Decidability and Universality of Quasiminimal Subshifts
نویسنده
چکیده
We show that there exists a universal subshift having only a finite number of minimal subsystems, refuting a conjecture in [Delvenne, Kůrka, Blondel, ’05]. We then introduce the still smaller class of quasiminimal subshifts, having finitely many subsystems in total. With N-actions, their theory essentially reduces to the theory of minimal systems, but with Zactions, the class is much larger. We construct a universal system in this class, and show examples of such systems generated by substitutions. In fact, while [Delvenne, Kůrka, Blondel, ’05] shows that minimal systems have a decidable model-checking problem for regular languages, we show that one (proper nontrivial) subsystem, which can be chosen more or less arbitrarily among minimal systems, is enough to make this problem Σ01complete, even when also restricting to a small subclass of the regular languages. On the other hand, the halting problem (given two clopen sets U,V , is V reachable from U in some number of steps?) is decidable for quasiminimal subshifts with any amount of subsystems. We also consider these questions with the additional restriction of countability. For countable quasiminimal subshifts, the model-checking problem of regular languages becomes Σ01-complete at 2 subsystems, and the model-checking problem of starfree languages at 4 subsystems. We also investigate the boundary of the result of [Delvenne, Kůrka, Blondel, ’05] in another direction, and show that the model-checking problem of context-free languages is not decidable even for minimal Π01 subshifts. We also give an example of a minimal subshift whose language is Σ01-complete (for Turing reductions) but not Π01.
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عنوان ژورنال:
- J. Comput. Syst. Sci.
دوره 89 شماره
صفحات -
تاریخ انتشار 2017