We use finite model theory to prove:Let m ≥ 2. Then: If there exists k such that NP ⊆ σmTIME ∩ ΠmTIME, then for every r there exists kr such that PNP[nr] ⊆ σmTIME ∩ ΠmTIME; If there exists a superpolynomial time-constructible function f such that NTIME ⊆ Σpm ∪ Πpm, then additionally PNP[nr] ⊈ Σpm ∪ Πpm.This strengthens a result by Mocas [M96] that for any r, PNP[nr] ⊈ NEXP.In addition, we use FM-truth definitions to give a simple sufficient (...) condition for the σ11 arity hierarchy to be strict over finite models. (shrink)

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second‐order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing (...) a direct logical characterization of rudimentary languages and a representation result of second‐order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity. (shrink)

We give a plausible-sounding conjecture involving the number of n-equivalence classes of structures of size m which would imply that the complement of a spectrum is also a spectrum.

Ash's functions Nσ ,k count the number of k -equivalence classes of σ -structures of size n . Some conditions on their asymptotic behavior imply the long standing spectrum conjecture. We present a new condition which is equivalent to this conjecture and we discriminate some easy and difficult particular cases.

Dependence logic and its many variants are new logics that aim at establishing a unified logical theory of dependence and independence underlying seemingly unrelated subjects. The area of dependence logic has developed rapidly in the past few years. We will give a short introduction to dependence logic and review some of the recent developments in the area.

The spectrum of a first-order sentence is the set of cardinalities of its finite models. Given a spectrum S and a function f, it is not always clear whether or not the image of S under f is also a spectrum. In this paper, we consider questions of this form for functions that increase very quickly and for functions that increase very slowly. Roughly speaking, we prove that the class of all spectra is closed under functions that increase arbitrarily quickly, (...) but it is not closed under some natural slowly increasing functions. (shrink)