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From: Marko Kääramees <Marko.Kaaramees@tt...>  20091116 16:47:18

Hi, I use Reduce and its package Redlog for simplifying the boolean combination of linear inequalities. I found that some quite obvious simplifications work only one way in the redlog. An example script load_package redlog; rlset pasf; c := (x = 1 or x >= 2) and (x = 6 or x <= 5); rlsimpl c; simplifies to x  1 >= 0 and (x  5 <= 0 or x  6 = 0) The first disjunction with >= is simplified, but the other with <= is not. Could somebody explain the reasons please or guide if it can be tuned somehow. I am working in the finite domains (bounded integers) and generate lots of combinations of linear inequalities automatically, which should be simplified and converted to a normal form to handle complexity. Although Redlog seems to do exactly what I need, it is not too difficult to get to the situation where the system eats up all the memory or does not seem to finish the computation any time soon. Redlog seems to be quite general tool accepting more complex arithmetics and formulas. Could anybody point to some other tool which seems to be more suitable for such tasks? Best regards, marko 
From: Dr. Thomas Sturm <sturm@re...>  20091116 17:58:27
Attachments:
smime.p7s

Am 16.11.2009 um 17:25 schrieb Marko Kääramees: > > Hi, > > I use Reduce and its package Redlog for simplifying the boolean > combination of linear inequalities. I found that some quite obvious > simplifications work only one way in the redlog. An example script > > load_package redlog; > rlset pasf; > c := (x = 1 or x >= 2) and (x = 6 or x <= 5); > rlsimpl c; > > simplifies to > > x  1 >= 0 and (x  5 <= 0 or x  6 = 0) > > The first disjunction with >= is simplified, but the other with <= is > not. Could somebody explain the reasons please or guide if it can be > tuned somehow. This indeed puzzled me at first glance. In fact there is no simplification at all implemented that would discover this situation. What happens is the following (I am simplifying a bit): The atomic formula simplifier has some odd policy about deciding between strict (i.e., ">", "<") and weak (i.e. ">=", "<=") orderings to represent certain constraints. That might be tuned to support the quantifier elimination; I would have to look into that in more detail. Now x>=2 internally becomes x>1, and "x=1 or x>1" is something obvious to contract by literal comparison of terms. In contrast, x<=5 is *not* turned into x<6, and the possible simplification is not recognised. > I am working in the finite domains (bounded integers) and > generate lots of combinations of linear inequalities automatically, > which should be simplified and converted to a normal form to handle > complexity. > > Although Redlog seems to do exactly what I need, it is not too difficult > to get to the situation where the system eats up all the memory or does > not seem to finish the computation any time soon. Redlog seems to be > quite general tool accepting more complex arithmetics and formulas. Recall that Presburger arithmetic is triply exponential (lower bound on size of the output and thus also on time). > > Could anybody point to some other tool which seems to be more suitable > for such tasks? If you do not use any addition, then software focussing on on discrete orderings might be more efficient. Redlog does not offer this at present.  Thomas Sturm Departamento de Matematicas, Estadistica y Computacion Universidad de Cantabria, Santander, Spain Avda. Los Castros s/n, Room 1072, +34 693 251058 http://personales.unican.es/sturmt/ 
From: Dr. Thomas Sturm <sturm@re...>  20091116 18:35:56
Attachments:
smime.p7s

Am 16.11.2009 um 18:58 schrieb Dr. Thomas Sturm: > The atomic formula simplifier has some odd policy about deciding between strict (i.e., ">", "<") and weak (i.e. ">=", "<=") orderings to represent certain constraints. That might be tuned to support the quantifier elimination; I would have to look into that in more detail. The implementor of PASF has just confirmed that the selection of strong vs. weak orderings is so that the absolute values of the occurring integers are minimised since these are crucial for the complexity of the elimination. So now you at least understand what is going on. Sorry to have no better news.  Thomas Sturm Departamento de Matematicas, Estadistica y Computacion Universidad de Cantabria, Santander, Spain Avda. Los Castros s/n, Room 1072, +34 693 251058 http://personales.unican.es/sturmt/ 
From: Marko Kääramees <Marko.Kaaramees@tt...>  20091118 13:18:49

Thomas, thank you for the insight. marko Dr. Thomas Sturm wrote: > Am 16.11.2009 um 17:25 schrieb Marko Kääramees: > > >> Hi, >> >> I use Reduce and its package Redlog for simplifying the boolean >> combination of linear inequalities. I found that some quite obvious >> simplifications work only one way in the redlog. An example script >> >> load_package redlog; >> rlset pasf; >> c := (x = 1 or x >= 2) and (x = 6 or x <= 5); >> rlsimpl c; >> >> simplifies to >> >> x  1 >= 0 and (x  5 <= 0 or x  6 = 0) >> >> The first disjunction with >= is simplified, but the other with <= is >> not. Could somebody explain the reasons please or guide if it can be >> tuned somehow. >> > > This indeed puzzled me at first glance. In fact there is no simplification at all implemented that would discover this situation. What happens is the following (I am simplifying a bit): > > The atomic formula simplifier has some odd policy about deciding between strict (i.e., ">", "<") and weak (i.e. ">=", "<=") orderings to represent certain constraints. That might be tuned to support the quantifier elimination; I would have to look into that in more detail. > > Now x>=2 internally becomes x>1, and "x=1 or x>1" is something obvious to contract by literal comparison of terms. In contrast, x<=5 is *not* turned into x<6, and the possible simplification is not recognised. > > >> I am working in the finite domains (bounded integers) and >> generate lots of combinations of linear inequalities automatically, >> which should be simplified and converted to a normal form to handle >> complexity. >> >> Although Redlog seems to do exactly what I need, it is not too difficult >> to get to the situation where the system eats up all the memory or does >> not seem to finish the computation any time soon. Redlog seems to be >> quite general tool accepting more complex arithmetics and formulas. >> > > Recall that Presburger arithmetic is triply exponential (lower bound on size of the output and thus also on time). > > >> >> Could anybody point to some other tool which seems to be more suitable >> for such tasks? >> > > If you do not use any addition, then software focussing on on discrete orderings might be more efficient. Redlog does not offer this at present. > > 