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82
Simplification by cooperating decision procedures
 ACM Transactions on Programming Languages and Systems
, 1979
"... A method for combining decision procedures for several theories into a single decision procedure for their combination is described, and a simplifier based on this method is discussed. The simplifier finds a normal form for any expression formed from individual variables, the usual Boolean connectiv ..."
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Cited by 396 (1 self)
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A method for combining decision procedures for several theories into a single decision procedure for their combination is described, and a simplifier based on this method is discussed. The simplifier finds a normal form for any expression formed from individual variables, the usual Boolean connectives, the equality predicate =, the conditional function ifthenelse, the integers, the arithmetic functions and predicates +,, and _<, the Lisp functions and predicates car, cdr, cons, and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. If the expression is a theorem it is simplified to the constant true, so the simplifier can be used as a decision procedure for the quantifierfree theory containing these functions and predicates. The simplifier is currently used in the Stanford Pascal Verifier.
Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations
, 1997
"... We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, i ..."
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Cited by 134 (5 self)
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We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
Hybrid Logics: Characterization, Interpolation and Complexity
 Journal of Symbolic Logic
, 1999
"... Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We sho ..."
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Cited by 101 (35 self)
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Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of EhrenfeuchtFrasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rstorder logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...
Toward Logic Tailored for Computational Complexity
 COMPUTATION AND PROOF THEORY
, 1984
"... Whereas firstorder logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the firstorder theory of finite structures and alternatives to firstorder logic, especially polynomial time logic. ..."
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Cited by 75 (6 self)
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Whereas firstorder logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the firstorder theory of finite structures and alternatives to firstorder logic, especially polynomial time logic.
Lazy abstraction with interpolants
 In Proc. CAV, LNCS 4144
, 2006
"... Abstract. We describe a model checker for infinitestate sequential programs, based on Craig interpolation and the lazy abstraction paradigm. On device driver benchmarks, we observe a speedup of up to two orders of magnitude relative to a similar tool using predicate abstraction. 1 ..."
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Cited by 75 (6 self)
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Abstract. We describe a model checker for infinitestate sequential programs, based on Craig interpolation and the lazy abstraction paradigm. On device driver benchmarks, we observe a speedup of up to two orders of magnitude relative to a similar tool using predicate abstraction. 1
A practical and complete approach to predicate refinement
 In Tools and Algorithms for the Construction and Analysis of Systems, LNCS 3920
, 2006
"... Abstract. Predicate abstraction is a method of synthesizing the strongest inductive invariant of a system expressible as a Boolean combination of a given set of atomic predicates. A predicate selection method can be said to be complete for a given theory if it is guaranteed to eventually find atomic ..."
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Cited by 57 (6 self)
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Abstract. Predicate abstraction is a method of synthesizing the strongest inductive invariant of a system expressible as a Boolean combination of a given set of atomic predicates. A predicate selection method can be said to be complete for a given theory if it is guaranteed to eventually find atomic predicates sufficient to prove a given property, when such exist. Current heuristics are incomplete, and often diverge on simple examples. We present a practical method of predicate selection that is complete in the above sense. The method is based on interpolation and uses a “split prover”, somewhat in the style of structurebased provers used in artificial intelligence. We show that it allows the verification of a variety of simple programs that cannot be verified by existing software model checkers. 1
Lower Bounds to the Size of ConstantDepth Propositional Proofs
, 1994
"... 1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp ..."
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Cited by 54 (6 self)
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1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp(O(log 2 n)) but such that every depth d refutation must have the size at least exp(n\Omega\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...
PartitionBased Logical Reasoning for FirstOrder and Propositional Theories
 Artificial Intelligence
, 2000
"... In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with ..."
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Cited by 51 (8 self)
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In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward messagepassing and using backward messagepassing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our messagepassing ...
Interpolantbased transition relation approximation
 In CAV 05: ComputerAided Verification, LNCS 3576
, 2005
"... Abstract. In predicate abstraction, exact image computation is problematic, requiring in the worst case an exponential number of calls to a decision procedure. For this reason, software model checkers typically use a weak approximation of the image. This can result in a failure to prove a property, ..."
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Cited by 38 (3 self)
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Abstract. In predicate abstraction, exact image computation is problematic, requiring in the worst case an exponential number of calls to a decision procedure. For this reason, software model checkers typically use a weak approximation of the image. This can result in a failure to prove a property, even given an adequate set of predicates. We present an interpolantbased method for strengthening the abstract transition relation in case of such failures. This approach guarantees convergence given an adequate set of predicates, without requiring an exact image computation. We show empirically that the method converges more rapidly than an earlier method based on counterexample analysis. 1
Applications of Craig interpolants in model checking
 In TACAS’2005: Tools and Algorithms for the Construction and Analysis of Systems, LNCS 3440
, 2005
"... Abstract. A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B. An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and pro ..."
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Cited by 32 (0 self)
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Abstract. A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B. An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and proof systems. We will discuss a number of applications of this concept in finite and infinitestate model checking. 1