Unbounded Recursion and Non-size-increasing Functions

We investigate the computing power of function algebras defined by means of unbounded recursion on notation. We introduce two function algebras which contain respectively the regressive logspace computable functions and the non-size-increasing logspace computable functions. However, such algebras are unlikely to be contained in the set of logspace computable functions because this is equivalent to L = P. Finally, we introduce a function algebra based on simultaneous recursion on notation for the non-size-increasing functions computable in polynomial time and linear space.