Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?

Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula $\phi$ and an atom $x$, whether $x$ is true in some cardinality-minimal model of $\phi$. This problem is easy for the Horn fragment, but, as we will show in this paper, remains $\Theta_2$-complete (and thus $\mathrm{NP}$-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems

Belief merging within fragments of propositional logic

Recently, belief change within the framework of fragments of propositional logic has gained increasing attention. Previous works focused on belief contraction and belief revision on the Horn fragment. However, the problem of belief merging within fragments of propositional logic has been neglected so far. This paper presents a general approach to define new merging operators derived from existing ones such that the result of merging remains in the fragment under consideration. Our approach is not limited to the case of Horn fragment but applicable to any fragment of propositional logic characterized by a closure property on the sets of models of its formulae. We study the logical properties of the proposed operators in terms of satisfaction of merging postulates, considering in particular distance-based merging operators for Horn and Krom fragments.Comment: To appear in the Proceedings of the 15th International Workshop on Non-Monotonic Reasoning (NMR 2014

Complexity of Generalized Satisfiability Counting Problems

AbstractThe class of generalized satisfiability problems, introduced in 1978 by Schaefer, presents a uniform way of studying the complexity of satisfiability problems with special conditions. The complexity of each decision and counting problem in this class depends on the involved logical relations. In 1979, Valiant defined the class #P, the class of functions definable as the number of accepting computations of a polynomial-time nondeterministic Turing machine. Clearly, all satisfiability counting problems belong to this class #P. We prove a Dichotomy Theorem for generalized satisfiability counting problems. That is, if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time, otherwise this function is #P-complete. This gives us a comparison between decision and counting generalized satisfiability problems. We can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already #P-complete. Moreover, taking advantage of a similar dichotomy result proved in 1978 by Schaefer for generalized satisfiability decision problems, we get as a corollary the implication that the counting counterpart of each NP-complete generalized satisfiability decision problem is #P-complete

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

Expected number of locally maximal solutions for random Boolean CSPs

International audienceFor a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs

06401 Abstracts Collection -- Complexity of Constraints

From 01.10.06 to 06.10.06, the Dagstuhl Seminar 06401 ``Complexity of Constraints\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. 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

Complexity Classifications for logic-based Argumentation

We consider logic-based argumentation in which an argument is a pair (Fi,al), where the support Fi is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by De) that entails the claim al (a formula). We study the complexity of three central problems in argumentation: the existence of a support Fi ss De, the validity of a support and the relevance problem (given psi is there a support Fi such that psi ss Fi?). When arguments are given in the full language of propositional logic these problems are computationally costly tasks, the validity problem is DP-complete, the others are SigP2-complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints build upon a fixed finite set of Boolean relations Ga (the constraint language). We show that according to the properties of this language Ga, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete or SigP2-complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or SigP2-complete. These last two classifications are obtained by means of algebraic tools

On P completeness of some counting problems

We prove that the counting problems #1-in-3Sat, #Not-All-Equal 3Sat and #3-Colorability, whose decision counterparts have been the most frequently used in proving NP-hardness of new decision problems, are #P-complete. On one hand, the explicit #P-completeness proof of #1-in-3Sat could be useful to prove complexity results within unification theory. On the ither hand, the fact that #3-Colorability is #P-complete allows us to deduce immediately that the enumerative versions of a large class of NP-complete problems are #P-complete. Moreover, our proofs shed some new light on the interest of exhibiting linear reductions between NP problems