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.

Mindfulness-based training programs are highly established in competitive and recreational sports. One of the best-known approaches is the Mindfulness-Acceptance-Commitment Approach by Gardner and Moore), which integrates mindfulness aspects of awareness, non-judgmental attitude, and focus. Based on these aspects, Thienot and colleagues developed and validated an English language sport-specific questionnaire, the so-called Mindfulness Inventory for Sport, for the assessment of mindfulness skills in athletes. The aim of this study is to psychometrically test a German language version of the MIS. To assess (...) the psychometric properties, the MIS-D was examined in an online survey with an integrated test–retest design for reliability, validity, and measurement invariance. The present results support the psychometric quality of the German language version of the MIS. Necessary replications should among others focus on checking the measurement invariance for further relevant subgroups. (shrink)

1. Die Lehre vom Primat der praktischen Vernunft.

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 $D_\sigma$. (shrink)

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/Müller and Niggl. (shrink)

Zentrale Denkfiguren der von Wolfgang Klafki begründeten kritisch-konstruktiven Erziehungswissenschaft werden in diesem Buch aufgegriffen und in ihrem Innovationswert für die wissenschaftliche Pädagogik reflektiert. Namhafte Autorinnen und Autoren unterschiedlicher Subdisziplinen und Generationen vertiefen Klafkis Ansatz theoretisch wie methodisch und öffnen Problemstellungen, die bisher nicht bearbeitet worden sind.

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)

We investigate a simply typed term system ℘R ω aimed at defining partial primitive recursive functionals over arbitrary Scott domains . A hierarchy of complexity classes R n ω for functionals definable in ℘R ω is given based on a hierarchy of term classes ℘R n ωpn denoting the n th class of so-called prenormal terms . They come into play by the key observation that every term t can be transformed by what we call higher type modularization as a (...) kind of inversion of normalization into an αβη equal term t ′ having almost no structural complexity. However, it turns out that normalization of a prenormal term may increase its structural complexity with respect to the classes ℘R n ωpn , and conversely, ground type modularization being still possible may reduce it. Thus the structural complexity of a prenormal term t defined as the least n with t ϵ ℘R n ωpn depends strongly on the representation of t . We present a measure denoted μ = n rel v , ϱ for prenormal terms t to be read as t is of complexity n with valued free variables v and valued type π . It is shown that μ is stable on αβη equal terms and furthermore, μ ⩽ min {n |∃t′ ϵ ℘R n ω pn .t′ = αβη t} . Moreover, if t is in a certain μ- normal form 2, the estimate above is even true with equality, that is μ yields the structural complexity of the maximal modularization of t , clearly the best a purely structural measure can do. μ-normal forms 2 do not always exist. The counterexample we give, however, clearly shows that μ does not only take into account the structural complexity of a prenormal term but also the nature and computationl complexity of the algorithm it represents. (shrink)

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...) of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)