The relationship between fixpoint logic and the infinitary logic L/sub infinity omega //sup omega / with a finite number of variables is studied. It is observed that the equivalence of two finite structures with respect to L/sub infinity omega //sup omega / is expressible in fixpoint logic. As a first application of this, a normal-form theorem for L infinity /sub omega //sup omega / on finite structures is obtained. The relative expressive power of first-order logic, fixpoint logic, and L/sub infinity omega //sup omega / on arbitrary classes of finite structures is examined. A characterization of when L/sub infinity omega //sup omega / collapses to first-order logic on an arbitrary class of finite structures is given.
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