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.

We consider small-weight 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) Tree-like CP proofs cannot polynomially simulate non-tree-like CP proofs. (2) Tree-like CP proofs and Bounded-depth-Frege 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...

In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. 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 message-passing and using backward message-passing 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 message-passing ...

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) complexity-theoretic 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...

We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove

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 constant-depth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss

We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6= monotone-P. 2. For every i 1, monotone-NC i 6= monotone-NC 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 (fan-in 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotone-NC 1 from monotone-NC 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...

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 interpolant-based 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

We prove an exponential lower bound for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution from (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Putnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...

this paper we are interested in systems that use uses polynomials instead of boolean formulas. From the previous list this includes the Nullstellensatz refutations. Recently a stronger system using polynomials was proposed, the polynomial calculus, also called the Groebner calculus [9]. The proof systems form a similar hierarchy as the complexity classes or classes of circuits in the computational complexity, but there is no direct relation between the two hierarchies. Recently a new method was found which makes it possible to prove lower bounds on the length of proofs for some propositional proof systems using lower bounds on circuit complexity. This method is based on proving computationally efficient versions of Craig's interpolation theorem for the proof system in question [14, 18]. For appropriate tautologies the interpolation theorem