Computational Properties of Partial Non-deterministic Matrices and Their Logics.

Incorporating non-determinism and partiality in the traditional notion of logical matrix semantics has proven to be decisive in a myriad of recent compositionality results in logic. However, several important properties which are known to be computable for finite matrices have not been studied in this wider context of partial non-deterministic matrices (PNmatrices). This paper is dedicated to understanding how this generalization of the considered semantical structures affects the computational properties of basic problems regarding their induced logics, in particular their sets of theorems. We will show that the landscape is quite rich, as some problems keep their computational status, for others the complexity increases, and for a few decidability is lost. Namely, we show that checking if the logics defined by two finite PNmatrices have the same theorems is undecidable. This latter result is obtained by reduction from the undecidable problem of checking universality of term-DAG-automata.