### Paper:

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TR05-126 | 5th November 2005 00:00
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#### Minimizing DNF Formulas and AC0 Circuits Given a Truth Table

**Abstract:**
For circuit classes R, the fundamental computational problem, Min-R,

asks for the minimum R-size of a boolean function presented as a truth

table. Prominent examples of this problem include Min-DNF, and

Min-Circuit (also called MCSP). We begin by presenting a new reduction

proving that Min-DNF is NP-complete. It is significantly simpler than

the known reduction of Masek, which is from Circuit-SAT. We then present

a more complex, approximation-preserving reduction, that yields new

inapproximability results for Min-DNF. Finally, we extend known hardness

results for Min-TC0 to obtain new hardness results for Min-AC0, under

cryptographic assumptions.

### Comment(s):

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Comment #1 to TR05-126 | 25th July 2021 10:55
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#### Correction to the statement of Lemma 14

Authors:

Eric Allender
Accepted on: 25th July 2021 10:55

Keywords:

**Comment:**
Lemma 14 states that there is a partition system with $m = O(2^h h \log t)$. However, if we make use of the construction given by Lemma 3.2 of [Lund, Yannakakis; JACM 1994] we actually obtain the bound $m = O(2^{2h}t^2}$.

The bound is used a few lines after Lemma 14, in giving a bound for the size of the entire universe: $n^{O(k)} 2^{O(h)}$. This bound still holds, using the corrected value in the statement of Lemma 14.

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Comment #2 to TR05-126 | 25th July 2021 14:01
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#### LaTeX error makes the previous comment unreadable.

Authors:

Eric Allender
Accepted on: 25th July 2021 14:01

Keywords:

**Comment:**
Trying again. (ECCC doesn't have a "preview" option for comments.)

Lemma 14 states that there is a partition system with $m=O(2^h h \log t)$. However, if we make use of the construction given by Lemma 3.2 of [Lund, Yannakakis; JACM 1994] we actually obtain the bound $m = O(2^{2h} t^2)$.

The bound is used a few lines after Lemma 14, in giving a bound for the size of the entire universe: $n^{O(k)} 2^{O(h)}$. This bound still holds, using the corrected value in the statement of Lemma 14.