Monotonicity Constraints in Characterizations of PSPACE

A celebrated contribution of Bellantoni and Cook was a function algebra to capture FPTIME. This algebra uses recursion on notation. Later, Oitavem showed that including primitive recursion, an algebra is obtained which captures FPSPACE. The main results of this paper concern variants of the latter algebra. First, we show that iteration can replace primitive recursion. Then, we consider the results of imposing a monotonicity constraint on the primitive recursion or iteration. We find that in the case of iteration, the power of the algebra shrinks to FPTIME. More interestingly, with primitive recursion, we obtain a new implicit characterisation of the polynomial hierarchy (FPH). The idea to consider these monotonicity constraints arose from the results on write-once tapes for Turing machines. We review this background and also note a new machine characterisation of ∆2 , that similarly to our function algebras, arises by combining monotonicity constraints with a known characterisation of PSPACE.