Abstract.
The paper builds on both a simply typed term system \({\cal PR}^\omega\) and a computation model on Scott domains via so-called parallel typed while programs (PTWP). The former provides a notion of partial primitive recursive functional on Scott domains \(D_\rho\) supporting a suitable concept of parallelism. Computability on Scott domains seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (scvr) is not reducible to partial primitive recursion. So extensions \({\cal PR}^{\omega e}\) and PTWP\(^e\) are studied that are closed under scvr. The twist are certain type 1 Gödel recursors \({\cal R}_k\) for simultaneous partial primitive recursion. Formally, \({\cal R}_k\vec{g}\vec{H}xi\) denotes a function \(f_i \in D_{\iota\to\iota}\), however, \({\cal R}_k\) is modelled such that \(f_i\) is finite, or in other words, a partial sequence. As for PTWP\(^e\), the concept of type \(\iota\to\iota\) writable variables is introduced, providing the possibility of creating and manipulating partial sequences. It is shown that the PTWP\(^e\)-computable functionals coincide with those definable in \({\cal PR}^{\omega e}\) plus a constant for sequential minimisation. In particular, the functionals definable in \({\cal PR}^{\omega e}\) denoted \({\cal R}^{\omega e}\) can be characterised by a subclass of PTWP\(^e\)-computable functionals denoted \({\rm PPR}^{\omega e}\). Moreover, hierarchies of strictly increasing classes \({\cal R}^{\omega e}_n\) in the style of Heinermann and complexity classes \({\rm PPR}^{\omega e}_n\) are introduced such that \(\forall n\ge 0. {\cal R}^{\omega e}_n ={\rm PPR}^{\omega e}_n\). These results extend those for \({\cal PR}^\omega\) and PTWP [Nig94]. Finally, scvr is employed to define for each type \(\sigma\) the enumeration functional \(E^\sigma\) of all finite elements of \(D_\sigma\).
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received January 30, 1996
Rights and permissions
About this article
Cite this article
Niggl, KH. A restricted computation model on Scott domains and its partial primitive recursive functionals. Arch Math Logic 37, 443–481 (1998). https://doi.org/10.1007/s001530050108
Issue Date:
DOI: https://doi.org/10.1007/s001530050108