Suppose that for a psychological investigation, tasks including linear-recursive 6-term number sequences of maximal recursion degree r=3 are to be utilized; the sequences shall not be constant, and their members as well as the continuations sought shall consist of natural numbers. The problem to analyze is how to construct appropriate tasks that are solvable at all and, moreover, that are solvable by some uniquely determined solution.
The problem introduced above actually occurred in a certain research context of knowledge psychology. We will analyze this specific problem with the purpose of carrying out an exemplary mathematical task-type analysis with respect to the basic problems of solvability and uniqueness of solution.
Apparently for a (non-constant) 6-term sequence the necessary condition of Corollary 4.6 for the uniqueness of a recursion formula within is not satisfied because . From that follows: If there exist recursion formulae within then there even exist infinitely many; furthermore, in this case, there may exist also (possibly infinitely many) recursion formulae in , and possibly even in . Moreover, according to Theorem 4.9 each number sequence that is linear-recursive of degree 1 or 2 is also linear-recursive with infinitely many recursion formulae of the higher degree(s) until 3 inclusively.
Restricting the maximal permissible recursion degree for 6-term number sequences to r=3 does not provide sufficient control of the (non-)uniqueness problem. Therefore, according to the approach proposed in 4.15 b) we additionally impose some restrictions for the domain of permissible recursion formulae. From a psychological viewpoint the following restriction for is suggested:
5.1 Restriction (R).
Formally, the Restriction (R) implies that only the following domains
for the recursion formulae of different degrees
are to be taken into account:
The restriction in is introduced only for the
purpose that a formula can be assigned
uniquely to the domain with the consequence that
; the restrictions for
and are introduced correspondingly.
Thus, the above domains are pairwise disjoint.
Furthermore, the following abbreviations will be used in this section:
Observing these conventions, we next formulate the special type of number sequence task obtained from the general type of task in 3.1 by imposing the restrictions for the recursion degree and the permitted recursion formulae.
5.2 Special type of number sequence task.
Given some (non-constant) 6-term sequence of natural numbers and the maximal permissible degree of recursion.
As remarked following 3.2, the solvability for some number sequence task is usually ensured by constructing the sequence according to a certain permitted formula. Thus the more serious problem in our context is the (non-)uniqueness problem for the solutions. The following analyses therefore concentrate on the (non-)uniqueness problem for our special type of number sequence task.
5.3 The problem of uniqueness of solution.
Let be a 6-term sequence. Assume that for some there exists an such that is -recursive and is the onefold -extension of .
Introducing the apparently rather strong Restriction (R) for the domain of permissible recursion formulae may suggest two hypotheses concerning our (non-)uniqueness problem for solutions - a ''strong'' hypothesis and a ''weak'' hypothesis.
With these hypotheses as guidelines we will analyze the uniqueness problem in the following section.