Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences [J.-P. Ressayre, Formal languages defined by the underlying structure of their words, J. Symbolic Logic 53 (4) (1988) 1009-1026]. Another stretching theorem of Finkel and Ressayre implies that one can decide, for a given local sentence ϕ and an ordinal α < ωω, whether ϕ has a model of order type α. This result is very similar to Buchi's one who proved that the monadic second order theory of the structure (α, <), for a countable ordinal α, is decidable. It is in fact an extension of that result, as shown in [O. Finkel, finite languages, Theoret. Comput. Sci. 255 (1-2) (2001) 223-261] by considering the expressive power of monadic sentences and of local sentences over languages of words of length α. The aim of this paper is twofold. We wish first to attract the reader's attention on these powerful decidability results proved using methods of model theory and which should find some applications in computer science and we prove also here several additional results on local sentences.The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal ωα (the αth infinite cardinal), whether a local sentence ϕ has a model of order type ωα.Secondly we show that this result cannot be extended to the general case. Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type ω2 is not determined by the axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesis.Next we prove that for all integers n, p ≥ 1, if n < p then the local theory of ωn, i.e. the set of local sentences having a model of order type ωn, is recursive in the local theory of ωp and also in the local theory of α where α is any ordinal of cofinality ωn.
['Olivier Finkel']
On decidability properties of local sentences