Re: [Haskell-cafe] Solving integer equations in Haskell
On Oct 17, 2012, at 3:35 AM, Justin Paston-Cooper paston.coo...@gmail.com wrote: Thanks for all the informative replies. SBV seems the simplest solution right now, and speed isn't too much of an issue here. Anything under 20 seconds per solution should be bearable. I'm happy to announce the SMT based linear equation solver library: http://hackage.haskell.org/package/linearEqSolver You can use it get solutions over Integers only, or over Rationals if so needed. Functions are provided to extract one solution, or all possible solutions (as a lazy list). The latter variant is useful for underspecified systems. Regarding performance: SMT solvers are very good at solving such equations, so aside from the overhead of calling out to an external program, it should be fairly fast. I'd expect no practical instance to come to anywhere near the 20 second limit you've mentioned. For most practical instances, the process switch overhead would dominate computation time, which should be negligible. Let me know if you find otherwise. -Levent. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Solving integer equations in Haskell
Thanks for all the informative replies. SBV seems the simplest solution right now, and speed isn't too much of an issue here. Anything under 20 seconds per solution should be bearable. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Solving integer equations in Haskell
For Linear integer equations, I think you want http://hackage.haskell.org/packages/archive/agum/2.4/doc/html/Algebra-AbelianGroup-IntLinEq.html The algorithm used to find solutions is described in Vol. 2 of The Art of Computer Programming / Seminumerical Alorithms, 2nd Ed., 1981, by Donald E. Knuth, pg. 327. John On Mon, Oct 15, 2012 at 10:39 PM, Levent Erkok erk...@gmail.com wrote: On Mon, Oct 15, 2012 at 9:00 AM, Johannes Waldmann waldm...@imn.htwk-leipzig.de wrote: Justin Paston-Cooper paston.cooper at gmail.com writes: Can anyone suggest a library written in Haskell which can solve equations of the form xM(transpose(x)) = y, where x should be an integer vector, M is an integer matrix and y is an integer? when in doubt, use brute force: write this as a constraint system (in QF_NIA or QF_BV logics) and solve with Z3. As Johannes mentioned, you can indeed use SBV/Z3 to solve such problems. Indeed, there's an existing example for showing how to solve Diophantine equations this way: http://hackage.haskell.org/packages/archive/sbv/2.3/doc/html/Data-SBV-Examples-Existentials-Diophantine.html The technique described there can be used to solve the problem you've described; or systems of arbitrary constraint equations in general with minor tweaks. Having said that, using an SMT solver for this problem may not necessarily be the fastest route. The general purpose nature of SMT solving, while sound and complete for this class of problems, are not necessarily the most efficient when there are existing fast algorithms. In particular, you should check out John Ramsdell's cmu package: http://hackage.haskell.org/package/cmu. In particular see: http://hackage.haskell.org/packages/archive/cmu/1.8/doc/html/Algebra-CommutativeMonoid-LinDiophEq.html While the approach here only applies to linear diophantine equations, you might be able to adapt it to your particular needs. -Levent. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Solving integer equations in Haskell
Hello, Can anyone suggest a library written in Haskell which can solve equations of the form xM(transpose(x)) = y, where x should be an integer vector, M is an integer matrix and y is an integer? I'm aware that Mathematica can do this, but I would like something written in Haskell. I haven't been sure of what exact keywords I should be searching for, hence my asking here. Thanks, Justin ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Solving integer equations in Haskell
Can anyone suggest a library written in Haskell which can solve equations of the form xM(transpose(x)) = y, where x should be an integer vector, M is an integer matrix and y is an integer? I'm aware that Mathematica can do this, but I would like something written in Haskell. I haven't been sure of what exact keywords I should be searching for, hence my asking here. You may interpret the function x ↦ x M transpose(x) as a quadratic form on the additive group of vectors. The group of integer vectors, together with such a quadratic form, is usually called a /lattice/ (not to be confused with its other meaning, of a set with a partial order and some meet/join operators). A vector x satisfying your equation is sometimes said to /represent/ the number y, with respect to the quadratic form. On hackage, a quick search gives the Lattices package, which seems related. (There is also the lattices package, but this is about the other lattices.) Incidentally, I also have use for this functionality, but specifically for vectors with 2 components (so the quadratic form is then binary). The equation is then a generalised Pell equation, and I think I have read somewhere how to solve it. If such code does not exist yet, I will probably write it, but this might not be very soon. Regards, Arie ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Solving integer equations in Haskell
Justin Paston-Cooper paston.cooper at gmail.com writes: Can anyone suggest a library written in Haskell which can solve equations of the form xM(transpose(x)) = y, where x should be an integer vector, M is an integer matrix and y is an integer? when in doubt, use brute force: write this as a constraint system (in QF_NIA or QF_BV logics) and solve with Z3. Use SBV to write the constraints programmatically http://hackage.haskell.org/package/sbv J.W. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Solving integer equations in Haskell
On Mon, Oct 15, 2012 at 9:00 AM, Johannes Waldmann waldm...@imn.htwk-leipzig.de wrote: Justin Paston-Cooper paston.cooper at gmail.com writes: Can anyone suggest a library written in Haskell which can solve equations of the form xM(transpose(x)) = y, where x should be an integer vector, M is an integer matrix and y is an integer? when in doubt, use brute force: write this as a constraint system (in QF_NIA or QF_BV logics) and solve with Z3. As Johannes mentioned, you can indeed use SBV/Z3 to solve such problems. Indeed, there's an existing example for showing how to solve Diophantine equations this way: http://hackage.haskell.org/packages/archive/sbv/2.3/doc/html/Data-SBV-Examples-Existentials-Diophantine.html The technique described there can be used to solve the problem you've described; or systems of arbitrary constraint equations in general with minor tweaks. Having said that, using an SMT solver for this problem may not necessarily be the fastest route. The general purpose nature of SMT solving, while sound and complete for this class of problems, are not necessarily the most efficient when there are existing fast algorithms. In particular, you should check out John Ramsdell's cmu package: http://hackage.haskell.org/package/cmu. In particular see: http://hackage.haskell.org/packages/archive/cmu/1.8/doc/html/Algebra-CommutativeMonoid-LinDiophEq.html While the approach here only applies to linear diophantine equations, you might be able to adapt it to your particular needs. -Levent. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
