Copyright | (c) Levent Erkok |
---|---|

License | BSD3 |

Maintainer | erkokl@gmail.com |

Stability | stable |

Safe Haskell | None |

Language | Haskell2010 |

(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

- data Solver
- solveIntegerLinearEqs :: Solver -> [[Integer]] -> [Integer] -> IO (Maybe [Integer])
- solveIntegerLinearEqsAll :: Solver -> Int -> [[Integer]] -> [Integer] -> IO [[Integer]]
- solveRationalLinearEqs :: Solver -> [[Rational]] -> [Rational] -> IO (Maybe [Rational])
- solveRationalLinearEqsAll :: Solver -> Int -> [[Rational]] -> [Rational] -> IO [[Rational]]

# 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.

Solvers that SBV is aware of

# Solutions over Integers

solveIntegerLinearEqs Source #

:: Solver | SMT Solver to use |

-> [[Integer]] | Coefficient matrix (A) |

-> [Integer] | Result vector (b) |

-> IO (Maybe [Integer]) | A solution to |

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

`>>>`

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

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`

:

`>>>`

Nothing`solveIntegerLinearEqs Z3 [[1], [1]] [2, 3]`

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

solveIntegerLinearEqsAll Source #

:: 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 |

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:

`>>>`

[[-47,-2,30],[-34,0,22],[-21,2,14]]`solveIntegerLinearEqsAll Z3 3 [[2, 3, 4],[6, -3, 9]] [20, -6]`

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

# Solutions over Rationals

solveRationalLinearEqs Source #

:: Solver | SMT Solver to use |

-> [[Rational]] | Coefficient matrix (A) |

-> [Rational] | Result vector (b) |

-> IO (Maybe [Rational]) | A solution to |

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

`>>>`

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

solveRationalLinearEqsAll Source #

:: 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 |

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:

`>>>`

[[0 % 1,10 % 3],[3 % 4,17 % 6],[3 % 2,7 % 3]]`solveRationalLinearEqsAll Z3 3 [[2.4, 3.6]] [12]`

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