Which CNF boolean formulas blow up exponentially at conversion to DNF?

Question

If I'm correct, some boolean formulas in CNF require exponential size when being converted to an equivalent DNF version (and vice versa).

But what is an example of such a formula (and is there a general way to capture their structure - if the first question is too easy, and I'm just too stupid to see it)?

In this post, the parity function is discussed as an example of a function which blows up exponentially when converted to both CNF and DNF from a non-NF version: Formulas for which any equivalent CNF formula has exponential length

But what about just CNF <=> DNF? And how frequent are these "bad" formulas? (Very rare, I'd suspect, but can it be proved?)

Answer 1

The classical example is $$(x_1 \lor y_1) \land (x_2 \lor y_2) \land \cdots \land (x_n \lor y_n)$$ which blows up to $2^n$ terms when converted to a DNF.

Most functions have CNF and DNF complexity very large – $\Theta(2^n/n)$ if I remember correctly – and so for random functions there is no blow-up for trivial reasons.

Miltersen, Radhakrishnan and Wegener construct a monotone function which has an exponential gap.