linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two, which were used for proving exponential lower bounds for conventional DPLL algorithms. We prove exponential lower bounds on the running time of DPLL with splitting

to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the ℓ1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty

We investigate the combination of propositional SAT checkers with constraint solvers for domain-specific theories such as linear arithmetic, arrays, lists and the combination thereof. Our procedure realizes a lazy approach to satisfiability checking of propositional constraint formulas

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease

. This is due to the fact that Semidefinite Programming can be used to decide in polynomial time (up to a prescribed precision) whether a polynomial can be rewritten as a sum of squares of linear combinations of monomials coming from a specified set. We investigate this approach in the context of Satisfiability

has size at least 2 2 k 2 −1. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree,..a a. i.e. every clause of F is used only once in the resolution tree

We restrict attention to the validity problem for unquantified disjunctions of literals (possibly negated atomic formulae) over the domain of integers, or what is just as good, the satisfiability problem for unquantified conjunctions. When = is the only predicate symbol and all function symbols

reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof

, but not both (except that CBJ has been combined with the unit-clause rule). The diÆculty is that more advanced reasoning derives new clauses, but does not identify which backtrackable assumptions were relevant for the derivation. However CBJ needs this information. The linear-time decision procedure for binary

and uninterpreted functions (EUF ), linear arithmetic over the rationals (LA(Q)), and their combination-and they are successfully used within model checking tools. For the theory of linear arithmetic over the integers (LA(Z)), however, the problem of finding an interpolant is more challenging, and the task