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

Computational complexity of generators and nongenerators in algebra

We discuss the computational complexity of several prob- lems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),... all lie in the class P (NP)

Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)Comment: 72 page