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#lang debug racket
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(require racket/generator sugar/debug)
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(provide (all-defined-out))
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(struct $csp ([vars #:mutable]
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[constraints #:mutable]) #:transparent)
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(define (make-csp) ($csp null null))
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(define debug (make-parameter #false))
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(define (in-cartesian xss)
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(in-generator (let loop ([xss xss][args empty])
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(if (null? xss)
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(yield (reverse args))
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(for ([x (in-list (car xss))])
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(loop (cdr xss) (cons x args)))))))
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(struct $var (name vals) #:transparent)
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(define $var-name? symbol?)
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(struct $constraint (names proc) #:transparent
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#:property prop:procedure
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(λ (constraint csp)
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(unless ($csp? csp)
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(raise-argument-error '$constraint-proc "$csp" csp))
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;; apply proc in many-to-many style
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(for/and ([args (in-cartesian (map (λ (cname) ($csp-vals csp cname)) ($constraint-names constraint)))])
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(apply ($constraint-proc constraint) args))))
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(define/contract (check-name-in-csp! caller csp name)
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(symbol? $csp? $var-name? . -> . void?)
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(define names (map $var-name ($csp-vars csp)))
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(unless (memq name names)
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(raise-argument-error caller (format "one of these existing csp var names: ~v" names) name)))
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(define (nary-constraint? constraint n)
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(= n (constraint-arity constraint)))
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(define/contract (unary-constraint? constraint)
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($constraint? . -> . boolean?)
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(nary-constraint? constraint 1))
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(define/contract (binary-constraint? constraint)
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($constraint? . -> . boolean?)
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(nary-constraint? constraint 2))
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(define/contract (add-vars! csp names [vals-or-procedure empty])
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(($csp? (listof $var-name?)) ((or/c (listof any/c) procedure?)) . ->* . void?)
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(for/fold ([vars ($csp-vars csp)]
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#:result (set-$csp-vars! csp vars))
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([name (in-list names)])
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(when (memq name (map $var-name vars))
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(raise-argument-error 'add-vars! "var that doesn't exist" name))
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(append vars
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(let ([vals (if (procedure? vals-or-procedure)
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(vals-or-procedure)
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vals-or-procedure)])
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(list ($var name vals))))))
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(define/contract (add-var! csp name [vals-or-procedure empty])
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(($csp? $var-name?) ((or/c (listof any/c) procedure?)) . ->* . void?)
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(add-vars! csp (list name) vals-or-procedure))
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(define/contract (add-constraints! csp proc namess [proc-name #false])
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(($csp? procedure? (listof (listof $var-name?))) ((or/c #false $var-name?)) . ->* . void?)
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(set-$csp-constraints! csp (append ($csp-constraints csp)
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(for/list ([names (in-list namess)])
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(for ([name (in-list names)])
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(check-name-in-csp! 'add-constraint! csp name))
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($constraint names (if proc-name
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(procedure-rename proc proc-name)
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proc))))))
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(define/contract (add-pairwise-constraint! csp proc var-names [proc-name #false])
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(($csp? procedure? (listof $var-name?)) ($var-name?) . ->* . void?)
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(add-constraints! csp proc (combinations var-names 2) proc-name))
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(define/contract (add-constraint! csp proc var-names [proc-name #false])
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(($csp? procedure? (listof $var-name?)) ($var-name?) . ->* . void?)
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(add-constraints! csp proc (list var-names) proc-name))
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(define/contract (no-solutions? csp)
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($csp? . -> . boolean?)
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(zero? (state-count csp)))
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(struct inconsistency-signal () #:transparent)
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(define/contract (apply-unary-constraint csp constraint)
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($csp? unary-constraint? . -> . $csp?)
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(match-define ($constraint (list cname) proc) constraint)
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(define new-csp ($csp (for/list ([var (in-list ($csp-vars csp))])
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(match-define ($var name vals) var)
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(if (eq? name cname)
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;; special rule: use promise for a constant value
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;; to skip the filtering
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($var name (if (promise? proc)
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(force proc)
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(filter proc vals)))
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var))
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;; once the constraint is applied, it can go away
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;; ps this is not the same as an "assigned" constraint
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;; because the var may still have multiple values
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(remove constraint ($csp-constraints csp))))
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(when (no-solutions? new-csp) (raise (inconsistency-signal)))
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new-csp)
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(define/contract (make-nodes-consistent csp)
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($csp? . -> . $csp?)
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(for/fold ([csp csp])
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([constraint (in-list ($csp-constraints csp))]
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#:when (unary-constraint? constraint))
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(apply-unary-constraint csp constraint)))
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(define/contract ($csp-vals csp name)
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($csp? $var-name? . -> . (listof any/c))
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(check-name-in-csp! '$csp-vals csp name)
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(for/first ([var (in-list ($csp-vars csp))]
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#:when (eq? name ($var-name var)))
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($var-vals var)))
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(struct $arc (name constraint) #:transparent)
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(define/contract (reduce-domains-by-arc csp arc)
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($csp? $arc? . -> . $csp?)
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(match-define ($arc name ($constraint names constraint-proc)) arc)
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(match-define (list other-name) (remove name names))
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(define proc (if (eq? name (first names)) ; name is on left
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constraint-proc ; so val stays on left
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(λ (val other-val) (constraint-proc other-val val)))) ; otherwise reverse arg order
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(define (satisfies-arc? val)
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(for/or ([other-val (in-list ($csp-vals csp other-name))])
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(proc val other-val)))
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(apply-unary-constraint csp ($constraint (list name)
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(procedure-rename
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satisfies-arc?
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(string->symbol (format "~a-arc-to-~a" (object-name proc) other-name))))))
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(define/contract (binary-constraints->arcs constraints)
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((listof binary-constraint?) . -> . (listof $arc?))
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(for*/list ([constraint (in-list constraints)]
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[name (in-list ($constraint-names constraint))])
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($arc name constraint)))
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(define/contract (terminating-at arcs name)
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((listof $arc?) $var-name? . -> . (listof $arc?))
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(for/list ([arc (in-list arcs)]
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#:when (eq? name (second ($constraint-names ($arc-constraint arc)))))
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arc))
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(define/contract (constraint-assigned? csp constraint)
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($csp? $constraint? . -> . any/c)
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(for/and ([name (in-list ($constraint-names constraint))])
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(memq name (map $var-name (assigned-vars csp)))))
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(define/contract (remove-assigned-constraints csp [arity #false])
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(($csp?) ((or/c #false exact-nonnegative-integer?)) . ->* . $csp?)
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($csp
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($csp-vars csp)
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(for/list ([constraint (in-list ($csp-constraints csp))]
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#:unless (and (if arity (= arity (constraint-arity constraint)) #true)
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(constraint-assigned? csp constraint)))
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constraint)))
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(define (remove-assigned-binary-constraints csp)
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(remove-assigned-constraints csp 2))
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(define/contract (ac-3 csp)
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($csp? . -> . $csp?)
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;; as described by AIMA @ 265
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(define all-arcs (binary-constraints->arcs (filter binary-constraint? ($csp-constraints csp))))
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(for/fold ([csp csp]
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[arcs all-arcs]
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#:result (remove-assigned-binary-constraints csp))
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([i (in-naturals)]
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#:break (empty? arcs))
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(match-define (cons arc other-arcs) arcs)
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(match-define ($arc name _) arc)
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(define reduced-csp (reduce-domains-by-arc csp arc))
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(values reduced-csp (if (= (length ($csp-vals csp name)) (length ($csp-vals reduced-csp name)))
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;; revision did not reduce the domain, so keep going
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other-arcs
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;; revision reduced the domain, so supplement the list of arcs
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(remove-duplicates (append (all-arcs . terminating-at . name) other-arcs))))))
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(define/contract (make-arcs-consistent csp)
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($csp? . -> . $csp?)
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;; csp is arc-consistent if every pair of variables (x y)
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;; has values in their domain that satisfy every binary constraint
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(ac-3 csp))
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(define/contract (var-assigned? var)
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($var? . -> . boolean?)
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(= 1 (remaining-values var)))
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(define/contract (solution-complete? csp)
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($csp? . -> . boolean?)
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(and (andmap var-assigned? ($csp-vars csp)) (empty? ($csp-constraints csp))))
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(define (assigned-helper csp) (partition var-assigned? ($csp-vars csp)))
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(define/contract (unassigned-vars csp)
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($csp? . -> . (listof $var?))
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(match-define-values (assigned unassigned) (assigned-helper csp))
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unassigned)
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(define/contract (assigned-vars csp)
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($csp? . -> . (listof $var?))
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(match-define-values (assigned unassigned) (assigned-helper csp))
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assigned)
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(define/contract (constraint-arity constraint)
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($constraint? . -> . exact-nonnegative-integer?)
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(length ($constraint-names constraint)))
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(define/contract (var-degree csp var)
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($csp? $var? . -> . exact-nonnegative-integer?)
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(for/sum ([constraint (in-list ($csp-constraints csp))]
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#:when (constraint-contains-name? constraint ($var-name var)))
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1))
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(define/contract (select-unassigned-var csp)
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($csp? . -> . $var?)
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(define uvars (unassigned-vars csp))
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(when (empty? uvars)
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(raise-argument-error 'select-unassigned-var "csp with unassigned vars" csp))
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;; minimum remaining values (MRV) rule
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(define uvars-by-rv (sort uvars < #:key remaining-values))
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(define minimum-remaining-values (remaining-values (first uvars-by-rv)))
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(match (takef uvars-by-rv (λ (var) (= minimum-remaining-values (remaining-values var))))
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[(list winning-uvar) winning-uvar]
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[(list mrv-uvars ...) ;; use degree as tiebreaker
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(argmax (λ (var) (var-degree csp var)) mrv-uvars)]))
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(define/contract (order-domain-values vals)
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((listof any/c) . -> . (listof any/c))
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;; todo: least constraining value sort
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vals)
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(define/contract (constraint-contains-name? constraint name)
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($constraint? $var-name? . -> . boolean?)
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(and (memq name ($constraint-names constraint)) #true))
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(define/contract (validate-assignments csp)
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($csp? . -> . $csp?)
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(define assigned-constraints (filter (λ (c) (constraint-assigned? csp c)) ($csp-constraints csp)))
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(for ([constraint (in-list (sort assigned-constraints < #:key constraint-arity))]
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#:unless (constraint csp))
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(raise (inconsistency-signal)))
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(reduce-constraint-arity (remove-assigned-constraints csp)))
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(define/contract (assign-val csp name val)
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($csp? $var-name? any/c . -> . $csp?)
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(define assignment-constraint ($constraint (list name) (delay (list val))))
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(validate-assignments (apply-unary-constraint csp assignment-constraint)))
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(define (reduce-arity proc pattern)
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(unless (match (procedure-arity proc)
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[(arity-at-least val) (<= val (length pattern))]
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[(? number? val) (= val (length pattern))])
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(raise-argument-error 'reduce-arity (format "list of length ~a, same as procedure arity" (procedure-arity proc)) pattern))
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(define reduced-arity-name (string->symbol (format "reduced-arity-~a" (object-name proc))))
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(define-values (id-names vals) (partition symbol? pattern))
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(define new-arity (length id-names))
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(procedure-rename
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(λ xs
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(unless (= (length xs) new-arity)
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(apply raise-arity-error reduced-arity-name new-arity xs))
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(apply proc (for/fold ([acc empty]
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[xs xs]
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[vals vals]
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#:result (reverse acc))
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([pat-item (in-list pattern)])
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(if (symbol? pat-item)
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(values (cons (car xs) acc) (cdr xs) vals)
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(values (cons (car vals) acc) xs (cdr vals))))))
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reduced-arity-name))
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(module+ test
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(require rackunit)
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(define f (λ (a b c d) (+ a b c d)))
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(check-equal? 10 ((reduce-arity f '(1 b c d)) 2 3 4))
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(check-equal? 10 ((reduce-arity f '(1 2 c d)) 3 4))
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(check-equal? 10 ((reduce-arity f '(1 2 3 d)) 4))
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(check-equal? 10 ((reduce-arity f '(1 b 3 d)) 2 4))
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(check-equal? 10 ((reduce-arity f '(a b 3 d)) 1 2 4)))
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(define/contract (reduce-constraint-arity csp [minimum-arity 3])
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(($csp?) (exact-nonnegative-integer?) . ->* . $csp?)
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(define (assigned-name? cname) (memq cname (map $var-name (assigned-vars csp))))
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(define (partially-assigned? constraint)
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(ormap assigned-name? ($constraint-names constraint)))
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($csp ($csp-vars csp)
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(for/list ([constraint (in-list ($csp-constraints csp))])
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(cond
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[(and (<= minimum-arity (constraint-arity constraint))
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(partially-assigned? constraint))
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(match-define ($constraint cnames proc) constraint)
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($constraint (filter-not assigned-name? cnames)
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;; pattern is mix of values and symbols (indicating variables to persist)
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(let ([reduce-arity-pattern (for/list ([cname (in-list cnames)])
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(if (assigned-name? cname)
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($csp-ref csp cname)
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cname))])
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(reduce-arity proc reduce-arity-pattern)))]
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[else constraint]))))
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(module+ test
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(define creduce (assign-val ($csp (list ($var 'a '(1 2 3)) ($var 'b '(2 3)) ($var 'c '(1 2 3 4 5))) (list ($constraint '(a b c) (procedure-rename (λ (a b c) (= (+ a b c) 4)) 'summer)))) 'a 1))
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(check-equal?
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(make-arcs-consistent (reduce-constraint-arity creduce))
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($csp (list ($var 'a '(1)) ($var 'b '(2)) ($var 'c '(1))) '())))
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;; todo: inferences between assignments
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(define/contract (infer csp)
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($csp? . -> . $csp?)
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(validate-assignments (make-arcs-consistent csp)))
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(define/contract (backtracking-solver csp)
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($csp? . -> . generator?)
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(generator ()
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(let backtrack ([csp (make-arcs-consistent (make-nodes-consistent csp))])
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(cond
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[(solution-complete? csp) (yield csp)]
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[else ;; we have at least 1 unassigned var
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(match-define ($var name vals) (select-unassigned-var csp))
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(for ([val (in-list (order-domain-values vals))])
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(with-handlers ([inconsistency-signal? void])
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(backtrack (infer (assign-val csp name val)))))]))))
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(define/contract (solve* csp [finish-proc values][solution-limit +inf.0])
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(($csp?) (procedure? integer?) . ->* . (listof any/c))
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(define solutions
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(for/list ([solution (in-producer (backtracking-solver csp) (void))]
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[idx (in-range solution-limit)])
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(finish-proc solution)))
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(unless (pair? solutions) (raise (inconsistency-signal)))
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solutions)
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(define/contract (solve csp [finish-proc values])
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(($csp?) (procedure?) . ->* . any/c)
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(first (solve* csp finish-proc 1)))
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(define ($csp-ref csp name) (first ($csp-vals csp name)))
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(define/contract (alldiff x y)
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(any/c any/c . -> . boolean?)
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(not (equal? x y)))
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(define/contract (alldiff= x y)
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(any/c any/c . -> . boolean?)
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(not (= x y)))
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(define/contract (remaining-values var)
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($var? . -> . exact-nonnegative-integer?)
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(length ($var-vals var)))
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(define/contract (state-count csp)
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($csp? . -> . exact-nonnegative-integer?)
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(for/product ([var (in-list ($csp-vars csp))])
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(remaining-values var))) |