Halpern and Pearl introduced a definition of actual causality; Eiter and Lukasiewicz showed that computing whether X = x is a cause of Y = y is NP-complete in binary models (where all variables can take on only two values) and Σ P 2 -complete in general models. In the final version of their paper, Halpern and Pearl slightly modified the definition of actual cause, in order to deal with problems pointed out by Hopkins and Pearl. As we show, this modification has a nontrivial impact on the complexity of computing whether X = x is a cause of Y = y. To characterize the complexity, a new family D P k , k = 1, 2, 3, . . ., of complexity classes is introduced, which generalizes the class D P introduced by Papadimitriou and Yannakakis (D P is just D P 1 ).We show that the complexity of computing causality under the updated definition is D P 2 -complete. Chockler and Halpern extended the definition of causality by introducing notions of responsibility and blame, and characterized the complexity of determining the degree of responsibility and blame using the original definition of causality. Here, we completely characterize the complexity using the updated definition of causality. In contrast to the results on causality, we show that moving to the updated definition does not result in a difference in the complexity of computing responsibility and blame.

Abstract. Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in state-of-the-art constrained sampling and counting tools, while still retaining theoretical guarantees.

We describe an incremental algorithm for computing interpolants for a pair ϕA, ϕB of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕA ∧ ϕB, and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube. This work is partially supported by the European Community under the call FP7-ICT-2009-5-project PINCETTE 257647. 1 There are efficient algorithms known to compute interpolants based on refutations in proof systems other than resolution (e.g., in Cutting Planes [Kra97]), but the one based on resolution is the canonical one from practical perspective.

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