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TR15-099 | 12th June 2015 12:49
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#### Satisfiability Algorithms and Lower Bounds for Boolean Formulas over Finite Bases

**Abstract:**
We give a \#SAT algorithm for boolean formulas over arbitrary finite bases. Let $B_k$ be the basis composed of all boolean functions on at most $k$ inputs. For $B_k$-formulas on $n$ inputs of size $cn$, our algorithm runs in time $2^{n(1-\delta_{c,k})}$ for $\delta_{c,k} = c^{-O(c^2k2^k)}$. We also show the average-case hardness of computing affine extractors using linear-size $B_k$-formulas.

We also give improved algorithms and lower bounds for formulas over finite unate bases, i.e., bases of functions which are monotone increasing or decreasing in each of the input variables.