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TR18-020 | 30th January 2018 08:33
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#### Computing the maximum using $(\min, +)$ formulas

**Abstract:**
We study computation by formulas over $(min, +)$. We consider the computation of $\max\{x_1,\ldots,x_n\}$

over $\mathbb{N}$ as a difference of $(\min, +)$ formulas, and show that size $n + n \log n$ is sufficient and necessary. Our proof also shows that any $(\min, +)$ formula computing the minimum of all sums of $n - 1$ out of $n$ variables must have $n \log n$ leaves; this too is tight. Our proofs use a

complexity measure for $(\min, +)$ functions based on minterm-like behaviour and on the entropy of an

associated graph.

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Comment #1 to TR18-020 | 27th March 2018 10:52
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#### (Min, Plus) is Not stronger than (Or, And)

**Abstract:**
We observe that a known structural property of (min,+) circuits (and formulas) implies that lower bounds on the monotone circuit/formula size remain valid also for (min,+) circuits, even when only nonnegative integer weights are allowed. So, the lower bound proved in ECCC TR18-020 can be alternatively derived from known lower bounds on the monotone formula complexity of the threshold-2 function.