We study a notion ofpartial primitive recursion (p.p.r.) including the concept ofparallelism in the context of partial continuous functions of type level one in the sense of [Krei], [Sco82], [Ers]. A variety of subrecursive hierarchies with respect top.p.r. is introduced and it turns out that they all coincide.

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 Müller 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 Müller 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. (shrink)

The paper studies a simply typed term system Mω providing a primitive recursive concept of parallelism in the sense of Plotkin. The system aims at defining and computing partial continuous functionals. Some connections between denotational and operational semantics → for Mω are investigated. It is shown that → is correct with respect to the denotational semantics. Conversely, → is complete in the sense that if a program denotes some number k, then it is reducible to the numeral nk. Restricting to (...) the primitive recursive kernel Rω of Mω, it is shown that → is strongly normalising with uniquely determined normal forms. The twist is the design of fixed point style conversion rules for constants accounting for parallelly bounded parallel search such that correctness and strong normalisation hold. Thereupon, minor alternations to → bring about that every reduction sequence for a program of Rω terminates either in a numeral nk if the program denotes k, or in the term 0 if the program denotes the “undefined” object. Thus, Rω can be considered a primitive recursive version of Plotkin's PA+·. (shrink)

A key problem in implicit complexity is to analyse the impact on program run times of nesting control structures, such as recursion in all finite types in functional languages or for-do statements in imperative languages.Three types of programs are studied. One type of program can only use ground type recursion. Another is concerned with imperative programs: ordinary loop programs and stack programs. Programs of the third type can use higher type recursion on notation as in functional programming languages.The present approach (...) to analysing run time yields various characterisations of fptime and all Grzegorczyk classes at and above the flinspace level. Central to this approach is the distinction between “top recursion” and “side recursion”. Top recursions are the only form of recursion which can cause an increase in complexity. Counting the depth of nested top recursions leads to several versions of “the μ-measure”. In this way, various recent solutions of key problems in implicit complexity are integrated into a uniform approach. (shrink)

A well-known result states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑n formulas is equivalent to the uniform reflection principle for ∑n + 1 formulas . We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑n formulas is equivalent to ω times iterated ∑n reflection principle. Moreover, for k < ω, k (...) times iterated ∑n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of ∑n induction rule.The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory by k nested applications of ∑n or Πn induction rules. In particular, the closure of a theory T under just one application of ∑1 induction rule is equivalent to T together with ∑1 reflection principle for each finite Π2 axiomatized subtheory of T.These results have closely parallel ones in the theory of subrecursive function classes. The rules under study correspond, in a canonical way, to natural closure operators on the classes of provably recursive functions. Thus, ∑1 induction rule precisely corresponds to the primitive recursive closure operator, and ∑1 collection rule, introduced below, corresponds to the elementary closure operator. (shrink)