Kolaitis and Kopparty have shown that for any first-order formula with
parity quantifiers over the language of graphs there is a family of
multi-variate polynomials of constant-degree that agree with the
formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$
vertices. The proof yields a bound on the degree of the polynomials
that is a tower of exponentials of height as large as the nesting
depth of parity quantifiers in the formula. We show that this
tower-type dependence on the depth of the formula is necessary. We
build a family of formulas of depth $q$ whose approximating
polynomials must have degree bounded from below by a tower of
exponentials of height proportional to $q$. Our proof has two main
parts. First, we adapt and extend the results by Kolaitis and Kopparty that describe the joint
distribution of the parity of the number of copies of small subgraphs
on a random graph to the setting of graphs of growing size. Secondly,
we analyse a variant of Karp's graph canonical labelling algorithm and
exploit its massive parallelism to get a formula of low depth that
defines an almost canonical pre-order on a random graph.

More details were added to the proof of Lemma 3.

Kolaitis and Kopparty have shown that for any first-order formula with parity quantifiers over the language of graphs there is a family of multi-variate polynomials of constant-degree that agree with the formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$ vertices. The proof bounds the degree of the polynomials by a tower of exponentials in the nesting depth of parity quantifiers in the formula. We show that this tower-type dependence is necessary. We build a family of formulas of depth $q$ whose approximating polynomials must have degree bounded from below by a tower of exponentials of height proportional to $q$. Our proof has two main parts. First, we adapt and extend known results describing the joint distribution of the parity of the number of copies of small subgraphs on a random graph to the setting of graphs of growing size. Second, we analyse a variant of Karp's graph canonical labeling algorithm and exploit its massive parallelism to get a formula of low depth that defines an almost canonical pre-order on a random graph.

Kolaitis and Kopparty have shown that for any first-order formula with
parity quantifiers over the language of graphs there is a family of
multi-variate polynomials of constant-degree that agree with the
formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$
vertices. The proof yields a bound on the degree of the polynomials
that is a tower of exponentials of height as large as the nesting
depth of parity quantifiers in the formula. We show that this
tower-type dependence on the depth of the formula is necessary. We
build a family of formulas of depth $q$ whose approximating
polynomials must have degree bounded from below by a tower of
exponentials of height proportional to $q$. Our proof has two main
parts. First, we adapt and extend known results describing the joint
distribution of the parity of the number of copies of small subgraphs
on a random graph to the setting of graphs of growing size. Secondly,
we analyse a variant of Karp's graph canonical labelling algorithm and
exploit its massive parallelism to get a formula of low depth that
defines an almost canonical pre-order on a random graph.