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Exercises Chapter 4

Exercise 4.1

Notice that we cannot tell whether the metacircular evaluator evaluates operands
from left to right or from right to left. Its evaluation order is inherited from
the underlying Lisp: If the arguments to cons in list-of-values are evaluated
from left to right, then list-of-values will evaluate operands from left to
right; and if the arguments to cons are evaluated from right to left, then
list-of-values will evaluate operands from right to left.

Write a version of list-of-values that evaluates operands from left to right
regardless of the order of evaluation in the underlying Lisp. Also write a
version of list-of-values that evaluates operands from right to left.


(define (rtl-list-of-values exps env)
  (if (no-operands? exps) '()
      (let* ((left (zeval (first-operand exps) env) )
             (right (rtl-list-of-values (rest-operands exps) env) ))
        (cons left right))))

Exercise 4.2

Louis Reasoner plans to reorder the cond clauses in eval so that the clause for
procedure applications appears before the clause for assignments. He argues that
this will make the interpreter more efficient: Since programs usually contain
more applications than assignments, definitions, and so on, his modified eval
will usually check fewer clauses than the original eval before identifying the
type of an expression.

1. What is wrong with Louis’s plan? (Hint: What will Louis’s evaluator do with the expression (define x 3)?)

2. Louis is upset that his plan didn’t work. He is willing to go to any lengths to make his evaluator recognize procedure applications before it checks for most other kinds of expressions. Help him by changing the syntax of the evaluated language so that procedure applications start with call. For example, instead of (factorial 3) we will now have to write (call factorial 3) and instead of (+ 1 2) we will have to write (call + 1 2).


#| Answer:
1: The order of operations will cause some variables to be undefined. As the
example suggests, define will be called with 'x' and '3' as arguments. `define'
cannot be made into a procedure because the arguments will be evaluated.

2. Only `application' need be changed:

(define application? (exp)
(tagged-list? exp 'call))
(define operator (exp) (cadr exp))
(define operands (exp) (cddr exp)) |#

Exercise 4.3

Rewrite eval so that the dispatch is done in data-directed style. Compare this
with the data-directed differentiation procedure of Exercise 2.73.
(You may use the car of a compound expression as the type of the expression, as
is appropriate for the syntax implemented in this section.)


(define-class <dispatch-table> ()
  (method-table #:init-value (make-hash-table)
                #:getter method-table))

(define (table-ordinal op type)
  (let ([opstr   (symbol->string op)]
        [typestr (symbol->string type)])
    (string-append opstr "/" typestr)))

(define-method (get (dt <dispatch-table>) op type)
  (if (and (symbol? op) (symbol? type))
      (hash-ref (method-table dt) (table-ordinal op type))

(define-method (put (dt <dispatch-table>) op type item)
  (hash-set! (method-table dt) (table-ordinal op type) item))

(define dispatch-tt (make <dispatch-table>))

(define (list-tag expr)
  "Extract the type of expression"
  (if (pair? expr) (car expr) #f))

(define (install-procedure p)
  (put dispatch-tt 'eval (car p) (cadr p)))

;; Install our procedures
 `([quote  ,(λ (expr env) (text-of-quotation expr))]
   [set!    ,eval-assignment]
   [define ,eval-definition]
   [if     ,eval-if]
   [lambda ,(λ (expr env) (make-procedure (lambda-parameters expr) (lambda-body expr) env))]
   [begin  ,(λ (expr env) (eval-sequence (begin-actions expr) env))]
   [cond   ,(λ (expr env) (zeval (cond->if expr) env))]))

(define (zeval expr env)
  (let ([dispatch-fn (get dispatch-tt 'eval (list-tag expr))])
     [(self-evaluating? expr)
     [(variable? expr)
      (lookup-variable-value expr env)]
     [(procedure? dispatch-fn)
      (dispatch-fn expr env)]
     [(application? expr)
      (zapply (zeval (operator expr) env)
              (list-of-values (operands expr) env))]
     [else (error "Bad Expression" expr)])))

(define (install-driver-loop evaluator fn)
  (put dispatch-tt 'driver-loop evaluator fn))

(install-driver-loop 'zeval base-driver-loop)

Exercise 4.4

Recall the definitions of the special forms and and or from Chapter 1:

`and': The expressions are evaluated from left to right. If any expression
evaluates to false, false is returned; any remaining expressions are not
evaluated. If all the expressions evaluate to true values, the value of the last
expression is returned. If there are no expressions then true is returned.
`or': The expressions are evaluated from left to right. If any expression
evaluates to a true value, that value is returned; any remaining expressions are
not evaluated. If all expressions evaluate to false, or if there are no
expressions, then false is returned.

Install `and' and `or' as new special forms for the evaluator by defining
appropriate syntax procedures and evaluation procedures eval-and and eval-or.
Alternatively, show how to implement and and or as derived expressions.


(define (disjunct exp)
  (if (null? (cdr exp)) 'false
      (cadr exp)))

(define (rest-disjunctions exp)
  (if (null? (cdr exp)) '()
      (cddr exp)))

;; or
(define (or? exp) (tagged-list? exp 'or))
(define (eval-or exp env) (eval-connective exp env true?))
(install-procedure `(or ,eval-or))

;; and
(define (and? exp) (tagged-list? exp 'and))
(define (eval-and exp env) (eval-connective exp env false?))
(install-procedure `(and ,eval-and))

(define (eval-connective exp env oper)
  "eval-connective evaluates the first part of an expression in the given
environment. If the result applied to `oper' is false, it continues to evaluate
until `(oper exp)' argument returns true or no arguments remain."
  (let ([disjunction (zeval (disjunct exp) env)]
        [rest-disjunctions (rest-disjunctions exp)])
    (if (or (oper disjunction) (null? rest-disjunctions)) disjunction
        (eval-connective (cons (operator exp) rest-disjunctions)

Exercise 4.5

Scheme allows an additional syntax for cond clauses, (⟨test⟩ => ⟨recipient⟩). If
⟨test⟩ evaluates to a true value, then ⟨recipient⟩ is evaluated. Its value must
be a procedure of one argument; this procedure is then invoked on the value of
the ⟨test⟩, and the result is returned as the value of the cond expression. For

 (cond ((assoc 'b '((a 1) (b 2))) => cadr)
 (else false))

returns 2.
Modify the handling of cond so that it supports this extended syntax.


(define (cond-is-pipe? exp)
  (if (pair? exp) (eq? (cadr exp) '=>) #f))

(define (cond-recipient exp) (caddr exp))

(define (expand-clauses clauses)
  (if (null? clauses) 'false
      (let ([first (car clauses)]
            [rest  (cdr clauses)])

        ;; check for =>
        (if (cond-is-pipe? first)
            (let ([test (cond-predicate first)])
              (make-if test
                       (list (cond-recipient first) test)
                       (expand-clauses rest)))

            ;; otherwise a normal cond applies
            (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)
                         (sequence->exp (cond-actions first))
                         (expand-clauses rest)))))))

Exercise 4.6

 Let expressions are derived expressions, because

    (let ((⟨var₁⟩ ⟨exp₁⟩) … (⟨varₙ⟩ ⟨expₙ⟩))

is equivalent to

    ((lambda (⟨var₁⟩ … ⟨varₙ⟩)

 Implement a syntactic transformation let->combination that reduces evaluating
 let expressions to evaluating combinations of the type shown above, and add the
 appropriate clause to eval to handle let expressions.


 ([let-bindings       cadr]
  [let-body           cddr]
  [let-binding-vars   (cut map car <...>)]
  [let-binding-exprs  (cut map cadr <...>)]
  [let-vars           (compose let-binding-vars let-bindings)]
  [let-exprs          (compose let-binding-exprs let-bindings)]))

(define (make-let->lambda vars exprs body)
  "Makes a let expression as ((lambda (vars) body) exprs)"
  (cons (make-lambda vars body) exprs))

(define (let->combination exp)
  (if (null? exp) 'false
      (let ([bindings (let-bindings exp)]
            [body     (let-body exp)])
        (make-let->lambda (let-binding-vars bindings)
                          (let-binding-exprs bindings)

(install-procedure `(let ,(λ (exp env) (zeval (let->combination exp) env))))

Exercise 4.7

`let*' is similar to `let', except that the bindings of the `let*' variables are
performed sequentially from left to right, and each binding is made in an
environment in which all of the preceding bindings are visible. For example

       (let* ((x 3)
       (y (+ x 2))
       (z (+ x y 5)))
       (* x z))

returns 39. Explain how a `let*' expression can be rewritten as a set of nested
let expressions, and write a procedure `let*->nested-lets' that performs this
transformation. If we have already implemented let (Exercise 4.6) and we want to
extend the evaluator to handle `let*', is it sufficient to add a clause to eval
whose action is

       (eval (let*->nested-lets exp) env)

       (let (x 3)
       (let (y 2)

       ((lambda (x) (lambda (y) 1) 2) 3)

or must we explicitly expand `let*' in terms of non-derived expressions?


;;; There is nothing preventing `let*' from being defined in terms of existing
;;; `let' expressions
 ([let*-body  caddr]
  [let*-inits cadr]))

;;;; This is a little funky here, I've replaced this with another version copied
;;;; online -- the only thing wrong with this is some monkeying around with the let
;;;; order
;; (define (let*->nested-let exp)
;;   (define (next exp)
;;     (list (operator exp) (cdadr exp) (caddr exp)))
;;   (if (null? exp) 'false
;;       (let ([bindings (let-bindings exp)]
;;             [body     (let-body exp)])
;;         (if (null? bindings) body
;;             (make-let->lambda
;;              (list (car (let-binding-vars bindings)))
;;              (list (car (let-binding-exprs bindings)))
;;              (let*->nested-let (next exp)))))))
(define (let*->nested-lets expr)
  (let ([inits (let*-inits expr)]
        [body (let*-body expr)])
    (define (next expr)
      (if (null? expr) body
          (list 'let (list (car expr)) (next (cdr expr)))))
    (next inits)))

(install-procedure `(let* ,(λ (exp env) (zeval (let*->nested-lets exp) env))))

Exercise 4.8

“Named let” is a variant of let that has the form

       (let ⟨var⟩ ⟨bindings⟩ ⟨body⟩)

The ⟨bindings⟩ and ⟨body⟩ are just as in ordinary let, except that ⟨var⟩ is
bound within ⟨body⟩ to a procedure whose body is ⟨body⟩ and whose parameters are
the variables in the ⟨bindings⟩. Thus, one can repeatedly execute the ⟨body⟩ by
invoking the procedure named ⟨var⟩. For example, the iterative Fibonacci
procedure (1.2.2) can be rewritten using named let as follows:

       (define (fib n)
       (let fib-iter ((a 1) (b 0) (count n))
       (if (= count 0)
       (fib-iter (+ a b)
       (- count 1)))))

Modify let->combination of Exercise 4.6 to also support named let.


(define (named-let? exp) (symbol? (cadr exp)))
 ([nlet-var cadr]
  [nlet-bindings caddr]
  [nlet-body cdddr]))

(define (make-named-let exp)
  ;; get prepped for that long let
  (let* ([body       (nlet-body exp)]
         [bindings   (nlet-bindings exp)]
         [vars       (let-binding-vars bindings)]
         [exprs      (let-binding-exprs bindings)]
         [fnname     (nlet-var exp)]
         [fn         (make-lambda vars body)])
     (let ,bindings
         (define ,fnname ,fn)

(define (let->combination exp)
  (if (null? exp) 'false
      (if (named-let? exp) (make-named-let exp)
          ;; otherwise we're processing a normal let
          (make-let->lambda (let-vars exp)
                            (let-exprs exp)
                            (let-body exp)))))

Exercise 4.9

Many languages support a variety of iteration constructs, such as `do', `for',
`while', and `until'. In Scheme, iterative processes can be expressed in terms of
ordinary procedure calls, so special iteration constructs provide no essential
gain in computational power. On the other hand, such constructs are often
convenient. Design some iteration constructs, give examples of their use, and
show how to implement them as derived expressions.


 ([while-cond cadr]
  [while-body caddr]))

(define (make-while exp)
  (let ([body (while-body exp)]
        [cond (while-cond exp)])
    (if (null? cond) 'false
         (let while-loop ()
           (if ,cond
               (begin ,body

(install-procedure `(while ,(λ (exp env) (zeval (make-while exp) env))))

Exercise 4.11 & Exercise 4.12

4.11: Instead of representing a frame as a pair of lists, we can represent a
frame as a list of bindings, where each binding is a name-value pair. Rewrite
the environment operations to use this alternative representation.

4.12: The procedures define-variable!, set-variable-value! and
lookup-variable-value can be expressed in terms of more abstract procedures for
traversing the environment structure. Define abstractions that capture the
common patterns and redefine the three procedures in terms of these

* Answer

(define (make-frame variables values)
  (map cons variables values))

(define (var-process var environment fn)
  (define (var-search env)
    (if (eq? env the-empty-environment) (begin
                                          ;;(pretty-print environment)
                                          (error "Unbound variable" var))
        (let* ([frame (first-frame env)]
               [entry (assoc var frame)])
          (if entry (fn frame entry)
              (var-search (enclosing-environment env))))))
  (var-search environment))

(define (lookup-variable-value var env)
  (var-process var env (λ (_frame entry) (cdr entry))))

(define (set-variable-value! var val env)
  (var-process var env (λ (frame entry) (set-cdr! entry val))))

(define (define-variable! var val env)
  (set-car! env (assoc-set! (first-frame env) var val)))

Exercise 4.13

Scheme allows us to create new bindings for variables by means of define, but
provides no way to get rid of bindings. Implement for the evaluator a special
form make-unbound! that removes the binding of a given symbol from the
environment in which the make-unbound! expression is evaluated. This problem is
not completely specified. For example, should we remove only the binding in the
first frame of the environment? Complete the specification and justify any
choices you make.


#| Spec:
`undefine' and `unset' are functions that set the name of the binding inside the
closest stack-frame to null. |#
(define (undefine-variable! var env)
  (var-process var env (λ (frame entry) (set-car! entry '()))))

(define (eval-undefinition exp env)
   (definition-variable exp) env)

(install-procedure `(undefine ,eval-undefinition))

Exercise 4.14

Eva Lu Ator and Louis Reasoner are each experimenting with the metacircular
evaluator. Eva types in the definition of map, and runs some test programs that
use it. They work fine. Louis, in contrast, has installed the system version of
map as a primitive for the metacircular evaluator. When he tries it, things go
terribly wrong. Explain why Louis’s map fails even though Eva’s works.


#| Solution:
Louis is trying to use a variable defined inside the *interpreters* stack
frame, not the *interpreTED* stack frame |#

Exercise 4.15

Given a one-argument procedure p and an object a, p is said to “halt” on a if
evaluating the expression (p a) returns a value (as opposed to terminating with
an error message or running forever). Show that it is impossible to write a
procedure halts? that correctly determines whether p halts on a for any
procedure p and object a. Use the following reasoning: If you had such a
procedure halts?, you could implement the following program:

       (define (run-forever)

       (define (try p)
       (if (halts? p p)

Now consider evaluating the expression (try try) and show that any possible
outcome (either halting or running forever) violates the intended behavior of


#| - - - - - Solution:
This problem is an abstract description of a thought experiment Turing conducted in the 1930s which would later be known as the 'halting problem'.

The problem has no solution for a similar reason to the 'liar' paradox:

Suppose it returns true -- `try' enters an endless loop, so it obviously doesn’t halt, while halts? returned true. The contrawise position is identical

Therefore there can be no solution to the problem |#

Exercise 4.16

In this exercise we implement the method just described for interpreting
internal definitions. We assume that the evaluator supports let (see Exercise

1. Change `lookup-variable-value' (4.1.3) to signal an error if the value it finds
is the symbol unassigned.
2. Write a procedure `scan-out-defines' that takes a procedure body and returns an
equivalent one that has no internal definitions, by making the transformation
described above.
3. Install `scan-out-defines' in the interpreter, either in make-procedure or in
procedure-body (see 4.1.3). Which place is better? Why?


;; 1. Solution
(define (simultaneous/lookup-variable-value var env)
  (var-process var env (λ (_f entry)
                         (if (eq? (cdr entry) '*unassigned*)
                             (error "Unassigned var: " var)
                             (cdr entry)))))

;; 2
(define (scan-out-defines expr)
  "Transform a procedure, returning an equivalent one with no internal
  (define has-define (find (λ (e) (and (pair? e) (eq? (car e) 'define)))
  (if has-define
       (λ (elt prev)
         (let ([bindings (let-bindings prev)]
               [body     (let-body prev)])
           ;; merge our (new) bindings & body
           (match elt
             [('define var val)
              `(let ((,var '*unassigned*)
                 (set! ,var ,val)
                 ;; we use ,@ to prevent recursive lists
             [_ `(let ,bindings ,@body ,elt)])))
       '(let ()) ;; we start with a basic let expression

;; simulatanous test
 (scan-out-defines '((define a 1)
                     (make-thing 1)
                     (define b 2)
                     (define c 3)
                     (make-thing a 1)))
 '(let ((c '*unassigned*)
        (b '*unassigned*)
        (a '*unassigned*))
    (set! c 3)
    (set! b 2)
    (set! a 1)
    (make-thing 1)
    (make-thing a 1)))

;; 3 -- I've selected make-procedure so that the conversion is done at
;; interpretation, rather than runtime.
(define (simultaneous/make-procedure parameters body env)
  (list 'procedure
        (scan-out-defines body)

Exercise 4.18

Consider an alternative strategy for scanning out definitions that translates
the example in the text to

       (lambda ⟨vars⟩
       (let ((u 'unassigned)
       (v 'unassigned))
       (let ((a ⟨e1⟩)
       (b ⟨e2⟩))
       (set! u a)
       (set! v b))

Here a and b are meant to represent new variable names, created by the
interpreter, that do not appear in the user’s program. Consider the solve
procedure from 3.5.4:

       (define (solve f y0 dt)
       (define y (integral (delay dy) y0 dt))
       (define dy (stream-map f y))

Will this procedure work if internal definitions are scanned out as shown in
this exercise? What if they are scanned out as shown in the text? Explain.


                                        ; - - - - - - Solution:
;; This wont work because the proxy-value of `y' (a) cannot be directly
;; referenced upon the definition of `dy'

Exercise 4.19

Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator are arguing about the desired
result of evaluating the expression

       (let ((a 1))
       (define (f x)
       (define b (+ a x))
       (define a 5)
       (+ a b))
       (f 10))

Ben asserts that the result should be obtained using the sequential rule for
define: `b' is defined to be 11, then `a' is defined to be 5, so the result is
16. Alyssa objects that mutual recursion requires the simultaneous scope rule
for internal procedure definitions, and that it is unreasonable to treat
procedure names differently from other names. Thus, she argues for the mechanism
implemented in Exercise 4.16. This would lead to a being unassigned at the time
that the value for `b' is to be computed. Hence, in Alyssa’s view the procedure
should produce an error. Eva has a third opinion. She says that if the
definitions of `a' and `b' are truly meant to be simultaneous, then the value 5
for `a' should be used in evaluating b. Hence, in Eva’s view `a' should be 5, `b'
should be 15, and the result should be 20. Which (if any) of these viewpoints do
you support? Can you devise a way to implement internal definitions so that they
behave as Eva prefers?


#| Solution
I like Alyssas view, although Ben's dominates most thinking.

;; Eva's view can be easily supported by swapping the order within the `let'
;; quasiquote of `set!' and `@,body'

Exercise 4.20 (lol)

Because internal definitions look sequential but are actually simultaneous, some
people prefer to avoid them entirely, and use the special form letrec instead.
Letrec looks like let, so it is not surprising that the variables it binds are
bound simultaneously and have the same scope as each other. The sample procedure
f above can be written without internal definitions, but with exactly the same
meaning, as

       (define (f x)
           (lambda (n)
             (if (= n 0)
             (odd? (- n 1)))))
                 (lambda (n)
                   (if (= n 0)
                     (even? (- n 1))))))
       ⟨rest of body of f⟩))

`letrec' expressions, which have the form

       (letrec ((⟨var₁⟩ ⟨exp₁⟩) … (⟨varₙ⟩ ⟨expₙ⟩))

are a variation on let in which the expressions ⟨expₖ⟩ that provide the initial
values for the variables ⟨varₖ⟩ are evaluated in an environment that includes
all the letrec bindings. This permits recursion in the bindings, such as the
mutual recursion of even? and odd? in the example above, or the evaluation of 10
factorial with

       (lambda (n)
       (if (= n 1)
       (* n (fact (- n 1)))))))
       (fact 10))

1. Implement letrec as a derived expression, by transforming a letrec expression
into a let expression as shown in the text above or in Exercise 4.18. That is,
the letrec variables should be created with a let and then be assigned their
values with set!.

2. Louis Reasoner is confused by all this fuss about internal definitions. The
way he sees it, if you don’t like to use define inside a procedure, you can just
use let. Illustrate what is loose about his reasoning by drawing an environment
diagram that shows the environment in which the ⟨rest of body of f⟩ is evaluated
during evaluation of the expression (f 5), with f defined as in this exercise.
Draw an environment diagram for the same evaluation, but with let in place of
letrec in the definition of f.


;; 1.
(define (letrec->let expr)
       ,(map ; generate the (binding . '*unassigned) let binds
         (λ (v) (list v ''*unassigned))
         (map car (cadr expr)))

     ,@(map ; generate the `set!' expressions
        (λ (bind) `(set! ,(car bind) ,(cadr bind)))
        (let-bindings expr))

     ;; and merge our existing body
     ,@(let-body expr))))

  `(letrec ((a (lambda () (b)))
            (b (lambda () (a))))
     (x a)
     (x b)
     (x c)))

 '(let ((a '*unassigned)
        (b '*unassigned))
    (set! a (lambda () (b)))
    (set! b (lambda () (a)))
    (x a)
    (x b)
    (x c)))

(install-procedure `(letrec ,(λ (exp env) (zeval (letrec->let exp) env))))
;; 2.
;; The main problem with Louis's reasoning is that the environment that `let' is
;; being evaluating against is actually expressed in the form of a `lambda' whoses
;; actual function bodies are being passed in as arguments (in the case of (f x)),
;; this means that the lexical scope of `even?' cannot see that of `odd?' and
;; versa.

Exercise 4.21

Amazingly, Louis’s intuition in Exercise 4.20 is correct. It is indeed possible
to specify recursive procedures without using letrec (or even define), although
the method for accomplishing this is much more subtle than Louis imagined. The
following expression computes 10 factorial by applying a recursive factorial

       ((lambda (n)
       ((lambda (fact) (fact fact n))
       (lambda (ft k)
       (if (= k 1)
       (* k (ft ft (- k 1)))))))

1. Check (by evaluating the expression) that this really does compute
factorials. Devise an analogous expression for computing Fibonacci numbers.

Consider the following procedure, which includes mutually recursive internal

       (define (f x)
       (define (even? n)
       (if (= n 0)
       (odd? (- n 1))))
       (define (odd? n)
       (if (= n 0)
       (even? (- n 1))))
       (even? x))

2. Fill in the missing expressions to complete an alternative definition of f,
which uses neither internal definitions nor letrec:

       (define (f x)
       ((lambda (even? odd?)
       (even? even? odd? x))
       (lambda (ev? od? n)
       (if (= n 0)
       (od? ⟨??⟩ ⟨??⟩ ⟨??⟩)))
       (lambda (ev? od? n)
       (if (= n 0)
       (ev? ⟨??⟩ ⟨??⟩ ⟨??⟩)))))

* Answer

;; 1.
;; It does indeed produce Factorials
(define funk-fibonacci
  (λ (n) ;; (it's a fibonacci number)
    ((λ (fib) (fib fib n))
     (λ (fb k)
       (match k
         [0 1]
         [1 1]
         [_ (+ (fb fb (- k 1))
               (fb fb (- k 2)))])))))

;; 2.
(define (feven-4.21 x)
  ((λ (even? odd?)
     (even? even? odd? x))
   (λ (ev? od? n)
     (if (= n 0)
         (od? ev? od? (- n 1))))
   (λ (ev? od? n)
     (if (= n 0)
         (ev? ev? od? (- n 1))))))

(assert (= (funk-fibonacci 4) 5))
(assert (feven-4.21 4))

Exercise 4.22

Extend the evaluator in this section to support the special form let. (See Exercise 4.6.)


(install-analyze-procedure `(let ,(λ (exp) (analyze (let->combination exp)))))

Exercise 4.25

Suppose that (in ordinary applicative-order Scheme) we define unless as shown
above and then define factorial in terms of unless as

       (define (factorial n)
       (unless (= n 1)
       (* n (factorial (- n 1)))
What happens if we attempt to evaluate (factorial 5)? Will our definitions work
in a normal-order language?


;; Solution:

;; Applicative order languages will cause an infinite loop with the definition
;; of `unless' provided by SICP -- on the other hand a normal-order language will
;; do just fine

Exercise 4.26

Ben Bitdiddle and Alyssa P. Hacker disagree over the importance of lazy
evaluation for implementing things such as unless. Ben points out that it’s
possible to implement unless in applicative order as a special form. Alyssa
counters that, if one did that, unless would be merely syntax, not a procedure
that could be used in conjunction with higher-order procedures. Fill in the
details on both sides of the argument. Show how to implement unless as a derived
expression (like cond or let), and give an example of a situation where it might
be useful to have unless available as a procedure, rather than as a special


(define (zv-unless condition consequent alternative)
  (if condition alternative

 ([unless-predicate cadr]
  [unless-alternative caddr]))

(define (unless-consequent exp)
  (if (not (null? (cdddr exp)))
      (cadddr exp)

(define (unless->if exp)
   (unless-predicate exp)
   (unless-consequent exp)
   (unless-alternative exp)))

(define (analyze-unless exp)
  (let ([pproc (analyze (unless-predicate exp))]
        [cproc (analyze (unless-consequent exp))]
        [aproc (analyze (unless-alternative exp))])
    (lambda (env)
      (if (true? (pproc env))
          (cproc env)
          (aproc env)))))

(install-procedure `(unless ,(λ (exp env) (zeval (unless->if exp) env))))
(install-analyze-procedure `(unless ,analyze-unless))

;; this functions utility is for lazy bums who cant type `not'
;; Various evaluator utils

Exercise 4.27

Suppose we type in the following definitions to the lazy evaluator:

(define count 0)
(define (id x) (set! count (+ count 1)) x)
(define w (id (id 10)))

Give the missing values in the following sequence of interactions, and explain
your answers.

;;; L-Eval input:

;;; L-Eval value:

;;; L-Eval input:

;;; L-Eval value:

;;; L-Eval input:

;;; L-Eval value:


Exercise 4.28

lazy-eval uses actual-value rather than leval to evaluate the operator before passing
it to apply, in order to force the value of the operator. Give an example that
demonstrates the need for this forcing.


;; Solution
;; Any function using a lambda as an argument will fail -- the operands are
;; not forced and when trying to apply them you will attempt to apply a thunk
;; instead of a "real" value.

Exercise 4.29

Exhibit a program that you would expect to run much more slowly without
memoization than with memoization. Also, consider the following interaction,
where the id procedure is defined as in Exercise 4.27 and count starts at 0:

       (define (square x) (* x x))

       ;;; L-Eval input:
       (square (id 10))

       ;;; L-Eval value:

       ;;; L-Eval input:

       ;;; L-Eval value:

Give the responses both when the evaluator memoizes and when it does not.


;; Solutions

;; The canonical example of a function sped up by memoization is factorial --
;; each of the components can be reused n^2 times

#| Memoized:
;;; L-Eval input:
(square (id 10))

;;; L-Eval value:

;;; L-Eval input:

;;; L-Eval value:

#| Raw:

;;; L-Eval input:
(square (id 10))

;;; L-Eval value:

;;; L-Eval input:

;;; L-Eval value:


Exercise 4.30

Cy D. Fect, a reformed C programmer, is worried that some side effects may never
take place, because the lazy evaluator doesn’t force the expressions in a
sequence. Since the value of an expression in a sequence other than the last one
is not used (the expression is there only for its effect, such as assigning to a
variable or printing), there can be no subsequent use of this value (e.g., as an
argument to a primitive procedure) that will cause it to be forced. Cy thus
thinks that when evaluating sequences, we must force all expressions in the
sequence except the final one. He proposes to modify eval-sequence from 4.1.1 to
use actual-value rather than eval:

       (define (eval-sequence exps env)
       (cond ((last-exp? exps)
       (eval (first-exp exps) env))
       (actual-value (first-exp exps)
       (eval-sequence (rest-exps exps)

1. Ben Bitdiddle thinks Cy is wrong. He shows Cy the for-each procedure
described in Exercise 2.23, which gives an important example of a sequence with
side effects:

       (define (for-each proc items)
       (if (null? items)
       (begin (proc (car items))
       (for-each proc
       (cdr items)))))

He claims that the evaluator in the text (with the original eval-sequence)
handles this correctly:

       ;;; L-Eval input:
       (lambda (x) (newline) (display x))
       '(57 321 88))

;;; L-Eval value:

Explain why Ben is right about the behavior of for-each.

2. Cy agrees that Ben is right about the for-each example, but says that that’s
not the kind of program he was thinking about when he proposed his change to
eval-sequence. He defines the following two procedures in the lazy evaluator:

       (define (p1 x)
       (set! x (cons x '(2)))

       (define (p2 x)
       (define (p e) e x)
       (p (set! x (cons x '(2)))))

What are the values of (p1 1) and (p2 1) with the original eval-sequence?
What would the values be with Cy’s proposed change to eval-sequence?

3. Cy also points out that changing eval-sequence as he proposes does not affect
the behavior of the example in part a. Explain why this is true.

4. How do you think sequences ought to be treated in the lazy evaluator? Do you
like Cy’s approach, the approach in the text, or some other approach?


#| Solutions
1. In `for-each's case, the procedure is called every time, all primitive
procedures are called -- even if they are the last.

leval: (p1 1) => (1 2); (p2 1) => 1
(`e' is never evaluated -- it's a compound procedure)
actual-value: (p1 1) => (1 2); (p2 1) => (1 2)

3. There is no difference -- primitive procedures are called either way

4. Side effects are a critical aspect of computer programming -- a lazy computer
system needs to execute them in a manner consistent with our expectations of a
basic interpreter -- Cy's approach is the winner. |#

Exercise 4.31

The approach taken in this section is somewhat unpleasant, because it makes an
incompatible change to Scheme. It might be nicer to implement lazy evaluation as
an upward-compatible extension, that is, so that ordinary Scheme programs will
work as before. We can do this by extending the syntax of procedure declarations
to let the user control whether or not arguments are to be delayed. While we’re
at it, we may as well also give the user the choice between delaying with and
without memoization. For example, the definition

(define (f a (b lazy) c (d lazy-memo))

would define f to be a procedure of four arguments, where the first and third
arguments are evaluated when the procedure is called, the second argument is
delayed, and the fourth argument is both delayed and memoized. Thus, ordinary
procedure definitions will produce the same behavior as ordinary Scheme, while
adding the lazy-memo declaration to each parameter of every compound procedure
will produce the behavior of the lazy evaluator defined in this section. Design
and implement the changes required to produce such an extension to Scheme. You
will have to implement new syntax procedures to handle the new syntax for
define. You must also arrange for eval or apply to determine when arguments are
to be delayed, and to force or delay arguments accordingly, and you must arrange
for forcing to memoize or not, as appropriate.


(define (perpetual-thunk? obj) (tagged-list? obj 'always-thunk))
(define (delay-it-perpetually exp env)
  (list 'always-thunk exp env))

(define (list-of-resolved-args parameters arguments env)
   (λ (p a)
     (match p
       [(_ 'lazy)       (delay-it-perpetually a env)]
       [(_ 'lazy-memo)  (delay-it a env)]
       [_               a]))

(define (force-it obj)
  "This is just a memoizing version of `force-it'"
  (define (fetch-result e)
    (actual-value (thunk-exp e) (thunk-env e)))
  (match obj
     (let ([result (fetch-result obj)])
       (set! obj `(evaluated-thunk ,result))
    [evaluated-thunk? (thunk-value obj)]
    [perpetual-thunk? (fetch-result obj)]
    [_ obj]))

(define (fetch-parameter p) (if (pair? p) (car p) p))
(define (extract-parameters fn)
  (map fetch-parameter (procedure-parameters fn)))

(define (capply procedure arguments env)
  "capply is a combined `apply' function -- determining which parameters are
lazy, memoized or raw and supplying them to the function at hand"
  ;;(format #t "procedure: ~a\narguments: ~a\nenv: ~a" procedure arguments env)
  (cond ((primitive-procedure? procedure)
          (list-of-arg-values arguments env)))  ; changed
        ((compound-procedure? procedure)
          (procedure-body procedure)

           (extract-parameters procedure)
           (list-of-resolved-args (procedure-parameters procedure)
                                  env) ; changed
           (procedure-environment procedure))))
        (else (error "Unknown procedure type: APPLY" procedure))))

;; Install our new driver-loop
(define (combined-driver-loop)
  (prompt-for-input ";; Strict/Lazy (ceval) input: ")
  (let* ((input (read))
         (output (actual-value input the-global-environment)))
    (announce-output output-prompt)
    (user-print output))

(install-driver-loop 'ceval combined-driver-loop)

Exercise 4.35

Write a procedure an-integer-between that returns an integer between two given
bounds. This can be used to implement a procedure that finds Pythagorean
triples, i.e., triples of integers ( i , j , k ) between the given bounds such
that i ≤ j and i 2 + j 2 = k 2 , as follows:

       (define (a-pythagorean-triple-between low high)
       (let ((i (an-integer-between low high)))
       (let ((j (an-integer-between i high)))
       (let ((k (an-integer-between j high)))
       (require (= (+ (* i i) (* j j))
       (* k k)))
       (list i j k)))))


 '(define (an-integer-between low high)
    (require (< low high))
    (amb low
         (an-integer-between (+ 1 low) high))))

Exercise 4.36

Exercise 3.69 discussed how to generate the stream of all Pythagorean triples,
with no upper bound on the size of the integers to be searched. Explain why
simply replacing an-integer-between by an-integer-starting-from in the procedure
in Exercise 4.35 is not an adequate way to generate arbitrary Pythagorean
triples. Write a procedure that actually will accomplish this. (That is, write a
procedure for which repeatedly typing try-again would in principle eventually
generate all Pythagorean triples.)


(append! primitive-procedures `((sqrt ,sqrt)))
 '(define (pyth-triple)
    (define k (an-integer-starting-from 1))
    (define j (an-integer-between 1 k))
    (define i (an-integer-between 1 j))
    (require (= (+ (* i i) (* j j))
                (* k k)))
    (list i j k)))

#| Solution Explanation:
Depth first search means that an infinite `amb' loop will never progress past
the first 'amb' |#

Exercise 4.37

Ben Bitdiddle claims that the following method for generating Pythagorean
triples is more efficient than the one in Exercise 4.35. Is he correct? (Hint:
Consider the number of possibilities that must be explored.)

       (define (a-pythagorean-triple-between low high)
       (let ((i (an-integer-between low high))
       (hsq (* high high)))
       (let ((j (an-integer-between i high)))
       (let ((ksq (+ (* i i) (* j j))))
       (require (>= hsq ksq))
       (let ((k (sqrt ksq)))
       (require (integer? k))
       (list i j k))))))


;; It seems to be -- Ben's method only uses N^2 space, while the .35 method uses
;; N^3 space, while also throwing away tons of 'impossible' results.

Exercise 4.38

Modify the multiple-dwelling procedure to omit the requirement that Smith and
Fletcher do not live on adjacent floors. How many solutions are there to this
modified puzzle?


#| Solution

;; There are 5 distinct solutions

 '(define (multiple-dwelling)
    (define baker (amb 1 2 3 4 5))
    (define cooper (amb 1 2 3 4 5))
    (define fletcher (amb 1 2 3 4 5))
    (define miller (amb 1 2 3 4 5))
    (define smith (amb 1 2 3 4 5))
     (distinct? (list baker cooper fletcher
                      miller smith)))
    (require (not (= baker 5)))
    (require (not (= cooper 1)))
    (require (not (= fletcher 5)))
    (require (not (= fletcher 1)))
    (require (> miller cooper))
     (not (= (abs (- fletcher cooper)) 1)))
    (list (list 'baker baker)
          (list 'cooper cooper)
          (list 'fletcher fletcher)
          (list 'miller miller)
          (list 'smith smith))))

Exercise 4.39

Does the order of the restrictions in the multiple-dwelling procedure affect the
answer? Does it affect the time to find an answer? If you think it matters,
demonstrate a faster program obtained from the given one by reordering the
restrictions. If you think it does not matter, argue your case.


#| Solution:
The order of the restrictions can matter regarding runtime invoking the
'failure' continuation through a `require' statement or otherwise can prevent a
great deal of work from being performed.

TODO: Exercise 4.40

In the multiple dwelling problem, how many sets of assignments are there of
people to floors, both before and after the requirement that floor assignments
be distinct? It is very inefficient to generate all possible assignments of
people to floors and then leave it to backtracking to eliminate them. For
example, most of the restrictions depend on only one or two of the person-floor
variables, and can thus be imposed before floors have been selected for all the
people. Write and demonstrate a much more efficient nondeterministic procedure
that solves this problem based upon generating only those possibilities that are
not already ruled out by previous restrictions. (Hint: This will require a nest
of `let' expressions.)


Exercise 4.41

Write an ordinary Scheme program to solve the multiple dwelling puzzle.


(define possible-floors '([baker (1 2 3 4)]
                          [cooper (2 3 4 5)]
                          [fletcher (2 3 4)]
                          [miller (3 4 5)]
                          [smith (1 2 3 4 5)]))

(define (solve-it floors)
  (define (valid? lst)
    (define (distinct? lst)
       ((null? lst) #t)
       ((null? (cdr lst)) #t)
        (and (not (member (car lst) (cdr lst)))
             (distinct? (cdr lst))))))
    (and (distinct? (map cdr lst))
         (> (assoc-ref lst 'miller)
            (assoc-ref lst 'cooper))
         ;; check that smith and fletcher are not adjacent
         (not (= 1 (abs (- (assoc-ref lst 'smith)
                           (assoc-ref lst 'fletcher)))))
         (not (= 1 (abs (- (assoc-ref lst 'cooper)
                           (assoc-ref lst 'fletcher)))))))

  (define (recursive-level lst acc)
    (if (null? lst) (if (valid? acc) (display acc))
        (let* ((top (car lst))
               (name (car top))
               (pfloors (cadr top)))
           (λ (elt)
             (recursive-level (cdr lst)
                              (cons (cons name elt) acc)))

  (recursive-level floors '()))

Exercise 4.42

 Solve the following "Liars" puzzle (from Phillips 1934):

 Five schoolgirls sat for an examination. Their parents–so they
 thought–showed an undue degree of interest in the result. They
 therefore agreed that, in writing home about the examination, each
 girl should make one true statement and one untrue one. The
 following are the relevant passages from their letters:

 - Betty: "Kitty was second in the examination. I was only

 - Ethel: "You'll be glad to hear that I was on top. Joan was

 - Joan: "I was third, and poor old Ethel was bottom."

 - Kitty: "I came out second. Mary was only fourth."

 - Mary: "I was fourth. Top place was taken by Betty."

What in fact was the order in which the five girls were placed?


(append! primitive-procedures `((and ,(λ (a b) (and a b)))))

;;(append! primitive-procedures `((or ,(λ (a b) (or a b)))))
(define (amb/eval-or expr)
  "Both `or' and `and' can be implemented as primitive procedures without
significant issue, `amb/eval-or' exists to advance my knowledge of the
  (let ([disjunction (amb/analyze (disjunct expr))]
        [rest-disjunctions (rest-disjunctions expr)])
    (λ (env succeed fail)
       (λ (value fail2)
         (if (true? value) (succeed value fail2)
             (if (null? rest-disjunctions) (succeed #f fail2)
                 ((amb/eval-or (cons 'or rest-disjunctions)) env succeed fail))))
(amb/install-procedure `(or ,amb/eval-or))

 '(define (xor a b)
    (and (or a b)
         (not (and a b)))))

 '(define (require-or p1 p2)
    (require (xor p1 p2))))

 '(define (solve-liars)
    (define betty (amb 1 2 3 4 5))
    (define ethel (amb 1 2 3 4 5))
    (define joan  (amb 1 2 3 4 5))
    (define kitty (amb 1 2 3 4 5))
    (define mary  (amb 1 2 3 4 5))

     (distinct? (list betty ethel joan kitty mary)))

    ;; betty
    (require-or (= kitty 2) (= betty 3))
    ;; ethel
    (require-or (= ethel 1) (= joan 2))
    ;; joan
    (require-or (= joan 3) (= ethel 5))
    ;; kitty
    (require-or (= kitty 2) (= mary 4))
    ;; mary
    (require-or (= mary 4) (= 1 betty))
     (list 'betty betty)
     (list 'ethel ethel)
     (list 'joan joan)
     (list 'kitty kitty)
     (list 'mary mary))))

Exercise 4.43

Use the `amb' evaluator to solve the following puzzle

Mary Ann Moore's father has a yacht and so has each of his four friends: Colonel
Downing, Mr. Hall, Sir Barnacle Hood, and Dr. Parker. Each of the five also has
one daughter and each has named his yacht after a daughter of one of the others.
Sir Barnacle's yacht is the Gabrielle, Mr. Moore owns the Lorna; Mr. Hall the
Rosalind. The Melissa, owned by Colonel Downing, is named after Sir Barnacle's
daughter. Gabrielle's father owns the yacht that is named after Dr. Parker's
daughter. Who is Lorna's father?

Try to write the program so that it runs efficiently. Also determine how many
solutions there are if we are not told that Mary Ann's last name is Moore.


║ Individual  ║ Yacht      ║
║ Downing     ║ Melissa    ║
║ Hall        ║ Rosalind   ║
║ Moore       ║ Lorna      ║
║ Hood        ║ Gabrielle  ║
Gabrielle's father owns the yacht that is named after Dr. Parker's daughter
║ Parker's Daughter      ║ Gabrielle's Father's Ship ║
The Melissa, owned by Colonel Downing, is named after Sir Barnacle's daughter.
║ Barnacle    ║ (Daughter: Melissa)       ║

(append! primitive-procedures `((eq? ,eq?)))
(append! primitive-procedures `((cadr ,cadr)))
(append! primitive-procedures `((caddr ,caddr)))
 '(define (sailors)
    (define father car)
    (define daughter cadr)
    (define yacht caddr)
    (define (different-names father)
      (not (eq? (daughter father)
                (yacht father))))

    (let ((moore   (list 'moore
          (hood    (list 'hood
          (hall    (list 'hall
                         (amb 'gabrielle 'lorna)
          (downing (list 'downing
                         (amb 'gabrielle 'lorna 'rosalind)
          (parker (list 'parker
                        (amb 'gabrielle 'lorna 'rosalind)

      (let ((gabrielle-father (amb hall downing parker))
            (lorna-father     (amb hall downing parker)))
        (require (eq? (daughter gabrielle-father) 'gabrielle))
        (require (eq? (daughter lorna-father) 'lorna))
        (require (different-names lorna-father))
        (require (different-names gabrielle-father))
        (require (eq? (daughter parker)
                      (yacht gabrielle-father)))

TODO: Exercise 4.44

Exercise 2.42 described the “eight-queens puzzle” of placing queens on a
chessboard so that no two attack each other. Write a nondeterministic program to
solve this puzzle.



Exercise 4.45

Programs designed to accept natural language as input usually start by
  attempting to "parse" the input, that is, to match the input against some
  grammatical structure. For example, we might try to recognize simple sentences
  consisting of an article followed by a noun followed by a verb, such as "The cat
  eats." To accomplish such an analysis, we must be able to identify the parts of
  speech of individual words. We could start with some lists that classify various
          (append! primitive-procedures `((set! ,(λ (x y) (set! x y)))))
          (append! primitive-procedures `((memq ,memq)))
          (map amb/infuse
          '((define nouns '(noun student professor cat class))
          (define verbs '(verb studies lectures eats sleeps))
          (define articles '(article the a))
          (define prepositions '(prep for to in by with))
          (define unparsed '())
          (define (parse-sentence)
          (list 'sentence
          (define (parse-word word-list)
          (require (not (null? unparsed)))
          (require (memq (car unparsed) (cdr word-list)))
          (let ((found-word (car unparsed)))
          (set! unparsed (cdr unparsed))
          (list (car word-list) found-word)))
          (define (parse input)
          (set! unparsed input)
          (let ((sent (parse-sentence)))
          (require (null? unparsed))
          (define (parse-prepositional-phrase)
          (list 'prep-phrase
          (parse-word prepositions)
          (define (parse-verb-phrase)
          (define (maybe-extend verb-phrase)
          (amb verb-phrase
          (maybe-extend (list 'verb-phrase
          (maybe-extend (parse-word verbs)))
          (define (parse-simple-noun-phrase)
          (list 'simple-noun-phrase
          (parse-word articles)
          (parse-word nouns)))
          (define (parse-noun-phrase)
          (define (maybe-extend noun-phrase)
          (amb noun-phrase
          (maybe-extend (list 'noun-phrase
          (maybe-extend (parse-simple-noun-phrase)))))
 With the grammar given above, the following sentence can be parsed in five
 different ways: "the professor lectures to the student in the class with the
 cat." Give the five parses and explain the differences in shades of meaning
 among them.


(parse '(the professor lectures to the student in the class with the cat)) ~>

(simple-noun-phrase (article the) (noun professor))
(verb lectures)
(prep to)
(simple-noun-phrase (article the) (noun student))))
(prep-phrase (prep in) (simple-noun-phrase (article the) (noun class))))
(prep-phrase (prep with) (simple-noun-phrase (article the) (noun cat)))))

(simple-noun-phrase (article the) (noun professor))
(verb lectures)
(prep-phrase (prep to) (simple-noun-phrase (article the) (noun student))))
(prep in)
(simple-noun-phrase (article the) (noun class))
(prep with)
(simple-noun-phrase (article the) (noun cat)))))))

(simple-noun-phrase (article the) (noun professor))
(verb lectures)
(prep to)
(simple-noun-phrase (article the) (noun student))
(prep in)
(simple-noun-phrase (article the) (noun class))))))
(prep-phrase (prep with) (simple-noun-phrase (article the) (noun cat)))))

(simple-noun-phrase (article the) (noun professor))
(verb lectures)
(prep to)
(simple-noun-phrase (article the) (noun student))
(prep-phrase (prep in) (simple-noun-phrase (article the) (noun class))))
(prep with)
(simple-noun-phrase (article the) (noun cat)))))))

(simple-noun-phrase (article the) (noun professor))
(verb lectures)
(prep to)
(simple-noun-phrase (article the) (noun student))
(prep in)
(simple-noun-phrase (article the) (noun class))
(prep with)
(simple-noun-phrase (article the) (noun cat))))))))) |#

Exercise 4.46

The evaluators in sections *Note and do not determine what order operands are
evaluated in. We will see that the `amb' evaluator evaluates them from left to
right. Explain why our parsing program wouldn't work if the operands were
evaluated in some other order.


#| parse-word advances the *unparsed* list from left to right, because it knows
that the parser works this way. Without being sure of the order, we couldn’t
implement parse-word. |#

Exercise 4.49

Alyssa P. Hacker is more interested in generating interesting sentences than in
parsing them. She reasons that by simply changing the procedure `parse-word' so
that it ignores the "input sentence" and instead always succeeds and generates
an appropriate word, we can use the programs we had built for parsing to do
generation instead. Implement Alyssa's idea, and show the first half-dozen or so
sentences generated.


(map amb/infuse
       (define (generate) (generate-sentence))
       (define (generate-sentence)
         (list 'sentence
       (define (generate-word word-list)
         (list (car word-list) (amb word-list)))
       (define (generate-prepositional-phrase)
         (list 'prep-phrase
               (generate-word prepositions)
       (define (generate-verb-phrase)
         (define (maybe-extend verb-phrase)
           (amb verb-phrase
                (maybe-extend (list 'verb-phrase
         (maybe-extend (generate-word verbs)))
       (define (generate-simple-noun-phrase)
         (list 'simple-noun-phrase
               (generate-word articles)
               (generate-word nouns)))
       (define (generate-noun-phrase)
         (define (maybe-extend noun-phrase)
           (amb noun-phrase
                (maybe-extend (list 'noun-phrase
         (maybe-extend (generate-simple-noun-phrase)))))

Exercise 4.50

Implement a new special form ramb that is like amb except
that it searches alternatives in a random order, rather than
from left to right. Show how this can help with Alyssa’s
problem in Exercise 4.49.


(define (shuffle lst)
  "Returns a randomly re-ordered copy of `lst'"
  (if (< (length lst) 1) lst
      (let ([item (list-ref lst (random (length lst)))])
        (cons item (shuffle (delete item lst))))))

(define (amb/analyze-ramb exp)
  (amb/analyze-amb (cons
                    (car exp)
                    (shuffle (amb/choices exp)))))

Exercise 4.51

Implement a new kind of assignment called permanent-set!
that is not undone upon failure. For example, we can choose
two distinct elements from a list and count the number of
trials required to make a successful choice as follows:

       (define count 0)
       (let ((x (an-element-of '(a b c)))
       (y (an-element-of '(a b c))))
       (permanent-set! count (+ count 1))
       (require (not (eq? x y)))
       (list x y count))

       ;;; Starting a new problem
       ;;; Amb-Eval value:
       (a b 2)

       ;;; Amb-Eval input:

     ;;; Amb-Eval value:
     (a c 3)

What values would have been displayed if we had used set!
here rather than permanent-set!?


(append! primitive-procedures `((shuffle ,shuffle)))

(define (amb/analyze-permanent-assignment exp)
  "The execution procedure for permanent assignment is essentially a
redaction of the components of `analyze-assignment' that restore the value
of the old variable proceeding from a failed assignment"
  (let ([var (definition-variable exp)]
        (vproc (amb/analyze (definition-value exp))))
    (lambda (env succeed fail)
       (λ (val fail2)
         (set-variable-value! var val env)
          ;; a failure continues without restoring the old value
          (λ () (fail2))))

(amb/install-procedure `(permanent-set! ,amb/analyze-permanent-assignment))

 '(define (cps)
    (define count 0)
    (let ((x (car (amb '(a b c))))
          (y (car (amb '(z b c)))))
      (permanent-set! count (+ count 1))
      (require (not (eq? x y)))
      (list x y count))))

Exercise 4.52

Implement a new construct called `if-fail' that permits the
user to catch the failure of an expression. `if-fail' takes
two expressions. It evaluates the first expression as usual
and returns as usual if the evaluation succeeds. If the
evaluation fails, however, the value of the second
expression is returned, as in the following example:

       ;;; Amb-Eval input:
       (let ((x (an-element-of '(1 3 5))))
       (require (even? x))

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

       ;;; Amb-Eval input:
       (let ((x (an-element-of '(1 3 5 8))))
       (require (even? x))

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

* Answer

(define (amb/analyze-if-fail exp)
  (let ((clause      (amb/analyze (cadr exp)))
        (alternative (caddr exp)))
    (λ (env succeed fail)
      (clause env
              ;; success, just suplly the result
              (λ (value fail2) (succeed value fail2))
              ;; fail, send our alternative
              (λ () (succeed alternative fail))))))

(amb/install-procedure `(if-fail ,amb/analyze-if-fail))

Exercise 4.53

With `permanent-set!' as described in *Note4.51 and
`if-fail' as in *Note Exercise 4.52, what will be the result
of evaluating

(let ((pairs '()))
(if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110))))
(permanent-set! pairs (cons p pairs))


#| Solution:
It prints:

((8 35) (3 110) (3 20))

Although the let form always fails (it calls (amb) as its
last statement), the pairs get added into pairs, because
permanent-set! doesn’t roll assignments back from failed
paths. |#

Exercise 4.54

If we had not realized that require could be implemented as an ordinary
procedure that uses `amb', to be defined by the user as part of a
nondeterministic program, we would have had to implement it as a special
form. This would require syntax procedures

        (define (require? exp)
        (tagged-list? exp 'require))

        (define (require-predicate exp)
        (cadr exp))

and a new clause in the dispatch in analyze

        ((require? exp) (analyze-require exp))

as well the procedure analyze-require that handles require expressions.
Complete the following definition of analyze-require.

        (define (analyze-require exp)
        (let ((pproc (analyze
        (require-predicate exp))))
        (lambda (env succeed fail)
        (pproc env
        (lambda (pred-value fail2)
        (if ⟨??⟩
        (succeed 'ok fail2)))


(define (require-predicate exp)
  (cadr exp))
(define (amb/analyze-require exp)
  (let ([pproc (analyze (require-predicate exp))])
    (λ (env succeed fail)
       (λ (value fail2)
         (if (not (true? value))
             (succeed 'ok fail2))))

Exercise 4.55

Give simple queries that retrieve the following information from the data

- all people supervised by Ben Bitdiddle;
- the names and jobs of all people in the accounting division;
- the names and addresses of all people who live in Slumerville.


#| Solution:

- (supervisor ?x (Bitdiddle Ben))
- (job ?name (accounting . ?other))
- (address ?name (Slumerville . ?rest))

Exercise 4.56

Formulate compound queries that retrieve the following information:

a. the names of all people who are supervised by Ben Bitdiddle,
together with their addresses;

b. all people whose salary is less than Ben Bitdiddle's,
together with their salary and Ben Bitdiddle's salary;

c. all people who are supervised by someone who is not in the
computer division, together with the supervisor's name and


#| Solutions:
a. (and (supervisor ?name (Bitdiddle Ben))
(address ?name . ?rest))

b. (and (salary ?person ?salary)
(salary (Bitdiddle Ben) ?ben-salary)
(lisp-value > ?salary ?ben-salary))

c. (and (supervisor ?supervisor ?supervisee)
(job ?supervisor ?job))

Exercise 4.57

Define a rule that says that person 1 can replace person 2 if either person
1 does the same job as person 2 or someone who does person 1’s job can also
do person 2’s job, and if person 1 and person 2 are not the same person.
Using your rule, give queries that find the following:

- all people who can replace Cy D. Fect;
- all people who can replace someone who is being paid more than they are,
  together with the two salaries.


 '(rule (replaceable ?p1 ?p2)
        (and (or (and (job ?p1 ?p1post)
                      (job ?p2 ?p1post))
                 (and (job ?p1 ?p1post)
                      (job ?p2 ?p2post)
                      (can-do-job ?p1post ?p2post)))
             (not (same ?p1 ?p2)))))

#| Solution:
- all people who can replace Cy D. Fect
=> (replaceable ?t (Fect Cy D))
- all people who can replace someone who is being paid more than they are,
together with the two salaries.
=> (and (replaceable ?p1 ?p2)
(salary ?p1 ?s1)
(salary ?p2 ?s2)
(lisp-value > ?s2 ?s1))


Exercise 4.58

Define a rule that says that a person is a "big shot" in a division if the
person works in the division but does not have a supervisor who works in
the division.


 '(rule (big-shot ?p1)
         (job ?p1 (?dept . ?job))
          (not (supervisor ?p1 ?p2))
          (and (supervisor ?p1 ?p2)
               (not (job ?p2 (?dept . ?job2))))))))

Exercise 4.59

Ben Bitdiddle has missed one meeting too many. Fearing that his habit of
forgetting meetings could cost him his job, Ben decides to do something
about it. He adds all the weekly meetings of the firm to the Microshaft
data base by asserting the following:


(map query/infuse
     '((meeting accounting (Monday 9am))
       (meeting administration (Monday 10am))
       (meeting computer (Wednesday 3pm))
       (meeting administration (Friday 1pm))))

Each of the above assertions is for a meeting of an entire division.
Ben also adds an entry for the company-wide meeting that spans all the
divisions. All of the company's employees attend this meeting.

(query/infuse '(meeting whole-company (Wednesday 4pm)))


a. On Friday morning, Ben wants to query the data base for all the meetings
that occur that day. What query should he use?

b. Alyssa P. Hacker is unimpressed. She thinks it would be much more useful
to be able to ask for her meetings by specifying her name. So she designs a
rule that says that a person's meetings include all `whole-company'
meetings plus all meetings of that person's division. Fill in the body of
Alyssa's rule.

(rule (meeting-time ?person ?day-and-time)

c. Alyssa arrives at work on Wednesday morning and wonders what meetings
she has to attend that day. Having defined the above rule, what query
should she make to find this out?

;; Solutions:
;; a. (meeting ?meeting (Friday ?time))
;; b.
 '(rule (meeting-time ?person ?day-and-time)
         (meeting whole-company ?day-and-time)
          (meeting ?division ?day-and-time)
          (job ?person (?division . ?job))))))

;; c. (meeting-time (Hacker Alyssa P) (Wednesday . ?x))

Exercise 4.60

By giving the query

       (lives-near ?person (Hacker Alyssa P))

Alyssa P. Hacker is able to find people who live near her, with
whom she can ride to work. On the other hand, when she tries to
find all pairs of people who live near each other by querying

       (lives-near ?person-1 ?person-2)

she notices that each pair of people who live near each other is
listed twice; for example,

       (lives-near (Hacker Alyssa P) (Fect Cy D))
       (lives-near (Fect Cy D) (Hacker Alyssa P))

Why does this happen? Is there a way to find a list of people who
live near each other, in which each pair appears only once?


#| Solution:
The reason this occurs it that there is no 'ordering' on the results
returned, e.g (Dewitt) (Louis) is equally valid to (Louis) (Dewitt) when
the query analyzer searches through valid responses.

 '(rule (lives-near-only ?person-1 ?person-2)
         (address ?person-1 (?town . ?rest-1))
         (address ?person-2 (?town . ?rest-2))
         (not (same ?person-1 ?person-2))
         (lisp-value (lambda (x y)
                       (string<=? (symbol->string (car x))
                                  (symbol->string (car y)))) ?person-1 ?person-2))))

Exercise 4.61

The following rules implement a `next-to' relation that finds adjacent
elements of a list:

(rule (?x next-to ?y in (?x ?y . ?u)))

(rule (?x next-to ?y in (?v . ?z))
(?x next-to ?y in ?z))

What will the response be to the following queries?

(?x next-to ?y in (1 (2 3) 4))

(?x next-to 1 in (2 1 3 1))


;; Solution:
;; The rules written as-is are incompatible with the existing system, a s, I've resolved them in the following way:

(query/infuse '(rule (next-to ?x ?y in (?x ?y . ?u))))
(query/infuse '(rule (next-to ?x ?y in (?v . ?z))
                     (next-to ?x ?y in ?z)))

;; (next-to ?x ?y in (1 (2 3) 4)) => ((1 (2 3) 4) ((2 3) 4))
;; (next-to ?x 1 in (2 1 3 1)) => ((3 1)  (2 1))

Exercise 4.62

Define rules to implement the `last-pair' operation of *Note Exercise 2-17,
which returns a list containing the last element of a nonempty list. Check
your rules on queries such as `(last-pair (3) ?x)', `(last-pair (1 2 3)
?x)', and `(last-pair (2 ?x) (3))'. Do your rules work correctly on queries
such as `(last-pair ?x (3))' ?


;; Solution, it does not work correctly on results that yield an infinite
;; nubmer of answers;
(map query/infuse
     '((rule (last-pair (?h) (?h)))
       (rule (last-pair (?h . ?t) ?last)
             (last-pair ?t ?last))))

Exercise 4.63

The following data base (see Genesis 4) traces the genealogy of the
descendants of Ada back to Adam, by way of Cain:


(map query/infuse
     '((son Adam Cain)
       (son Cain Enoch)
       (son Enoch Irad)
       (son Irad Mehujael)
       (son Mehujael Methushael)
       (son Methushael Lamech)
       (wife Lamech Ada)
       (son Ada Jabal)
       (son Ada Jubal)))

Formulate rules such as

"If S is the son of f, and f is the son of
G, then S is the grandson of G" and "If W is the wife of M, and S
is the son of W, then S is the son of M"

(which was supposedly more true in biblical times than today) that will
enable the query system to find the grandson of Cain; the sons of Lamech;
the grandsons of Methushael. (See *Note Exercise 4-69 for some rules to
deduce more complicated relationships.)

 '(rule (grandparent ?g ?f ?s)
        (and (son ?g ?f) (son ?f ?s))))

 '(rule (son-of ?w ?m ?s)
        (and (wife ?m ?w) (son ?w ?s))))

Exercise 4.64

Louis Reasoner mistakenly deletes the `outranked-by' rule
(section *Note 4.4.1) from the data base. When he realizes this,
he quickly reinstalls it. Unfortunately, he makes a slight change
in the rule, and types it in as


#+beginverse (query/infuse '(rule (loop-outranked-by ?staff-person ?boss) (or (supervisor ?staff-person ?boss) (and (outranked-by ?middle-manager ?boss) (supervisor ?staff-person ?middle-manager)))))

Just after Louis types this information into the system, DeWitt Aull comes by to find out who outranks Ben Bitdiddle. He issues the query

(outranked-by (Bitdiddle Ben) ?who)

After answering, the system goes into an infinite loop. Explain why.


The problem with this query is the order in which applicable rules are specified.

After finding the previous `supervisor' rule applied; that is, the rule conclusion `(supervisor (Bitdiddle Ben) ?boss)' successfully unifies with the query pattern `(supervisor ?staff-person ?boss)' to produce a frame in which `?boss' is bound to `?who'. The interpreter proceeds to evaluate the next rule: `(outranked-by ?middle-manager ?boss)' in the same frame. One answer appears in the database. The `outranked-by' rule is also applicable, so the evaluator again evaluators the `outranked-by' rule body which is equivalent to `(outranked-by ?middle-manager ?who)' |# #+endsrc

Exercise 4.65

Cy D. Fect, looking forward to the day when he will rise in the
organization, gives a query to find all the wheels
(using the `wheel' rule of section *Note 4-4-1)

(wheel ?who)

To his surprise, the system responds

;;; Query results:
(wheel (Warbucks Oliver))
(wheel (Bitdiddle Ben))
(wheel (Warbucks Oliver))
(wheel (Warbucks Oliver))
(wheel (Warbucks Oliver))

Why is Oliver Warbucks listed four times?


#| Solution:

The body of `wheel' contains a query pattern
`(supervisor ?x ?middle-manager)' that can be satisfied by a
variety of different values, each of those values contributes
a match to the rule |#

Exercise 4.66

Ben has been generalizing the query system to provide statistics about
the company. For example, to find the total salaries of all the
computer programmers one will be able to say

(sum ?amount
(and (job ?x (computer programmer))
(salary ?x ?amount)))

In general, Ben's new system allows expressions of the form

(accumulation-function <VARIABLE>

where `accumulation-function' can be things like `sum', `average', or
`maximum'. Ben reasons that it should be a cinch to implement this. He
will simply feed the query pattern to `qeval'. This will produce a
stream of frames. He will then pass this stream through a mapping
function that extracts the value of the designated variable from each
frame in the stream and feed the resulting stream of values to the
accumulation function. Just as Ben completes the implementation and is
about to try it out, Cy walks by, still puzzling over the `wheel'
query result in exercise *Note Exercise 4-65. When Cy shows Ben the
system's response, Ben groans, "Oh, no, my simple accumulation scheme
won't work!"

What has Ben just realized? Outline a method he can use to salvage the


#| Solution
Although the query supplied doesn't demonstrate it, it is possible to
have multiple frames returns. |#

Exercise 4.67

Devise a way to install a loop detector in the query system so as to
avoid the kinds of simple loops illustrated in the text and in *NoteExercise 4.64.

The general idea is that the system should maintain some sort of
history of its current chain of deductions and should not begin
processing a query that it is already working on. Describe what kind
of information (patterns and frames) is included in this history, and
how the check should be made. (After you study the details of the
query-system implementation in section *Note 4.4.4, you may want to
modify the system to include your loop detector.)


;; Because the rule variable contains the execution information, we can
;; lookup if a rule with the exact same parameters has been called before.
(define executed-rules '())
(define (previously-executed? rule)
  (member rule executed-rules))
(define (add-executed-rule! rule)
  (set! executed-rules (cons rule executed-rules)))

(define (noloop/apply-a-rule rule query-pattern query-frame)
  (if (previously-executed? rule) stream-null
      (let* ([clean-rule (rename-variables-in rule)] ; alpha-conversion
             [unify-result (unify-match query-pattern
                                        (conclusion clean-rule)
        (if (eq? unify-result 'failed) stream-null
              (add-executed-rule! rule)
              (qeval (rule-body clean-rule)

;; The following are just driver extensions to permit for `noloop' to work
(define (noloop/query-driver-loop)
  (set! executed-rules '())
  (set! apply-a-rule noloop/apply-a-rule)

(define (noloop/query/eval expr)
  (set! executed-rules '())
  (set! apply-a-rule noloop/apply-a-rule)
  (let ((q (query-syntax-process expr)))
      (λ (frame)
        (instantiate q frame (λ (v f) (contract-question-mark v))))
      (qeval q (singleton-stream '()))))))

(install-driver-loop 'qeval-noloop noloop/query-driver-loop)

Exercise 4.68

Define rules to implement the `reverse' operation of *Note Exercise 2-18,
which returns a list containing the same elements as a given list in
reverse order. (Hint: Use `append-to-form'.) Can your rules answer both
`(reverse (1 2 3) ?x)' and `(reverse ?x (1 2 3))' ?


;; This is one solution that doesn't use append-to-form, but doesn't
;; generate proper lists
(map query/infuse
     '((rule (bad-reverse () ()))
       (rule (bad-reverse (?car . ?cdr) (?cdr1 . ?car))
             (bad-reverse ?cdr ?cdr1))))

(map query/infuse
     '((rule (reverse () ()))
       (rule (reverse ?x ?y)
              (append-to-form (?car) ?rest ?x)
              (append-to-form ?rev (?car) ?y)
              (reverse ?rest ?rev)))))

Exercise 4.69

Beginning with the data base and the rules you formulated in Exercise 4.63,
devise a rule for adding “greats” to a grandson relationship. This should
enable the system to deduce that Irad is the great-grandson of Adam, or
that Jabal and Jubal are the great-great-great-great-great-grandsons of
Adam. (Hint: Represent the fact about Irad, for example, as ((great
grandson) Adam Irad). Write rules that determine if a list ends in the word
grandson. Use this to express a rule that allows one to derive the
relationship ((great . ?rel) ?x ?y), where ?rel is a list ending in
grandson.) Check your rules on queries such as ((great grandson) ?g ?ggs)
and (?relationship Adam Irad).


I cannot find a solution to this

Exercise 4.70

What is the purpose of the `let' bindings in the procedures
`add-assertion!' and `add-rule!' ? What would be wrong with the following
implementation of `add-assertion!' ? Hint: Recall the definition of the
infinite stream of ones in section *Note 3.5.2: `(define ones (cons-stream 1 ones))'.

(define (add-assertion! assertion)
(store-assertion-in-index assertion)
(cons-stream assertion THE-ASSERTIONS))

* Answer

#| Solution is non-indexable |#

Exercise 4.71

Louis Reasoner wonders why the simple-query and disjoin procedures
( are implemented using explicit delay operations, rather than
being defined as follows:

(define (simple-query query-pattern frame-stream)
(lambda (frame)
(find-assertions query-pattern frame)
(apply-rules query-pattern frame)))

(define (disjoin disjuncts frame-stream)
(if (empty-disjunction? disjuncts)
(qeval (first-disjunct disjuncts)
(disjoin (rest-disjuncts disjuncts)

Can you give examples of queries where these simpler definitions would lead
to undesirable behavior?


Exercise 4.72

Why do disjoin and stream-flatmap interleave the streams rather than simply
append them? Give examples that illustrate why interleaving works better.
(Hint: Why did we use interleave in 3.5.3?)


#| Solution:
`interleave' provides a way to achieve 'diagonalization', that is to say so
that every element in an infinite list can be reached, rather than focusing
on either an input stream or a database |#

Exercise 4.73

Why does flatten-stream use delay explicitly? What would be wrong with
defining it as follows:

(define (flatten-stream stream)
(if (stream-null? stream)
(interleave (stream-car stream)
(stream-cdr stream)))))


#| Solution:
Because it would loop forever - the evaluation strategy of a scheme
interpreter dictates that arguments are passed in their 'fully evaluated'

TODO Exercise 4.74

Alyssa P. Hacker proposes to use a simpler version of stream-flatmap in
negate, lisp-value, and find-assertions. She observes that the procedure
that is mapped over the frame stream in these cases always produces either


TODO Exercise 4.75

Implement for the query language a new special form called `unique'.
`Unique' should succeed if there is precisely one item in the data base
satisfying a specified query. For example,

(unique (job ?x (computer wizard)))

should print the one-item stream

(unique (job (Bitdiddle Ben) (computer wizard)))

since Ben is the only computer wizard, and

(unique (job ?x (computer programmer)))

should print the empty stream, since there is more than one
computer programmer. Moreover,

(and (job ?x ?j) (unique (job ?anyone ?j)))

should list all the jobs that are filled by only one person, and
the people who fill them.

There are two parts to implementing `unique'. The first is to
write a procedure that handles this special form, and the second
is to make `qeval' dispatch to that procedure. The second part is
trivial, since `qeval' does its dispatching in a data-directed
way. If your procedure is called `uniquely-asserted', all you
need to do is

(put 'unique 'qeval uniquely-asserted)

and `qeval' will dispatch to this procedure for every query whose
`type' (`car') is the symbol `unique'.

The real problem is to write the procedure `uniquely-asserted'.
This should take as input the `contents' (`cdr') of the `unique'
query, together with a stream of frames. For each frame in the
stream, it should use `qeval' to find the stream of all extensions
to the frame that satisfy the given query. Any stream that does
not have exactly one item in it should be eliminated. The
remaining streams should be passed back to be accumulated into one
big stream that is the result of the `unique' query. This is
similar to the implementation of the `not' special form.

Test your implementation by forming a query that lists all people
who supervise precisely one person.


TODO Exercise 4.76

Our implementation of `and' as a series combination of queries
is elegant, but it is inefficient because in processing
the second query of the `and' we must scan the data
base for each frame produced by the first query. If the data base
has n elements, and a typical query produces a number
of output frames proportional to n (say n/k),
then scanning the data base for each frame produced by the first
query will require n2/k calls to the pattern matcher. Another
approach would be to process the two clauses of the `and'
separately, then look for all pairs of output frames that are
compatible. If each query produces n/k output frames, then this
means that we must perform n2/k2 compatibility checks–a factor
of k fewer than the number of matches required in our current

Devise an implementation of `and' that uses this strategy. You
must implement a procedure that takes two frames as inputs, checks
whether the bindings in the frames are compatible, and, if so,
produces a frame that merges the two sets of bindings. This
operation is similar to unification.


TODO Exercise 4.77

In section *Note 4-4-3:: we saw that `not' and
`lisp-value' can cause the query language to give "wrong" answers
if these filtering operations are applied to frames in which
variables are unbound. Devise a way to fix this shortcoming. One
idea is to perform the filtering in a "delayed" manner by
appending to the frame a "promise" to filter that is fulfilled
only when enough variables have been bound to make the operation
possible. We could wait to perform filtering until all other
operations have been performed. However, for efficiency's sake, we
would like to perform filtering as soon as possible so as to cut
down on the number of intermediate frames generated.


TODO Exercise 4.78

Redesign the query language as a nondeterministic
program to be implemented using the evaluator of section *Note
4-3::, rather than as a stream process. In this approach, each
query will produce a single answer (rather than the stream of all
answers) and the user can type `try-again' to see more answers.
You should find that much of the mechanism we built in this
section is subsumed by nondeterministic search and backtracking.
You will probably also find, however, that your new query language
has subtle differences in behavior from the one implemented here.
Can you find examples that illustrate this difference?


TODO Exercise 4.79

When we implemented the Lisp evaluator in section *Note 4-1, we
saw how to use local environments to avoid name conflicts between
the parameters of procedures. For example, in evaluating

(define (square x)
(* x x))

(define (sum-of-squares x y)
(+ (square x) (square y)))

(sum-of-squares 3 4)

there is no confusion between the `x' in `square' and the `x' in
`sum-of-squares', because we evaluate the body of each procedure
in an environment that is specially constructed to contain
bindings for the local variables. In the query system, we used a
different strategy to avoid name conflicts in applying rules.
Each time we apply a rule we rename the variables with new names
that are guaranteed to be unique. The analogous strategy for the
Lisp evaluator would be to do away with local environments and
simply rename the variables in the body of a procedure each time
we apply the procedure.

Implement for the query language a rule-application method that
uses environments rather than renaming. See if you can build on
your environment structure to create constructs in the query
language for dealing with large systems, such as the rule analog
of block-structured procedures. Can you relate any of this to the
problem of making deductions in a context (e.g., "If I supposed
that P were true, then I would be able to deduce A and B.") as a
method of problem solving? (This problem is open-ended. A good
answer is probably worth a Ph.D.)