The $\mu$ -measure as a tool for classifying computational complexity

Abstract

Two simply typed term systems \(\sf {PR}_1\) and \(\sf {PR}_2\) are considered, both for representing algorithms computing primitive recursive functions. \(\sf {PR}_1\) is based on primitive recursion, \(\sf {PR}_2\) on recursion on notation. A purely syntactical method of determining the computational complexity of algorithms in \(\sf {PR}_i\), called $\mu$-measure, is employed to uniformly integrate traditional results in subrecursion theory with resource-free characterisations of sub-elementary complexity classes. Extending the Schwichtenberg and Muller characterisation of the Grzegorczyk classes \({\mathcal{E}}_n\) for \(n\ge 3\), it is shown $\mathcal{E}_{n+1} = \mathcal{R}^n_1\( for \)n\ge 1\(, where \)\mathcal{R}^n_i$ denotes the \emph{\(n\)th modified Heinermann class} based on \(\mu\). The proof does not refer to any machine-based computation model, unlike the Schwichtenberg and Muller proofs. This is due to the notion of modified recursion lying on top of each other provided by \(\mu\). By Ritchie's result, \(\mathcal{R}^1_1\) characterises the linear-space computable functions. Using the same method, a short and straightforward proof is presented, showing that \(\mathcal{R}^1_2\) characterises the polynomial time computable functions. Furthermore, the classes \(\mathcal{R}^n_2\) and \(\mathcal{R}^n_1\) coincide at and above level 2.