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Idealized strategies

We now describe what we mean by an idealized strategy. An idealized strategy uses the Categorize and Decompose operations. While an idealized strategy cannot be implemented on a computer, approximations of it can be implemented and the use of these approximations is the core of our paper.

An idealized strategy is an iterative procedure which would invoke Categorize on the pair of sets of polynomial equations C and tex2html_wrap_inline4190 ( tex2html_wrap_inline4190 is initially empty), allow creation of motivated unknowns, allow for unknowns to be redeclared as knowns and allow for human intervention. Human intervention would consist of selecting particular equations which are nice in some way (e.g., this is a Ricatti equation), adding them to the collection tex2html_wrap_inline4190 and possibly selecting a motivated unknown y as described above. This procedure would terminate when the ideal generated by tex2html_wrap_inline4314 equals the ideal generated by tex2html_wrap_inline4316 . This, of course, is equivalent to the condition that tex2html_wrap_inline4212 is empty for each tex2html_wrap_inline4320 . In other words, an idealized strategy terminates when all of the polynomial equations which can be derived algebraically and involve unknowns can be derived algebraically from the equations in tex2html_wrap_inline4190 together with the equations which do not involve unknowns.

If the reader would indulge us, we now discuss a more visual way of thinking about idealized strategies. For ease of exposition, the discussion in this paragraph will ignore the possibility of introducing motivated unknowns, the redeclaration of unknowns as knowns or human intervention. For each pair of collections of polynomials C and tex2html_wrap_inline4190 , suppose that one has an idealized display:


When one begins using an idealized strategy, C would be the collections of equations coming from the problem at hand and tex2html_wrap_inline4190 would be initially empty. During each iteration of a strategy, one would choose a number of polynomials from the ideal tex2html_wrap_inline4198 generated by tex2html_wrap_inline4316 , (that is, below the dark line) and place them in tex2html_wrap_inline4190 (that is, above the dark line). The idealized display would then adjust itself so that what appears in tex2html_wrap_inline4328 above the line and what appears below the line corresponds to the new value of tex2html_wrap_inline4190 . When there are no equations below the dark line, the idealized strategy is complete. If tex2html_wrap_inline4376 and, for tex2html_wrap_inline4378 , tex2html_wrap_inline4380 is produced from C and tex2html_wrap_inline4384 using one step of an idealized strategy and each equation which is in tex2html_wrap_inline4380 but not in tex2html_wrap_inline4384 has at most 1 (motivated) unknown, then we say that tex2html_wrap_inline4392 is derivable from C by an idealized strategy.

A question about strategies is

Which classical results can be derived using an idealized strategy?

This question gives an abstract statement of what we address in this paper.

One intends, of course, to move beyond this question to the derivation of new theorems. However, in the early stages of the subject, we think that the most urgent task is to understand classical theorems from this viewpoint. Evaluating these new and unusual techniques on a new and not well understood problem gives less of an idea of their strength than evaluating them on a well understood problem. Typically, if we derive a new theorem using symbol manipulation, then one can go back and produce a proof by hand, using ideas gotten from the symbolic manipulation.

next up previous contents
Next: New derivations of classical Up: A highly idealized picture Previous: Basic ``idealized'' operations

Wed Jul 3 10:27:42 PDT 1996