  We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an
arithmetical theory based on basic logic which is weaker than intuitionistic
logic. We show that the class of the provably total recursive functions of BA
is a proper sub-class of the primitive recursive functions. Three extensions of
BA, called BA+U, BA_c and EBA are investigated with relation to their provably
total recursive functions. It is shown that the provably total recursive
functions of these three extensions of BA are exactly the primitive recursive
functions. Moreover, among other things, it is shown that the well-known MRDP
theorem does not hold in BA, BA+U, BA_c, but holds in EBA.
