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Description

We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search.

Our first main result is a simple deterministic algorithm running in time $2^{n - Omega(n)}$ for satisfiability of formulae of linear size in $n$, where $n$ is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound.

Our second main result is a deterministic algorithm running in time $2^{n - Omega(n/log(n)}$ for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are ``structured'', in a certain precise sense, the algorithm can be modified to run in time $2^{n - Omega(n)}$.

To the best of our knowledge, no non-trivial algorithms were known for these problems before.

As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size $2^{n - Omega(n)}$. As a consequence, we get strong superlinear {it average-case} formula size lower bounds for the Parity function.

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