Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Improved Homomorphism Preservation Theorem From Lower Bounds in
  Circuit Complexity   Previous work of the author [39] showed that the Homomorphism Preservation
Theorem of classical model theory remains valid when its statement is
restricted to finite structures. In this paper, we give a new proof of this
result via a reduction to lower bounds in circuit complexity, specifically on
the AC$^0$ formula size of the colored subgraph isomorphism problem. Formally,
we show the following: if a first-order sentence $\Phi$ of quantifier-rank $k$
is preserved under homomorphisms on finite structures, then it is equivalent on
finite structures to an existential-positive sentence $\Psi$ of quantifier-rank
$k^{O(1)}$. Quantitatively, this improves the result of [39], where the upper
bound on the quantifier-rank of $\Psi$ is a non-elementary function of $k$.

31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics