The computational complexity of structure-based causality

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\ Sigma_2^P-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 by Hopkins and Pearl. As we show, this modification has a nontrivial impact on the complexity of computing actual cause. To characterize the complexity, a new family D_k^P, k= 1, 2, 3, ..., of complexity classes is introduced, which generalizes the class DP introduced by Papadimitriou and Yannakakis (DP is just D_1^P). %joe2 %We show that the complexity of computing causality is \D_2-complete %under the new definition. Chockler and Halpern \citeyear{CH04} extended the We show that the complexity of computing causality under the updated definition is $D_2^P$-complete. Chockler and Halpern extended the definition of causality by introducing notions of responsibility and blame. The complexity of determining the degree of responsibility and blame using the original definition of causality was completely characterized. Again, we show that changing the definition of causality affects the complexity, and completely characterize it using the updated definition.Comment: Appears in AAAI 201

IC3-Guided Abstraction

Abstract-Localization is a powerful automated abstraction-refinement technique to reduce the complexity of property checking. This process is often guided by SATbased bounded model checking, using counterexamples obtained on the abstract model, proofs obtained on the original model, or a combination of both to select irrelevant logic. In this paper, we propose the use of bounded invariants obtained during an incomplete IC3 run to derive higher-quality abstractions for complex problems. Experiments confirm that this approach yields significantly smaller abstractions in many cases, and that the resulting abstract models are often easier to verify

Generalized Counterexamples to Liveness Properties

Abstract-We consider generalized counterexamples in the context of liveness property checking. A generalized counterexample comprises only a subset of values necessary to establish the existence of a concrete counterexample. While useful in various ways even for safety properties, the length of a generalized liveness counterexample may be exponentially shorter than that of a concrete counterexample, entailing significant potential algorithmic benefits. One application of this concept extends the k-LIVENESS proof technique of [1] to enable failure detection. The resulting algorithm is simple, and poses negligible overhead to k-LIVENESS in practice. We additionally propose dedicated algorithms to search for generalized liveness counterexamples, and to manipulate generalized counterexamples to and from concrete ones. Experiments confirm the capability of these techniques to detect failures more efficiently than existing techniques for various benchmarks

The local counting function of operators of Dirac and Laplace type

Let PP be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various PP-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the mollified spectral counting function of Dirac type operators in terms of geometric quantities and characterize those Dirac type operators for which this coefficient vanishes

Exploiting Isomorphic Subgraphs in SAT

While static symmetry breaking has been explored in the SAT community for decades, only as of 2010 research has focused on exploiting the same discovered symmetry dynamically, during the run of the SAT solver, by learning extra clauses. The two methods are distinct and not compatible. The former may prune solutions, whereas the latter does not – it only prunes areas of the search that are guaranteed not to have solutions, like standard conflict clauses. Both approaches, however, require what we call full symmetry, namely a propositionally-consistent mapping σ between the literals, such that σ(φ) ≡ φ, where here ≡ means syntactic equivalence modulo clause ordering and literal ordering within the clauses. In this article we show that such full symmetry is not a necessary condition for adding extra clauses: isomorphism between possibly-overlapping subgraphs of the colored incidence graph is sufficient. While finding such subgraphs is a computationally hard problem, there are many cases in which they can be detected a priori by analyzing the high-level structure of the problem from which the CNF was derived. We demonstrate this principle with several well-known problems.2204211