ANLDE system description
The Web-SynDic system works with homogeneous ANLDE systems
(ANLDE systems in short).
The ANLDE system is a system of nonnegative linear
Diophantine equations
I x = A x
where matrix A is a matrix with non-negative integers and matrix I
is a (0,1)-integer matrix with the following property: each unknown has at most
one non-zero coefficient in matrix I.
See the User Manual
(section "ANLDE system theory") for more formal definition.
Samples of ANLDE systems
Copy (with or without #-comment lines) any of these ANLDE systems into the
clipboard, move to corresponding processing
a single
ANLDE system, or
a set of ANLDE systems forms
and paste back the ANLDE system(s) to the input form(s).
Simple ANLDE system 1
# A very simple case
x = y
Simple ANLDE system 2
This is the same as x = y,
but in the standard ANLDE form.
x + y = 2*y
Simple ANLDE system 3
This system also is an ANLDE system;
it has only zero-solution.
# It is satisfied by the only solution x=y=z=0
x + y + z = 0
Simple ANLDE system 4
Another example of a system with only zero-solution.
# It is satisfied by the only solution x=y=0
x = 2*y
y = x
Simple ANLDE system 5
This is slightly more complex description of x=y.
# (2x2)-ANLDE system with
# non-trivial solutions
x = y
y = x
Typical (2x4)-ANLDE system 1
This simple example is from Dmitry Korzun's PhD Thesis.
# It has 2 minimal solutions
x1 + x2 = 2*x1 + 3*x3
x3 + x4 = x1 + 2*x2 + x3
Typical (2x4)-ANLDE system 2
The same as previous one, but not in the standard ANLDE form.
x2 = x1 + 3*x3
x4 = x1 + 2*x2
ANLDE system with one equation
Only unit coefficient on the left-hand side;
arbitrary positive integers on the right-hand side.
# It has 40 minimal solutions!
x1 + x2 + x3 = 5*y1 + 2*y2 + y3 + 3*y4
In the standard ANLDE form this equation is rewritten as:
x1 + x2 + x3 + y1 + y2 + y3 + y4 = 5*y1 + 2*y2 + y3 + 3*y4
Also some ANLDE systems or a set of ANLDE
systems are hard for processing using alternative solvers, but not
for the syntactic algorithm.
One ANLDE system
# many basis solutions
# hard for slopes
# easy for syntactic & lp_solve
x0 +x2 = x2 + c + 10*x1
y1 + y2 = 10*x2 + c
z1 + z2 = 2*c + y1
x1 = 2*c + z1
c = 2*x2 + 9*y1
# small basis
# hard for slopes
# easy for syntactic & lp_solve
x1 = 9*x3 + 12*x8 + 14*x9 + 4*x10
x3 + x10 = 5*x8 + 3*x9 + 7*x10
x8 + x4 = 9*x8 + 8*x9 + 13*x10
x5 = 13*x8 + 9*x9 + 10*x10 + 2*x6
x9 + x7 = 6*x8 + 10*x9 + 14*x10
x6 = x8 + x9 + x10
# simple basis
# hard for slopes
# easy for syntactic & lp_solve
x1 = 29*x1 + 40*x3 + 61*x4 + 56*x10 + 19*x11
x2 + x11 = 82*x1 + x2 + 27*x3 + 12*x4 + 13*x10 + 48*x11
x3 = 42*x1 + 13*x3 + 19*x4 + 58*x10 + 15*x11
x4 = 74*x1 + 35*x3 + 71*x4 + 27*x10 + 59*x11
x5 + x9 + x10 = 32*x1 + 39*x2 + 39*x4 + x5 + x9 + 51*x10 + 63*x11
x6 = 98*x1 + 3*x3 + 19*x4 + x6 + 91*x10 + 27*x11
x7 = 55*x1 + 20*x3 + 66*x4 + x7 + 17*x10 + 75*x11
x8 = 84*x1 + 98*x3 + 2*x4 + x8 + 96*x10 + 12*x11
# many basis solutions
# hard for slopes
# easy for syntactic & lp_solve
x1 + x9 + x10 = 6*x9 + 4*x10 + 4*x8 + 2*x11 + 3*x12 + 3*x13 + 2*x14
x8 + x2 = 2*x9 + 2*x10 + 4*x8 + x11 + 2*x12 + 2*x13 + 3*x14 + x3
x3 = 6*x9 + 4*x10 + 3*x8 + 4*x11 + 3*x12 + 5*x13 + x14
x13 + x14 + x4 = 3*x9 + 4*x10 + 5*x8 + 3*x11 + 7*x13 + 3*x14
x11 + x12 + x5 = 9*x9 + 6*x10 + x8 + 3*x11 + 6*x12 + x13 + 2*x14
x6 = 7*x9 + 5*x10 + x8 + 3*x11 + 7*x12 + 6*x13 + 9*x14
x7 = 6*x9 + 4*x10 + 5*x8 + 5*x11 + 9*x12 + 8*x13 + 5*x14
# easy for syntactic
# hard for lp_solve
# unsolvable for slopes
x1 = x8 + 17*x11 + 15*x12 + 8*x13 + 16*x14 + 37*x15
x2 = x8 + 43*x11 + 3*x12 + 49*x13 + 41*x14 + 43*x15
x3 = x8 + 27*x11 + 56*x12 + 49*x13 + 59*x14 + 11*x15
x14 + x4 = x8 + 5*x11 + 49*x12 + 55*x13 + 39*x14 + 34*x15
x12 + x5 = x8 + 28*x11 + 35*x12 + 4*x13 + 55*x14 + 21*x15
x15 + x6 = x8 + 22*x11 + 51*x12 + 34*x13 + 34*x14 + 23*x15
x7 = x8 + 45*x11 + 33*x12 + 49*x13 + 36*x14 + 27*x15
x8 = 48*x11 + 37*x12 + 12*x13 + 36*x14 + 35*x15
x9 = 90*x11 + 40*x12 + 40*x13 + 44*x14 + 36*x15
x11 + x13 + x10 = 96*x11 + 35*x12 + 75*x13 + 34*x14 + 35*x15
A set of ANLDE systems
# hard for slopes, but not for syntactic & lp_solve
x1 = 9*x3 + 12*x8 + 14*x9 + 4*x10
x3 = 5*x8 + 3*x9 + 6*x10
x8 + x4 = 9*x8 + 8*x9 + 13*x10
x5 = 13*x8 + 9*x9 + 10*x10 + 2*x6
x9 + x7 = 6*x8 + 10*x9 + 14*x10
x6 = x8 + x9
x10 = 2*x6
%
x1 = 9*x3 + 12*x8 + 14*x9 + 4*x10
x3 + x10 = 5*x8 + 3*x9 + 7*x10
x8 + x4 = 9*x8 + 8*x9 + 13*x10
x5 = 13*x8 + 9*x9 + 10*x10 + 2*x6
x9 + x7 = 6*x8 + 10*x9 + 14*x10
x6 = x8 + x9 + x10
%
x1 + x2 = 2*x1 + 3*x3
x3 + x4 = x1 + 2*x4 + x3
# hard for lp_solve & slopes, but not for syntactic
x1 = 9*x3 + 12*x8 + 14*x9 + 4*x10
x3 = 5*x8 + 3*x9 + 6*x10
x8 + x4 = 9*x8 + 8*x9 + 13*x10
x5 = 13*x8 + 9*x9 + 10*x10 + 2*x6
x9 + x7 = 6*x8 + 10*x9 + 14*x10
x6 = x8 + x9
x10 = 2*x6
%
x1 = 9*x3 + 12*x8 + 14*x9 + 4*x10
x3 + x10 = 5*x8 + 3*x9 + 7*x10
x8 + x4 = 9*x8 + 8*x9 + 13*x10
x5 = 13*x8 + 9*x9 + 10*x10 + 2*x6
x9 + x7 = 6*x8 + 10*x9 + 14*x10
x6 = x8 + x9 + x10
%
x1 = x8 + 17*x11 + 15*x12 + 8*x13 + 16*x14 + 37*x15
x2 = x8 + 43*x11 + 3*x12 + 49*x13 + 41*x14 + 43*x15
x3 = x8 + 27*x11 + 56*x12 + 49*x13 + 59*x14 + 11*x15
x14 + x4 = x8 + 5*x11 + 49*x12 + 55*x13 + 39*x14 + 34*x15
x12 + x5 = x8 + 28*x11 + 35*x12 + 4*x13 + 55*x14 + 21*x15
x15 + x6 = x8 + 22*x11 + 51*x12 + 34*x13 + 34*x14 + 23*x15
x7 = x8 + 45*x11 + 33*x12 + 49*x13 + 36*x14 + 27*x15
x8 = 48*x11 + 37*x12 + 12*x13 + 36*x14 + 35*x15
x9 = 90*x11 + 40*x12 + 40*x13 + 44*x14 + 36*x15
x11 + x13 + x10 = 96*x11 + 35*x12 + 75*x13 + 34*x14 + 35*x15
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