Subrecursive functions on partial sequences

Abstract

The paper studies a domain theoretical notion of primitive recursion over partial sequences in the context of Scott domains. Based on a non-monotone coding of partial sequences, this notion supports a rich concept of parallelism in the sense of Plotkin. The complexity of these functions is analysed by a hierarchy of classes \({\cal E}^{\bot}_n\) similar to the Grzegorczyk classes. The functions considered are characterised by a function algebra \({\cal R}^{\bot}\) generated by continuity preserving operations starting from computable initial functions. Its layers \({\cal R}^{\bot}_n\) are related to those above by showing \(\forall n \ge 2.{\cal E}^{\bot}_{n+1} ={\cal R}^{\bot}_n\), thus generalising results of Schwichtenberg/Muller and Niggl.