FindInstance boolean program is slow, but fast with BooleanConvert

FindInstance can be incredibly slow to solve a boolean program, but it can be significantly improved by using BooleanConvert to convert the constraints to "conjunctive normal form".

For example, here is a boolean program with some random constraints. My laptop can't solve it in 10 seconds.

vars = Array[x, 50];
SeedRandom[1234];
const = And @@ Table[Or[
     Xor[RandomChoice[vars], RandomChoice[vars]],
     Xor[RandomChoice[vars], RandomChoice[vars]]], 100];


TimeConstrained[FindInstance[const, vars, Booleans], 10, "Timeout!"]

(* Timeout! *)

Converting the constraints to "conjunctive normal form" helps FindInstance find a solution almost instantly:

AbsoluteTiming@FindInstance[ BooleanConvert[const,"CNF"], vars, Booleans ] // Short

(* {0.014715, {{x[1] -> False, x[2] -> False, <<46>>, x[49] -> False, x[50] -> False}}} *)

Wolfram's 4-color map-coloring example uses this trick; without BooleanConvert, it takes forever to run.

Does anyone know why this is so? I wish the FindInstance documentation would've mentioned this trick.

Related: 1, 2

Update: A colleague pointed out that "conjunctive normal form" has the appealing property that it allows short-circuiting, which allows the satisfiability solver to prune away large branches of the tree of candidate variable choices. For example,

BooleanConvert[Nand[x, y]~And~Xor[y, z], "CNF"]

converts the expression into the AND of a bunch of OR expressions:

(! x || ! y) && (! y || ! z) && (y || z)

When testing a candidate set of values for x,y,z, the solver can evaluate each OR expression in turn and stop as soon as it encounters one that evaluates to FALSE. For example, once it concludes that x=TRUE, y=TRUE makes the first OR clause FALSE, it can prune off the lower levels of the tree and doesn't need to try either value of z.

Comments

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