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649
Lower bounds for splittings by linear combinations ∗
, 2014
"... A typical DPLL algorithm for the Boolean satisfiability problem splits the input problem into two by assigning the two possible values to a variable; then it simplifies the two resulting formulas. In this paper we consider an extension of the DPLL paradigm. Our algorithms can split by an arbitrary l ..."
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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
Rank-sparsity incoherence for matrix decomposition
, 2010
"... Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is intractable ..."
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Cited by 230 (21 self)
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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
Lemmas on Demand for Satisfiability Solvers
, 2002
"... 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 by iterativel ..."
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Cited by 42 (5 self)
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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
The Complexity of Generalized Satisfiability for Linear Temporal Logic
"... 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 ..."
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Cited by 20 (10 self)
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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
Sums of squares, satisfiability and maximum satisfiability
- In SAT 2005
, 2005
"... Recently the Mathematical Programming community showed a renewed interest in Hilbert’s Positivstellensatz. The reason for this is that global optimization of polynomials in IR[x1,..., xn] is N P-hard, while the question whether a polynomial can be written as a sum of squares has tractable aspects. T ..."
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Cited by 2 (0 self)
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. 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
UNSATISFIABLE LINEAR CNF FORMULAS ARE LARGE AND COMPLEX
, 2010
"... Abstract. We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k 2 4 k 4 k 8e 2 k 2 clauses is clauses, and on the other hand, any linear k-CNF formula with at most satisfiable. The upper bound ..."
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Cited by 2 (0 self)
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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
Two easy theories whose combination is hard
, 1977
"... 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 are ..."
Abstract
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Cited by 41 (0 self)
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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
Some 3CNF properties are hard to test
- In Proc. 35th ACM Symp. on Theory of Computing
, 2003
"... Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of n-bit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that th ..."
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Cited by 59 (10 self)
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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
Combining preorder and postorder resolution in a satisfiability solver
- Electronic Notes in Discrete Mathematics
, 2001
"... Recently, the classical backtracking-search satisability algorithm of Davis, Put-nam, Loveland and Logemann (DPLL) has been enhanced in two distinct directions: with more advanced reasoning methods, or with con
ict-directed back-jumping (CBJ). Previous methods used one idea or the other, but not bot ..."
Abstract
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Cited by 7 (0 self)
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, 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
Efficient interpolant generation in satisfiability modulo theories,” in
- Proc. TACAS, ser. LNCS 4963.
, 2008
"... Abstract. The problem of computing Craig interpolants in SAT and SMT has recently received a lot of interest, mainly for its applications in formal verification. Efficient algorithms for interpolant generation have been presented for some theories of interest -including that of equality and uninter ..."
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Cited by 42 (7 self)
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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
Results 1 - 10
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649