Dynamic topological logic interpreted over minimal systems
This investigation of a propositional logical system has a chance to be interesting to mainstream mathematicians. Dynamic topological logic studies dynamic topological systems in a modal-temporal propositional language with connectives \(\square\) (interior of a set), \(\circ\) (next) and \(*\) (always in future). A dynamic system is minimal if no proper subset of the carrier space is invariant (under a continuous function interpreting the \(\circ\) connective). The author proves that the logic \(\mathrm {DTL}_\mathcal{M}\) of minimal systems in this language is decidable. This is established via a kind of finite model property using Kruskal's lemma. Since this lemma has complexity far exceeding one of the first-order arithmetic, \(\mathrm {DTL}_\mathcal{M}\) may turn out to be the first example of a useful propositional system with impredicative complexity. On the negative side, \(\mathrm {DTL}_\mathcal{M}\) in not Kripke-complete, since the unit circle under the action of an irrations rotation refutes the formula \({*}\square\, p\to \square {*}p\), which is valid for dynamic systems based on Aleksandrov spaces. The main part of the paper is taken up by the completeness proof for systems of Kripke models called here nondeterministic quasimodels. A non-primitive recursive lower complexity bound is established by embedding DTL with finite iterations into \(\mathrm {DTL}_\mathcal{M}\). Although decidability implies axiomatizability, no familiar-style recursive axiomatization for \(\mathrm {DTL}_\mathcal{M}\) is known. The general propositional logic DTL of dynamic topological systems was proved by the author to be recursively enumerable, but no familiar-style recursive axiomatization is known either. However, the author proves that the schema \(\exists \square\phi\to \forall\neg*\neg\phi\) axiomatizes \(\mathrm {DTL}_\mathcal{M}\) over DTL.
scientific article; zbMATH DE number 6018335
scientific article