Simple solvers for simple constraint-satisfaction problems. It uses the forward-checking + conflict-directed backjumping algorithm described in @link["http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.225.3123&rep=rep1&type=pdf"]{@italic{Hybrid Algorithms for the Constraint Satisfaction Problem}} by Patrick Prosser. Plus other improvements of my own devising.
A @deftech{constraint-satisfaction problem} (often shortened to @deftech{CSP}) has two ingredients. The first is a set of @deftech{variables}, each associated with a set of possible values (called its @deftech{domain}). The other is a set of @deftech{constraints} —a fancy word for @italic{rules} —that describe relationships among the variables.
When we select a value for each variable, we have what's known as an @deftech{assignment} or a @deftech{state}. Solving a CSP means finding an assignment that @deftech{satisfies} all the constraints. A CSP may have any number of solution states (including zero).
Even if the name isnew, the idea of a CSP is probably familiar. For instance, many brain teasers —like Sudoku or crosswords or logic puzzles —are really just constraint-satisfaction problems. (Indeed, you can use this package to ruin all of them.)
When the computer solves a CSP, it's using an analogous process of deductive reasoning to eliminate impossible assignments, eventually converging on a solution (or determining that no solution exists).
We use CSP variables to represent the values in the triple. We insert each one with @racket[add-var!], where each variable has a @tech{symbol} for its name and a list of values for its domain:
Then we need our constraint. We make a function called @racket[valid-triple?] that tests three values to see if they qualify as a Pythagorean triple. Then we insert this function as a constraint using @racket[add-constraint!], passing as arguments 1) the function we want to use for the constraint, and 2) a list of variable names that the constraint applies to.
Notice that the argument names used within the constraint function (@racket[x] @racket[y] @racket[z]) have nothing to do with the CSP variable names that are passed to the function @racket['(a b c)]. This makes sense —we might want constraints that apply the same function to different groups of CSP variables. What's important is that the @tech{arity} of the constraint function matches the number of variable names, and that the variable names are ordered correctly (the first variable will become the first argument to the constraint function, and so on).
``But that's just the 5--12--13 triple, doubled.'' True. Suppose we want to ensure that the values in our solution have no common factors. We add a new @racket[coprime?] constraint:
``But really there's only two solutions —the values for @racket[a] and @racket[b] are swapped in the other two.'' Fair enough. We might say that this problem is @deftech{symmetric} relative to variables @racket[a] and @racket[b], because they have the same domains and are constrained the same way. We can break the symmetry by adding a constraint that forces @racket[a] to be less than or equal to @racket[b]:
Nothing changes. Why not? Because of the existing @racket[valid-triple?] constraint, @racket[c] is necessarily going to be larger than @racket[a] and @racket[b]. So it always meets this constraint too. It's good practice to not duplicate constraints between the same sets of variables — the ``belt and suspenders'' approach just adds work for no benefit.
We should use @racket[solve*] with care. It can't finish untilthe CSP solver examines every possible assignment of values in the problem, which can be a big number. Specifically, it's the product of the domain sizes of each variable, which in this case is 40 × 40 × 40 = 64,000. This realm of possible assignments is also known as the CSP's @deftech{state space}. We can also get this number from @racket[state-count]:
It's easy for a CSP to have a state count in the zillions. For this reason we can supply @racket[solve*] with an optional argument that will only generate a certain number of solutions:
Of course, even when we use ordinary @racket[solve], we don't know how many assignments it will have to try before it finds a solution. If the problem is impossible, even @racket[solve] will have to visit the entire state space before it knows for sure. For instance, let's see what happens if we add a constraint that's impossible to meet:
``Dude, are you kidding me? I can write a much shorter loop to do the same thing—"
@my-examples[
(for*/list ([a (in-range 10 50)]
[b (in-range 10 50)]
#:when (<= a b)
[c (in-range 10 50)]
#:when (and (coprime? a b c) (valid-triple? a b c)))
(map cons '(a b c) (list a b c)))
]
Yes, I agree that in this toy example, the CSP approach is overkill. The variables are few enough, the domains small enough, and the constraints simple enough, that a loop is more concise. Also, with only 64,000 possibilities in the state space, this sort of brute-force approach is cheap & cheerful.
@section{Second example}
But what about a more complicated problem —like a Sudoku? A Sudoku has 81 squares, each of which can hold the digits 1 through 9. The goal in Sudoku is to fill the grid so that no row, no column, and no ``box'' (a 3 × 3 subgroup of cells) has a duplicate digit. About 25 of the squares are filled in at the start, so the size of the state space is therefore:
Well over a zillion, certainly. Let's optimistically suppose that the 3.7GHz processor in your computer takes one cycle to check an assignment. There are 31,557,600 seconds in a year, so the brute-force method will only take this many years:
@my-examples[
(define states (expt 9 (- 81 25)))
(define states-per-second (* 3.7 1e9))
(define seconds-per-year 31557600)
(/ states states-per-second seconds-per-year)
]
@section{Another interlude}
``Dude, are you serious? The JMAXX Sudoku Solver runs three to four times faster—''
@racketblock[
;; TK
]
Yes, I agree that an algorithm custom-tailored to the problem will likely beat the CSP solver, which is necessarily general-purpose.
But let's consider the labor involved. To write something like the JMAXX Sudoku Solver, we'd need a PhD in computer science, and the time to explain not just the rules of Sudoku to the computer, but the process for solving a Sudoku.
By contrast, when we use a CSP, @italic{all we need are the rules}. The CSP solver does the rest. In this way, a CSP gives us an alternative, simpler way to explain Sudoku to the computer, just like regular expressions are an alternate way of expressing string patterns. And if the CSP solver is half a second slower, that seems like a reasonable tradeoff.
@margin-note{Daring minds might even consider a CSP solver to be a kind of domain-specific language.}
Create a new CSP. Variables and constraints can be added to the CSP by passing them as arguments. Or you can create an empty CSP and then add variables and constraints imperatively (e.g., with @racket[add-var!] or @racket[add-constraint!]).
Imperatively add a new variable called @racket[_name] to the CSP with permissible values listed in @racket[_domain]. The solution to a CSP is a list of pairs where each variable has been assigned a value from its domain.
@racket[add-vars!] is the same, but adds multiple variables that have the same domain.
Imperatively add a new constraint. The constraint applies the function @racket[_func] to the list of variable names given in @racket[_names]. The return value of @racket[_func] does not need to be a Boolean, but any return value other than @racket[#false] is treated as if it were @racket[#true].
Similar to @racket[add-constraint!], but it takes a two-arity procedure @racket[_func] and adds it as a constraint between each pair of names in @racket[_names].
Why? CSPs are more efficient with lower-arity constraints (roughly, because you can rule out invalid values sooner). So usually, decomposing a larger-arity constraint into a group of smaller ones is a good idea.
For instance, suppose you have three variables, and you want them to end up holding values that are coprime. Your constraint function is @racket[coprime?]. This function is variadic (meaning, it can take any number of arguments) so you could use @racket[add-constraint!] like so:
@racketblock[
(add-constraint! my-csp coprime? '(a b c))
]
But because the comparison can be done two at a time, we could write this instead:
@racketblock[
(add-pairwise-constraint! my-csp coprime? '(a b c))
]
Which would be equivalent to:
@racketblock[
(add-constraint! my-csp coprime? '(a b))
(add-constraint! my-csp coprime? '(b c))
(add-constraint! my-csp coprime? '(a c))
]
Still, @racket[add-pairwise-constraint!] doesn't substitute for thoughtful constraint design. For instance, suppose instead we want our variables to be strictly increasing. This time, our constraint function is @racket[<]:
@racketblock[
(add-constraint! my-csp < '(a b c))
]
And we could instead write:
@racketblock[
(add-pairwise-constraint! my-csp < '(a b c))
]
Which would become:
@racketblock[
(add-constraint! my-csp < '(a b))
(add-constraint! my-csp < '(b c))
(add-constraint! my-csp < '(a c))
]
This is better, but also overkill, because if @racket[(< a b)] and @racket[(< b c)], then by transitivity, @racket[(< a c)] is necessarily true. So this is a case where pairwise expands into more constraints than we actually need. This will not produce any wrong solutions, but especially on larger lists of variables, it creates unnecessary work that my slow down the solution search.
Return all the solutions for the CSP. If there are none, returns @racket[null]. The optional @racket[_count] argument returns a certain number of solutions (or fewer, if not that many solutions exist)
Number of possible variable assignments for @racket[_prob], otherwise known as the state space. This is the product of the domain sizes of each variable. So a CSP that assigns five variables, each of which can have the values @racket["a-z"], has a state count of @racket[(expt 5 26)] = @racket[1490116119384765625].
Creates an undirected graph (using Racket's @racket[graph] library) where each CSP variable is represented in the graph as a vertex, and each constraint between any pair of variables is represented as an edge.
Whether CSP will be decomposed into independent subproblems (if possible), because smaller CSPs are typically easier to solve than larger ones (and then the component solutions are reassembled into a larger solution).
