linearEqSolver-2.0: Use SMT solvers to solve linear systems over integers and rationals

Math.LinearEquationSolver

Description

(The linear equation solver library is hosted at http://github.com/LeventErkok/linearEqSolver. Comments, bug reports, and patches are always welcome.)

Solvers for linear equations over integers and rationals. Both single solution and all solution variants are supported.

Synopsis

# Available SMT solvers

Note that while we allow all SMT-solvers supported by SBV to be used, not all will work. In particular, the backend solver will need to understand unbounded integers and rationals. Currently, the following solvers provide the required capability: Z3, CVC4, and MathSAT. Passing other instances will result in an "unsupported" error, though this can of course change as the SBV package itself evolves.

data Solver :: * #

Solvers that SBV is aware of

Constructors

 Z3 Yices Boolector CVC4 MathSAT ABC

Instances

 Methods MethodstoEnum :: Int -> Solver #enumFrom :: Solver -> [Solver] #enumFromThen :: Solver -> Solver -> [Solver] #enumFromTo :: Solver -> Solver -> [Solver] #enumFromThenTo :: Solver -> Solver -> Solver -> [Solver] # MethodsshowsPrec :: Int -> Solver -> ShowS #showList :: [Solver] -> ShowS #

# Solutions over Integers

Arguments

 :: Solver SMT Solver to use -> [[Integer]] Coefficient matrix (A) -> [Integer] Result vector (b) -> IO (Maybe [Integer]) A solution to Ax = b, if any

Solve a system of linear integer equations. The first argument is the matrix of coefficients, known as A, of size mxn. The second argument is the vector of results, known as b, of size mx1. The result will be either Nothing, if there is no solution, or Just x -- such that Ax = b holds. (Naturally, the result x will be a vector of size nx1 in this case.)

Here's an example call, to solve the following system of equations:

    2x + 3y + 4z = 20
6x - 3y + 9z = -6
2x      +  z = 8

>>> solveIntegerLinearEqs Z3 [[2, 3, 4],[6, -3, 9],[2, 0, 1]] [20, -6, 8]
Just [5,6,-2]


The first argument picks the SMT solver to use. Valid values are z3 and cvc4. Naturally, you should have the chosen solver installed on your system.

In case there are no solutions, we will get Nothing:

>>> solveIntegerLinearEqs Z3 [, ] [2, 3]
Nothing


Note that there are no solutions to this second system as it stipulates the unknown is equal to both 2 and 3. (Overspecified.)

Arguments

 :: Solver SMT Solver to use -> Int Maximum number of solutions to return, in case infinite -> [[Integer]] Coefficient matrix (A) -> [Integer] Result vector (b) -> IO [[Integer]] All solutions to Ax = b

Similar to solveIntegerLinearEqs, except in case the system has an infinite number of solutions, then it will return the number of solutions requested. (Note that if the system is underspecified, then there are an infinite number of solutions.) So, the result can be empty, a singleton, or precisely the number requested, last of which indicates there are an infinite number of solutions.

Here's an example call, where we underspecify the system and hence there are multiple (in this case an infinite number of) solutions. Here, we ask for the first 3 elements for testing purposes.

    2x + 3y + 4z = 20
6x - 3y + 9z = -6


We have:

>>> solveIntegerLinearEqsAll Z3 3 [[2, 3, 4],[6, -3, 9]] [20, -6]
[[-8,4,6],[-21,2,14],[-34,0,22]]


The solutions you get might differ, depending on what the solver returns. (Though they'll be correct!)

# Solutions over Rationals

Arguments

 :: Solver SMT Solver to use -> [[Rational]] Coefficient matrix (A) -> [Rational] Result vector (b) -> IO (Maybe [Rational]) A solution to Ax = b, if any

Solve a system of linear equations over rationals. Same as the integer version solveIntegerLinearEqs, except it takes rational coefficients and returns rational results.

Here's an example call, to solve the following system of equations:

    2.4x + 3.6y = 12
7.2x - 5y   = -8.5

>>> solveRationalLinearEqs Z3 [[2.4, 3.6],[7.2, -5]] [12, -8.5]
Just [245 % 316,445 % 158]


Arguments

 :: Solver SMT Solver to use -> Int Maximum number of solutions to return, in case infinite -> [[Rational]] Coefficient matrix (A) -> [Rational] Result vector (b) -> IO [[Rational]] All solutions to Ax = b

Solve a system of linear equations over rationals. Similar to solveRationalLinearEqs, except if the system is underspecified, then returns the number of solutions requested.

Example system:

    2.4x + 3.6y = 12


In this case, the system has infinitely many solutions. We can compute three of them as follows:

>>> solveRationalLinearEqsAll Z3 3 [[2.4, 3.6]] 
[[5 % 1,0 % 1],[13 % 2,(-1) % 1],[8 % 1,(-2) % 1]]


The solutions you get might differ, depending on what the solver returns. (Though they'll be correct!)