1,023 research outputs found
Closing the Gap Between Short and Long XORs for Model Counting
Many recent algorithms for approximate model counting are based on a
reduction to combinatorial searches over random subsets of the space defined by
parity or XOR constraints. Long parity constraints (involving many variables)
provide strong theoretical guarantees but are computationally difficult. Short
parity constraints are easier to solve but have weaker statistical properties.
It is currently not known how long these parity constraints need to be. We
close the gap by providing matching necessary and sufficient conditions on the
required asymptotic length of the parity constraints. Further, we provide a new
family of lower bounds and the first non-trivial upper bounds on the model
count that are valid for arbitrarily short XORs. We empirically demonstrate the
effectiveness of these bounds on model counting benchmarks and in a
Satisfiability Modulo Theory (SMT) application motivated by the analysis of
contingency tables in statistics.Comment: The 30th Association for the Advancement of Artificial Intelligence
(AAAI-16) Conferenc
An Approximation Algorithm for #k-SAT
We present a simple randomized algorithm that approximates the number of
satisfying assignments of Boolean formulas in conjunctive normal form. To the
best of our knowledge this is the first algorithm which approximates #k-SAT for
any k >= 3 within a running time that is not only non-trivial, but also
significantly better than that of the currently fastest exact algorithms for
the problem. More precisely, our algorithm is a randomized approximation scheme
whose running time depends polynomially on the error tolerance and is mildly
exponential in the number n of variables of the input formula. For example,
even stipulating sub-exponentially small error tolerance, the number of
solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For
4-CNF input the bound increases to O(1.6155^n).
We further show how to obtain upper and lower bounds on the number of
solutions to a CNF formula in a controllable way. Relaxing the requirements on
the quality of the approximation, on k-CNF input we obtain significantly
reduced running times in comparison to the above bounds
Bit-Vector Model Counting using Statistical Estimation
Approximate model counting for bit-vector SMT formulas (generalizing \#SAT)
has many applications such as probabilistic inference and quantitative
information-flow security, but it is computationally difficult. Adding random
parity constraints (XOR streamlining) and then checking satisfiability is an
effective approximation technique, but it requires a prior hypothesis about the
model count to produce useful results. We propose an approach inspired by
statistical estimation to continually refine a probabilistic estimate of the
model count for a formula, so that each XOR-streamlined query yields as much
information as possible. We implement this approach, with an approximate
probability model, as a wrapper around an off-the-shelf SMT solver or SAT
solver. Experimental results show that the implementation is faster than the
most similar previous approaches which used simpler refinement strategies. The
technique also lets us model count formulas over floating-point constraints,
which we demonstrate with an application to a vulnerability in differential
privacy mechanisms
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