As soon as linear inequations are involved, projection gets more
demanding complexity wise. The current clp(Q,R) version uses a
Fourier-Motzkin algorithm for the projection of linear inequalities.
The choice of a suitable algorithm is somewhat dependent on the number
of variables to be eliminated, the total number of
variables, and other factors. It is quite easy to produce problems
of moderate size where the elimination step takes some time. For
example, when the dimension of the projection is 1, you might be better
off computing the supremum and the infimum of the remaining
variable instead of eliminating n-1 variables via
implicit projection.
In order to make answers as concise as possible, redundant constraints are removed by the system as well. In the following set of inequalities, half of them are redundant.
% library('clpqr/examples/eliminat')
example(2, [X0,X1,X2,X3,X4]) :-
{
+87*X0 +52*X1 +27*X2 -54*X3 +56*X4 =< -93,
+33*X0 -10*X1 +61*X2 -28*X3 -29*X4 =< 63,
-68*X0 +8*X1 +35*X2 +68*X3 +35*X4 =< -85,
+90*X0 +60*X1 -76*X2 -53*X3 +24*X4 =< -68,
-95*X0 -10*X1 +64*X2 +76*X3 -24*X4 =< 33,
+43*X0 -22*X1 +67*X2 -68*X3 -92*X4 =< -97,
+39*X0 +7*X1 +62*X2 +54*X3 -26*X4 =< -27,
+48*X0 -13*X1 +7*X2 -61*X3 -59*X4 =< -2,
+49*X0 -23*X1 -31*X2 -76*X3 +27*X4 =< 3,
-50*X0 +58*X1 -1*X2 +57*X3 +20*X4 =< 6,
-13*X0 -63*X1 +81*X2 -3*X3 +70*X4 =< 64,
+20*X0 +67*X1 -23*X2 -41*X3 -66*X4 =< 52,
-81*X0 -44*X1 +19*X2 -22*X3 -73*X4 =< -17,
-43*X0 -9*X1 +14*X2 +27*X3 +40*X4 =< 39,
+16*X0 +83*X1 +89*X2 +25*X3 +55*X4 =< 36,
+2*X0 +40*X1 +65*X2 +59*X3 -32*X4 =< 13,
-65*X0 -11*X1 +10*X2 -13*X3 +91*X4 =< 49,
+93*X0 -73*X1 +91*X2 -1*X3 +23*X4 =< -87
}.
Consequently, the answer consists of the system of nine non-redundant inequalities only:
clp(q) ?- use_module(library('clpqr/examples/elimination')).
clp(q) ?- example(2, [X0,X1,X2,X3,X4]).
{X0-2/17*X1-35/68*X2-X3-35/68*X4>=5/4},
{X0-73/93*X1+91/93*X2-1/93*X3+23/93*X4=<-29/31},
{X0-29/25*X1+1/50*X2-57/50*X3-2/5*X4>=-3/25},
{X0+7/39*X1+62/39*X2+18/13*X3-2/3*X4=<-9/13},
{X0+2/19*X1-64/95*X2-4/5*X3+24/95*X4>=-33/95},
{X0+2/3*X1-38/45*X2-53/90*X3+4/15*X4=<-34/45},
{X0-23/49*X1-31/49*X2-76/49*X3+27/49*X4=<3/49},
{X0+44/81*X1-19/81*X2+22/81*X3+73/81*X4>=17/81},
{X0+9/43*X1-14/43*X2-27/43*X3-40/43*X4>=-39/43}
The projection (the shadow) of this polyhedral set into the X0,X1
space can be computed via the implicit elimination of non-query
variables:
clp(q) ?- example(2, [X0,X1|_]).
{X0+2619277/17854273*X1>=-851123/17854273},
{X0+6429953/16575801*X1=<-12749681/16575801},
{X0+19130/1213083*X1>=795400/404361},
{X0-1251619/3956679*X1>=21101146/3956679},
{X0+601502/4257189*X1>=220850/473021}
Projection is quite a powerful concept that leads to surprisingly terse executable specifications of nontrivial problems like the computation of the convex hull from a set of points in an n-dimensional space: Given the program
% library('clpqr/examples/elimination')
conv_hull(Points, Xs) :-
lin_comb(Points, Lambdas, Zero, Xs),
zero(Zero),
polytope(Lambdas).
polytope(Xs) :-
positive_sum(Xs, 1).
positive_sum([], Z) :- {Z=0}.
positive_sum([X|Xs], SumX) :-
{ X >= 0, SumX = X+Sum },
positive_sum(Xs, Sum).
zero([]).
zero([Z|Zs]) :- {Z=0}, zero(Zs).
lin_comb([], [], S1, S1).
lin_comb([Ps|Rest], [K|Ks], S1, S3) :-
lin_comb_r(Ps, K, S1, S2),
lin_comb(Rest, Ks, S2, S3).
lin_comb_r([], _, [], []).
lin_comb_r([P|Ps], K, [S|Ss], [Kps|Ss1]) :-
{ Kps = K*P+S },
lin_comb_r(Ps, K, Ss, Ss1).
we can post the following query:
clp(q) ?- conv_hull([ [1,1], [2,0], [3,0], [1,2], [2,2] ], [X,Y]).
{Y=<2},
{X+1/2*Y=<3},
{X>=1},
{Y>=0},
{X+Y>=2}
This answer is easily verified graphically:
|
2 - * *
|
|
1 - *
|
|
0 -----|----*----*----
1 2 3
The convex hull program directly corresponds to the mathematical definition of the convex hull. What does the trick in operational terms is the implicit elimination of the Lambdas from the program formulation. Please note that this program does not limit the number of points or the dimension of the space they are from. Please note further that quantifier elimination is a computationally expensive operation and therefore this program is only useful as a benchmark for the projector and not so for the intended purpose.