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In previous papers [4, 6] we have claimed that for each function which is efficiently Turing computable
there exists a deep and sparse network which approximates it arbitrarily well. We also claimed a key role
for compositional sparsity in this result. Though the general claims are correct some of our statements
may have been imprecise and thus potentially misleading. In this short paper we wish to formally restate
our claims and provide definitions and proofs.
