## Random Graphs

- B. Bollobas
- Academic Press
- 1985

- Published 1999 in Arch. Math. Log.

Numerous results about capturing complexity classes of queries by means of logical languages work for ordered structures only, and deal with non-generic, or order-dependent, queries. Recent attempts to improve the situation by characterizing wide classes of nite models where linear order is deenable by certain simple means have not been very promising, as certain commonly believed conjectures were recently refuted (Dawar's Conjecture). We take on another approach that has to do with normalization of a given order (rather than with deening a linear order from scratch). To this end, we show that normalizability of linear order is a strictly weaker condition than deenability (say, in the least xpoint logic), and still allows for extending Immerman-Vardi-style results to generic queries. It seems to be the weakest such condition. We then conjecture that linear order is nor-malizable in the least xpoint logic for any nitely axiomatizable class of rigid structures. Truth of this conjecture, which is a strengthened version of Stolboushkin's conjecture, would have the same practical implications as Dawar's Conjecture. Finally, we suggest a series of reductions of the two conjectures to specialized classes of graphs, which we believe should simplify further work.

@article{Stolboushkin1999NormalizableLO,
title={Normalizable linear orders and generic computations in finite models},
author={Alexei P. Stolboushkin and Michael A. Taitslin},
journal={Arch. Math. Log.},
year={1999},
volume={38},
pages={257-271}
}