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General type of task, solvability and uniqueness of solution

The further considerations in this paper concern number sequence tasks where a finite initial segment of a linear-recursive sequence of rational numbers is given and one or more subsequent members as a ''correct'' continuation of the sequence are sought.

Many number sequence tasks in psychological applications do not demand that the subject give an explicit rule or formula but only a ''correct'', ''regular'' or ''logical'' continuation or completion of the presented number sequence. For our analysis, however, the detailed version of a number sequence task is essential. Referring to the previously definined notions, we first give a precise and explicit formulation of the general type of linear-recursive number sequence task. We presuppose that the type of rules sought are specified in the task instruction (linear recursion formulae), however, that no essential restriction is imposed on the domain of recursion formulae; that is, the domain for the recursion formulae in this general type of task is tex2html_wrap_inline1687. After the formulation of the general type of linear-recursive number sequence task, we will describe the problems of existence and uniqueness of solutions for this type of number sequence task. In Section 4, these problems will be discussed in further detail.

3.1 The general type of linear-recursive number sequence task.
Given a (non-constant) n-term sequence tex2html_wrap_inline1595 of rational numbers tex2html_wrap_inline1589, a maximal permissible degree tex2html_wrap_inline1695 of recursion, and the number k for a k-fold extension of tex2html_wrap_inline1591.

I.
Find tex2html_wrap_inline1703 and tex2html_wrap_inline1705 so that tex2html_wrap_inline1591 is tex2html_wrap_inline1519-recursive.
II.
Calculate tex2html_wrap_inline1711 so that tex2html_wrap_inline1713 is the k-fold tex2html_wrap_inline1519-extension of tex2html_wrap_inline1591.

One easily sees that for tex2html_wrap_inline1721 each n-term sequence is linear-recursive of degree tex2html_wrap_inline1725 because a solution tex2html_wrap_inline1727 for
displaymath1683
is given, for instance, by
displaymath1684
Generally, when some recursion formula for a given sequence is found, the subsequent members can be calculated. Not for each sequence tex2html_wrap_inline1595, however, does there exist an tex2html_wrap_inline1733 such that tex2html_wrap_inline1591 is tex2html_wrap_inline1519-recursive with tex2html_wrap_inline1739 (Example: The sequence tex2html_wrap_inline1741 is not linear-recursive of some degree r<5). Therefore, the mathematical analysis of a number sequence task basically involves examining the problem of solvability.

3.2 The problem of solvability.

Given a (non-constant) n-term number sequence tex2html_wrap_inline1595 and a maximal permissible degree tex2html_wrap_inline1695 of recursion. How can it be determined whether the sequence task of type 3.1 is solvable ?

In practical application the solvability of a task of type 3.1 can be guaranteed simply by constructing the number sequence task according to an appropriate recursion formula of some degree tex2html_wrap_inline1703. Nevertheless, even if the solvability of a given sequence task of type 3.1 is assured and there exists some solution, there is another more profound problem with the sequence task, namely the problem of uniqueness of the solution.

3.3 The problem of uniqueness of solution.

Let tex2html_wrap_inline1595 be an n-term sequence for which an tex2html_wrap_inline1739 exists such that tex2html_wrap_inline1591 is tex2html_wrap_inline1519-recursive and tex2html_wrap_inline1763 is the k-fold tex2html_wrap_inline1519-extension.

I.
Are r and tex2html_wrap_inline1519 uniquely determined ?
If that is the case, then also tex2html_wrap_inline1713 is (via r, tex2html_wrap_inline1519 ) uniquely determined.
II.
Suppose, tex2html_wrap_inline1779 are not unique, i.e. for some tex2html_wrap_inline1781 there exist tex2html_wrap_inline1783 and tex2html_wrap_inline1785, tex2html_wrap_inline1787, such that tex2html_wrap_inline1591 is tex2html_wrap_inline1519-recursive as well as tex2html_wrap_inline1793-recursive, and let tex2html_wrap_inline1795 be the tex2html_wrap_inline1519-extension and tex2html_wrap_inline1799 be the tex2html_wrap_inline1793-extension of tex2html_wrap_inline1591.
The question that needs to be answered is: Are tex2html_wrap_inline1805 and tex2html_wrap_inline1807 identic ?

The formulation of the uniqueness problem indicates fundamental problems in the application of number sequence tasks in psychology that are by no means artificial:

In Sections 5 and 6 we will discuss the uniqueness problem 3.3 in connection with a special type of natural-number sequences where the domain tex2html_wrap_inline1527 for the permissible recursion formulae is substantially restricted. First, in Section 4 some general results concerning the existence and uniqueness problems of recursion formulae for given number sequences are derived; that is, we will concentrate on the uniqueness problems of type 3.3 I. and leave a more general treatment of problems of type 3.3 II. for another time.


next up previous
Next: Existence and uniqueness of Up: MPR-online 1998Vol.3, No.1 Previous: Linear-recursive number sequences: definitions

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