P. Beame

Publications (20)5.08 Total impact

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  Available from: ArXiv

  Article: Towards Understanding and Harnessing the Potential of Clause Learning

  P. Beame, H. Kautz, A. Sabharwal
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  ABSTRACT: 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 precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the well-studied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and Davis-Putnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speed-ups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas.
  06/2011;
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  Available from: psu.edu

  Conference Proceeding: A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness

  P. Beame, T. Pitassi, N. Segerlind, A. Wigderson
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  ABSTRACT: We prove that corruption, one of the most powerful measures used to analyze 2-party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to analyze the complexity of the widely-studied set disjointness problem in the usual "number-on-the-forehead" (NOF) model of multiparty communication complexity.
  Computational Complexity, 2005. Proceedings. Twentieth Annual IEEE Conference on; 07/2005
• Conference Proceeding: Memoization and DPLL: formula caching proof systems

  P. Beame, R. Impagliazzo, T. Pitassi, N. Segerlind
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  ABSTRACT: A fruitful connection between algorithm design and proof complexity is the formalization of the DPLL approach to satisfiability testing in terms of tree-like resolution proofs. We consider extensions of the DPLL approach that add some version of memoization, remembering formulas the algorithm has previously shown unsatisfiable. Various versions of such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability (S. M. Majercik et al., 1998; F. Bacchus et al., 2003). We formalize this method, and characterize the strength of various versions in terms of proof systems. These proof systems seem to be both new and simple, and have a rich structure. We compare their strength to several studied proof systems: tree-like resolution, regular resolution, general resolution, and Res(k). We give both simulations and separations.
  Computational Complexity, 2003. Proceedings. 18th IEEE Annual Conference on; 08/2003
• Article: Cv

  Xiaodong Sun, P. Beame, M. Saks, X. Sun, E. Vee, Super-linear Time-space, Tradeoff Lower Bounds
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  ABSTRACT: F14.46> polynomials, Exposition. Math. 16(3) (1998), 263--270. Talks Space lower bounds for distance approximation in the data stream model Nov. 20, 2001, DIMACS Discrete Math/Theory of Computing Seminar, Rutgers Univ., NJ Jan. 21, 2002, Institute for Advanced Study, Princeton, NJ Computing the Unmeasured: An Algebraic Approach to Internet Mapping April 2001, IEEE Infocom 2001, Anchorage, AK. Super-linear Time-space Tradeoff Lower Bounds for Randomized Computation November 12, 2000, FOCS 2000, Redondo Beach, CA. Explicit Interpolation Sets Using Perfect Hash Families October 12, 2000, DIMACS Mixer, AT&T Labs, Florham Park, NJ. Experience at Rutgers ffl Teaching assistant, Fall 1995 --- Spring 1999 and Spring 2001 --- Fall 2001 ffl Research assistant, Fall 1999 --- Fall 2000 ffl Tutoring at Rutgers Learning Resource Center, Spring 1996 --- Fall 1997 Awarded College Reading & Learning Associati
  02/2002;
• Conference Proceeding: Time-space tradeoffs, multiparty communication complexity, andnearest-neighbor problems

  P. Beame, E. Vee
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  ABSTRACT: The first non-trivial time-space tradeoff lower bounds have been shown for decision problems in P using notions derived from the study of two-party communication complexity. These results are proven directly for branching programs, natural generalizations of decision trees to directed graphs that provide elegant models of both non-uniform time T and space S simultaneously. We develop a new lower bound criterion, based on extending two-party communication complexity ideas to multiparty communication complexity. Applying this criterion to an explicit Boolean function based on a multilinear form over F ₂. for suitable s, we show lower bounds that yield T = Ω(n log² n) when S ⩽ n^1-ε log |D| for large input domain D. Finally, we develop lower bounds for nearest-neighbor problems involving n data points in a variety of d-dimensional metric spaces
  Computational Complexity, 2002. Proceedings. 17th IEEE Annual Conference on; 02/2002
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  Available from: psu.edu

  Conference Proceeding: Bounded-depth Frege lower bounds for weaker pigeonhole principles

  J. Buresh-Oppenheim, P. Beame, T. Pitassi, R. Raz, A. Sabharwal
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  ABSTRACT: We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHP_n^m where m = (1 + 1/polylog n)n. This lower bound qualitatively matches the known quasipolynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument.
  Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on; 02/2002
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  Available from: psu.edu

  Article: Optimizing symbolic model checking for statecharts

  W. Chan, R.J. Anderson, P. Beame, D.H. Jones, D. Notkin, W.E. Warner
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  ABSTRACT: Symbolic model checking based on binary decision diagrams is a powerful formal verification technique for reactive systems. In this paper, we present various optimizations for improving the time and space efficiency of symbolic modal checking for systems specified as statecharts. We used these techniques in our analyses of the models of a collision avoidance system and a fault-tolerant electrical power distribution (EPD) system, both used on commercial aircraft. The techniques together reduce the time and space requirements by orders of magnitude, making feasible some analysis that was previously intractable. We also elaborate on the results of verifying the EPD model. The analysis disclosed subtle modeling and logical flaws not found by simulation
  IEEE Transactions on Software Engineering 03/2001; · 1.98 Impact Factor
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  Available from: psu.edu

  Conference Proceeding: Resolution complexity of independent sets in random graphs

  P. Beame, R. Impagliazzo, A. Sabharwal
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  ABSTRACT: We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and Δn edges does not contain an independent set of size k. For randomly chosen graphs with constant Δ, we show that such proofs almost surely require size exponential in n. Further, for Δ=o(n^1/5) and any k⩽n/5, we show that these proofs almost surely require size 2(n^δ) for some global constant δ>0, even though the largest independent set in graphs with Δ≈n^1/5 is much smaller than n/5. Our result shows that almost all instances of the independent set problem are hard for resolution. It also provides a lower bound on the running time of a certain class of search algorithms for finding a largest independent set in a given graph
  Computational Complexity, 16th Annual IEEE Conference on, 2001.; 02/2001
• Conference Proceeding: Super-linear time-space tradeoff lower bounds for randomized computation

  P. Beame, M. Saks, Xiaodong Sun, E. Vee
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  ABSTRACT: We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by M. Ajtai (1999) in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for deterministic Boolean branching programs computing an explicit function based on quadratic forms over GF(2). Our results also give a quantitative improvement over those given by Ajtai. Ajtai shows, for certain specific functions, that any branching program using space S=o(n) requires time T that is superlinear. The functional form of the superlinear bound is not given in his paper, but optimizing the parameters in his arguments gives T= Ω(n log log n/log log log n) for S=0(n^1-ε). For the same functions considered by Ajtai, we prove a time-space tradeoff of the form T=Ω(n√(log(n/S)/log log(n/S))). In particular for space 0(n^1-ε), this improves the lower bound on time to Ω(n√(log n/log log n))
  Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on; 02/2000
• Conference Proceeding: Decoupling synchronization from local control for efficientsymbolic model checking of statecharts

  W. Chan, R.J. Anderson, P. Beame, D.H. Jones, D. Notkin, W.E. Warner
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  ABSTRACT: Symbolic model checking is a powerful formal verification technique for reactive systems. We address the problem of symbolic model checking for software specifications written as statecharts. We concentrate on how the synchronization of statecharts relates to the efficiency of model checking. We show that statecharts synchronized in an oblivious manner, such that the synchronization and the local control are decoupled, tend to be easier for symbolic analysis. Based on this insight, the verification of some non-oblivious systems can be optimized by a simple, transparent modification to the model to separate the synchronization from the local control. The technique enabled the analysis of the statecharts model of a fault tolerant electrical power distribution system developed by the Boeing Commercial Airplane Group. The results disclosed subtle modeling and logical flaws not found by simulation
  Software Engineering, 1999. Proceedings of the 1999 International Conference on; 02/1999
• Conference Proceeding: Time-space tradeoffs for branching programs

  P. Beame, M. Saks, J.S. Thathachar
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  ABSTRACT: We obtain the first non-trivial time-space tradeoff lower bound for functions f: {0,1}ⁿ→{0,1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1+ε)n, for some constant ε>0. We also give the first separation result between the syntactic and semantic read-k models for k>1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show a time-space tradeoff result on the more general R-way branching program model: for any k, we give a function that requires exponential size to be computed by length kn q-way branching programs, for some q=q(k)
  Foundations of Computer Science, 1998. Proceedings.39th Annual Symposium on; 12/1998
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  Available from: psu.edu

  Article: Model checking large software specifications

  W. Chan, R.J. Anderson, P. Beame, S. Burns, F. Modugno, D. Notkin, J.D. Reese
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  ABSTRACT: In this paper, we present our experiences in using symbolic model checking to analyze a specification of a software system for aircraft collision avoidance. Symbolic model checking has been highly successful when applied to hardware systems. We are interested in whether model checking can be effectively applied to large software specifications. To investigate this, we translated a portion of the state-based system requirements specification of Traffic Alert and Collision Avoidance System II (TCAS II) into input to a symbolic model checker (SMV). We successfully used the symbolic model checker to analyze a number of properties of the system. We report on our experiences, describing our approach to translating the specification to the SMV language, explaining our methods for achieving acceptable performance, and giving a summary of the properties analyzed. Based on our experiences, we discuss the possibility of using model checking to aid specification development by iteratively applying the technique early in the development cycle. We consider the paper to be a data point for optimism about the potential for more widespread application of model checking to software systems
  IEEE Transactions on Software Engineering 08/1998; · 1.98 Impact Factor
• Article: Improved depth lower bounds for small distance connectivity

  P. Beame, R. Impagliazzo, T. Pitassi
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  ABSTRACT: We consider the problem of determining, given a graph G with specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fan-in circuits requires depth , improving on a depth lower bound of when given by Ajtai (1989), Bellantoni et al. (1992). More generally, we obtain an improved size-depth tradeoff lower bound for the problem for all .¶The key to our technique is a new form of “switching lemma” which combines some of the features of iteratively shortening terms due to Furst et al. (1984) and Ajtai (1983) with the features of switching lemma arguments introduced by Yao (1985), Håstad (1987), and Cai (1986) that have been the methods of choice for subsequent results.
  Computational Complexity 01/1998; 7(4):325-345. · 1.12 Impact Factor
• Conference Proceeding: Simplified and improved resolution lower bounds

  P. Beame, T. Pitassi
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  ABSTRACT: We give simple new lower bounds on the lengths of resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which non-trivial lower bounds are known. For example, we show that with probability approaching 1, any resolution refutation of a randomly chosen 3-CNF formula with at most n^6/5-ε clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between resolution proof size and maximum clause size
  Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on; 11/1996
• Conference Proceeding: Improved depth lower bounds for small distance connectivity

  P. Beame, R. Impagliazzo, T. Pitassi
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  ABSTRACT: We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomial-size unbounded fan-in circuits, requires depth Ω(loglogk), improving on a depth lower bound of n(log*k) when k=log^O(1) n. In addition we show that there is a constant c such that for k⩽logn, any depth d unbounded fan-in circuit for this problem requires size at least n^ckεd where ε_d=φ^-2d/3 and φ is the golden mean. This latter result improves on an n^Ω(log(d+3k)) bound where log⁽ⁱ) is the i-fold composition of log with itself. The key to our technique is a new form of switching lemma which combines some of the features of iteratively shortening terms due to Furst, Saxe, and Sipser (1981) and Ajtai (1983) with the kinds of switching lemma arguments introduced by Yao (1985), Hastad (1986), and Cai (1986) that have been the methods of choice for subsequent results
  Foundations of Computer Science, 1995. Proceedings., 36th Annual Symposium on; 11/1995
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  Available from: psu.edu

  Conference Proceeding: Lower bounds on Hilbert's Nullstellensatz and propositional proofs

  P. Beame, R. Impagliazzo, J. Krajicek, T. Pitassi, P. Pudlak
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  ABSTRACT: The weak form of the Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q_i(x¯)=0, does not have a solution in the algebraic closure iff 1 is in the ideal generated by the polynomials Q_i(x¯). We shall prove a lower bound on the degrees of polynomials P_i(x¯) such that Σ _i P_i(x¯)Q_i(x¯)=1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into q-element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count_q ^N. Ajtai (1988) proved recently that, whenever p, q are two different primes, the propositional formulas Count_q^qn+1 do not have polynomial size, constant-depth Frege proofs from instances of Count_p^m, m≠0 (mod p). We give a new proof of this theorem based on the lower bound for the Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This results in an exact characterization of when Count_q can be proven efficiently from Count_p, for all p and q
  Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on; 12/1994
• Conference Proceeding: An exponential separation between the matching principle and the pigeonhole principle

  P. Beame, T. Pitassi
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  ABSTRACT: The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size bounded-depth Frege proofs. M. Ajtai (1990) previously showed that the matching principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential, and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes. In particular, oracle separations between the complexity classes PPA and PPAD and between PPA and PPP follow from our techniques
  Logic in Computer Science, 1993. LICS '93., Proceedings of Eighth Annual IEEE Symposium on; 07/1993
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  Available from: cs.washington.edu

  Conference Proceeding: Communication-space tradeoffs for unrestricted protocols

  P. Beame, M. Tompa, P. Yan
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  ABSTRACT: Communicating branching programs are introduced, and a general technique for demonstrating communication-space tradeoffs for pairs of communicating branching programs is developed. The technique is used to prove communication-space tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. For example any pair of communicating Boolean branching programs that computes matrix-vector products over GF(2) requires communication-space product Ω( n ²). These are the first examples of communication-space tradeoffs on a completely general model of communicating processes
  Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on; 11/1990
• Conference Proceeding: Time-space tradeoffs for undirected graph traversal

  P. Beame, A. Borodin, P. Raghavan, W.L. Ruzzo, M. Tompa
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  ABSTRACT: Time-space tradeoffs for traversing undirected graphs are proved. One of these tradeoffs is a quadratic lower bound on a deterministic model that closely matches the probabilistic upper bound of A.Z. Broder et al. (1989). The models used are variants of S.A. Cook and C.W. Rackoff's (1980) jumping automata for graphs. Some open problems are stated
  Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on; 11/1990
• Article: Low overhead parallel schedules for task graphs

  R. Anderson, P. Beame, W. Ruzzo
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  ABSTRACT: An abstract is not available.