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88
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 135 (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.
Bounded model checking
, 2009
"... Besides Equivalence Checking [KK97, KPKG02] the most important industrial application of SAT is currently Bounded Model Checking (BMC) [BCCZ99]. Both techniques are used for formal hardware verification in the context of electronic design automation (EDA), but have successfully been applied to many ..."
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Cited by 89 (2 self)
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Besides Equivalence Checking [KK97, KPKG02] the most important industrial application of SAT is currently Bounded Model Checking (BMC) [BCCZ99]. Both techniques are used for formal hardware verification in the context of electronic design automation (EDA), but have successfully been applied to many other domains as well. In this chapter, we focus on BMC. In practice, BMC is mainly used for falsification resp. testing, which is concerned with violations of temporal properties. However, the original paper on BMC [BCCZ99] already discussed extensions that can prove properties. A considerable part of this chapter discusses these complete extensions, which are often called “unbounded ” model checking techniques, even though they are build upon the same principles as plain BMC. Two further related applications, in which BMC becomes more and more important, are automatic test case generation for closing coverage holes, and disproving redundancy in designs. Most of the techniques discussed in this chapter transfer to this more general setting as well, even though our focus is on property
Lower Bounds for Cutting Planes Proofs with Small Coefficients
, 1995
"... We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the cl ..."
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Cited by 75 (19 self)
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We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Treelike CP proofs cannot polynomially simulate nontreelike CP proofs. (2) Treelike CP proofs and BoundeddepthFrege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...
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 ...
On Interpolation and Automatization for Frege Systems
, 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 49 (7 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 Frege. As a corollary, we obtain that TC 0 Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
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
Separation of the Monotone NC Hierarchy
, 1999
"... We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC i+1 . 3. More generally: For any integer function D( ..."
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Cited by 34 (0 self)
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We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fanin 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotoneNC 1 from monotoneNC 2 was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In...
On the Automatizability of Resolution and Related Propositional Proof Systems
, 2002
"... We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove ..."
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Cited by 34 (6 self)
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We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove
Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting
"... We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, us ..."
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Cited by 32 (9 self)
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We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constantdepth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss
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