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Suppose we have an algorithm for a decision problem with $n$ bit inputs that runs in $DTIME[f(n)]$ is there ways to convert to circuits of $O(f(n))$ size with AND, OR and NOT gates?

How about when we go from circuits to programs?

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The Cook–Levin theorem shows how to construct a circuit of size $f(n)^{O(1)}$. I'm not sure what's the best exponent.

The opposite direction is impossible, since circuits of size 1 can compute the following language: $$ \{ w : \text{the $|w|$th Turing machine halts on the empty input} \}. $$ More generally, circuits of size 1 can compute any language in which the answer depends only on the length of the input.

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  • $\begingroup$ 'circuits of size 1 can compute the following language:....' why? $\endgroup$ – T.... May 23 at 7:16
  • $\begingroup$ That's a nice question for you. $\endgroup$ – Yuval Filmus May 23 at 7:21
  • $\begingroup$ Dont get it that looks undecidable. $\endgroup$ – T.... May 23 at 7:22
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    $\begingroup$ @Brout It's a map from length to a non-computable predicate. That's the most standard example for non-uniformity allowing for strictly more than r.e. languages. $\endgroup$ – dkaeae May 23 at 7:22
  • $\begingroup$ @dkaee Is there reference for this (it is new to me)? $\endgroup$ – T.... May 23 at 7:32

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