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

Arthur Milchior (1, 2)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 LACL - Laboratoire d'Algorithmique Complexité et Logique

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.

Domaines: Mathématiques [math]; Logique [math.LO]

Arthur Milchior. Spectra and satisfiability for logics with successor and a unary func- tion. Mathematical Logic Quarterly, In press. ⟨hal-01676718⟩