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In this paper we prove an exponential lower bound on the size of bounded-depth Frege proofs for the pigeonhole principle (PHP). We also obtain an ~(log log rz)-depth lower bound for any polynomial-sized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact complexity of the PHP, as Sam Buss has constructed… (More)

An important extension of satisfiability testing is model-counting, a task that corresponds to problems such as probabilistic reasoning and computing the permanent of a Boolean matrix. We recently introduced Cachet, an exact model-counting algorithm that combines formula caching, clause learning, and component analysis. This paper reports on experiments… (More)

Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant real-world problems , such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first… (More)

- Tian Sang, Paul Beame, Henry Kautz
- 2005

Over the past decade general satisfiability testing algorithms have proven to be surprisingly effective at solving a wide variety of constraint satisfaction problem, such as planning and scheduling (Kautz and Selman 2003). Solving such NP-complete tasks by " compilation to SAT " has turned out to be an approach that is of both practical and theoretical… (More)

In this paper we present our results and experiences of using symbolic model checking to study the specification of an aircraft collision avoidance system. Symbolic model checking has been highly successful when applied to hardware systems. We are interested in the question of whether or not model checking techniques can be applied to large software… (More)

We present optimal depth Boolean circuits (depth O(log n)) for integer division, powering, and multiple products. We also show that these three problems are of equivalent uniform depth and space complexity. In addition, we describe an algorithm for testing divisibility that is optimal for both depth and space. AMS(MOS) subject classification. 68Q 1.… (More)

Over the past decade general satisfiability testing algorithms have proven to be surprisingly effective at solving a wide variety of constraint satisfaction problem, such as planning and scheduling (Kautz and Selman 2003). Solving such NP-complete tasks by " compilation to SAT " has turned out to be an approach that is of both practical and theoretical… (More)

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and… (More)

- Paul Beame
- 1996

The weak form of the Hilbert's Nullstellensatz says that a system of algebraic equations over a eld, Q i (x) = 0, does not have a solution in the algebraic closure ii 1 is in the ideal generated by the polynomi-als Q i (x). We shall prove a lower bound on the degrees of polynomials P i (x) such that P i P i (x)Q i (x) = 1. This result has the following… (More)