Disjunctive Logic Programs Research Articles

Backdoors to Normality for Disjunctive Logic Programs

The main reasoning problems for disjunctive logic programs are complete for the second level of the polynomial hierarchy and hence considered harder than the same problems for normal (i.e., disjunction-free) programs, which are on the first level. We propose a new exact method for solving the disjunctive problems which exploits the small distance of a disjunctive programs from being normal. The distance is measured in terms of the size of a smallest “backdoor to normality,” which is the smallest number of atoms whose deletion makes the program normal. Our method consists of three phases. In the first phase, a smallest backdoor is computed. We show that this can be done using an efficient algorithm for computing a smallest vertex cover of a graph. In the second phase, the backdoor is used to transform the logic program into a quantified Boolean formula (QBF) where the number of universally quantified variables equals the size of the backdoor and where the total size of the quantified Boolean formula is quasilinear in the size of the given logic program. The quasilinearity is achieved by means of a characterization of the least model of a Horn program in terms of level numberings. In a third phase, the universal variables are eliminated using universal expansion yielding a propositional formula. The blowup in the last phase is confined to a factor that is exponential in the size of the backdoor but linear in the size of the quantified Boolean formula. By checking the satisfiability of the resulting formula with a S at solver (or by checking the satisfiability of the quantified Boolean formula by a Q bf -S at solver), we can decide the A sp reasoning problems on the input program. In consequence, we have a transformation from A sp problems to propositional satisfiability where the combinatorial explosion, which is expected when transforming a problem from the second level of the polynomial hierarchy to the first level, is confined to a function of the distance to normality of the input program. In terms of parameterized complexity, the transformation is fixed-parameter tractable. We complement this result by showing that (under plausible complexity-theoretic assumptions) such a fixed-parameter tractable transformation is not possible if we consider the distance to tightness instead of distance to normality.

Characterization of logic program revision as an extension of propositional revision

AbstractWe address the problem of belief revision of logic programs (LPs), i.e., how to incorporate to a LP P a new LP Q. Based on the structure of SE interpretations, Delgrande et al. (2008. Proc. of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR'08), 411–421; 2013b. Proc. of the 12th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'13), 264–276) adapted the well-known AGM framework (Alchourrón et al. 1985. Journal of Symbolic Logic 50, 2, 510–530) to LP revision. They identified the rational behavior of LP revision and introduced some specific operators. In this paper, a constructive characterization of all rational LP revision operators is given in terms of orderings over propositional interpretations with some further conditions specific to SE interpretations. It provides an intuitive, complete procedure for the construction of all rational LP revision operators and makes easier the comprehension of their semantic and computational properties. We give a particular consideration to LPs of very general form, i.e., the generalized logic programs (GLPs). We show that every rational GLP revision operator is derived from a propositional revision operator satisfying the original AGM postulates. Interestingly, the further conditions specific to GLP revision are independent from the propositional revision operator on which a GLP revision operator is based. Taking advantage of our characterization result, we embed the GLP revision operators into structures of Boolean lattices, that allow us to bring to light some potential weaknesses in the adapted AGM postulates. To illustrate our claim, we introduce and characterize axiomatically two specific classes of (rational) GLP revision operators which arguably have a drastic behavior. We additionally consider two more restricted forms of LPs, i.e., the disjunctive logic programs (DLPs) and the normal logic programs (NLPs) and adapt our characterization result to disjunctive logic program and normal logic program revision operators.

A Syntax-Independent Approach to Forgetting in Disjunctive Logic Programs

In this paper, we present an approach to forgetting in disjunctive logic programs, where forgetting an atom from a program amounts to a reduction in the signature of that program. Notably, the approach is syntax-independent, so that if two programs are strongly equivalent, then the result of forgetting a given atom in each program is also strongly equivalent. Our central definition of forgetting is abstract: forgetting an atom from program P is characterised by the set of those SE consequences of P that do not mention the atom to be forgotten. We provide an equivalent, syntactic, characterization in which forgetting an atom p is given by those rules in the program that do not mention p, together with rules obtained by a single inference step from those rules that do mention p. Forgetting is shown to have appropriate properties; in particular, answer sets are preserved in forgetting an atom. As well, forgetting an atom via the syntactic characterization results in a modest (at worst quadratic) blowup in the program size. Finally, we provide a prototype implementation of this approach to forgetting.

On Elementary Loops and Proper Loops for Disjunctive Logic Programs

This paper proposes an alternative definition of elementary loops and extends the notion of proper loops for disjunctive logic programs. Different from normal logic programs, the computational complexities of recognizing elementary loops and proper loops for disjunctive programs are coNP-complete. To address this problem, we introduce weaker versions of both elementary loops and proper loops and provide polynomial time algorithms for identifying them respectively. On the other hand, based on the notion of elementary loops, the class of Head-Elementary-loop-Free (HEF) programs was presented, which can be turned into equivalent normal logic programs by shifting head atoms into bodies. However, the problem of recognizing an HEF program is coNP-complete. Then we present a subclass of HEF programs which generalizes the class of Head-Cycle-Free programs and provide a polynomial time algorithm to identify them. At last, some experiments show that both elementary loops and proper loops could be replaced by their weak versions in practice.

Possibilistic nested logic programs and strong equivalence

In this paper, the class of possibilistic nested logic programs is introduced. These possibilistic logic programs allow us to use nested expressions in the bodies and heads of their rules. By considering a possibilistic nested logic program as a possibilistic theory, a construction of a possibilistic logic programing semantics based on answer sets for nested logic programs and the proof theory of possibilistic logic is defined. In order to define a general method for computing the possibilistic answer sets of a possibilistic nested program, the idea of equivalence between possibilistic nested programs is explored. By considering properties of equivalence between possibilistic programs, a process of transforming a possibilistic nested logic program into a possibilistic disjunctive logic program is defined. Given that our approach is an extension of answer set programming, we also explore the concept of strong equivalence between possibilistic nested logic programs. To this end, we introduce the concept of poss SE-models. Therefore, we show that two possibilistic nested logic programs are strong equivalents whenever they have the same poss SE-models.The expressiveness of the possibilistic nested logic programs is illustrated by a scenario from the medical domain. In particular, we exemplify how possibilistic nested logic programs are expressive enough for capturing medical guidelines which are pervaded by vagueness and qualitative information.

Defeasibility in answer set programs with defaults and argumentation rules

Defeasible reasoning has been studied extensively in the last two decades and many different and dissimilar approaches are currently on the table. This multitude of ideas has made the field hard to navigate and the different techniques hard to compare. Our earlier work on Logic Programming with Defaults and Argumentation Theories (LPDA) introduced a degree of unification into the approaches that rely on the well-founded semantics. The present work takes this idea further and introduces ASPDA (Answer Set Programs via Argumentation Rules) - a unifying framework for defeasibility of disjunctive logic programs under the Answer Set Programming (ASP). Since the well-founded and the answer set semantics underlie almost all existing approaches to defeasible reasoning in Logic Programming, LPDA and ASPDA together can be seen as an attempt to unify most of those approaches. In addition to ASPDA, we obtained a number of interesting and non-trivial results. First, we show that ASPDA is reducible to ordinary ASP programs. Second, we study reducibility of ASPDA to the non-disjunctive case and show that head-cycle-free ASPDA programs reduce to the non-disjunctive case - similarly to head-cycle-free ASP programs, but through a more complex transformation. We also shed light on the relationship between ASPDA and some of the earlier theories such as Defeasible Logic and LPDA.

Backdoors to tractable answer set programming

Answer Set Programming (ASP) is an increasingly popular framework for declarative programming that admits the description of problems by means of rules and constraints that form a disjunctive logic program. In particular, many AI problems such as reasoning in a nonmonotonic setting can be directly formulated in ASP. Although the main problems of ASP are of high computational complexity, complete for the second level of the Polynomial Hierarchy, several restrictions of ASP have been identified in the literature, under which ASP problems become tractable.In this paper we use the concept of backdoors to identify new restrictions that make ASP problems tractable. Small backdoors are sets of atoms that represent “clever reasoning shortcuts” through the search space and represent a hidden structure in the problem input. The concept of backdoors is widely used in theoretical investigations in the areas of propositional satisfiability and constraint satisfaction. We show that it can be fruitfully adapted to ASP. We demonstrate how backdoors can serve as a unifying framework that accommodates several tractable restrictions of ASP known from the literature. Furthermore, we show how backdoors allow us to deploy recent algorithmic results from parameterized complexity theory to the domain of answer set programming.

A new rational algorithm for view updating in relational databases

The dynamics of belief and knowledge is one of the major components of any autonomous system that should be able to incorporate new pieces of information. In order to apply the rationality result of belief dynamics theory to various practical problems, it should be generalized in two respects: first it should allow a certain part of belief to be declared as immutable; and second, the belief state need not be deductively closed. Such a generalization of belief dynamics, referred to as base dynamics, is presented in this paper, along with the concept of a generalized revision algorithm for knowledge bases (Horn or Horn logic with stratified negation). We show that knowledge base dynamics has an interesting connection with kernel change via hitting set and abduction. In this paper, we show how techniques from disjunctive logic programming can be used for efficient (deductive) database updates. The key idea is to transform the given database together with the update request into a disjunctive (datalog) logic program and apply disjunctive techniques (such as minimal model reasoning) to solve the original update problem. The approach extends and integrates standard techniques for efficient query answering and integrity checking. The generation of a hitting set is carried out through a hyper tableaux calculus and magic set that is focused on the goal of minimality.

Disjunctive logic programs with existential quantification in rule heads

AbstractWe 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.

A game semantics for disjunctive logic programming

Denotational semantics of logic programming and its extensions (by allowing negation, disjunctions, or both) have been studied thoroughly for many years. In 1998, a game semantics was given to definite logic programs by Di Cosmo, Loddo, and Nicolet, and a few years later it was extended to deal with negation by Rondogiannis and Wadge. Both approaches were proven equivalent to the traditional semantics. In this paper we define a game semantics for disjunctive logic programs and prove soundness and completeness with respect to the minimal model semantics of Minker. The overall development has been influenced by the games studied for PCF and functional programming in general, in the styles of Abramsky–Jagadeesan–Malacaria and Hyland–Ong–Nickau.

Backdoors to Normality for Disjunctive Logic Programs

Over the last two decades, propositional satisfiability (SAT) has become one of the most successful and widely applied techniques for the solution of NP-complete problems. The aim of this paper is to investigate theoretically how SAT can be utilized for the efficient solution of problems that are harder than NP or co-NP. In particular, we consider the fundamental reasoning problems in propositional disjunctive answer set programming (ASP), BRAVE REASONING and SKEPTICA REASONING, which ask whether a given atom is contained in at least one or in all answer sets, respectively. Both problems are located at the second level of the Polynomial Hierarchy and thus assumed to be harder than NP or co-NP. One cannot transform these two reasoning problems into SAT in polynomial time, unless the Polynomial Hierarchy collapses. We show that certain structural aspects of disjunctive logic programs can be utilized to break through this complexity barrier, using new techniques from Parameterized Complexity. In particular, we exhibit transformations from BRAVE and SKEPTICAL REASONING to SAT that run in time $O(2^k n^2)$ where k is a structural parameter of the instance and n the input size. In other words, the reduction is fixed-parameter tractable for parameter k. As the parameter k we take the size of a smallest backdoor with respect to the class of normal (i.e., disjunction-free) programs. Such a backdoor is a set of atoms that when deleted makes the program normal. In consequence, the combinatorial explosion, which is expected when transforming a problem from the second level of the Polynomial Hierarchy to the first level, can now be confined to the parameter k, while the running time of the reduction is polynomial in the input size n, where the order of the polynomial is independent of k. We show that such a transformation is not possible if we consider backdoors with respect to tightness instead of normality. We think that our approach is applicable to many other hard combinatorial problems that lie beyond NP or co-NP, and thus significantly enlarge the applicability of SAT.

Computing Loops with at Most One External Support Rule

A consequence of a logic program under answer set semantics is one that is true for all answer sets. This article considers using loop formulas to compute some of these consequences in order to increase the efficiency of answer set solvers. Since computing loop formulas are in general intractable, we consider only loops with either no external support or at most one external support, as their loop formulas are either unit or binary clauses. We show that for disjunctive logic programs, loop formulas of loops with no external support can be computed in polynomial time, and that an iterative procedure using unit propagation on these formulas and the program completion computes the well-founded models in the case of normal logic programs and the least fixed point of a simplification operator used by DLV for disjunctive logic programs. For loops with at most one external support, their loop formulas can be computed in polynomial time for normal logic programs, but are NP-hard for disjunctive programs. So for normal logic programs, we have a procedure similar to the iterative one for loops without any external support, but for disjunctive logic programs, we present a polynomial approximation algorithm. All these algorithms have been implemented, and our experiments show that for certain logic programs, the consequences computed by our algorithms can significantly speed up current ASP solvers cmodels, clasp, and DLV.

Rough Description Logic Programs

The Semantic Web is an extension of the current World Wide Web, and aims to help computers to understand and process web information automatically. In recent years, the integration ontologies and rules has become a central topic in the Semantic Web. Therefore, many researchers have focused their study on investigating the combination of answer set programming with description logics for the semantic web. However, these can not deal with uncertainty and inexactness. To address this problem, we propose tightly coupled rough description logic programs (or simply rough dl-programs) under the answer set semantics, which can model uncertain, inexact information, and can deal with non monotonic reasoning at the same time. To our knowledge, this is the first such approach. First of all, we define the syntax and semantics of rough dl-program KB= ( L,P ), which is a tight integration of disjunctive logic program under the answer set semantics, rough set theory and rough description logic. Then, we present some reasoning problems of rough dl-program. Finally, we show some semantic properties of rough dl-program under the answer set semantics.

Ordered completion for first-order logic programs on finite structures

In this paper, we propose a translation from normal first-order logic programs under the stable model semantics to first-order sentences on finite structures. The translation is done through, what we call, ordered completion which is a modification of Clarkʼs completion with some auxiliary predicates added to keep track of the derivation order. We show that, on finite structures, classical models of the ordered completion of a normal logic program correspond exactly to the stable models of the program. We also extend this result to normal programs with constraints and choice rules. From a theoretical viewpoint, this work clarifies the relationships between normal logic programming under the stable model semantics and classical first-order logic. It follows that, on finite structures, every normal program can be defined by a first-order sentence if new predicates are allowed. This is a tight result as not every normal logic program can be defined by a first-order sentence if no extra predicates are allowed or when infinite structures are considered. Furthermore, we show that the result cannot be extended to disjunctive logic programs, assuming that NP ≠ coNP . From a practical viewpoint, this work leads to a new type of ASP solver by grounding on a programʼs ordered completion instead of the program itself. We report on a first implementation of such a solver based on several optimization techniques. Our experimental results show that our solver compares favorably to other major ASP solvers on the Hamiltonian Circuit program, especially on large domains.

Progression Semantics for Disjunctive Logic Programs

In this paper, we extend the progression semantics for first-order disjunctive logic programs and show that it coincides with the stable model semantics. Based on it, we further show how disjunctive answer set programming is related to Satisfiability Modulo Theories.

Semantics for Possibilistic Disjunctive Programs

AbstractIn this paper, a possibilistic disjunctive logic programming approach for modeling uncertain, incomplete, and inconsistent information is defined. This approach introduces the use of possibilistic disjunctive clauses, which are able to capture incomplete information and states of a knowledge base at the same time. By considering a possibilistic logic program as a possibilistic logic theory, a construction of a possibilistic logic programming semantic based on answer sets and the proof theory of possibilistic logic is defined. It shows that this possibilistic semantics for disjunctive logic programs can be characterized by a fixed-point operator. It is also shown that the suggested possibilistic semantics can be computed by a resolution algorithm and the consideration of optimal refutations from a possibilistic logic theory. In order to manage inconsistent possibilistic logic programs, a preference criterion between inconsistent possibilistic models is defined. In addition, the approach of cuts for restoring consistency of an inconsistent possibilistic knowledge base is adopted. The approach is illustrated in a medical scenario.

Nested Preferences in Answer Set Programming

In this paper, we define a class of nested logic programs, called Nested Logic Programs with Ordered Disjunction (LPODs +), which makes it possible to specify conditional (qualitative) preferences by means of nested preference statements. To this end, we augment the syntax of Logic Programs with Ordered Disjunction (LPODs) to capture more general expressions. We define the LPODs + semantics in a simple way and we extend most of the results of LPODs showing how our approach generalizes the LPODs framework in a proper way. We also show how the LPODs + semantics can be computed in terms of a translation procedure that maps a nested ordered disjunction program (OD +-program) into a disjunctive logic program.

Trichotomy and dichotomy results on the complexity of reasoning with disjunctive logic programs

AbstractWe present trichotomy results characterizing the complexity of reasoning with disjunctive logic programs. To this end, we introduce a certain definition schema for classes of programs based on a set of allowed arities of rules. We show that each such class of programs has a finite representation, and for each of the classes definable in the schema, we characterize the complexity of the existence of an answer set problem. Next, we derive similar characterizations of the complexity of skeptical and credulous reasoning with disjunctive logic programs. Such results are of potential interest. On the one hand, they reveal some reasons responsible for the hardness of computing answer sets. On the other hand, they identify classes of problem instances, for which the problem is “easy” (in P) or “easier than in general” (in NP). We obtain similar results for the complexity of reasoning with disjunctive programs under the supported-model semantics.

Outlier detection for simple default theories

It was noted recently that the framework of default logics can be exploited for detecting outliers. Outliers are observations expressed by sets of literals that feature unexpected properties. These observations are not explicitly provided in input (as it happens with abduction) but, rather, they are hidden in the given knowledge base. Unfortunately, in the two related formalisms for specifying defaults — Reiter's default logic and extended disjunctive logic programs — the most general outlier detection problems turn out to lie at the third level of the polynomial hierarchy. In this note, we analyze the complexity of outlier detection for two very simple classes of default theories, namely NU and DNU, for which the entailment problem is solvable in polynomial time. We show that, for these classes, checking for the existence of an outlier is anyway intractable. This result contributes to further showing the inherent intractability of outlier detection in default reasoning.

Disjunctive ASP with functions: Decidable queries and effective computation

AbstractQuerying over disjunctive ASP with functions is a highly undecidable task in general. In this paper we focus on disjunctive logic programs with stratified negation and functions under the stable model semantics (ASPfs). We show that query answering in this setting is decidable, if the query is finitely recursive (ASPfsfr). Our proof yields also an effective method for query evaluation. It is done by extending the magic set technique to ASPfsfr. We show that the magic-set rewritten program is query equivalent to the original one (under both brave and cautious reasoning). Moreover, we prove that the rewritten program is also finitely ground, implying that it is decidable. Importantly, finitely ground programs are evaluable using existing ASP solvers, making the class of ASPfsfr queries usable in practice.