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We compare the expressiveness of two extensions of monadic second-order logic(MSO) over the class of finite structures. The first, counting monadicsecond-order logic (CMSO), extends MSO with first-order modulo-countingquantifiers, allowing the expression of queries like ``the number of elementsin the structure is even''. The second extension allows the use of anadditional binary predicate, not contained in the signature of the queriedstructure, that must be interpreted as an arbitrary linear order on itsuniverse, obtaining order-invariant MSO.  While it is straightforward that every CMSO formula can be translated into anequivalent order-invariant MSO formula, the converse had not yet been settled.Courcelle showed that for restricted classes of structures both order-invariantMSO and CMSO are equally expressive, but conjectured that, in general,order-invariant MSO is stronger than CMSO.  We affirm this conjecture by presenting a class of structures that isorder-invariantly definable in MSO but not definable in CMSO.

Order-Invariant MSO is Stronger than Counting MSO in the Finite