  We study variants of Buss's theories of bounded arithmetic axiomatized by
induction schemes disallowing the use of parameters, and closely related
induction inference rules. We put particular emphasis on $\hat\Pi^b_i$
induction schemes, which were so far neglected in the literature. We present
inclusions and conservation results between the systems (including a witnessing
theorem for $T^i_2$ and $S^i_2$ of a new form), results on numbers of instances
of the axioms or rules, connections to reflection principles for quantified
propositional calculi, and separations between the systems.
