Bounded variable logics: two, three, and more

Abstract.

Consider the bounded variable logics $L^k_{\infty\omega}$ (with k variable symbols), and $C^k_{\infty\omega}$ (with k variables in the presence of counting quantifiers $\exists^{\geq m}$). These fragments of infinitary logic $L_{\infty\omega}$ are well known to provide an adequate logical framework for some important issues in finite 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 invariants for $\equiv_{C^3}$, in particular, would imply a positive answer to the tempting conjecture that fixed-point logic with counting captures $\bigcup_k$ Ptime $\cap C^k_{\infty\omega}$.