Bounded variable logics: two, three, and more

Consider the bounded variable logics L k 1! (with k variable symbols), and C k 1! (with k variables in the presence of counting quantiiers 9 >m). These fragments of innnitary logic L 1! are well known to provide an adequate logical framework for some important issues in nite model theory. This paper deals with a translation that associates equivalence of structures in the k-variable fragments with bisimulation equivalence between derived structures. Apart from a uniform and intuitively appealing treatment of these equivalences , this approach relates some interesting issues for the case of an arbitrary number of variables to the case of just three variables. Invertibility of the in-variants for C 3 , in particular, would imply a positive answer to the tempting conjecture that xed-point logic with counting captures S k Ptime \ C k 1! .