model admitted by the stable model semantics may be circularly justified. We show that the gap between the two semantics can be closed by transforming a weight constraint program to a strongly satisfiable one so that no circular models may be generated under the current implementation of the stable model semantics. We further demonstrate the close relationship between the two semantics by formulating a transformation from weight constraint programs to logic programs with nested expressions, which preserves the answer set semantics. Our study on the semantics leads to an investigation of a methodological issue, namely, the possibility of compact representation of aggregate programs by weight constraint programs. We show that almost all standard aggregates can be encoded by weight constraints compactly. This makes it possible to compute the answer sets of aggregate programs using the answer set programming solvers for weight constraint programs. This approach is compared experimentally with the ones where aggregates are handled more explicitly, which show that the weight constraint encoding of aggregates enables a competitive approach to answer set computation for aggregate programs.

Abstract We consider disjunctive logic programs without function symbols but with existential quantification in rule heads, under the semantics of general stable models. There are at least two interesting prospects in these programs. The first is that a program can be made more succinct by using existential variables, and the second is on the potential in representing defeasible ontological knowledge by these logic programs. This paper studies some of the properties of these programs. First, we show a simple yet intuitive definition of stable models for these programs that does not resort to second-order logic. Second, the stable models of these programs can be characterized by an extension of progression for disjunctive programs, which provides a native characterization of justification for stable models. We then study the decidability issue. While the stable model existence problem for safe disjunctive programs is decidable, with existential quantification allowed in rule heads the problem becomes undecidable. We identify an interesting decidable fragment by exploring a new notion of stratification over existential quantification.

Logic programs with abstract constraint atoms proposed by Marek and Truszczynski are very general logic programs.They are general enough to captureaggregate logic programs as well asrecently proposed description logic programs.In this paper, we propose a well-founded semantics for basic logic programs with arbitrary abstract constraint atoms, which are sets of rules whose heads have exactly one atom. Weshow that similar to the well-founded semanticsof normal logic programs, it has many desirable properties such as that it can becomputed in polynomial time, and is always correct with respect to theanswer set semantics. This paves the way for using our well-founded semanticsto simplify these logic programs. We also show how our semantics can be applied toaggregate logic programs and description logic programs, and compare itto the well-founded semantics already proposed for these logic programs.

We present a tight integration of SAT with CP, called SAT(gc), which embeds global constraints into SAT. A prototype is implemented by integrating the state of the art SAT solver ZCHAFF and the generic constraint solver GECODE. Experiments are carried out for benchmarks from puzzle domains and planning domains to reveal insights in compact representation, solving effectiveness, and novel usability of the new framework.

We present a new approach to characterizing the semantics for the integration of rules and first-order logic in general, and description logics in particular, based on a circumscription characterization of answer set programming, introduced earlier by Lin and Zhou. We show that both Rosati's semantics based on NM-models and Lukasiewicz's answer set semantics can be characterized by circumscription, and the difference between the two can be seen as a matter of circumscription policies. This approach leads to a number of new insights. First, we rebut a criticism on Lukasiewicz's semantics for its inability to reason for negative consequences. Second, our approach leads to a spectrum of possible semantics based on different circumscription policies, and shows a clear picture of how they are related. Finally, we show that the idea of this paper can be applied to first-order general stable models.

Level mapping and loop formulas are two different means to justify and characterize answer sets for normal logic programs. Both of them specify conditions under which a supported model is an answer set. Though serving a similar purpose, in the past the two have been studied largely in isolation with each other. In this paper, we study level mapping and loop formulas for weight constraint and aggregate (logic) programs. We show that, for these classes of programs, loop formulas can be devised from level mapping characterizations. First, we formulate a level mapping characterization of stable models and show that it leads to a new formulation of loop formulas for arbitrary weight constraint programs, without using any new atoms. This extends a previous result on loop formulas for weight constraint programs, where weight constraints contain only positive literals. Second, since aggregate programs are closely related to weight constraint programs, we further use level mapping to characterize the underlying answer set semantics based on which we formulate loop formulas for aggregate programs. The main result is that for aggregate programs not involving the inequality comparison operator, the dependency graphs can be built in polynomial time. This compares to the previously known exponential time method.

Top-k ranking for an uncertain database is to rank tuples in it so that the best k of them can be determined. The problem has been formalized under the unified approach based on parameterized ranking functions (PRFs) and the possible world semantics. Given a PRF, one can always compute the ranking function values of all the tuples to determine the top-k tuples, which is a formidable task for large databases. In this paper, we present a general approach to pruning for the framework based on PRFs. We show a mathematical manipulation of possible worlds which reveals key insights in the part of computation that may be pruned and how to achieve it in a systematic fashion. This leads to concrete pruning methods for a wide range of ranking functions. We show experimentally the effectiveness of our approach.

Level mapping and loop formulas are two different means to justify and characterize answer sets for normal logic programs. Both of them specify conditions under which a supported model is an answer set. Though serving a similar purpose, in the past the two have been studied largely in isolation with each other. In this paper, we study level mapping and loop formulas for weight constraint and aggregate (logic) programs. We show that, for these classes of programs, loop formulas can be devised from level mapping characterizations. First, we formulate a level mapping characterization of stable models and show that it leads to a new formulation of loop formulas for arbitrary weight constraint programs, without using any new atoms. This extends a previous result on loop formulas for weight constraint programs, where weight constraints contain only positive literals. Second, since aggregate programs are closely related to weight constraint programs, we further use level mapping to characterize the underlying answer set semantics based on which we formulate loop formulas for aggregate programs. The main result is that for aggregate programs not involving the inequality comparison operator, the dependency graphs can be built in polynomial time. This compares to the previously known exponential time method.

Abstract Description Logic Programs (dl-programs) proposed by Eiter et al. constitute an elegant yet powerful formalism for the integration of answer set programming with description logics, for the Semantic Web. In this paper, we generalize the notions of completion and loop formulas of logic programs to description logic programs and show that the answer sets of a dl-program can be precisely captured by the models of its completion and loop formulas. Furthermore, we propose a new, alternative semantics for dl-programs, called the canonical answer set semantics, which is defined by the models of completion that satisfy what are called canonical loop formulas. A desirable property of canonical answer sets is that they are free of circular justifications. Some properties of canonical answer sets are also explored.

Abstract This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond–Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al. (2007), which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended to logic programs with constraint atoms.

Lookahead is a well-known constraint propagation technique for DPLL-based SAT and answer set solvers. Despite its space pruning power, it can also slow down the search, due to its high overhead. In this paper, this twofold effect is analyzed. On one side, we give characterizations of the problems for which the cause for the reduction of search efficiency shows clearly. On the other we show that problem instances that lie in the phase transition regions often significantly benefit from the use of lookahead. Our analysis leads to a proposal of adaptive lookahead, which performs lookahead according to the learned information during the search. Adaptive lookahead is implemented in one of the best-known answer set solvers, smodels. Our experiments show that adaptive lookahead adapts well to different search environments it is going through.

In this paper, a new representation is presented for the Maximum Quartet Consistency (MQC) problem, where solving the MQC problem becomes searching for an ultrametric matrix that satisfies a maximum number of given quartet topologies. A number of structural properties of the MQC problem in this new representation are characterized through formulating into answer set programming, a recent powerful logic programming tool for modeling and solving search problems. Using these properties, a number of optimization techniques are proposed to speed up the search process. The experimental results on a number of simulated data sets suggest that the new representation, combined with answer set programming, presents a unique perspective to the MQC problem.

In rule-based systems, goal-oriented computations correspond naturally to the possible ways that an observation may be explained. In some applications, we need to compute explanations for a series of observations with the same domain. The question arises as to whether previously computed answers can be recycled. A “yes” answer could result in substantial savings of repeated computations. For systems based on classical logic, the answer is yes. For nonmonotonic systems, however, one tends to believe that the answer should be no , since recycling is a form of adding information. In this article, we show that computed answers can always be recycled, in a nontrivial way, for the class of rewrite procedures proposed earlier by the authors for logic programs with negation. We present some experimental results on an encoding of the logistics domain.

This article studies an implementation methodology for partial and disjunctive stable models where partiality and disjunctions are unfolded from a logic program so that an implementation of stable models for normal (disjunction-free) programs can be used as the core inference engine. The unfolding is done in two separate steps. First, it is shown that partial stable models can be captured by total stable models using a simple linear and modular program transformation. Hence, reasoning tasks concerning partial stable models can be solved using an implementation of total stable models. Disjunctive partial stable models have been lacking implementations which now become available as the translation handles also the disjunctive case. Second, it is shown how total stable models of disjunctive programs can be determined by computing stable models for normal programs. Thus an implementation of stable models of normal programs can be used as a core engine for implementing disjunctive programs. The feasibility of the approach is demonstrated by constructing a system for computing stable models of disjunctive programs using the SMODELS system as the core engine. The performance of the resulting system is compared to that of DLV, which is a state-of-the-art system for disjunctive programs.

Evolution is an important subarea of study in biological science, where given a set of species, the goal is to reconstruct their evolutionary history, or phylogeny. Many kinds of data associated with the species can be deployed for this task and many reconstruction methods have been proposed and examined in the literature. One very recent approach is to build a local phylogeny for every subset of 4 species, which is called a quartet for these 4 species, and then to assemble a phylogeny for the whole set of species satisfying these predicted quartets. In general, those predicted quartets might not always agree each other; and thus the objective function becomes to satisfy a maximum number of predicted quartets. This is the well-known maximum quartet consistency (MQC) problem, which is studied by a lot of researchers in the last two decades. We present a new equivalent representation for the MQC problem, that is, to search for an ultrametric matrix to satisfy the maximum number of those predicted quartets. We examine a few number of structural properties of the MQC problem in this new representation, through formulating it into answer set programming (ASP), a recent powerful logic programming tool for modeling and solving searching problems. The efficiency and usefulness of our approach are confirmed by our computational experiments on the artificial data as well as two real datasets.

Most existing formalizations treat belief change as a single‐step process, and ignore several problems that become important when a theory, or belief state, is revised over several steps. This paper identifies these problems, and argues for the need to retain all of the multiple possible outcomes of a belief change step, and for a framework in which the effects of a belief change step persist as long as is consistently possible. To demonstrate that such a formalization is indeed possible, we develop a framework, which uses the language of PJ‐default logic (Delgrande and Jackson 1991) to represent a belief state, and which enables the effects of a belief change step to persist by propagating belief constraints. Belief change in this framework maps one belief state to another, where each belief state is a collection of theories given by the set of extensions of the PJ‐default theory representing that belief state. Belief constraints do not need to be separately recorded; they are encoded as clearly identifiable components of a PJ‐default theory. The framework meets the requirements for iterated belief change that we identify and satisfies most of the AGM postulates (Alchourrón, Gärdenfors, and Makinson 1985) as well.

We present a new characterization of termination of general logic programs. Most existing termination analysis approaches rely on some static information about the structure of the source code of a logic program, such as modes/types, norms/level mappings, models/interargument relations, and the like. We propose a dynamic approach that employs some key dynamic features of an infinite (generalized) SLDNF-derivation, such as repetition of selected subgoals and recursive increase in term size. We also introduce a new formulation of SLDNF-trees, called generalized SLDNF-trees. Generalized SLDNF-trees deal with negative subgoals in the same way as Prolog and exist for any general logic programs.

This paper proposes a formalism for nonmonotonic reasoning based on prioritized argumentation. We argue that nonmonotonic reasoning in general can be viewed as selecting monotonic inferences by a simple notion of priority among inference rules. More importantly, these types of constrained inferences can be specified in a knowledge representation language where a theory consists of a collection of rules of first order formulas and a priority among these rules. We recast default reasoning as a form of prioritized argumentation and illustrate how the parameterized formulation of priority may be used to allow various extensions and modifications to default reasoning. We also show that it is possible, but more difficult, to express prioritized argumentation by default logic: Even some particular forms of prioritized argumentation cannot be represented modularly by defaults under the same language.

In this paper, we present a new approach to the problem of revising extended programs; we base this approach on the coherence theory initially advocated by Gardenfors for belief revision. Our approach resolves contradiction by removing only conflicting information, not the believed source of it, and therefore, keeps information loss minimal. Furthermore, since there is no need to search for problematic assumptions, as is done in the traditional assumption-removal approach, our approach provides a skeptical revision semantics that is tractable. We define the skeptical and credulous coherence semantics and show that both semantics can be characterized in terms of the fixpoint semantics of a revised program using a simple program-revision technique. These semantics provide a suitable framework for knowledge and belief revision in the context of logic programs. Semantical properties and advantages of the proposed revision semantics are also analyzed.

In parallel matrix manipulation operations, some data patterns need to be accessed in one memory cycle without conflict. Investigating the frequently used data patterns, we propose a powerful skewing scheme which allows most frequently used data patterns of N elements, including rows, columns, diagonals, blocks with various shapes, points scattered over various blocks, and chessboards with various shapes, to be accessed in one memory cycle. We also propose simple methods to combine different skewing schemes into a single parallel storage system such that all the frequently used data patterns of N elements (the above patterns plus folded lines, two-pairs, and column-pairs) can be accessed in one memory cycle. The storage sytem uses N memory modules where N is any (even or odd) power of two. Address generation in the system need only exclusive-or operations and can be completed in constant time. The storage scheme allows different sized matrices to be processed efficiently on large scale systems by using the skewing scheme designed according to the size of the system, i.e. address generation mechanism is independent of the size of the matrices to be processed. Data alignment requirements (connecting each memory module to a proper processor) can be easily realized on a general-purpose interconnecting network, such as a hypercube.

The relationship between Reiter's default logic and general logic programs, which may contain negative subgoals in rule bodies, has been discussed in the literature by translating logic programs to default logic theories. Here, we present a method to translate some default logic theories to general logic programs, and study the extensions of default logic theories with the stable model semantics of logic programming. Based on the translation method, we show that another semantics of logic programming, the well‐founded semantics, can be used to define a new version of default logic, which is more cautious than the original one. This enables the existing proof procedures for the well‐founded semantics to perform default inference. We also study the property of cumulative monotonicity for both default logic theories and general logic programs under the two different semantics. As a direct application of the translation method, logic programs can be used to make default inference for semantic networks with exceptions. La relation entre la logique par défaut de Reiter et les programmes logiques généraux, qui peuvent comporter des sous‐buts négatifs dans les ensembles de règies, a déjàété discutée en traduisant les programmes logiques en théories de la logique par défaut. Dans cet article, les auteurs proposent une méthode pour traduire certaines théories de la logique par defaut en programmes logiques généraux et étudient les extensions des theories de la logique par defaut avec la sémantique de modéle stable de la programmation logique. En se basant sur la méthode de traduction, les auteurs démontrent qu'une autre sémantique de la programmation logique, la sémantique bien fondée, peut ětre utilisée pour définir une nouvelle version de la logique par défaut, qui est plus prudente que la première. Cela permet aux procédures de preuve existantes de la sémantique bien fondée d' effectuer l' inférence par défaut. Les auteurs examinent également les caracteristiques de la monotonicité cumulative pour les theories de la logique par défaut et les programmes logiques généraux selon les deux différentes sémantiques. Une application directe de la méthode de traduction est la possibilityé d' utiliser les programmes logiques pour effectuer de l' inférence par défaut pour les réseaux sémantiques avec exceptions.

The central issue of this paper concerns the truth value undefined in Przymusinsi's 3-valued formalization of nonmonotonic reasoning and logic programming. We argue that this formalization can lead to the problem of unintended semantics and loss of disjunctive information. We modify the formalization by proposing two general principles for logic program semantics: justifiability and minimal undefinedness. The former is shown to be a general property for almost all logic program semantics, and the latter requires the use of the undefined only when it is necessary. We show that there are three types of information embedded in the undefined: the disjunctive, the factoring, and the “difficult-to-be-assigned”. In the modified formalization, the first two can be successfully identified and branched into multiple models. This leaves only the “difficult-to-be-assigned” as the undefined. It is shown that the truth value undefined is needed only for a very special type of programs whose practicality is yet to be evidenced.

The paradigm of equational programming potentially possesses all the features provided by Prolog-like languages. In addition, the ability to reason about equations, which is not provided by Prolog, can be accommodated by equational languages. In this paper, we propose an extended equational programming paradigm, and describe an equational logic programming language which is an extension of the equational language defined in [Hoff82]. Semantic foundations for the extension are discussed. The extended language is a powerful logic programming language in the sense of Prolog and thus enjoys the programming features that Prolog possesses. Furthermore, it provides an ability to solve equations, which captures the essential power of equational programming.

A novel lazy evaluation mechanism, pattern-driven lazy reduction, is developed that serves as a unifying evaluation mechanism for both functional and logic programs. The reduction of a function call can be viewed as “semantically” unifying the function call with the left hand side of a defining equation, and applying the unifier to the right hand side. Lazy reduction is achieved by the pattern which the function call matches against. Function reductions are actually “driven” by patterns in this sense. It is shown that this evaluation mechanism works well for both functional programs and logic programs that involve “executable” functions. As a result, logic programs can be enhanced with (1) the availability of a functional computing environment where there is no notion of backtracking, thus alleviating the degree of control difficulties typically encountered in logic programs, and (2) the ability to terminate “infinite computations” without the introduction of complex control issues at the user-level. On the other hand, functional programs can be equipped with the power of logic programming languages, e.g., Prolog.