  In the framework of computable queries in Database Theory, there are many
examples of queries to (properties of) relational database instances that can
be expressed by simple and elegant third order logic ($\mathrm{TO}$) formulae.
In many of those properties the expressive power of $\mathrm{TO}$ is not
required, but the equivalent second order logic ($\mathrm{SO}$) formulae can be
very complicated or unintuitive. From the point of view of the study of highly
expressive query languages, it is then relevant to identify fragments of
$\mathrm{TO}$ (and, in general, of higher-order logics of order $\geq 3$) which
do have an $\mathrm{SO}$ equivalent formula. In this article we investigate
this precise problem as follows. Firstly, we define a general schema of
$\exists$TO formulas which consists of existentially quantifying a third-order
linear digraph of polynomial length, that is, a sequence of structures that
represents a computation, by explicitly stating which operations are the ones
which can be involved in the construction of a given structure in the sequence,
when applied to the previous one. Then we give a constructive proof of the fact
that all $\exists$TO sub formulas of that schema can be translated into an
equivalent SO formula. Secondly, aiming to formally characterize the fragment
of TO which can be translated to SO, we define a restriction of TO, which we
denote TO$^{P}$, for polynomial TO, and we give a constructive proof on the
fact that it collapses to SO. We define TO$^{P}$ as the fragment of TO where
valuations can assign to TO relation variables only TO relations whose
cardinalities are bounded by a polynomial that depends on the quantifier.
Moreover, we define a similar restriction for every higher order logic of order
$i \geq 4$, which we denote $\mathrm{HO}^{i,P}$, and give a constructive proof
of the fact that for all $i \geq 4$, $\mathrm{HO}^{i,P}$ collapses to SO.
