Abstract
In several previous works the construction of a computationally hard function with respect to a certain class of algorithms or Boolean circuits has been used to derive small pseudo-random spaces. In this paper, we revert this connection by presenting two new direct relations between the efficient construction of pseudo-random (both two-sided and one-sided) sets for Boolean affine spaces and the explicit construction of Boolean functions having hard branching program complexity
In the case of 1-read branching programs (1-Br.Pr.), we show that the construction of non trivial (i.e. of cardinality 2o(n)) discrepancy sets (i.e. two-sided pseudo-random sets) for Boolean affine spaces of dimension greater than n/2 yield a set of explicit Boolean functions having very hard 1-Br.Pr. size. By combining the best known construction of ∈-biased sample spaces for linear tests and a simple “Reduction” Lemma, we derive the required discrepancy set and obtain a Boolean function in P having 1-Br.Pr. size not smaller than \(2^{n - O\left( {log^2 n} \right)} \) and a Boolean function in DTIME\(\left( {2^{O\left( {log^2 n} \right)} } \right)\) having 1-Br.Pr. size not smaller than 2n-O(log n). The latter bound is optimal and both of them are exponential improvements over the best previously known lower bound that was \(2^{n - 3n^{1/2} } \)[21].
As for non deterministic syntactic k-read branching programs (k-Br.Pr.), we introduce a new method to derive explicit, exponential lower bounds that involves the construction of hitting sets (one-sided pseudo-random sets) for affine spaces of dimension o(n=2). Using an appropriate “orthogonal” representation of small Boolean affine spaces, we efficiently construct these hitting sets thus obtaining an explicit Boolean function in P that has k-Br.Pr. size not smaller than \(2^{n^{1 - o\left( 1 \right)} } \) for any \(k = o\left( {\frac{{\log n}}{{\log \log n}}} \right)\). This improves over the previous best known lower bounds given in [8,11,17] for some range of k
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Andreev, A.E., Baskakov, J.L., Clementi, A.E.F., Rolim, J.D.P. (1999). Small Pseudo-Random Sets Yield Hard Functions: New Tight Explicit Lower Bounds for Branching Programs. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds) Automata, Languages and Programming. Lecture Notes in Computer Science, vol 1644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48523-6_15
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DOI: https://doi.org/10.1007/3-540-48523-6_15
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