Complexity of Non-Monotonic Logics

Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in inferences from a given knowledge base but in computing possible explanations for an observation with respect to a given knowledge base. Complexity results for different reasoning tasks for propositional variants of these logics have been studied already in the nineties. In recent years, however, a renewed interest in complexity issues can be observed. One current focal approach is to consider parameterized problems and identify reasonable parameters that allow for FPT algorithms. In another approach, the emphasis lies on identifying fragments, i.e., restriction of the logical language, that allow more efficient algorithms for the most important reasoning tasks. In this survey we focus on this second aspect. We describe complexity results for fragments of logical languages obtained by either restricting the allowed set of operators (e.g., forbidding negations one might consider only monotone formulae) or by considering only formulae in conjunctive normal form but with generalized clause types. The algorithmic problems we consider are suitable variants of satisfiability and implication in each of the logics, but also counting problems, where one is not only interested in the existence of certain objects (e.g., models of a formula) but asks for their number.Comment: To appear in Bulletin of the EATC

The Complexity of Reasoning for Fragments of Autoepistemic Logic

Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given knowledge base. To the best of our knowledge this is the first paper analyzing the counting problem for autoepistemic logic

10061 Abstracts Collection -- Circuits, Logic, and Games

From 07/02/10 to 12/02/10, the Dagstuhl Seminar 10061 ``Circuits, Logic, and Games \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Limits of Preprocessing

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a complexity theoretic assumption, none of the considered problems can be reduced by polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, such as induced width or backdoor size. Our results provide a firm theoretical boundary for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: This is a slightly longer version of a paper that appeared in the proceedings of AAAI 201

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

Well-Founded Semantics for Extended Logic Programs with Dynamic Preferences

The paper describes an extension of well-founded semantics for logic programs with two types of negation. In this extension information about preferences between rules can be expressed in the logical language and derived dynamically. This is achieved by using a reserved predicate symbol and a naming technique. Conflicts among rules are resolved whenever possible on the basis of derived preference information. The well-founded conclusions of prioritized logic programs can be computed in polynomial time. A legal reasoning example illustrates the usefulness of the approach.Comment: See http://www.jair.org/ for any accompanying file