@ -48,11 +48,12 @@ But there's still some finesse and artistry involved in setting up the CSP, espe
@section{First example}
Suppose we wanted to find @link["http://www.friesian.com/pythag.htm"]{Pythagorean triples} with sides between 10 and 49, inclusive.
Suppose we wanted to find @link["http://www.friesian.com/pythag.htm"]{Pythagorean triples} with sides between 1 and 29, inclusive.
First we create a new CSP called @racket[triples], using @racket[make-csp]:
@examples[#:label #f #:eval my-eval
(require csp)
(define triples (make-csp))
]
@ -60,9 +61,9 @@ First we create a new CSP called @racket[triples], using @racket[make-csp]:
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:
@examples[#:label #f #:eval my-eval
(add-var! triples 'a (range 10 50))
(add-var! triples 'b (range 10 50))
(add-var! triples 'c (range 10 50))
(add-var! triples 'a (range 1 30))
(add-var! triples 'b (range 1 30))
(add-var! triples 'c (range 1 30))
]
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.
@ -82,7 +83,7 @@ Finally we call @racket[solve], which finds a solution (if it exists):
(solve triples)
]
``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 that's just the 3--4--5 triangle, tripled.'' True. Suppose we want to ensure that the values in our solution have no common factors. We add a new @racket[coprime?] constraint:
@examples[#:label #f #:eval my-eval
(require math/number-theory)
@ -95,13 +96,13 @@ We @racket[solve] again to see the new result:
(solve triples)
]
Perhaps we're curious to see how many of these triples exist. We use @racket[solve*] to find all four solutions:
Perhaps we're curious to see how many of these triples exist. We use @racket[solve*] to find all 10 solutions:
@examples[#:label #f #:eval my-eval
(solve* triples)
]
``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]:
``But really there's only five 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]:
@examples[#:label #f #:eval my-eval
(add-constraint! triples <= '(a b))
@ -153,9 +154,9 @@ The whole example in one block:
(define triples (make-csp))
(add-var! triples 'a (range 10 50))
(add-var! triples 'b (range 10 50))
(add-var! triples 'c (range 10 50))
(add-var! triples 'a (range 1 30))
(add-var! triples 'b (range 1 30))
(add-var! triples 'c (range 1 30))
(define (valid-triple? x y z)
(= (expt z 2) (+ (expt x 2) (expt y 2))))
@ -174,10 +175,10 @@ The whole example in one block:
``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)]
(for*/list ([a (in-range 1 30)]
[b (in-range 1 30)]
#:when (<= a b)
[c (in-range 10 50)]
[c (in-range 1 30)]
#:when (and (coprime? a b c) (valid-triple? a b c)))
