Monadicity of Non-deterministic Logical Matrices is Undecidable

TL;DR

This paper proves that determining monadicity in finite non-deterministic logical matrices is undecidable, contrasting with the decidability in deterministic cases, impacting the computability of logical properties.

Contribution

It establishes the undecidability of monadicity in finite non-deterministic matrices, extending the understanding of logical matrix properties and their computational limits.

Findings

Monadicity is decidable for finite deterministic matrices.

Monadicity becomes undecidable in the non-deterministic case.

Undecidability is shown via reduction from the halting problem.

Abstract

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.