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Order-invariant formulas access an ordering on a structure's universe, butthe model relation is independent of the used ordering. Order invariance isfrequently used for logic-based approaches in computer science. Order-invariantformulas capture unordered problems of complexity classes and they model theindependence of the answer to a database query from low-level aspects ofdatabases. We study the expressive power of order-invariant monadicsecond-order (MSO) and first-order (FO) logic on restricted classes ofstructures that admit certain forms of tree decompositions (not necessarily ofbounded width).  While order-invariant MSO is more expressive than MSO and, even, CMSO (MSOwith modulo-counting predicates), we show that order-invariant MSO and CMSO areequally expressive on graphs of bounded tree width and on planar graphs. Thisextends an earlier result for trees due to Courcelle. Moreover, we show thatall properties definable in order-invariant FO are also definable in MSO onthese classes. These results are applications of a theorem that shows how tolift up definability results for order-invariant logics from the bags of agraph's tree decomposition to the graph itself.

Order Invariance on Decomposable Structures