Expressive equivalence of least and inflationary fixed-point logic

We study the relationship between least and inflationary fixed-point logic. In 1986, Gurevich and Shelah proved that in the restriction to finite structures, the two logics have the same expressive power. On infinite structures however, the question whether there is a formula in IFP not equivalent to any LFP-formula was left open.

In this paper, we answer the question negatively, i.e. we show that the two logics are equally expressive on arbitrary structures. We give a syntactic translation of IFP-formulae to LFP-formulae such that the two formulae are equivalent on all structures.

As a consequence of the proof we establish a close correspondence between the LFP-alternation hierarchy and the IFP-nesting depth hierarchy. We also show that the alternation hierarchy for IFP collapses to the first level, i.e. the complement of any inflationary fixed point is itself an inflationary fixed point.     

Torsion of differentials on toric varieties

We introduce an invariant for semigroups with cancellation property. When the semigroup equals the set of lattice points in a rational, polyhedral cone, then this invariant describes the torsion of the differential sheaf on the associated toric variety. Finally, as an example, we present the case of two-dimensional cones (corresponding to two-dimensional cyclic quotient singularities).     

Diffusive clustering in an infinite system of hierarchically interacting diffusions

Summary We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).