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4.3.3 amb 評価器の実装

問題を解いていく前に、先に4.3.3節に進み amb 評価器を実装する。

以下が4.1.7節の評価器をベースにして、4.3.3節の解説に基づき実装した amb 評価器。

(define true #t)
(define false #f)

;; 基盤の apply への参照を apply-in-underlying-scheme へ退避させる（こうすることで、基盤の apply に apply-in-underlying-scheme という名前でアクセスできる）。
(define apply-in-underlying-scheme apply)

;;;; apply の定義
(define (apply procedure arguments)
  (cond ((primitive-procedure? procedure)
         (apply-primitive-procedure procedure arguments))
        ((compound-procedure? procedure)
         (eval-sequence
           (procedure-body procedure)
           (extend-environment
             (procedure-parameters procedure)
             arguments
             (procedure-environment procedure))))
        (else
          (error
            "Unknown procedure type -- APPLY" procedure))))

;;;; eval の定義
(define (eval exp env)
  ((analyze exp) env))

(define (analyze exp)
  (cond ((self-evaluating? exp)
         (analyze-self-evaluating exp))
        ((quoted? exp) (analyze-quoted exp))
        ((variable? exp) (analyze-variable exp))
        ((assignment? exp) (analyze-assignment exp))
        ((definition? exp) (analyze-definition exp))
        ((if? exp) (analyze-if exp))
        ((lambda? exp) (analyze-lambda exp))
        ((begin? exp) (analyze-sequence (begin-actions exp)))
        ((cond? exp) (analyze (cond->if exp)))
        ((let? exp) (analyze (let->combination exp)))
        ((amb? exp) (analyze-amb exp))
        ((application? exp) (analyze-application exp))
        (else
          (error "Unknown expression type -- ANALYZE" exp))))

;;;; 手続きの引数
(define (list-of-values exps env)
  (if (no-operands? exps)
      '()
      (cons (eval (first-operand exps) env)
            (list-of-values (rest-operands exps) env))))

;;;; 条件式
(define (eval-if exp env)
  (if (true? (eval (if-predicate exp) env))
      (eval (if-consequent exp) env)
      (eval (if-alternative exp) env)))

;;;; 並び
(define (eval-sequence exps env)
  (cond ((last-exp? exps) (eval (first-exp exps) env))
        (else (eval (first-exp exps) env)
              (eval-sequence (rest-exps exps) env))))

;;;; 代入と定義
(define (eval-assignment exp env)
  (set-variable-value! (assignment-variable exp)
                       (eval (assignment-value exp) env)
                       env)
  'ok)

(define (eval-definition exp env)
  (define-variable! (definition-variable exp)
                    (eval (definition-value exp) env)
                    env)
  'ok)

;;;; 4.1.2 式の表現

;;;; 自己評価式は数と文字だけ
(define (self-evaluating? exp)
  (cond ((number? exp) true)
        ((string? exp) true)
        (else false)))

(define (analyze-self-evaluating exp)
  (lambda (env succeed fail)
          (succeed exp fail)))

;;;; 変数は記号で表現
(define (variable? exp) (symbol? exp))

(define (analyze-variable exp)
  (lambda (env succeed fail)
          (succeed (lookup-variable-value exp env)
                   fail)))

;;;; クォート式は (quote <text-of-quotation>) の形
(define (quoted? exp)
  (tagged-list? exp 'quote))

(define (text-of-quotation exp) (cadr exp))

(define (analyze-quoted exp)
  (let ((qval (text-of-quotation exp)))
       (lambda (env succeed fail)
               (succeed qval fail))))

(define (tagged-list? exp tag)
  (if (pair? exp)
      (eq? (car exp) tag)
      false))

;;;; 代入は (set! <var> <value>) の形
(define (assignment? exp)
  (tagged-list? exp 'set!))

(define (assignment-variable exp) (cadr exp))

(define (assignment-value exp) (caddr exp))

(define (analyze-assignment exp)
  (let ((var (assignment-variable exp))
        (vproc (analyze (assignment-value exp))))
       (lambda (env succeed fail)
               (vproc env
                      (lambda (val fail2) ; *1*
                              (let ((old-value
                                      (lookup-variable-value var env)))
                                   (set-variable-value! var val env)
                                   (succeed 'ok
                                            (lambda () ; *2*
                                                    (set-variable-value! var
                                                                         old-value
                                                                         env)
                                                    (fail2)))))
                      fail))))

;;;; 定義
(define (definition? exp)
  (tagged-list? exp 'define))

(define (definition-variable exp)
  (if (symbol? (cadr exp))
      (cadr exp)
      (caadr exp)))

(define (definition-value exp)
  (if (symbol? (cadr exp))
      (caddr exp)
      (make-lambda (cdadr exp)      ; 仮パラメタ
                   (cddr exp))))    ; 本体

(define (analyze-definition exp)
  (let ((var (definition-variable exp))
        (vproc (analyze (definition-value exp))))
       (lambda (env succeed fail)
               (vproc env
                      (lambda (val fail2)
                              (define-variable! var val env)
                              (succeed 'ok fail2))
                      fail))))

;;;; lambda 式は記号 lambda で始まるリスト
(define (lambda? exp) (tagged-list? exp 'lambda))

(define (lambda-parameters exp) (cadr exp))

(define (lambda-body exp) (cddr exp))

(define (make-lambda parameters body)
  (cons 'lambda (cons parameters body)))

(define (analyze-lambda exp)
  (let ((vars (lambda-parameters exp))
        (bproc (analyze-sequence (lambda-body exp))))
       (lambda (env succeed fail)
               (succeed (make-procedure vars bproc env)
                        fail))))

;;;; 条件式
(define (if? exp) (tagged-list? exp 'if))

(define (if-predicate exp) (cadr exp))

(define (if-consequent exp) (caddr exp))

(define (if-alternative exp)
  (if (not (null? (cdddr exp)))
      (cadddr exp)
      'false))

(define (make-if predicate consequent alternative)
  (list 'if predicate consequent alternative))

(define (analyze-if exp)
  (let ((pproc (analyze (if-predicate exp)))
        (cproc (analyze (if-consequent exp)))
        (aproc (analyze (if-alternative exp))))
       (lambda (env succeed fail)
               (pproc env
                      ;; pred-value を得るための
                      ;; 述語の評価の成功継続
                      (lambda (pred-value fail2)
                              (if (true? pred-value)
                                  (cproc env succeed fail2)
                                  (aproc env succeed fail2)))
                      ;; 述語の評価の失敗継続
                      fail))))

;;;; begin
(define (begin? exp) (tagged-list? exp 'begin))

(define (begin-actions exp) (cdr exp))

(define (last-exp? seq) (null? (cdr seq)))

(define (first-exp seq) (car seq))

(define (rest-exps seq) (cdr seq))

(define (sequence->exp seq)
  (cond ((null? seq) seq)
        ((last-exp? seq) (first-exp seq))
        (else (make-begin seq))))

(define (make-begin seq) (cons 'begin seq))

(define (analyze-sequence exps)
  (define (sequentially a b)
    (lambda (env succeed fail)
            (a env
               ;; a を呼び出す時の成功継続
               (lambda (a-value fail2)
                       (b env succeed fail2))
               ;; a を呼び出す時の失敗継続
               fail)))
  (define (loop first-proc rest-procs)
    (if (null? rest-procs)
        first-proc
        (loop (sequentially first-proc (car rest-procs))
              (cdr rest-procs))))
  (let ((procs (map analyze exps)))
       (if (null? procs)
           (error "Empty sequence -- ANALYZE"))
       (loop (car procs) (cdr procs))))

;;;; 手続き作用
(define (application? exp) (pair? exp))

(define (operator exp) (car exp))

(define (operands exp) (cdr exp))

(define (no-operands? ops) (null? ops))

(define (first-operand ops) (car ops))

(define (rest-operands ops) (cdr ops))

(define (analyze-application exp)
  (let ((pproc (analyze (operator exp)))
        (aprocs (map analyze (operands exp))))
       (lambda (env succeed fail)
               (pproc env
                      (lambda (proc fail2)
                              (get-args aprocs
                                        env
                                        (lambda (args fail3)
                                                (execute-application
                                                  proc args succeed fail3))
                                        fail2))
                      fail))))

(define (get-args aprocs env succeed fail)
  (if (null? aprocs)
      (succeed '() fail)
      ((car aprocs) env
                    ;; この aproc の成功継続
                    (lambda (arg fail2)
                            (get-args (cdr aprocs)
                                      env
                                      ;; get-args の再帰呼び出しの
                                      ;; 成功継続
                                      (lambda (args fail3)
                                              (succeed (cons arg args)
                                                       fail3))
                                      fail2))
                    fail)))

(define (execute-application proc args succeed fail)
  (cond ((primitive-procedure? proc)
         (succeed (apply-primitive-procedure proc args)
                  fail))
        ((compound-procedure? proc)
         ((procedure-body proc)
          (extend-environment (procedure-parameters proc)
                              args
                              (procedure-environment proc))
          succeed
          fail))
        (else
          (error
            "Unknown procedure type -- EXECUTE-APPLICATION"
            proc))))

;;;; cond 式
(define (cond? exp) (tagged-list? exp 'cond))

(define (cond-clauses exp) (cdr exp))

(define (cond-else-clause? clause)
  (eq? (cond-predicate clause) 'else))

(define (cond-predicate clause) (car clause))

(define (cond-actions clause) (cdr clause))

(define (cond->if exp)
  (expand-clauses (cond-clauses exp)))

(define (expand-clauses clauses)
  (if (null? clauses)
      false
      (let ((first (car clauses))
            (rest (cdr clauses)))
           (if (cond-else-clause? first)
               (if (null? rest)
                   (sequence->exp (cond-actions first))
                   (error "ELSE clause isn't last -- COND->IF"
                          clauses))
               (make-if (cond-predicate first)
                        (let ((action (cond-actions first))
                              (predicate (cond-predicate first)))
                             (if (eq? (car action) '=>)
                                 (list (cadr action) predicate)
                                 (sequence->exp action)))
                        (expand-clauses rest))))))

;;;; let 式
(define (let? exp) (tagged-list? exp 'let))

(define (let-clauses exp) (cdr exp))

(define (let-bindings clauses) (car clauses))

(define (let-body clauses) (cdr clauses))

(define (let->combination exp)
  (if (pair? (car (let-clauses exp)))
      (expand-let-clauses (let-clauses exp))
      (expand-named-let-clauses (let-clauses exp))))

(define (expand-let-clauses clauses)
  (if (null? (let-bindings clauses))
      'false
      (cons (make-lambda (map car (let-bindings clauses)) (let-body clauses))
            (map cadr (let-bindings clauses)))))

;;;; 名前付きlet
(define (named-let-var clauses) (car clauses))

(define (named-let-bindings clauses) (cadr clauses))

(define (named-let-body clauses) (caddr clauses))

(define (expand-named-let-clauses clauses)
  (make-begin
    (list
      (list 'define (cons (named-let-var clauses)
                          (map car (named-let-bindings clauses)))
            (named-let-body clauses))
      (cons (named-let-var clauses)
            (map cadr (named-let-bindings clauses))))))

;;;; amb
(define (amb? exp) (tagged-list? exp 'amb))

(define (amb-choices exp) (cdr exp))

(define (ambeval exp env succeed fail)
  ((analyze exp) env succeed fail))

(define (analyze-amb exp)
  (let ((cprocs (map analyze (amb-choices exp))))
       (lambda (env succeed fail)
               (define (try-next choices)
                 (if (null? choices)
                     (fail)
                     ((car choices) env
                                    succeed
                                    (lambda ()
                                            (try-next (cdr choices))))))
               (try-next cprocs))))

;;;; 4.1.3 評価器のデータ構造

;;;; 述語のテスト
(define (true? x)
  (not (eq? x false)))

(define (false? x)
  (eq? x false))

;;;; 手続きの表現
(define (scan-out-defines body)
  (define (iter exp vars sets exps)
    (if (null? exp)
        (list (reverse vars) (reverse sets) (reverse exps))
        (if (definition? (car exp))
            (iter (cdr exp) (cons (list (definition-variable (car exp)) ''*unassigned*) vars) (cons (list 'set! (definition-variable(car exp)) (definition-value (car exp))) sets) exps)
            (iter (cdr exp) vars sets (cons (car exp) exps)))))
  (define (include-define? exp)
    (if (null? exp)
        false
        (if (definition? (car exp))
            true
            (include-define? (cdr exp)))))
  (if (include-define? body)
      (let ((var-val-exp-list (iter body '() '() '())))
           (list (cons 'let (cons (car var-val-exp-list) (append (cadr var-val-exp-list) (caddr var-val-exp-list))))))
      body))

(define (make-procedure parameters body env)
  (list 'procedure parameters body env))

(define (compound-procedure? p)
  (tagged-list? p 'procedure))

(define (procedure-parameters p) (cadr p))

(define (procedure-body p) (caddr p))

(define (procedure-environment p) (cadddr p))

;;;; 環境に対する操作
(define (enclosing-environment env) (cdr env))

(define (first-frame env) (car env))

(define the-empty-environment '())

; フレーム
(define (make-frame variables values)
  (cons variables values))

(define (frame-variables frame) (car frame))

(define (frame-values frame) (cdr frame))

(define (add-binding-to-frame! var val frame)
  (set-car! frame (cons var (car frame)))
  (set-cdr! frame (cons val (cdr frame))))

(define (extend-environment vars vals base-env)
  (if (= (length vars) (length vals))
      (cons (make-frame vars vals) base-env)
      (if (< (length vars) (length vals))
          (error "Too many arguments supplied" vars vals)
          (error "Too few arguments supplied" vars vals))))

(define (lookup-variable-value var env)
  (define (env-loop env)
    (define (scan vars vals)
      (cond ((null? vars)
             (env-loop (enclosing-environment env)))
            ((eq? var (car vars))
             (if (eq? '*unassigned* (car vals))
                 (error "Unassigned variable -- LOOKUP-VARIABLE-VALUE" var)
                 (car vals)))
            (else (scan (cdr vars) (cdr vals)))))
    (if (eq? env the-empty-environment)
        (error "Unbound variable" var)
        (let ((frame (first-frame env)))
             (scan (frame-variables frame)
                   (frame-values frame)))))
  (env-loop env))

(define (set-variable-value! var val env)
  (define (env-loop env)
    (define (scan vars vals)
      (cond ((null? vars)
             (env-loop (enclosing-environment env)))
            ((eq? var (car vars))
             (set-car! vals val))
            (else (scan (cdr vars) (cdr vals)))))
    (if (eq? env the-empty-environment)
        (error "Unbound variable -- SET!" var)
        (let ((frame (first-frame env)))
             (scan (frame-variables frame)
                   (frame-values frame)))))
  (env-loop env))

(define (define-variable! var val env)
  (let ((frame (first-frame env)))
       (define (scan vars vals)
         (cond ((null? vars)
                (add-binding-to-frame! var val frame))
               ((eq? var (car vars))
                (set-car! vals val))
               (else (scan (cdr vars) (cdr vals)))))
       (scan (frame-variables frame)
             (frame-values frame))))

;;;; 4.1.4 評価器をプログラムとして走らせる
(define primitive-procedures
  (list (list 'car car)
        (list 'cdr cdr)
        (list 'cons cons)
        (list 'null? null?)
        (list 'list list)
        (list 'memq memq)
        (list 'member member)
        (list 'assoc assoc)
        (list 'not not)
        (list '+ +)
        (list '- -)
        (list '* *)
        (list '/ /)
        (list '= =)
        (list '< <)
        (list '> >)
        (list 'abs abs)
        (list 'remainder remainder)
        (list 'print print)
        ;; 基本手続きが続く
        ))

(define (primitive-procedure-names)
  (map car
       primitive-procedures))

(define (primitive-procedure-objects)
  (map (lambda (proc) (list 'primitive (cadr proc)))
       primitive-procedures))

(define (setup-environment)
  (let ((initial-env
          (extend-environment (primitive-procedure-names)
                              (primitive-procedure-objects)
                              the-empty-environment)))
       (define-variable! 'true true initial-env)
       (define-variable! 'false false initial-env)
       initial-env))

(define the-global-environment (setup-environment))

(define (primitive-procedure? proc)
  (tagged-list? proc 'primitive))

(define (primitive-implementation proc) (cadr proc))

(define (apply-primitive-procedure proc args)
  (apply-in-underlying-scheme
    (primitive-implementation proc) args))


;;;; 基盤の Lisp システムの"読み込み-評価-印字"ループをモデル化する"駆動ループ(driver loop)"を用意する。
(define input-prompt ";;; Amb-Eval input:")
(define output-prompt ";;; Amb-Eval value:")

(define (driver-loop)
  (define (internal-loop try-again)
    (prompt-for-input input-prompt)
    (let ((input (read)))
         (if (eq? input 'try-again)
             (try-again)
             (begin
               (newline)
               (display ";;; Starting a new problem ")
               (ambeval input
                        the-global-environment
                        ;; ambeval 成功
                        (lambda (val next-alternative)
                                (announce-output output-prompt)
                                (user-print val)
                                (internal-loop next-alternative))
                        ;; ambeval 失敗
                        (lambda ()
                                (announce-output
                                  ";;; There are no more values of")
                                (user-print input)
                                (driver-loop)))))))
  (internal-loop
    (lambda ()
            (newline)
            (display ";;; There is no current problem")
            (driver-loop))))

(define (prompt-for-input string)
  (newline) (newline) (display string) (newline))

(define (announce-output string)
  (newline) (display string) (newline))

(define (user-print object)
  (if (compound-procedure? object)
      (display (list 'compound-procedure
                     (procedure-parameters object)
                     (procedure-body object)
                     '<procedure-env>))
      (display object)))

(driver-loop)

amb の実行結果

;;; Amb-Eval input:
(list (amb 1 2 3) (amb 'a 'b))

;;; Starting a new problem 
;;; Amb-Eval value:
(1 a)

;;; Amb-Eval input:
try-again

;;; Amb-Eval value:
(1 b)

;;; Amb-Eval input:
try-again

;;; Amb-Eval value:
(2 a)

prime-sum-pair の実行結果

;;; Amb-Eval input:
(define (require p)
  (if (not p) (amb)))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (an-element-of items)
  (require (not (null? items)))
  (amb (car items) (an-element-of (cdr items))))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (prime-sum-pair list1 list2)
  (let ((a (an-element-of list1))
        (b (an-element-of list2)))
       (require (prime? (+ a b)))
       (list a b)))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (square n)
  (* n n))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (smallest-divisor n)
  (find-divisor n 2))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (find-divisor n test-divisor)
  (cond ((> (square test-divisor) n) n)
        ((divides? test-divisor n) test-divisor)
        (else (find-divisor n (+ test-divisor 1)))))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (divides? a b)
  (= (remainder b a) 0))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(define (prime? n)
  (= n (smallest-divisor n)))

;;; Starting a new problem
;;; Amb-Eval value:
ok

;;; Amb-Eval input:
(prime-sum-pair '(1 3 5 8) '(20 35 110))

;;; Starting a new problem
;;; Amb-Eval value:
(3 20)

;;; Amb-Eval input:
try-again

;;; Amb-Eval value:
(3 110)

;;; Amb-Eval input:
try-again

;;; Amb-Eval value:
(8 35)

;;; Amb-Eval input:
try-again

;;; There are no more values of
(prime-sum-pair '(1 3 5 8) '(20 35 110))

;;; Amb-Eval input:
(prime-sum-pair '(19 27 30) '(11 36 58))

;;; Starting a new problem
;;; Amb-Eval value:
(30 11)

ちゃんと動作しているようだ。

計算機プログラムの構造と解釈

posted with amazlet at 08.11.07

ジェラルド・ジェイ サスマン ジュリー サスマン ハロルド エイブルソン
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