An Approach to Forgetting in Disjunctive Logic Programs that Preserves Strong Equivalence

In this paper we investigate forgetting in disjunctive logic programs, where forgetting an atom from a program amounts to a reduction in the signature of that program. The goal is to provide an approach that is syntax-independent, in that if two programs are strongly equivalent, then the results of forgetting an atom in each program should also be strongly equivalent. Our central definition of forgetting is impractical but satisfies this goal: Forgetting an atom is characterised by the set of SE consequences of the program that do not mention the atom to be forgotten. We then provide an equivalent, practical definition, wherein forgetting an atom $p$ is given by those rules in the program that don't mention $p$, together with rules obtained by a single inference step from rules that do mention $p$. Forgetting is shown to have appropriate properties; as well, the finite characterisation results in a modest (at worst quadratic) blowup. Finally we have also obtained a prototype implementation of this approach to forgetting.Comment: In: Proceedings of 15th International Workshop on Non-Monotonic Reasonin

A finite-valued solver for disjunctive fuzzy answer set programs

Fuzzy Answer Set Programming (FASP) is a declarative programming paradigm which extends the flexibility and expressiveness of classical Answer Set Programming (ASP), with the aim of modeling continuous application domains. In contrast to the availability of efficient ASP solvers, there have been few attempts at implementing FASP solvers. In this paper, we propose an implementation of FASP based on a reduction to classical ASP. We also develop a prototype implementation of this method. To the best of our knowledge, this is the first solver for disjunctive FASP programs. Moreover, we experimentally show that our solver performs well in comparison to an existing solver (under reasonable assumptions) for the more restrictive class of normal FASP programs

Disjunctive Logic Programs with Inheritance

The paper proposes a new knowledge representation language, called DLP<, which extends disjunctive logic programming (with strong negation) by inheritance. The addition of inheritance enhances the knowledge modeling features of the language providing a natural representation of default reasoning with exceptions. A declarative model-theoretic semantics of DLP< is provided, which is shown to generalize the Answer Set Semantics of disjunctive logic programs. The knowledge modeling features of the language are illustrated by encoding classical nonmonotonic problems in DLP<. The complexity of DLP< is analyzed, proving that inheritance does not cause any computational overhead, as reasoning in DLP< has exactly the same complexity as reasoning in disjunctive logic programming. This is confirmed by the existence of an efficient translation from DLP< to plain disjunctive logic programming. Using this translation, an advanced KR system supporting the DLP< language has been implemented on top of the DLV system and has subsequently been integrated into DLV.Comment: 28 pages; will be published in Theory and Practice of Logic Programmin

Modularity Aspects of Disjunctive Stable Models

Practically all programming languages allow the programmer to split a program into several modules which brings along several advantages in software development. In this paper, we are interested in the area of answer-set programming where fully declarative and nonmonotonic languages are applied. In this context, obtaining a modular structure for programs is by no means straightforward since the output of an entire program cannot in general be composed from the output of its components. To better understand the effects of disjunctive information on modularity we restrict the scope of analysis to the case of disjunctive logic programs (DLPs) subject to stable-model semantics. We define the notion of a DLP-function, where a well-defined input/output interface is provided, and establish a novel module theorem which indicates the compositionality of stable-model semantics for DLP-functions. The module theorem extends the well-known splitting-set theorem and enables the decomposition of DLP-functions given their strongly connected components based on positive dependencies induced by rules. In this setting, it is also possible to split shared disjunctive rules among components using a generalized shifting technique. The concept of modular equivalence is introduced for the mutual comparison of DLP-functions using a generalization of a translation-based verification method

On finitely recursive programs

Disjunctive finitary programs are 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 is highly undecidable. In this paper we prove that a larger class of programs, called finitely recursive programs, preserves most of the good properties of finitary programs under the stable model semantics, namely: (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 program P is finitely recursive, then the partial stable models determined by any smooth splitting omega-sequence converge to a stable model of P.Comment: 26 pages, Preliminary version in Proc. of ICLP 2007, Best paper awar

NP Datalog: a Logic Language for Expressing NP Search and Optimization Problems

This paper presents a logic language for expressing NP search and optimization problems. Specifically, first a language obtained by extending (positive) Datalog with intuitive and efficient constructs (namely, stratified negation, constraints and exclusive disjunction) is introduced. Next, a further restricted language only using a restricted form of disjunction to define (non-deterministically) subsets (or partitions) of relations is investigated. This language, called NP Datalog, captures the power of Datalog with unstratified negation in expressing search and optimization problems. A system prototype implementing NP Datalog is presented. The system translates NP Datalog queries into OPL programs which are executed by the ILOG OPL Development Studio. Our proposal combines easy formulation of problems, expressed by means of a declarative logic language, with the efficiency of the ILOG System. Several experiments show the effectiveness of this approach.Comment: To appear in Theory and Practice of Logic Programming (TPLP

An Effective Fixpoint Semantics for Linear Logic Programs

In this paper we investigate the theoretical foundation of a new bottom-up semantics for linear logic programs, and more precisely for the fragment of LinLog that consists of the language LO enriched with the constant 1. We use constraints to symbolically and finitely represent possibly infinite collections of provable goals. We define a fixpoint semantics based on a new operator in the style of Tp working over constraints. An application of the fixpoint operator can be computed algorithmically. As sufficient conditions for termination, we show that the fixpoint computation is guaranteed to converge for propositional LO. To our knowledge, this is the first attempt to define an effective fixpoint semantics for linear logic programs. As an application of our framework, we also present a formal investigation of the relations between LO and Disjunctive Logic Programming. Using an approach based on abstract interpretation, we show that DLP fixpoint semantics can be viewed as an abstraction of our semantics for LO. We prove that the resulting abstraction is correct and complete for an interesting class of LO programs encoding Petri Nets.Comment: 39 pages, 5 figures. To appear in Theory and Practice of Logic Programmin

Semantical Characterizations and Complexity of Equivalences in Answer Set Programming

In recent research on non-monotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P union R and Q union R have the same answer sets for any other program R. This property strengthens equivalence of P and Q with respect to answer sets (which is the particular case for R is the empty set), and has its applications in program optimization, verification, and modular logic programming. In this paper, we consider more liberal notions of strong equivalence, in which the actual form of R may be syntactically restricted. On the one hand, we consider uniform equivalence, where R is a set of facts rather than a set of rules. This notion, which is well known in the area of deductive databases, is particularly useful for assessing whether programs P and Q are equivalent as components of a logic program which is modularly structured. On the other hand, we consider relativized notions of equivalence, where R ranges over rules over a fixed alphabet, and thus generalize our results to relativized notions of strong and uniform equivalence. For all these notions, we consider disjunctive logic programs in the propositional (ground) case, as well as some restricted classes, provide semantical characterizations and analyze the computational complexity. Our results, which naturally extend to answer set semantics for programs with strong negation, complement the results on strong equivalence of logic programs and pave the way for optimizations in answer set solvers as a tool for input-based problem solving.Comment: 58 pages, 6 tables. The contents were partially published in: Proceedings 19th International Conference on Logic Programming (ICLP 2003), pp. 224-238, LNCS 2916, Springer, 2003; and Proceedings 9th European Conference on Logics in Artificial Intelligence (JELIA 2004), pp. 161-173, LNCS 3229, Springer, 200

Trichotomy and Dichotomy Results on the Complexity of Reasoning with Disjunctive Logic Programs

We 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. To appear in Theory and Practice of Logic Programming (TPLP)Comment: 24 pages To appear in Theory and Practice of Logic Programming (TPLP

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 Skeptical 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.Comment: A short version will appear in the Proceedings of the Proceedings of the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary version of the paper was presented on the workshop Answer Set Programming and Other Computing Paradigms (ASPOCP 2012), 5th International Workshop, September 4, 2012, Budapest, Hungar