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On dynamic topological and metric logics. (English) Zbl 1114.03026
Summary: We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems $$\langle W,f\rangle$$, where $$W$$ is a topological space and $$f$$ a homeomorphism on $$W$$. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits $$\{w, f(w), f^{2}(w),\dots\}$$ of points $$w \in W$$. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems $$\langle W,f\rangle$$, where $$W$$ is a metric space and $$f$$ and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius $$a$$’, for $$a \in \mathbb {Q}^+$$. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.

##### MSC:
 03B60 Other nonclassical logic 03B45 Modal logic (including the logic of norms) 03B44 Temporal logic 03B25 Decidability of theories and sets of sentences 54H20 Topological dynamics (MSC2010)
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