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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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