Spectra and satisfiability for logics with successor and a unary func- tion

Abstract : We investigate the expressive power of two logics, both with the successor function: first-order logic with an un-interpreted function, and existential monadic second order logic — that is first-order logic over words —, with multiplication by a constant b. We prove that all b-recognizable sets are spectra of those logics. Furthermore, it is proven that some encoding of the set of halting times of a non-deterministic 2-counter automaton is also a spectrum. This yields undecidability of the finite satisfiability problem for those logics. Finally, it is shown that first-order logic with one uninterpreted function and successor can encode quickly increasing functions, such as the Knuth's up-arrows.
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Arthur Milchior. Spectra and satisfiability for logics with successor and a unary func- tion. Mathematical Logic Quarterly, Wiley, In press. 〈hal-01676718〉

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