It is known that WS1S can be decided by a DFA. Is this the strongest arithmetic theory decidable by a DFA? What happens when the automata class is extended to include DPDAs or PDAs?
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$\begingroup$ Question: how is going from a DFA to a PDA weakening? Aren’t PDAs capable of recognizing more languages, and therefore stronger? $\endgroup$ – D. Ben Knoble Jan 26 at 17:14
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$\begingroup$ @D.BenKnoble I was thinking of "weaker" in a different sense. Question edited. $\endgroup$ – danportin Jan 26 at 19:30
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$\begingroup$ The slides you link to do not provide a proof of regularity, and I find the claim that S1S can be decided by DFAs weird. How do you represent infinite sets? This is commonly done with $\omega$-automata. $\endgroup$ – Shaull Jan 26 at 19:38
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1$\begingroup$ @Shaull Right, it's WS1S that's regular. $\endgroup$ – danportin Jan 26 at 19:49
