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.
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.
