Mono	.NET	NuGet

GenSolver aims to solve a set of product and/or sum equations using the following general format:

Product Equation: y = x1 * x2
Sum Equation x1 = x3 + x4

A variable in an equation can also occur in another equation. So, for example x1 can be found in both the product equation as in the sum equation. These equations can be solved by giving each variable a value. When an equation contains n variables, then the equation can be solved when n-1 variables have a value.

For example, when x3 = 1, x4 = 1 and x2 = 3 then it follows that x1 becomes 2 and y becomes 6. Vice versa when y = 6, x1 = 2 and x3 = 1 then x4 is 1 and x2 is 2.

GenSolver can solve any combination of product and/or sum equations independent of the order in which variables are set.

But there is more. Each variable in an equation has the following properties:

• a name
• a list of possible values
• a minimum
• a maximum and
• an increment

A variable can be considered as a range of possible values. The following notation can be used to describe a variable:

< ... > meaning a variable that encompasses all rational numbers
< min.. > a variable restricted by values > min
[ min.. > a variable restricted by values >= min
< ..n.. > a variable in which each value is a multiple of n
< ..max > a variable restricted to values < max
< ..max ] a variable restricted to values <= max

So, in the aforementioned example, the variables actually are set to lists of values which happen to contain just one value. Therefore, the following scenario can also be solved by GenSolver:

y = [ 12 ] meaning that the only possible value in the y variable is 12, i.e. y = 12
x2 = [ 1..1.. > meaning a variable with only values >= 1 and an increment of 1 and
x1 = [ 1..1.. > a variable of values with only values >= 1 and an increment of 1.

It can be derived that each possible value in both x1 and x2 must be a divisor of 12. Solving the equation:

y [ 12 ] = x1 [ 1..1.. > * x2 [ 1..1.. >

Results in

x1 = [ 1,2,3,4,6,12 ] and
x2 = [ 1,2,3,4,6,12 ]

Another example is a sum equation in which each variable has a minimum > 0 like:

y < 0.. > = x1 < 0.. > + x2 < 0.. > + x3 < 0.. >

When the minimum of, for example, x2 is set to 1 then y will be at least 1 (as x1 and x3 will be > 0).

By specifying variables as lists of possible values with either a lower or upper boundary and/or a fixed increment, solving a sum or a product equation results in calculating the restrictions there are for each variable in the equation of possible values that will result in solving the equation. Given a sum equation or a product equation the starting point is that each variable in the equation can be set to any rational number:

y < ... > = x1 < .. > * x2 < ... >
x1 < ... > = x3 < .. > + x4 < ... >

When for example the lower limit of x3 is set to 1, then:

y < ... > = x1 < ... > * x2 < ... >
x1 < ... > = x3 [ 1.. > + x4 < ... >

This doesn't change (restrict) the rest of the variables. Also when the minimum of x2 is set to greater than 1, the rest of the variables remain untouched and still can take any value:

y < ... > = x1 < ... > * x2 < 1.. >
x1 < ... > = x3 [1 .. > + x4 < ... >

However, when in this equation the lower boundary of x4 is set to 2, then the rest of the variables will be restricted as well:

y < 3.. > = x1 [ 3.. > * x2 < 1.. >
x1 [ 3.. > = x3 [ 1.. > + x4 [2 .. >

Suddenly, both x1 and y cannot take on any value anymore but are restricted to values >= 3 and > 3, respectively.

GenSolver thus ensures that in any combination of product and/or sum equations as long as only one variable at a time is restricted and the resulting restrictions in other variables are respected, there is always a solution where each variable in the equations can just have one possible value, i.e. all the equations are in a solved state.

A more elaborate example and output of GenSolver can be found here.

The build can be started by running:

• build.sh on unix systems (tested on max os x)
• build.cmd on windows systems

Running build.cmd or ./build.sh is the same as running the command with the ALL argument. To list all build targets use the --listTargets or -lt argument.