Disjunctive Logic Programming (DLP) is an advanced formalism for knowledge representation and reasoning, which is very expressive in a precise mathematical sense: it allows one to express every property of finite structures that is decidable in the complexity class ΣP2 (NPNP). Thus, under widely believed assumptions, DLP is strictly more expressive than normal (disjunction-free) logic programming, whose expressiveness is limited to properties decidable in NP. Importantly, apart from enlarging the class of applications which can be encoded in the language, disjunction often allows for representing problems of lower complexity in a simpler and more natural fashion.This article presents the DLV system, which is widely considered the state-of-the-art implementation of disjunctive logic programming, and addresses several aspects. As for problem solving, we provide a formal definition of its kernel language, function-free disjunctive logic programs (also known as disjunctive datalog), extended by weak constraints, which are a powerful tool to express optimization problems. We then illustrate the usage of DLV as a tool for knowledge representation and reasoning, describing a new declarative programming methodology which allows one to encode complex problems (up to ΔP3-complete problems) in a declarative fashion. On the foundational side, we provide a detailed analysis of the computational complexity of the language of DLV, and by deriving new complexity results we chart a complete picture of the complexity of this language and important fragments thereof.Furthermore, we illustrate the general architecture of the DLV system, which has been influenced by these results. As for applications, we overview application front-ends which have been developed on top of DLV to solve specific knowledge representation tasks, and we briefly describe the main international projects investigating the potential of the system for industrial exploitation. Finally, we report about thorough experimentation and benchmarking, which has been carried out to assess the efficiency of the system. The experimental results confirm the solidity of DLV and highlight its potential for emerging application areas like knowledge management and information integration.

The DLV system for knowledge representation and reasoning

Extending and implementing the stable model semantics

Argumentation in artificial intelligence

This article surveys various complexity and expressiveness results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming.

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Complexity and expressive power of logic programming

We consider disjunctive Datalog, a powerful database query language based on disjunctive logic programming. Briefly, disjunctive Datalog is a variant of Datalog where disjunctions may appear in the rule heads; advanced versions also allow for negation in the bodies which can be handled according to a semantics for negation in disjunctive logic programming. In particular, we investigate three different semantics for disjunctive Datalog: the minimal model semantics the perfect model semantics, and the stable model semantics. For each of these semantics, the expressive power and complexity are studied. We show that the possibility variants of these semantics express the same set of queries. In fact, they precisely capture the complexity class ΣP2. Thus, unless the Polynomial Hierarchy collapses, disjunctive Datalog is more expressive that normal logic programming with negation. These results are not only of theoretical interest; we demonstrate that problems relevant in practice such as computing the optimal tour value in the Traveling Salesman Problem and eigenvector computations can be handled in disjunctive Datalog, but not Datalog with negation (unless the Polynomial Hierarchy collapses). In addition, we study modularity properties of disjunctive Datalog and investigate syntactic restrictions of the formalisms.

Disjunctive datalog

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.

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Propositional semantics for disjunctive logic programs

This paper addresses the problem of computing the minimal models of a given CNF propositional theory. We present two groups of algorithms. Algorithms in the first group are efficient when the theory is almost Horn, that is, when there are few non-Horn clauses and/or when the set of all literals that appear positive in any non-Horn clause is small. Algorithms in the other group are efficient when the theory can be represented as an acyclic network of low-arity relations. Our algorithms suggest several characterizations of tractable subsets for the problem of finding minimal models.

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On computing minimal models

Default reasoning using classical logic

We present a mapping from a class of default theories to sentences in propositional logic, such that each model of the latter corresponds to an extension of the former. Using this mapping we show that many properties of default theories can be determined by solving propositional satisfiability. In particular, we show how CSP techniques can be used to identify, analyze and solve tractable subsets of Reiter's default logic.

Default logic, propositional logic and constraints

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Ω1, Ω2,..., with the following properties: first, Ω1 is the class of all stratified knowledge bases; second, if a knowledge base Π is in Ωk, then Π has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Π, third, for an arbitrary knowledge base Π, we can find the minimum k such that Π belongs to Ωk in time polynomial in the size of Π, and, last, where κ is the class of all knowledge bases, it is the case that ∪i=1∞ Ωi= κ, that is, every knowledge base belongs to some class in the hierarchy.

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Rachel Ben-Eliyahu

Papers

Propositional semantics for disjunctive logic programs

On computing minimal models

Default reasoning using classical logic

Default logic, propositional logic and constraints

A hierarchy of tractable subsets for computing stable models