When Do Fixed Point Logics Capture Complexity Classes?

We give examples of classes of rigid structures which are of unbounded rigidity but Least xed point (Partial xed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that deenability of linear order in FO+LFP although suucient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very diier-ent for nonzero-ary queries. Next, we turn to the study of xed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of nite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some suu-cient conditions for PFP to capture PSPACE queries on classes of nite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to capture PTIME seems harder and remains open.