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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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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
A formal system for Euclid's Elements
, 2009
"... We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. ..."
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We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.
Canonization for Disjoint Unions of Theories
, 2003
"... If there exist ecient procedures (canonizers) for reducing terms of two rstorder theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, w ..."
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Cited by 13 (4 self)
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If there exist ecient procedures (canonizers) for reducing terms of two rstorder theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) cannot be combined in a similar fashion. These results are relevant to the widely used Shostak's method for combining decision procedures for theories. They provide the rst rigorous answers to the questions about the possibility of directly combining canonizers and solvers.
Verifying HeapManipulating Programs in an SMT Framework
, 2007
"... Automated software verification has made great progress recently, and a key enabler of this progress has been the advances in efficient, automated decision procedures suitable for verification (Boolean satisfiability solvers and satisfiabilitymodulotheories (SMT) solvers). Verifying general soft ..."
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Cited by 12 (1 self)
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Automated software verification has made great progress recently, and a key enabler of this progress has been the advances in efficient, automated decision procedures suitable for verification (Boolean satisfiability solvers and satisfiabilitymodulotheories (SMT) solvers). Verifying general software, however, requires reasoning about unbounded, linked, heapallocated data structures, which in turn motivates the need for a logical theory for such structures that includes unbounded reachability. So far, none of the available SMT solvers supports such a theory. In this paper, we present our integration of a decision procedure that supports unbounded heap reachability into an available SMT solver. Using the extended SMT solver, we can efficiently verify examples of heapmanipulating programs that we could not verify before.
ILP Modulo Theories
"... Abstract. We present Integer Linear Programming (ILP) Modulo Theories (IMT). An IMT instance is an Integer Linear Programming instance, where some symbols have interpretations in background theories. In previous work, the IMT approach has been applied to industrial synthesis and design problems with ..."
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Abstract. We present Integer Linear Programming (ILP) Modulo Theories (IMT). An IMT instance is an Integer Linear Programming instance, where some symbols have interpretations in background theories. In previous work, the IMT approach has been applied to industrial synthesis and design problems with realtime constraints arising in the development of the Boeing 787. Many other problems ranging from operations research to software verification routinely involve linear constraints and optimization. Thus, a general ILP Modulo Theories framework has the potential to be widely applicable. The logical next step in the development of IMT and the main goal of this paper is to provide theoretical underpinnings. This is accomplished by means of BC(T), the Branch and Cut Modulo T abstract transition system. We show that BC(T) provides a sound and complete optimization procedure for the ILP Modulo T problem, as long as T is a decidable, stablyinfinite theory. We compare a prototype of BC(T) against leading SMT solvers. 1
Decision Procedures for Region Logic
"... Abstract. Region logic is Hoare logic for objectbased programs. It features local reasoning with frame conditions expressed in terms of sets of heap locations. This paper studies tableaubased decision procedures for RL, the quantifierfree fragment of the assertion language. This fragment combines ..."
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Abstract. Region logic is Hoare logic for objectbased programs. It features local reasoning with frame conditions expressed in terms of sets of heap locations. This paper studies tableaubased decision procedures for RL, the quantifierfree fragment of the assertion language. This fragment combines sets and (functional) images with the theories of arrays and partial orders. The procedures are of practical interest because they can be integrated efficiently into the satisfiability modulo theories (SMT) framework. We provide a semidecision procedure for RL and its implementation as a theory plugin inside the SMT solver Z3. We also provide a decision procedure for an expressive fragment of RL termed restrictedRL. We prove that deciding satisfiability of restrictedRL formulas is NPcomplete. Both procedures are proven sound and complete. Preliminary performance results indicate that the semidecision procedure has the potential toscale to large input formulas. 1
Combination of Convex Theories: Modularity, Deduction Completeness, and Explanation
, 2008
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1ILP Modulo Data
"... Abstract—The vast quantity of data generated and captured every day has led to a pressing need for tools and processes to organize, analyze and interrelate this data. Automated reasoning and optimization tools with inherent support for data could enable advancements in a variety of contexts, from da ..."
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Abstract—The vast quantity of data generated and captured every day has led to a pressing need for tools and processes to organize, analyze and interrelate this data. Automated reasoning and optimization tools with inherent support for data could enable advancements in a variety of contexts, from databacked decision making to dataintensive scientific research. To this end, we introduce a decidable logic aimed at database analysis. Our logic extends quantifierfree Linear Integer Arithmetic with operators from Relational Algebra, like selection and cross product. We provide a scalable decision procedure that is based on the BC(T) architecture for ILP Modulo Theories. Our decision procedure makes use of database techniques. We also experimentally evaluate our approach, and discuss potential applications. I.
The Inez Mathematical Programming Modulo Theories Framework
"... Abstract. Our Mathematical Programming Modulo Theories (MPMT) constraint solving framework extends Mathematical Programming technology with techniques from the field of Automated Reasoning, e.g., solvers for firstorder theories. In previous work, we used MPMT to synthesize system architectures fo ..."
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Abstract. Our Mathematical Programming Modulo Theories (MPMT) constraint solving framework extends Mathematical Programming technology with techniques from the field of Automated Reasoning, e.g., solvers for firstorder theories. In previous work, we used MPMT to synthesize system architectures for Boeing’s Dreamliner and we studied the theoretical aspects of MPMT by means of the Branch and Cut Modulo T (BC(T)) transition system. BC(T) can be thought of as a blueprint for MPMT solvers. This paper provides a more practical and algorithmic view of BC(T). We elaborate on the design and features of Inez, our BC(T) constraint solver. Inez is an opensource, freely available superset of the OCaml programming language that uses the SCIP Branch and Cut framework to extend OCaml with MPMT capability. Inez allows users to write programs that arbitrarily interweave general computation with MPMT constraint solving. 1
Propositional Satisfiability: Algorithms and Applications
, 2008
"... In the first part of this paper we survey a number of algorithms for solving the propositional satisfiability problem (SAT). We dedicate a large amount of attention to the nonclausal SAT algorithms, that is, the algorithms that work on arbitrary propositional formulas, and to the circuit SAT algori ..."
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In the first part of this paper we survey a number of algorithms for solving the propositional satisfiability problem (SAT). We dedicate a large amount of attention to the nonclausal SAT algorithms, that is, the algorithms that work on arbitrary propositional formulas, and to the circuit SAT algorithms that work on Boolean circuit representation of formulas. We also discuss some of the nonmainstream clausal SAT algorithms. The second part of this paper discusses some of the practical applications of SAT, particularly to Bounded Model Checking and to Satisfiability Modulo Theories.