Toniann Pitassi

Simplified and improved resolution lower bounds

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/sup 6/5-/spl epsiv// 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.

Exponential lower bounds for the pigeonhole principle

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 Ω(loglogn)-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 S. Buss has constructed polynomial-size, logn-depth Frege proofs for the PHP. The main lemma in our proof can be viewed as a general Håstad-style Switching Lemma for restrictions that are partial matchings. Our lower bounds for the pigeonhole principle improve on previous superpolynomial lower bounds.

Stochastic Boolean Satisfiability

Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSAT, which can function for probabilistic domains as SAT does for deterministic domains. It shows the connection between SSAT and well-studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSAT instances. These algorithms are validated on random SSAT formulae generated under the fixed-clause model. In spite of the large complexity gap between SSAT (PSPACE) and SAT (NP), the paper suggests that much of what we have learned about SAT transfers to the probabilistic domain.

Algorithms and complexity results for #SAT and Bayesian inference

Bayesian inference is an important problem with numerous applications in probabilistic reasoning. Counting satisfying assignments is a closely related problem of fundamental theoretical importance. In this paper, we show that plain old DPLL equipped with memorization (an algorithm we call #DPLLCache) can solve both of these problems with time complexity that is at least as good as state-of-the-art exact algorithms, and that it can also achieve the best known time-space tradeoff. We then proceed to show that there are instances where #DPLLCache can achieve an exponential speedup over existing algorithms.

Lower bounds on Hilbert's Nullstellensatz and propositional proofs

The weak form of the Hilberts Nullstellensatz says that a system of algebraic equations over a field, Q/sub i/(x~)=0, does not have a solution in the algebraic closure iff 1 is in the ideal generated by the polynomials Q/sub i/(x~). We shall prove a lower bound on the degrees of polynomials P/sub i/(x~) such that /spl Sigma//sub i/ P/sub i/(x~)Q/sub 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/sub q//sup N/. Ajtai (1988) proved recently that, whenever p, q are two different primes, the propositional formulas Count/sub q//sup qn+1/ do not have polynomial size, constant-depth Frege proofs from instances of Count/sub p//sup m/, m/spl ne/0 (mod p). We give a new proof of this theorem based on the lower bound for the Hilberts 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/sub q/ can be proven efficiently from Count/sub p/, for all p and q.<<ETX>>

On Interpolation and Automatization for Frege Systems

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 TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-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-sized TC0-Frege. As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

The Limits of Two-Party Differential Privacy

We study differential privacy in a distributed setting where two parties would like to perform analysis of their joint data while preserving privacy for both datasets. Our results imply almost tight lower bounds on the accuracy of such data analyses, both for specific natural functions (such as Hamming distance) and in general. Our bounds expose a sharp contrast between the two-party setting and the simpler client-server setting (where privacy guarantees are one-sided). In addition, those bounds demonstrate a dramatic gap between the accuracy that can be obtained by differentially private data analysis versus the accuracy obtainable when privacy is relaxed to a computational variant of differential privacy. The first proof technique we develop demonstrates a connection between differential privacy and deterministic extraction from Santha-Vazirani sources. A second connection we expose indicates that the ability to approximate a function by a low-error differentially private protocol is strongly related to the ability to approximate it by a low communication protocol. (The connection goes in both directions.)

The Efficiency of Resolution and Davis--Putnam Procedures

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on the work of Clegg, Edmonds, and Impagliazzo in [Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, 1996, ACM, New York, 1996, pp. 174--183], we give an algorithm for unsatisfiability that when given an unsatisfiable formula of F finds a resolution proof of F. The runtime of our algorithm is subexponential in the size of the shortest resolution proof of F. Next, we investigate a class of backtrack search algorithms for producing resolution refutations of unsatisfiability, commonly known as Davis--Putnam procedures, and provide the first asymptotically tight average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL, we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is

On the complexity of unsatisfiability proofs for random k -CNF formulas

2^{\Theta(n(n/m)^{1/(k-2)})}

An exponential separation between regular and general resolution

with probability