From: fateman@gmail.com   
      
   While there are easy ways of sometimes bounding the size of the derivative of   
   an expression, you rapidly   
   run out of rules for more general "problems/solutions".  For instance, here is   
   a problem.   
   expand ((x+1)^(2^n)).   
   characterize the number of terms as a function of the integer n.   
      
   Simulation of random expression problems could (in general) provide   
   super-exponential growth in size.   
      
   Another study of "statistics" might be -- given a free on-line computer   
   algebra system that   
   does some task  (you can pick indefinite integration),  what is the   
   distribution of inputs   
   from  [random?] clients?     
   I suspect you will get (a) homework problems and (b) people just trying it out   
   to see if   
   it works.  People can ask ChatGPT to do math, and it sometimes gets the right   
   answer.   
   It sometimes doesn't.   
   A (much) earlier experiment  we ran at Berkeley TILU  collected some problems   
   (maybe a few hundred?) and mostly found that people were unlikely to master the   
   first stage of the problem:  getting the syntax right.  Thus we collected   
   stuff like   
   sin x, sinx, sin(x), Sin(x), Sin[x], SinX.   
      
   As for whether this is interesting or not, I would not expect simulation --   
   where you write   
   a program P to generate problems -- to reveal much other than the behavior of   
   program P.   
   RJF   
      
      
      
   On Wednesday, March 1, 2023 at 5:57:46 PM UTC-8, David Stork wrote:   
   > Are there any large-scale simulation studies of the statistics of symbolic   
   problem-solution pairs?    
   >    
   > Let's consider just symbolic integration of some class of functions, for   
   instance the class covered by the Risch algorithm (exponentials, logarithms,   
   trigonometric functions, addition, subtraction, multiplication, and division).   
   Suppose we quantify the    
   "size" of an integration problem by the number of leafs in its tree-based   
   representation, and likewise for the problem's anti-derivative.    
   >    
   > Problems of a given size may have solutions spanning a range of sizes, of   
   course. Thus there is some statistical distribution of the sizes, and thus a   
   statistical relation (perhaps correlation) between problem and solution sizes.    
   >    
   > So if we double the size of an integration problem will the size of its   
   solution double? or increase faster than linear? or slower than linear? What   
   about the variance in sizes? Or other statistics?    
   >    
   > Naturally there is an infinite number of integration problems of a given   
   size, so any simulation study will have inherent uncertainties. Nevertheless,   
   this seems like an interesting, and potentially fruitful, class of simulation   
   problems, for which we    
   must use large-scale computer-algebra systems.    
   >    
   > Anyone interested in working on this?    
   >    
   > --David G. Stork   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)