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, located at 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 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.

HEX-programs extend logic programs under the answer set semantics with external computations through external atoms. As reasoning from ground Horn programs with nonmonotonic external atoms of polynomial complexity is already on the second level of the polynomial hierarchy, minimality checking of answer set candidates needs special attention. To this end, we present an approach based on unfounded sets as a generalization of related techniques for ASP programs. The unfounded set detection is expressed as a propositional SAT problem, for which we provide two different encodings and optimizations to them. We then integrate our approach into a previously developed evaluation framework for HEX-programs, which is enriched by additional learning techniques that aim at avoiding the reconstruction of the same or related unfounded sets. Furthermore, we provide a syntactic criterion that allows one to skip the minimality check in many cases. An experimental evaluation shows that the new approach significantly decreases runtime.

A general game player is a system that can play previously unknown games just by being given their rules. For this purpose, the Game Description Language (GDL) has been developed as a high-level knowledge representation formalism to communicate game rules to players. In this paper, we address a fundamental limitation of state-of-the-art methods and systems for General Game Playing, namely, their being confined to deterministic games with complete information about the game state. We develop a simple yet expressive extension of standard GDL that allows for formalising the rules of arbitrary finite, n-player games with randomness and incomplete state knowledge. In the second part of the paper, we address the intricate reasoning challenge for general game-playing systems that comes with the new description language. We develop a full embedding of extended GDL into the Situation Calculus augmented by Scherl and Levesque's knowledge fluent. We formally prove that this provides a sound and complete reasoning method for players' knowledge about game states as well as about the knowledge of the other players.

The recently introduced series of description logics under the common moniker DL-Lite has attracted attention of the description logic and semantic web communities due to the low computational complexity of inference, on the one hand, and the ability to represent conceptual modeling formalisms, on the other. The main aim of this article is to carry out a thorough and systematic investigation of inference in extensions of the original DL-Lite logics along five axes: by (i) adding the Boolean connectives and (ii) number restrictions to concept constructs, (iii) allowing role hierarchies, (iv) allowing role disjointness, symmetry, asymmetry, reflexivity, irreflexivity and transitivity constraints, and (v) adopting or dropping the unique same assumption. We analyze the combined complexity of satisfiability for the resulting logics, as well as the data complexity of instance checking and answering positive existential queries. Our approach is based on embedding DL-Lite logics in suitable fragments of the one-variable first-order logic, which provides useful insights into their properties and, in particular, computational behavior.

In this paper, we investigate knowledge reasoning within a simple framework called knowledge structure. We use variable forgetting as a basic operation for one agent to reason about its own or other agents\ knowledge. In our framework, two notions namely agents\ observable variables and the weakest sufficient condition play important roles in knowledge reasoning. Given a background knowledge base and a set of observable variables for each agent, we show that the notion of an agent knowing a formula can be defined as a weakest sufficient condition of the formula under background knowledge base. Moreover, we show how to capture the notion of common knowledge by using a generalized notion of weakest sufficient condition. Also, we show that public announcement operator can be conveniently dealt with via our notion of knowledge structure. Further, we explore the computational complexity of the problem whether an epistemic formula is realized in a knowledge structure. In the general case, this problem is PSPACE-hard; however, for some interesting subcases, it can be reduced to co-NP. Finally, we discuss possible applications of our framework in some interesting domains such as the automated analysis of the well-known muddy children puzzle and the verification of the revised Needham-Schroeder protocol. We believe that there are many scenarios where the natural presentation of the available information about knowledge is under the form of a knowledge structure. What makes it valuable compared with the corresponding multi-agent S5 Kripke structure is that it can be much more succinct.

The survey propagation (SP) algorithm has been shown to work well on large instances of the random 3-SAT problem near its phase transition. It was shown that SP estimates marginals over covers that represent clusters of solutions. The SP-y algorithm generalizes SP to work on the maximum satisfiability (Max-SAT) problem, but the cover interpretation of SP does not generalize to SP-y. In this paper, we formulate the relaxed survey propagation (RSP) algorithm, which extends the SP algorithm to apply to the weighted Max-SAT problem. We show that RSP has an interpretation of estimating marginals over covers violating a set of clauses with minimal weight. This naturally generalizes the cover interpretation of SP. Empirically, we show that RSP outperforms SP-y and other state-of-the-art Max-SAT solvers on random Max-SAT instances. RSP also outperforms state-of-the-art weighted Max-SAT solvers on random weighted Max-SAT instances.

In the last two decades, modal and description logics have been applied to numerous areas of computer science, including knowledge representation, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In particular, many approaches have been proposed for efficiently handling the satisfiability of the core normal modal logic K(m), and of its notational variant, the description logic ALC. Although simple in structure, K(m)/ALC is computationally very hard to reason on, its satisfiability being PSPACE-complete. In this paper we start exploring the idea of performing automated reasoning tasks in modal and description logics by encoding them into SAT, so that to be handled by state-of-the-art SAT tools; as with most previous approaches, we begin our investigation from the satisfiability in K(m). We propose an efficient encoding, and we test it on an extensive set of benchmarks, comparing the approach with the main state-of-the-art tools available. Although the encoding is necessarily worst-case exponential, from our experiments we notice that, in practice, this approach can handle most or all the problems which are at the reach of the other approaches, with performances which are comparable with, or even better than, those of the current state-of-the-art tools.

Prime implicates and prime implicants have proven relevant to a number of areas of artificial intelligence, most notably abductive reasoning and knowledge compilation. The purpose of this paper is to examine how these notions might be appropriately extended from propositional logic to the modal logic K. We begin the paper by considering a number of potential definitions of clauses and terms for K. The different definitions are evaluated with respect to a set of syntactic, semantic, and complexity-theoretic properties characteristic of the propositional definition. We then compare the definitions with respect to the properties of the notions of prime implicates and prime implicants that they induce. While there is no definition that perfectly generalizes the propositional notions, we show that there does exist one definition which satisfies many of the desirable properties of the propositional case. In the second half of the paper, we consider the computational properties of the selected definition. To this end, we provide sound and complete algorithms for generating and recognizing prime implicates, and we show the prime implicate recognition task to be PSPACE-complete. We also prove upper and lower bounds on the size and number of prime implicates. While the paper focuses on the logic K, all of our results hold equally well for multi-modal K and for concept expressions in the description logic ALC.

Inspired by the recently introduced framework of AND/OR search spaces for graphical models, we propose to augment Multi-Valued Decision Diagrams (MDD) with AND nodes, in order to capture function decomposition structure and to extend these compiled data structures to general weighted graphical models (e.g., probabilistic models). We present the AND/OR Multi-Valued Decision Diagram (AOMDD) which compiles a graphical model into a canonical form that supports polynomial (e.g., solution counting, belief updating) or constant time (e.g. equivalence of graphical models) queries. We provide two algorithms for compiling the AOMDD of a graphical model. The first is search-based, and works by applying reduction rules to the trace of the memory intensive AND/OR search algorithm. The second is inference-based and uses a Bucket Elimination schedule to combine the AOMDDs of the input functions via the the APPLY operator. For both algorithms, the compilation time and the size of the AOMDD are, in the worst case, exponential in the treewidth of the graphical model, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs). We introduce the concept of semantic treewidth, which helps explain why the size of a decision diagram is often much smaller than the worst case bound. We provide an experimental evaluation that demonstrates the potential of AOMDDs.

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.