Partial Stable Models Research Articles

A Fixpoint Characterization of Three-Valued Disjunctive Hybrid MKNF Knowledge Bases

The logic of hybrid MKNF (minimal knowledge and negation as failure) is a powerful knowledge representation language that elegantly pairs ASP (answer set programming) with ontologies. Disjunctive rules are a desirable extension to normal rule-based reasoning and typically semantic frameworks designed for normal knowledge bases need substantial restructuring to support disjunctive rules. Alternatively, one may lift characterizations of normal rules to support disjunctive rules by inducing a collection of normal knowledge bases, each with the same body and a single atom in its head. In this work, we refer to a set of such normal knowledge bases as a head-cut of a disjunctive knowledge base. The question arises as to whether the semantics of disjunctive hybrid MKNF knowledge bases can be characterized using fixpoint constructions with head-cuts. Earlier, we have shown that head-cuts can be paired with fixpoint operators to capture the two-valued MKNF models of disjunctive hybrid MKNF knowledge bases. Three-valued semantics extends two-valued semantics with the ability to express partial information. In this work, we present a fixpoint construction that leverages head-cuts using an operator that iteratively captures three-valued models of hybrid MKNF knowledge bases with disjunctive rules. This characterization also captures partial stable models of disjunctive logic programs since a program can be expressed as a disjunctive hybrid MKNF knowledge base with an empty ontology. We elaborate on a relationship between this characterization and approximators in AFT (approximation fixpoint theory) for normal hybrid MKNF knowledge bases.

On the Semantics of Abstract Argumentation Frameworks: A Logic Programming Approach

AbstractRecently there has been an increasing interest in frameworks extending Dung’s abstract Argumentation Framework (AF). Popular extensions include bipolar AFs and AFs with recursive attacks and necessary supports. Although the relationships between AF semantics and Partial Stable Models (PSMs) of logic programs has been deeply investigated, this is not the case for more general frameworks extending AF.In this paper we explore the relationships between AF-based frameworks and PSMs. We show that every AF-based framework Δ can be translated into a logic program PΔ so that the extensions prescribed by different semantics of Δ coincide with subsets of the PSMs of PΔ. We provide a logic programming approach that characterizes, in an elegant and uniform way, the semantics of several AF-based frameworks. This result allows also to define the semantics for new AF-based frameworks, such as AFs with recursive attacks and recursive deductive supports.

On the Equivalence Between Abstract Dialectical Frameworks and Logic Programs

AbstractAbstract Dialectical Frameworks (ADFs) are argumentation frameworks where each node is associated with an acceptance condition. This allows us to model different types of dependencies as supports and attacks. Previous studies provided a translation from Normal Logic Programs (NLPs) toADFs and proved the stable models semantics for a normal logic program has an equivalent semantics to that of the correspondingADF. However, these studies failed in identifying a semantics forADFs equivalent to a three-valued semantics (as partial stable models and well-founded models) forNLPs. In this work, we focus on a fragment ofADFs, called Attacking Dialectical Frameworks (ADF+s), and provide a translation fromNLPs toADF+s robust enough to guarantee the equivalence between partial stable models, well-founded models, regular models, stable models semantics forNLPs and respectively complete models, grounded models, preferred models, stable models forADFs. In addition, we define a new semantics forADF+s, calledL-stable, and show it is equivalent to theL-stable semantics forNLPs.

Three-valued semantics for hybrid MKNF knowledge bases revisited

Knorr et al. (2011) [9] 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. The main results state that if a hybrid MKNF knowledge base has a three-valued MKNF model, its well-founded MKNF model exists, which is unique and can be computed by an alternating fixpoint construction. However, we show that these claims are erroneous. The goal of this paper is two-fold. First, we provide insight into why these claims fail to hold. This leads to a classification of hybrid MKNF knowledge bases into a hierarchy of four classes, of which the smallest one is what works for the well-founded semantics proposed by Knorr et al. After that, we characterize three-valued MKNF models into what we call stable partitions, which are similar to partial stable models in logic programming. The intuitive notion of stable partitions allows us to discover a new proof-theoretic method to determine whether a three-valued MKNF model exists for a given hybrid MKNF knowledge base. We can guess partial partitions and verify their stability by computing alternating fixpoints and by performing a consistency check. This algorithm can be applied to compute all three-valued MKNF models, as well as all two-valued ones originally formulated by Motik and Rosati for query answering. As a result, our work provides a uniform characterization of well-founded, two-valued, and all three-valued MKNF models, in terms of stable partitions and the alternating fixpoint construction.

On finitely recursive programs

AbstractDisjunctivefinitary programsare a class of logic programs admitting function symbols and hence infinite domains. They have very good computational properties; for example, ground queries are decidable, while in the general case the stable model semantics are Π11-hard. In this paper we prove that a larger class of programs, calledfinitely recursive programs, preserve most of the good properties of finitary programs under the stable model semantics, which are as follows: (i) finitely recursive programs enjoy a compactness property; (ii) inconsistency checking and skeptical reasoning are semidecidable; (iii) skeptical resolution is complete for normal finitely recursive programs. Moreover, we show how to check inconsistency and answer skeptical queries using finite subsets of the ground program instantiation. We achieve this by extending the splitting sequence theorem by Lifschitz and Turner: we prove that if the input programPis finitely recursive, then the partial stable models determined by any smooth splitting ω-sequence converge to a stable model ofP.

Partial equilibrium logic

Partial equilibrium logic (PEL) is a new nonmonotonic reasoning formalism closely aligned with logic programming under well-founded and partial stable model semantics. In particular it provides a logical foundation for these semantics as well as an extension of the basic syntax of logic programs. In this paper we describe PEL, study some of its logical properties and examine its behaviour on disjunctive and nested logic programs. In addition we consider computational features of PEL and study different approaches to its computation.

Comparison of Semantics of Disjunctive Logic Programs Based on Model-Equivalent Reduction

In this paper, it is shown that stable model semantics, perfect model semantics, and partial stable model semantics of disjunctive logic programs have the same expressive power with respect to the polynomial-time model-equivalent reduction. That is, taking perfect model semantics and stable model semantic as an example, any logic program P can be transformed in polynomial time to another logic program P′ such that perfect models (resp. stable models) of P 1-1 correspond to stable models (resp. perfect models) of P′, and the correspondence can be computed also in polynomial time. However, the minimal model semantics has weaker expressiveness than other mentioned semantics, otherwise, the polynomial hierarchy would collapse to NP.

Unfolding partiality and disjunctions in stable model semantics

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.

Abduction in logic programming: A new definition and an abductive procedure based on rewriting

A long outstanding problem for abduction in logic programming has been on how minimality might be defined. Without minimality, an abductive procedure is often required to generate exponentially many subsumed explanations for a given observation. In this paper, we propose a new definition of abduction in logic programming where the set of minimal explanations can be viewed as a succinct representation of the set of all explanations. We then propose an abductive procedure where the problem of generating explanations is formalized as rewriting with confluent and terminating rewrite systems. We show that these rewrite systems are sound and complete under the partial stable model semantics, and sound and complete under the answer set semantics when the underlying program is so-called odd-loop free. We discuss an application of abduction in logic programming to a problem in reasoning about actions and provide some experimental results.

Complexity and Expressive Power of Deterministic Semantics for DATALOG¬

Deterministic models are (partial) stable models which are not contradicted by any other stable model; i.e., M is a deterministic model if there is no stable model N such that M ∪ N is not an interpretation. For instanced the well-founded model, which coincides with the intersection of all partial stable models, is a deterministic model. As the well-founded model is deterministic and unique for each program, well-founded model semantics has been proposed as the canonical deterministic semantics for partial stable models. But the well-founded model is not the unique deterministic model; indeed the family of deterministic (partial stable) models is not, in general, a singleton and admits a minimum (the well-founded model) and a maximum, the max-deterministic model. The latter model is, another candidate for a deterministic semantics. The aim of this paper is to study the complexity and the expressive power of deterministic semantics. In coherence with the deterministic nature of the model, the expressive power of max-deterministic semantics is shown to be able to express problems with unique solutions, whereas the well-founded model only captures a proper subset of the queries computable in polynomial time, the so-called fixpoint queries.

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.

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.

Possible is certain'' is desirable and can be expressive

The partial stable models of a logic program form a class of models that include the (unique) well-founded model, total stable models and other two interesting subclasses: maximal stable models and least undefined stable models. As stable models different from the well-founded are not unique, DATALOG queries do not in general correspond to functions. The question is: what are the expressive powers of the various types of stable models when they are restricted to the class of all functional queries? The paper shows that this power does not go in practice beyond the one of stratified queries, except for least undefined stable models which, instead, capture the whole Boolean hierarchy BH. Finally, it is illustrated how the latter result can be used to design a ``functional'' language which, by means of a disciplined usage of negation, allows to achieve the desired level of expressiveness up to BH so that exponential time resolution is eventually enabled only for hard problems.

Graph theoretical structures in logic programs and default theories

In this paper we present a graph representation of logic programs and default theories. We show that many of the semantics proposed for logic programs with negation can be expressed in terms of notions emerging from graph theory, establishing in this way a link between the fields. Namely the stable models, the partial stable models, and the well-founded semantics correspond respectively to the kernels, semikernels and the initial acyclic part of an associated graph. This link allows us to consider both theoretical (existence, uniqueness) and computational problems (tractability, algorithms, approximations) from a more abstract and rather combinatorial point of view. It also provides a clear and intuitive understanding about how conflicts between rules are resolved within the different semantics. Furthermore, we extend the basic framework developed for logic programs to the case of Default Logic by introducing the notions of partial, deterministic and well-founded extensions for default theories. These semantics capture different ways of reasoning with a default theory.

Graph theoretical structures in logic programs and default theories

In this paper we present a graph representation of logic programs and default theories. We show that many of the semantics proposed for logic programs with negation can be expressed in terms of notions emerging from graph theory, establishing in this way a link between the fields. Namely the stable models, the partial stable models, and the well-founded semantics correspond respectively to the kernels, semikernels and the initial acyclic part of an associated graph. This link allows us to consider both theoretical (existence, uniqueness) and computational problems (tractability, algorithms, approximations) from a more abstract and rather combinatorial point of view. It also provides a clear and intuitive understanding about how conflicts between rules are resolved within the different semantics. Furthermore, we extend the basic framework developed for logic programs to the case of Default Logic by introducing the notions of partial, deterministic and well-founded extensions for default theories. These semantics capture different ways of reasoning with a default theory.

On the correctness of unfold/fold transformation of normal and extended logic programs

We show that the framework for unfold/fold transformation of logic programs, first proposed by Tamaki and Sato and later extended by various researchers, preserves various nonmonotonic semantics of normal logic programs, especially preferred extension, partial stable models, regular model, and stable theory semantics. The primary aim of this research is to adopt a uniform approach for every semantics of normal programs, and that is elegantly achieved through the notion of semantic kernel. Later, we show that this framework can also be applied to extended logic programs, preserving the answer set semantics.

A nondeterministic well-founded semantics

In order to define the meaning of a collection of rules forming a database or logic program, we have to consider a number of alternative interpretations. In the case of stratified and locally stratified programs, researchers agree that a single such interpretation captures the intended semantics [16]. Nevertheless, nonstratified datalog databases and logic programs can be seen as descriptions of a set of “alternative scenarios”. Sacca and Zaniolo [20] have shown that the nondeterministic nature of these programs can be used to model several useful queries such as those involving thechoice operator. They introducedpartial stable models, a semantics that exhibits the desired nondeterministic behavior. This work opened the problem of finding efficient ways to compute these nondeterministic scenarios. Papadimitriou and Yannakakis [18] introduced a tie-breaking procedure that nondeterministically computes fixpoints for some programs and has polynomial data complexity. However, this algorithm does not handle many programs includingchoice programs. In this paper, we introduce the notion ofwell-founded hypothesis, an intuitive account based on hypothetical reasoning that captures the same semantics as Sacca and Zaniolo's partial stable models. We introduce a notion of linearity that can be used in a skeptical or nondeterministic fashion. We show that the skeptical case corresponds to thewell-founded semantics and that the nondeterministic case computes a sound subclass of well-founded hypotheses. We show that this latter subclass has polynomial data complexity, correctly handleschoice programs, and is universally defined. We develop a simple nondeterministic procedure that computes these linear hypotheses, and we extend it to compute a strict superclass of the well-founded tie-breaking fixpoints in polynomial time.

On the equivalence of semantics for normal logic programs

Despite the frequent comment that there is no general agreement on the semantics of logic programs, this paper shows that a number of independently proposed extensions to the stable model semantics coincide: the regular model semantics proposed by You and Yuan, the partial stable model semantics by Saccà and Zaniolo, the preferential semantics by Dung, and a stronger version of the stable class semantics by Baral and Subrahmanian. We show that these equivalent semantics can be characterized simply as selecting a particular kind of stable classes, called normal alternating fixpoints. In addition, we indicate that almost all of the previously proposed semantic frameworks coincide with that of normal alternating fixpoints. Due to its simplicity and naturalness, the framework of normal alternating fixpoints offers great potential in the study of the semantics for various nonmonotonic systems.

Well-founded and stationary models of logic programs

We investigate the class ofstationary or partial stable models of normal logic programs. This important class of models includes all (total)stable models, and, moreover, thewell-founded model is always its smallest member. Stationary models have several natural fixed-point definitions and can be equivalently obtained as expansions or extensions of suitable autoepistemic or default theories. By taking a particular subclass of this class of models one can obtain different semantics of logic programs, including the stable semantics and the well-founded semantics. Stationary models can be also naturally extended to the class of all disjunctive logic programs. These features of stationary models designate them as an important class of models with applications reaching far beyond the realm of logic programming.

Short note preferred extensions are partial stable models

We show that two recently presented proposals for the semantics of normal logic programs, namely partial stable models of Saccá and Zaniolo and preferred extensions of Phan Minh Dung coincide.