A \emph{quasi-identity} is a formula of the form
\[
\forall x_1, \ldots, x_n \,:\, (s_1 (\overline{x}) \approx
t_1 (\overline{x}) \wedge
\cdots
\wedge s_k (\overline{x}) \approx t_k (\overline{x})) \rightarrow
u (\overline{x}) \approx v (\overline{x}),
\]
where $s_i,t_i,u,v$ are terms in the language of some algebra $\mathbf{A}$.
Such a quasi-identity is valid in $\mathbf{A}$ if
the solutions of
$s_1 \approx t_1 \wedge \cdots \wedge s_k \approx t_k$ are a subset of the solutions
of $u \approx v$.
Checking the validity of a quasi-identity in a given algebra
$\mathbf{A}$ is closely related to the \emph{identity checking} and
\emph{equation solving} problems that have already been considered
in a universal algebraic framework, for example by B.\ Larose and L.\ Z\'{a}dori.
For semigroups, the computational complexity of checking
the validity of a quasi-identity has been investigated by M.\ Volkov.
We determine its computational
complexity for finite Mal'cev algebras of finite type.
This is joint research with Simon Gr\"unbacher (JKU Linz). The
main result has also been presented at STACS 2023.