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15
A Fast LinearArithmetic Solver for DPLL(T)
, 2006
"... We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of the ..."
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Cited by 216 (10 self)
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We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of theory propagation. We also present a new and simple approach for solving strict inequalities. Experimental results show substantial performance improvements over existing tools that use other Simplexbased solvers in DPLL(T) decision procedures. The new solver is even competitive with stateoftheart tools specialized for the difference logic fragment.
Lazy Satisfiability Modulo Theories
 JOURNAL ON SATISFIABILITY, BOOLEAN MODELING AND COMPUTATION 3 (2007) 141Â224
, 2007
"... Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingl ..."
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Cited by 97 (38 self)
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Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingly important due to its applications in many domains in different communities, in particular in formal verification. An amount of papers with novel and very efficient techniques for SMT has been published in the last years, and some very efficient SMT tools are now available. Typical SMT (T) problems require testing the satisfiability of formulas which are Boolean combinations of atomic propositions and atomic expressions in T, so that heavy Boolean reasoning must be efficiently combined with expressive theoryspecific reasoning. The dominating approach to SMT (T), called lazy approach, is based on the integration of a SAT solver and of a decision procedure able to handle sets of atomic constraints in T (Tsolver), handling respectively the Boolean and the theoryspecific components of reasoning. Unfortunately, neither the problem of building an efficient SMT solver, nor even that
The ICS Decision Procedures for Embedded Deduction
, 2004
"... Automated theorem proving... linear arithmetic, and lists. The ground (i.e., quantifierfree) fragment of many combinations is decidable when the fully quantified combination is not, and practical experience indicates that automation of the ground case is adequate for most applications. Practical ex ..."
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Cited by 32 (6 self)
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Automated theorem proving... linear arithmetic, and lists. The ground (i.e., quantifierfree) fragment of many combinations is decidable when the fully quantified combination is not, and practical experience indicates that automation of the ground case is adequate for most applications. Practical experience also suggests several other desiderata for an effective deductive service. Some applications (e.g., construction of abstractions) invoke their deductive service a huge number of times in the course of a single calculation, so that performance of the service must be very good. Other applications such as proof search explore many variations on a formula (i.e., alternately asserting and denying various combinations of its premises), so the deductive service should not examine individual formulas in isolation, but should provide a rich application programming interface that supports incremental assertion, retraction, and querying of formulas. Other applications such as test case generation...
Policy ratification
 In POLICY ’05: Proceedings of the Sixth IEEE International Workshop on Policies for Distributed Systems and Networks (POLICY’05
, 2005
"... It is not sufficient to merely check the syntax of new policies before they are deployed in a system; policies need to be analyzed for their interactions with each other and with their local environment. That is, policies need to go through a ratification process. We believe policy ratification beco ..."
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Cited by 20 (3 self)
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It is not sufficient to merely check the syntax of new policies before they are deployed in a system; policies need to be analyzed for their interactions with each other and with their local environment. That is, policies need to go through a ratification process. We believe policy ratification becomes an essential part of system management as the number of policies in the system increases and as the system administration becomes more decentralized. In this paper, we focus on the basic tasks involved in policy ratification. To a large degree, these basic tasks can be performed independent of policy model and language and require little domainspecific knowledge. We present algorithms from constraint, linear, and logic programming disciplines to help perform ratification tasks. We provide an algorithm to efficiently assign priorities to the policies based on relative policy preferences indicated by policy administrators. Finally, with an example, we show how these algorithms have been integrated with our policy system to provide feedback to a policy administrator regarding potential interactions of policies with each other and with their deployment environment. 1
Integrating simplex with DPLL(T)
 CSL, SRI INTERNATIONAL
, 2006
"... We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of theor ..."
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Cited by 18 (4 self)
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We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of theory propagation. We also present a new and simple approach for solving strict inequalities. Experimental results show substantial performance improvements over existing tools that use other Simplexbased solvers in DPLL(T) decision procedures. The new solver is even competitive with stateoftheart tools specialized for the difference logic fragment.
An efficient nelsonoppen decision procedure for difference constraints over rationals
, 2005
"... Abstract. Nelson and Oppen provided a methodology for modularly combining decision procedures for individual theories to construct a decision procedure for a combination of theories. In addition to providing a check for satisfiability, the individual decision procedures need to provide additional fu ..."
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Cited by 13 (3 self)
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Abstract. Nelson and Oppen provided a methodology for modularly combining decision procedures for individual theories to construct a decision procedure for a combination of theories. In addition to providing a check for satisfiability, the individual decision procedures need to provide additional functionalities, including equality generation. In this paper, we propose a decision procedure for a conjunction of difference constraints over rationals (where the atomic formulas are of the form x ≤ y + c or x < y + c). The procedure extends any negative cycle detection algorithm (like the BellmanFord algorithm) to generate (1) equalities between all pair of variables, (2) produce proofs and (3) generates models that can be extended by other theories in a NelsonOppen framework. All the operations mentioned above can be performed with only a linear overhead to the cycle detection algorithm. 1
On Locally Minimal Nullstellensatz Proofs
 SATISFIABILITY MODULO THEORIES
, 2009
"... Hilbert’s weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically ..."
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Cited by 3 (1 self)
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Hilbert’s weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically using Gröbner bases and cofactorbased linear algebra techniques. However, these proof objects may contain redundant information: a proper subset of the equational assumptions used in these proofs may be sufficient to derive the unsatisfiability of the original polynomial system. For using Nullstellensatz techniques in SMTbased decision methods, a minimal proof object is often desired. With this in mind, we introduce a notion of locally minimal Nullstellensatz proofs and give idealtheoretic algorithms for their construction.
Simplex with Sum of Infeasibilities for SMT
"... Abstract—The de facto standard for stateoftheart real and integer linear reasoning within Satisfiability Modulo Theories (SMT) solvers is the Simplex for DPLL(T) algorithm given by Dutertre and de Moura. This algorithm works by performing a sequence of local optimization operations. While the alg ..."
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Cited by 2 (2 self)
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Abstract—The de facto standard for stateoftheart real and integer linear reasoning within Satisfiability Modulo Theories (SMT) solvers is the Simplex for DPLL(T) algorithm given by Dutertre and de Moura. This algorithm works by performing a sequence of local optimization operations. While the algorithm is generally efficient in practice, its local pivoting heuristics lead to slow convergence on some problems. More traditional Simplex algorithms minimize a global criterion to determine the feasibility of the input constraints. We present a novel Simplexbased decision procedure for use in SMT that minimizes the sum of infeasibilities of the constraints. Experimental results show that this new algorithm is comparable with or outperforms Simplex for DPLL(T) on a broad set of benchmarks. I.
Interpolant based Decision Procedure for QuantifierFree Presburger Arithmetic
, 2005
"... Recently, there have been two popular approaches for SATbased theorem proving — eager and lazy. Eager approaches are based on a satisfiability preserving translation to a Boolean formula, whereas the lazy approaches perform an incremental translation to SAT. Eager approaches are usually based on en ..."
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Cited by 1 (0 self)
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Recently, there have been two popular approaches for SATbased theorem proving — eager and lazy. Eager approaches are based on a satisfiability preserving translation to a Boolean formula, whereas the lazy approaches perform an incremental translation to SAT. Eager approaches are usually based on encoding integers as bitvectors and suffer from lack of structure and sometime very large size for the bitvectors. Lazy approaches suffer from large number of invocations of theory decision procedures and the complexity of the decision procedures for integer linear arithmetic. In this paper, we present a decision procedure for Quantifierfree Presburger arithmetic that is based on alternately under and overapproximating a formula. We use Boolean interpolants to compute the overapproximation. The algorithm seems to address the bottlenecks of both eager and lazy approaches, and improve on both. The algorithm consistently outperforms approaches based on eager and lazy methods on a set of verification benchmarks.