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A special type of linear-recursive number sequence tasks

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 psychologygif. 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 tex2html_wrap_inline2303 the necessary condition of Corollary 4.6 for the uniqueness of a recursion formula within tex2html_wrap_inline2305 is not satisfied because tex2html_wrap_inline2307. From that follows: If there exist recursion formulae within tex2html_wrap_inline2309 then there even exist infinitely many; furthermore, in this case, there may exist also (possibly infinitely many) recursion formulae in tex2html_wrap_inline2311, and possibly even in tex2html_wrap_inline2313. 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 tex2html_wrap_inline1527 of permissible recursion formulae. From a psychological viewpoint the following restriction for tex2html_wrap_inline1527 is suggested:

5.1 Restriction (R).

(i)
The recursion formulae have the maximal degree tex2html_wrap_inline2321.
(ii)
In each recursion formulae, whatever the degree tex2html_wrap_inline2323 may be, at most one recursion coefficient is unequal to 1. (Apart from that, the constant can take arbitrary rational values.)

Formally, the Restriction (R) implies that only the following domains for the recursion formulae of different degrees tex2html_wrap_inline2325 are to be taken into account:
displaymath2297

The restriction tex2html_wrap_inline2331 in tex2html_wrap_inline2333 is introduced only for the purpose that a formula tex2html_wrap_inline2335 can be assigned uniquely to the domain tex2html_wrap_inline2337 with the consequence that tex2html_wrap_inline2339; the restrictions for tex2html_wrap_inline2341 and tex2html_wrap_inline2343 are introduced correspondingly. Thus, the above domains are pairwise disjoint. Furthermore, the following abbreviations will be used in this section:
eqnarray518
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 tex2html_wrap_inline2347 of natural numbers tex2html_wrap_inline2349 and the maximal permissible degree tex2html_wrap_inline2351 of recursion.

I.
Find tex2html_wrap_inline2325 and tex2html_wrap_inline2355 so that tex2html_wrap_inline2303 is tex2html_wrap_inline1519-recursive.
II.
Compute tex2html_wrap_inline2361 so that tex2html_wrap_inline2363 is the onefold tex2html_wrap_inline1519-extension of tex2html_wrap_inline2303.

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 tex2html_wrap_inline2347 be a 6-term sequence. Assume that for some tex2html_wrap_inline2325 there exists an tex2html_wrap_inline2375 such that tex2html_wrap_inline2303 is tex2html_wrap_inline1519-recursive and tex2html_wrap_inline2381 is the onefold tex2html_wrap_inline1519-extension of tex2html_wrap_inline2303.

I.
Are r and tex2html_wrap_inline1519 uniquely determined ?
If that is the case, then also tex2html_wrap_inline2363 is (via r, tex2html_wrap_inline1519 ) uniquely determined.
II.
Suppose, tex2html_wrap_inline1779 are not unique, i.e. for some tex2html_wrap_inline2399 there exist tex2html_wrap_inline2401 and tex2html_wrap_inline2403 with tex2html_wrap_inline1787, such that tex2html_wrap_inline2303 is tex2html_wrap_inline1519-recursive as well as tex2html_wrap_inline1793-recursive, and let tex2html_wrap_inline2413 be the tex2html_wrap_inline1519-extension and tex2html_wrap_inline2417 be the tex2html_wrap_inline1793-extension of tex2html_wrap_inline2303.
The question that must be answered is: Are tex2html_wrap_inline2423 and tex2html_wrap_inline2425 identic ?

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.

5.4 Hypotheses.

I.
Because of the Restriction (R), for each sequence tex2html_wrap_inline2347 holds:
If tex2html_wrap_inline2429 is F-recursive for some tex2html_wrap_inline2433 at all, then F is uniquely determined (and thus the F-extension of tex2html_wrap_inline2303 as well).
If this hypothesis proves to be false, then at least:
II.
If for a given sequence tex2html_wrap_inline2347 there exist tex2html_wrap_inline2443 with tex2html_wrap_inline2445 such that tex2html_wrap_inline2303 is F-recursive as well as F'-recursive, and if tex2html_wrap_inline2413 is the F-extension and tex2html_wrap_inline2417 the F'-extension of tex2html_wrap_inline2303, then tex2html_wrap_inline2463 holds.

With these hypotheses as guidelines we will analyze the uniqueness problem in the following section.


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Next: Analysis of the uniqueness Up: MPR-online 1998Vol.3, No.1 Previous: Existence and uniqueness of

Methods of Psychological Research 1998 Vol.3 No.1
© 1998 Pabst Science Publishers