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We investigate the expressive power of certain fragments of second-order logic on finite structures. The fragments are second-order Horn logic, second-order Krom logic as well as a symmetric and a deterministic version of the latter. It is shown that all these logics collapse to their existential fragments. In the presence of successor relation they provide characterizations of polynomial time, deterministic and nondeterministic logspace and of the complement of symmetric logspace. Without successor relation these logics still can express certain problems that are complete in the corresponding complexity classes, but on the other hand they are strictly weaker than previously known logics for these classes and fail to express some very simple properties.