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#+TITLE: 4.2 - Variations on a Scheme — Lazy Evaluation
#+STARTUP: indent
#+OPTIONS: num:nil
#+BEGIN_QUOTE
Now that we have an evaluator expressed as a Lisp program, we can
experiment with alternative choices in language design simply by
modifying the evaluator.
#+END_QUOTE
Now that we're building our own scheme, we can try out alternate ways
of implementing underlying language features, like changing order of
evaluation, or how variables are bound.
* COMMENT Set up source file
#+BEGIN_SRC scheme :tangle yes
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; 4.2 - Variations on a Scheme — Lazy Evaluation
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(load "4-1.scheme")
#+END_SRC
* <<4.2.1>> Normal Order and Applicative Order
#+BEGIN_QUOTE
Scheme is an applicative-order language, namely, that all the
arguments to Scheme procedures are evaluated when the procedure is
applied. In contrast, normal-order languages delay evaluation of
procedure arguments until the actual argument values are
needed. Delaying evaluation of procedure arguments until the last
possible moment (e.g., until they are required by a primitive
operation) is called /lazy evaluation/.
#+END_QUOTE
From 1.1.5:
#+BEGIN_QUOTE
This alternative “fully expand and then reduce” evaluation method is
known as normal-order evaluation, in contrast to the “evaluate the
arguments and then apply” method that the interpreter actually uses,
which is called applicative-order evaluation.
#+END_QUOTE
** Exploiting lazy evaluation: ~Unless~
#+BEGIN_SRC scheme
(define (unless condition usual-value exceptional-value)
(if condition exceptional-value usual-value))
#+END_SRC
#+BEGIN_QUOTE
One can do useful computation, combining elements to form data
structures and operating on the resulting data structures, even if the
values of the elements are not known.
#+END_QUOTE
#+COMMENT: Find haskell article showing lazy evaluation
** 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
#+BEGIN_SRC scheme
(define (factorial n)
(unless (= n 1)
(* n (factorial (- n 1)))
1))
#+END_SRC
What happens if we attempt to evaluate ~(factorial 5)~? Will our
definitions work in a normal-order language?
----------------------------------------------------------------------
Evaluating this with applicative-order, attempting to evaluate
~(factorial 5)~ would recurse indefinitely, as it continues to
evaluate the recursion before reaching the terminating clause.
With normal-order, the recursion wouldn't be evaluated unless ~(= n
1)~, so the call should terminate successfully.
** 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 form.
----------------------------------------------------------------------
#+COMMENT: This implementation intentionally left blank
A situation where it could be useful to have ~unless~ available as a
procedure would be if there was some need to pass it as an argument to
some other method to parameterize flow control in a higher-order
procedure.
* <<4.2.2>> An Interpreter with Lazy Evaluation
** Modifying the evaluator
*** Eval
#+BEGIN_SRC scheme :tangle yes
(define (eval exp env)
(cond ((self-evaluating? exp)
exp)
((variable? exp)
(lookup-variable-value exp env))
((quoted? exp)
(text-of-quotation exp))
((assignment? exp)
(eval-assignment exp env))
((definition? exp)
(eval-definition exp env))
((if? exp)
(eval-if exp env))
((lambda? exp)
(make-procedure
(lambda-parameters exp)
(lambda-body exp)
env))
((begin? exp)
(eval-sequence
(begin-actions exp)
env))
((cond? exp)
(eval (cond->if exp) env))
((application? exp)
(apply (actual-value (operator exp) env)
(operands exp)
env))
(else
(error "Unknown expression
type: EVAL" exp))))
#+END_SRC
*** Apply
#+BEGIN_SRC scheme :tangle yes
(define (actual-value exp env)
(force-it (eval exp env)))
(define (apply procedure arguments env)
(cond ((primitive-procedure? procedure)
(apply-primitive-procedure
procedure
(list-of-arg-values arguments env))) ; changed
((compound-procedure? procedure)
(eval-sequence
(procedure-body procedure)
(extend-environment
(procedure-parameters procedure)
(list-of-delayed-args arguments env) ; changed
(procedure-environment procedure))))
(else
(error
"Unknown procedure type -- APPLY" procedure))))
#+END_SRC
*** Procedure Arguments
#+BEGIN_SRC scheme :tangle yes
(define (list-of-arg-values exps env)
(if (no-operands? exps)
'()
(cons (actual-value (first-operand exps) env)
(list-of-arg-values (rest-operands exps)
env))))
(define (list-of-delayed-args exps env)
(if (no-operands? exps)
'()
(cons (delay-it (first-operand exps) env)
(list-of-delayed-args (rest-operands exps)
env))))
#+END_SRC
*** Conditionals
#+BEGIN_SRC scheme :tangle yes
(define (eval-if exp env)
(if (true? (actual-value (if-predicate exp) env))
(eval (if-consequent exp) env)
(eval (if-alternative exp) env)))
#+END_SRC
*** driver-loop
#+BEGIN_SRC scheme :tangle yes
(define input-prompt ";;; L-Eval input:")
(define output-prompt ";;; L-Eval value:")
(define (driver-loop)
(prompt-for-input input-prompt)
(let ((input (read)))
(let ((output
(actual-value input the-global-environment)))
(announce-output output-prompt)
(user-print output)))
(driver-loop))
#+END_SRC
** Representing thunks
Essentially, a delayed object *plus* an environment to evaluate it in.
Memoization is achieved in ~force-it~ by changing the tag from ~thunk~
to ~evaluated-thunk~ the first time it is forced, saving the value,
and discarding the environment. Subsequent calls to ~force-it~ will
see the new tag, and simply return the stored value.
#+BEGIN_SRC scheme :tangle yes
(define (force-it obj)
(if (thunk? obj)
(actual-value (thunk-exp obj) (thunk-env obj))
obj))
(define (delay-it exp env)
(list 'thunk exp env))
(define (thunk? obj)
(tagged-list? obj 'thunk))
(define (thunk-exp thunk) (cadr thunk))
(define (thunk-env thunk) (caddr thunk))
(define (evaluated-thunk? obj)
(tagged-list? obj 'evaluated-thunk))
(define (thunk-value evaluated-thunk) (cadr evaluated-thunk))
(define (force-it obj)
(cond ((thunk? obj)
(let ((result (actual-value
(thunk-exp obj)
(thunk-env obj))))
(set-car! obj 'evaluated-thunk)
(set-car! (cdr obj) result) ; replace `exp' with its value
(set-cdr! (cdr obj) '()) ; forget unneeded `env'
result))
((evaluated-thunk? obj)
(thunk-value obj))
(else obj)))
#+END_SRC
** Exercise 4.27
Suppose we type in the following definitions to the lazy evaluator:
#+BEGIN_SRC scheme
(define count 0)
(define (id x)
(set! count (+ count 1))
x)
#+END_SRC
Give the missing values in the following sequence of interactions, and
explain your answers.
#+BEGIN_QUOTE
This exercise demonstrates that the interaction between lazy
evaluation and side effects can be very confusing. This is just what
you might expect from the discussion in *Note Chapter 3.
#+END_QUOTE
#+BEGIN_SRC scheme
(define w (id (id 10)))
#+END_SRC
-
#+BEGIN_SRC scheme
;;; L-Eval input:
count
;;; L-Eval value:
<RESPONSE>
#+END_SRC
- RESPONSE:: ~1~
The outer call to ~id~ is evaluated when passed to the primitive
~define~. The inner argument ~(id 10)~ is not evaluated at this
time.
-
#+BEGIN_SRC scheme
;;; L-Eval input:
w
;;; L-Eval value:
<RESPONSE>
#+END_SRC
- RESPONSE:: ~10~
The ~id~ of ~10~ is ~10~.
-
#+BEGIN_SRC scheme
;;; L-Eval input:
count
;;; L-Eval value:
<RESPONSE>
#+END_SRC
- RESPONSE:: ~2~
Evaluating ~w~ forces its evaluation, which evaluates ~(id
10)~. This increments count again, changing its value to ~2~.
** Exercise 4.28
~Eval~ uses ~actual-value~ rather than ~eval~ 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.
** 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 *Note Exercise 4-27:: and ~count~ starts at 0:
#+BEGIN_SRC scheme
(define (square x)
(* x x))
#+END_SRC
-
#+BEGIN_SRC scheme
;;; L-Eval input:
(square (id 10))
;;; L-Eval value:
<RESPONSE>
#+END_SRC
-
#+BEGIN_SRC scheme
;;; L-Eval input:
count
;;; L-Eval value:
<RESPONSE>
#+END_SRC
Give the responses both when the evaluator memoizes and when it
does not.
** 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 section *Note 4-1-1:: to use ~actual-value~
rather than ~eval~:
#+BEGIN_SRC scheme
(define (eval-sequence exps env)
(cond ((last-exp? exps) (eval (first-exp exps) env))
(else (actual-value (first-exp exps) env)
(eval-sequence (rest-exps exps) env))))
#+END_SRC
a. Ben Bitdiddle thinks Cy is wrong. He shows Cy the ~for-each~
procedure described in *Note Exercise 2-23::, which gives an
important example of a sequence with side effects:
#+BEGIN_SRC scheme
(define (for-each proc items)
(if (null? items)
'done
(begin (proc (car items))
(for-each proc (cdr items)))))
#+END_SRC
He claims that the evaluator in the text (with the original
~eval-sequence~) handles this correctly:
#+BEGIN_SRC scheme
;;; L-Eval input:
(for-each (lambda (x) (newline) (display x))
(list 57 321 88))
57
321
88
;;; L-Eval value:
done
#+END_SRC
Explain why Ben is right about the behavior of ~for-each~.
b. 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:
#+BEGIN_SRC scheme
(define (p1 x)
(set! x (cons x '(2)))
x)
(define (p2 x)
(define (p e)
e
x)
(p (set! x (cons x '(2)))))
#+END_SRC
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~?
c. 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.
d. 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?
** 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
#+BEGIN_SRC scheme
(define (f a (b lazy) c (d lazy-memo))
...)
#+END_SRC
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.
* <<4.2.3>> Streams as Lazy Lists
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