  The notion of non-deterministic logical matrix (where connectives are
interpreted as multi-functions) preserves many good properties of traditional
semantics based on logical matrices (where connectives are interpreted as
functions) whilst finitely characterizing a much wider class of logics, and has
proven to be decisive in a myriad of recent compositional results in logic.
Crucially, when a finite non-deterministic matrix satisfies monadicity
(distinct truth-values can be separated by unary formulas) one can
automatically produce an axiomatization of the induced logic. Furthermore, the
resulting calculi are analytical and enable algorithmic proof-search and
symbolic counter-model generation.
  For finite (deterministic) matrices it is well known that checking monadicity
is decidable. We show that, in the presence of non-determinism, the property
becomes undecidable. As a consequence, we conclude that there is no algorithm
for computing the set of all multi-functions expressible in a given finite
Nmatrix. The undecidability result is obtained by reduction from the halting
problem for deterministic counter machines.
