Circuits, Pebbling and Expressibility

We give characterizations of nondeterministic complexity classes such as NP and PSPACE and the classes in the polynomial time hierarchy in the two-person pebble game model [VT89]. These characterizations motivate the definitions of these classes using firstorder sentences extending the results in [Im82]. It is shown that the role-switches resource in the pebble games closely model the levels of the polynomial hierarchy. These characterizations are made possible by explicitly considering circuit-size in the pebbling characterizations and the size of the underlying universe in the first-order characterizations.