Monadicity of Non-deterministic Logical Matrices is Undecidable

Pedro Filipe
(Instituto de Telecomunicações - Instituto Superior Técnico)
Carlos Caleiro
(Instituto de Telecomunicações - Instituto Superior Técnico)
Sérgio Marcelino
(Instituto de Telecomunicações - Instituto Superior Técnico)

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.

In Andrzej Indrzejczak and Michał Zawidzki: Proceedings of the 10th International Conference on Non-Classical Logics. Theory and Applications (NCL 2022), Łódź, Poland, 14-18 March 2022, Electronic Proceedings in Theoretical Computer Science 358, pp. 55–67.
Published: 14th April 2022.

ArXived at: https://dx.doi.org/10.4204/EPTCS.358.5 bibtex PDF
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