|
In section 5.1 we saw how to
transform simple Scheme programs into descriptions of register
machines. We will now perform this transformation on a more complex
program, the metacircular evaluator of
sections 4.1.1-4.1.4,
which shows how
the behavior of a Scheme interpreter can be described in terms of the
procedures eval and apply.
The explicit-control
evaluator that we develop in this section shows how the underlying
procedure-calling and argument-passing mechanisms used in the
evaluation process can be described in terms of operations on
registers and stacks. In addition, the explicit-control evaluator can
serve as an implementation of a Scheme interpreter, written in a
language that is very similar to the native machine language of
conventional computers. The evaluator can be executed by the
register-machine simulator of section 5.2.
Alternatively, it can be used as a starting point for building a
machine-language implementation of a Scheme evaluator, or even a
special-purpose machine for evaluating Scheme expressions.
Figure 5.16 shows such a hardware implementation: a
silicon chip that acts as an evaluator for Scheme. The chip designers
started with the data-path and controller specifications for a
register machine similar to the evaluator described in this section
and used design automation programs to construct the
integrated-circuit layout.19
In designing the explicit-control evaluator, we must specify the
operations to be used in our register machine. We described the
metacircular evaluator in terms of abstract syntax, using procedures
such as quoted? and make-procedure. In implementing the
register machine, we could expand these procedures into sequences of
elementary list-structure memory operations, and implement these
operations on our register machine. However, this would make our
evaluator very long, obscuring the basic structure with
details. To clarify the presentation, we will include as primitive
operations of the register machine the syntax procedures given in
section 4.1.2 and the procedures for
representing environments and other run-time data given in
sections 4.1.3 and 4.1.4.
In order to completely specify an evaluator that could be programmed
in a low-level machine language or implemented in hardware, we would
replace these operations by more elementary operations, using the
list-structure implementation we described in
section 5.3.
|
|
Our Scheme evaluator register machine includes a stack and seven
registers: exp, env, val, continue, proc,
argl, and unev. Exp is used to hold the expression
to be evaluated, and env contains the environment in which the
evaluation is to be performed. At the end of an evaluation, val
contains the value obtained by evaluating the expression in the
designated environment. The continue register is used to
implement recursion, as explained in
section 5.1.4. (The evaluator needs to call
itself recursively, since evaluating an expression requires evaluating
its subexpressions.) The registers proc, argl, and unev are used in evaluating combinations.
We will not provide a data-path diagram to show how the registers and operations of the evaluator are connected, nor will we give the complete list of machine operations. These are implicit in the evaluator's controller, which will be presented in detail.
The central element in the evaluator is the sequence of instructions
beginning at eval-dispatch. This corresponds to the eval
procedure of the metacircular evaluator described in
section 4.1.1. When the controller starts at eval-dispatch, it evaluates the expression specified by exp in
the environment specified by env. When evaluation is complete,
the controller will go to the entry point stored in continue, and the
val register will hold the value of the expression. As with the
metacircular eval, the structure of eval-dispatch is a
case analysis on the syntactic type of the expression to be
evaluated.20
eval-dispatch
(test (op self-evaluating?) (reg exp))
(branch (label ev-self-eval))
(test (op variable?) (reg exp))
(branch (label ev-variable))
(test (op quoted?) (reg exp))
(branch (label ev-quoted))
(test (op assignment?) (reg exp))
(branch (label ev-assignment))
(test (op definition?) (reg exp))
(branch (label ev-definition))
(test (op if?) (reg exp))
(branch (label ev-if))
(test (op lambda?) (reg exp))
(branch (label ev-lambda))
(test (op begin?) (reg exp))
(branch (label ev-begin))
(test (op application?) (reg exp))
(branch (label ev-application))
(goto (label unknown-expression-type))
Numbers and strings (which are self-evaluating),
variables, quotations, and lambda expressions have no
subexpressions to be evaluated. For these, the evaluator simply
places the correct value in the val register and continues
execution at the entry point specified by continue. Evaluation
of simple expressions is performed by the following controller code:
ev-self-eval
(assign val (reg exp))
(goto (reg continue))
ev-variable
(assign val (op lookup-variable-value) (reg exp) (reg env))
(goto (reg continue))
ev-quoted
(assign val (op text-of-quotation) (reg exp))
(goto (reg continue))
ev-lambda
(assign unev (op lambda-parameters) (reg exp))
(assign exp (op lambda-body) (reg exp))
(assign val (op make-procedure)
(reg unev) (reg exp) (reg env))
(goto (reg continue))
Observe how ev-lambda uses the unev and exp
registers to hold the parameters and body of the lambda expression so
that they can be passed to the make-procedure operation, along
with the environment in env.
A procedure application is specified by a combination containing an
operator and operands. The operator is a subexpression whose value is
a procedure, and the operands are subexpressions whose values are the
arguments to which the procedure should be applied. The metacircular
eval handles applications by calling itself recursively to
evaluate each element of the combination, and then passing the results
to apply, which performs the actual procedure application. The
explicit-control evaluator does the same thing; these recursive calls
are implemented by goto instructions, together with use of the
stack to save registers that will be restored after the recursive call
returns. Before each call we will be careful to identify which
registers must be saved (because their values will be needed
later).21
We begin the evaluation of an application by evaluating the operator
to produce a procedure, which will later be applied to the evaluated
operands. To evaluate the operator, we move it to the exp
register and go to eval-dispatch. The environment in the env register is already the correct one in which to evaluate the
operator. However, we save env because we will need it later to
evaluate the operands. We also extract the operands into unev
and save this on the stack. We set up continue so that eval-dispatch will resume at ev-appl-did-operator after the
operator has been evaluated. First, however, we save the old value of
continue, which tells the controller where to continue after the
application.
ev-application
(save continue)
(save env)
(assign unev (op operands) (reg exp))
(save unev)
(assign exp (op operator) (reg exp))
(assign continue (label ev-appl-did-operator))
(goto (label eval-dispatch))
Upon returning from evaluating the operator subexpression, we proceed
to evaluate the operands of the combination and to accumulate the
resulting arguments in a list, held in argl. First we restore
the unevaluated operands and the environment. We initialize argl to an empty list. Then we assign to the proc register the
procedure that was produced by evaluating the operator. If there are
no operands, we go directly to apply-dispatch. Otherwise we
save proc on the stack and start the argument-evaluation
loop:22
ev-appl-did-operator
(restore unev) ; the operands
(restore env)
(assign argl (op empty-arglist))
(assign proc (reg val)) ; the operator
(test (op no-operands?) (reg unev))
(branch (label apply-dispatch))
(save proc)
Each cycle of the argument-evaluation loop evaluates an operand
from the list in unev and accumulates the result into argl.
To evaluate an operand, we place it in the exp register
and go to eval-dispatch, after setting continue so that
execution will resume with the argument-accumulation phase. But first
we save the arguments accumulated so far (held in argl), the
environment (held in env), and the remaining operands to be evaluated
(held in unev). A special case is made for the evaluation of the
last operand, which is handled at ev-appl-last-arg.
ev-appl-operand-loop
(save argl)
(assign exp (op first-operand) (reg unev))
(test (op last-operand?) (reg unev))
(branch (label ev-appl-last-arg))
(save env)
(save unev)
(assign continue (label ev-appl-accumulate-arg))
(goto (label eval-dispatch))
When an operand has been evaluated, the value is accumulated into the
list held in argl. The operand is then removed from the list of
unevaluated operands in unev, and the argument-evaluation continues.
ev-appl-accumulate-arg
(restore unev)
(restore env)
(restore argl)
(assign argl (op adjoin-arg) (reg val) (reg argl))
(assign unev (op rest-operands) (reg unev))
(goto (label ev-appl-operand-loop))
Evaluation of the last argument is handled differently. There is no
need to save the environment or the list of unevaluated operands
before going to eval-dispatch,
since they will not be required after the last operand is evaluated.
Thus, we return from the evaluation to a special entry point ev-appl-accum-last-arg, which restores the argument list, accumulates
the new argument, restores the saved procedure, and goes off to
perform the application.23
ev-appl-last-arg
(assign continue (label ev-appl-accum-last-arg))
(goto (label eval-dispatch))
ev-appl-accum-last-arg
(restore argl)
(assign argl (op adjoin-arg) (reg val) (reg argl))
(restore proc)
(goto (label apply-dispatch))
The details of the argument-evaluation loop determine the order in
which the interpreter evaluates the operands of a combination (e.g.,
left to right or right to left -- see
exercise 3.8). This order is not determined
by the metacircular evaluator, which inherits its control structure
from the underlying Scheme in which it is implemented.24 Because the first-operand
selector (used in ev-appl-operand-loop to extract successive operands
from unev) is implemented as car and the rest-operands selector is implemented as cdr, the
explicit-control evaluator will evaluate the operands of a combination
in left-to-right order.
The entry point apply-dispatch corresponds to the apply
procedure of the metacircular evaluator. By the time we get to apply-dispatch, the proc register contains the procedure to
apply and argl contains the list of evaluated arguments to which
it must be applied. The saved value of continue (originally
passed to eval-dispatch and saved at ev-application),
which tells where to return with the result of the procedure
application, is on the stack. When the application is complete, the
controller transfers to the entry point specified by the saved continue, with the result of the application in val. As with
the metacircular apply, there are 2 cases to consider. Either
the procedure to be applied is a primitive or it is a compound
procedure.
apply-dispatch
(test (op primitive-procedure?) (reg proc))
(branch (label primitive-apply))
(test (op compound-procedure?) (reg proc))
(branch (label compound-apply))
(goto (label unknown-procedure-type))
We assume that each primitive is implemented so as to obtain its
arguments from argl and place its result in val. To
specify how the machine handles primitives, we would have to provide a
sequence of controller instructions to implement each primitive and
arrange for primitive-apply to dispatch to the
instructions for the primitive identified by the
contents of proc. Since we are interested in the structure of
the evaluation process rather than the details of the primitives, we
will instead just use an apply-primitive-procedure operation
that applies the procedure in proc to the arguments in argl. For the purpose of simulating the evaluator with the simulator
of section 5.2 we use the procedure apply-primitive-procedure, which calls on the underlying Scheme
system to perform the application, just as we did for the metacircular
evaluator in section 4.1.4. After computing the
value of the primitive application, we restore continue and go
to the designated entry point.
primitive-apply
(assign val (op apply-primitive-procedure)
(reg proc)
(reg argl))
(restore continue)
(goto (reg continue))
To apply a compound procedure, we proceed just as with the
metacircular evaluator. We construct a frame that binds the
procedure's parameters to the arguments, use this frame to
extend the environment carried by the procedure, and evaluate in this
extended environment the sequence of expressions that forms the body
of the procedure. Ev-sequence, described below in
section 5.4.2, handles the evaluation
of the sequence.
compound-apply
(assign unev (op procedure-parameters) (reg proc))
(assign env (op procedure-environment) (reg proc))
(assign env (op extend-environment)
(reg unev) (reg argl) (reg env))
(assign unev (op procedure-body) (reg proc))
(goto (label ev-sequence))
Compound-apply is the only place in the interpreter where the
env register is ever assigned a new value. Just as in the
metacircular evaluator, the new environment is constructed from the
environment carried by the procedure, together with the argument list
and the corresponding list of variables to be bound.
The portion of the explicit-control evaluator at ev-sequence is
analogous to the metacircular evaluator's eval-sequence procedure. It
handles sequences of expressions in procedure bodies or in explicit
begin expressions.
Explicit begin expressions are evaluated by placing the sequence
of expressions to be evaluated in unev, saving continue on the
stack, and jumping to ev-sequence.
ev-begin
(assign unev (op begin-actions) (reg exp))
(save continue)
(goto (label ev-sequence))
The implicit sequences in procedure bodies are handled by jumping to
ev-sequence from compound-apply, at which point continue is already on the stack, having been saved at
ev-application.
The entries at ev-sequence
and ev-sequence-continue form a loop that
successively evaluates each expression in a sequence. The list of
unevaluated expressions is kept in unev. Before evaluating each
expression, we check to see if there are additional expressions to be
evaluated in the sequence. If so, we save the rest of the unevaluated
expressions (held in unev) and the environment in which these
must be evaluated (held in env) and call eval-dispatch to
evaluate the expression. The 2 saved registers are restored upon
the return from this evaluation, at ev-sequence-continue.
The final expression in the sequence is handled differently, at the
entry point ev-sequence-last-exp. Since there are no more
expressions to be evaluated after this one, we need not save unev or env before going to eval-dispatch. The value of
the whole sequence is the value of the last expression, so after the
evaluation of the last expression there is nothing left to do except
continue at the entry point currently held on the stack (which was saved
by ev-application or ev-begin.)
Rather than setting up continue to arrange for eval-dispatch to return here and then restoring continue from
the stack and continuing at that entry point, we restore continue from
the stack before going to eval-dispatch, so that eval-dispatch will continue at that entry point after evaluating the
expression.
ev-sequence
(assign exp (op first-exp) (reg unev))
(test (op last-exp?) (reg unev))
(branch (label ev-sequence-last-exp))
(save unev)
(save env)
(assign continue (label ev-sequence-continue))
(goto (label eval-dispatch))
ev-sequence-continue
(restore env)
(restore unev)
(assign unev (op rest-exps) (reg unev))
(goto (label ev-sequence))
ev-sequence-last-exp
(restore continue)
(goto (label eval-dispatch))
In chapter 1 we said that the process described by a procedure such as
(define (sqrt-iter guess x)
(if (good-enough? guess x)
guess
(sqrt-iter (improve guess x)
x)))
is an iterative process. Even though the procedure is syntactically
recursive (defined in terms of itself), it is not logically necessary
for an evaluator to save information in passing from one call to sqrt-iter to the next.25 An evaluator that can
execute a procedure such as sqrt-iter without requiring
increasing storage as the procedure continues to call itself is called
a tail-recursive evaluator. The metacircular implementation of
the evaluator in chapter 4 does not specify whether the evaluator is
tail-recursive, because that evaluator inherits its mechanism for
saving state from the underlying Scheme. With the explicit-control
evaluator, however, we can trace through the evaluation process to see
when procedure calls cause a net accumulation of information on the
stack.
Our evaluator is tail-recursive, because in order to evaluate the final
expression of a sequence we transfer directly to eval-dispatch without
saving any information on the stack. Hence, evaluating the final expression
in a sequence -- even if it is a procedure call (as in sqrt-iter, where
the if expression, which is the last expression in the procedure body,
reduces to a call to sqrt-iter) -- will not cause any information to be
accumulated on the stack.26
If we did not think to take advantage of the fact that it was
unnecessary to save information in this case, we might have
implemented eval-sequence by treating all the expressions in a
sequence in the same way -- saving the registers, evaluating the expression,
returning to restore the registers, and repeating this until all the
expressions have been evaluated:27
ev-sequence
(test (op no-more-exps?) (reg unev))
(branch (label ev-sequence-end))
(assign exp (op first-exp) (reg unev))
(save unev)
(save env)
(assign continue (label ev-sequence-continue))
(goto (label eval-dispatch))
ev-sequence-continue
(restore env)
(restore unev)
(assign unev (op rest-exps) (reg unev))
(goto (label ev-sequence))
ev-sequence-end
(restore continue)
(goto (reg continue))
This may seem like a minor change to our previous code for evaluation
of a sequence: The only difference is that we go through the
save-restore cycle for the last expression in a sequence as well as
for the
others. The interpreter will still give the same value for
any expression. But this change is fatal to the tail-recursive
implementation, because we must now return after evaluating the final
expression in a sequence in order to undo the (useless) register
saves. These extra saves will accumulate during a nest of procedure
calls. Consequently, processes such as sqrt-iter will require
space proportional to the number of iterations rather than requiring
constant space. This difference can be significant. For example,
with tail recursion, an infinite loop can be expressed using only the
procedure-call mechanism:
(define (count n)
(newline)
(display n)
(count (+ n 1)))
Without tail recursion, such a procedure would eventually run out of stack space, and expressing a true iteration would require some control mechanism other than procedure call.
As with the metacircular evaluator, special forms are handled by
selectively evaluating fragments of the expression. For an if
expression, we must evaluate the predicate and decide, based on the
value of predicate, whether to evaluate the consequent or the
alternative.
Before evaluating the predicate, we save the if expression
itself so that we can later extract the consequent or alternative. We
also save the environment, which we will need later in order to
evaluate the consequent or the alternative, and we save continue, which we will need later in order to return to the
evaluation of the expression that is waiting for the value of the if.
ev-if
(save exp) ; save expression for later
(save env)
(save continue)
(assign continue (label ev-if-decide))
(assign exp (op if-predicate) (reg exp))
(goto (label eval-dispatch)) ; evaluate the predicate
When we return from evaluating the predicate, we test whether it was
true or false and, depending on the result, place either the
consequent or the alternative in exp before going to eval-dispatch. Notice that restoring env and continue
here sets up eval-dispatch to have the correct environment and
to continue at the right place to receive the value of the if
expression.
ev-if-decide
(restore continue)
(restore env)
(restore exp)
(test (op true?) (reg val))
(branch (label ev-if-consequent))
ev-if-alternative
(assign exp (op if-alternative) (reg exp))
(goto (label eval-dispatch))
ev-if-consequent
(assign exp (op if-consequent) (reg exp))
(goto (label eval-dispatch))
Assignments are handled by ev-assignment, which is reached from
eval-dispatch with the assignment expression in exp. The code at ev-assignment first evaluates the value part of the expression and
then installs the new value in the environment. Set-variable-value! is assumed to be available as a machine
operation.
ev-assignment
(assign unev (op assignment-variable) (reg exp))
(save unev) ; save variable for later
(assign exp (op assignment-value) (reg exp))
(save env)
(save continue)
(assign continue (label ev-assignment-1))
(goto (label eval-dispatch)) ; evaluate the assignment value
ev-assignment-1
(restore continue)
(restore env)
(restore unev)
(perform
(op set-variable-value!) (reg unev) (reg val) (reg env))
(assign val (const ok))
(goto (reg continue))
Definitions are handled in a similar way:
ev-definition
(assign unev (op definition-variable) (reg exp))
(save unev) ; save variable for later
(assign exp (op definition-value) (reg exp))
(save env)
(save continue)
(assign continue (label ev-definition-1))
(goto (label eval-dispatch)) ; evaluate the definition value
ev-definition-1
(restore continue)
(restore env)
(restore unev)
(perform
(op define-variable!) (reg unev) (reg val) (reg env))
(assign val (const ok))
(goto (reg continue))
Exercise 5.23. Extend the evaluator to handle derived expressions such as cond,
let, and so on (section 4.1.2).
You may cheat
and assume that the syntax
transformers such as cond->if are available as machine
operations.28
Exercise 5.24. Implement cond as a new basic special form without
reducing it to if. You will have to construct a loop that tests
the predicates of successive cond clauses until you find one
that is true, and then use ev-sequence to evaluate the actions
of the clause.
Exercise 5.25. Modify the evaluator so that it uses normal-order evaluation, based on the lazy evaluator of section 4.2.
With the implementation of the explicit-control evaluator we come to the end of a development, begun in chapter 1, in which we have explored successively more precise models of the evaluation process. We started with the relatively informal substitution model, then extended this in chapter 3 to the environment model, which enabled us to deal with state and change. In the metacircular evaluator of chapter 4, we used Scheme itself as a language for making more explicit the environment structure constructed during evaluation of an expression. Now, with register machines, we have taken a close look at the evaluator's mechanisms for storage management, argument passing, and control. At each new level of description, we have had to raise issues and resolve ambiguities that were not apparent at the previous, less precise treatment of evaluation. To understand the behavior of the explicit-control evaluator, we can simulate it and monitor its performance.
We will install a driver loop in our evaluator machine. This plays
the role of the driver-loop procedure of
section 4.1.4. The evaluator will repeatedly print a
prompt, read an expression, evaluate the expression by going to eval-dispatch, and print the result. The following instructions form
the beginning of the explicit-control evaluator's controller
sequence:29
read-eval-print-loop
(perform (op initialize-stack))
(perform
(op prompt-for-input) (const ";;; EC-Eval input:"))
(assign exp (op read))
(assign env (op get-global-environment))
(assign continue (label print-result))
(goto (label eval-dispatch))
print-result
(perform
(op announce-output) (const ";;; EC-Eval value:"))
(perform (op user-print) (reg val))
(goto (label read-eval-print-loop))
When we encounter an error in a procedure (such as the unknown
procedure type error
indicated at apply-dispatch), we print an
error message and return to the driver loop.30
unknown-expression-type
(assign val (const unknown-expression-type-error))
(goto (label signal-error))
unknown-procedure-type
(restore continue) ; clean up stack (from apply-dispatch)
(assign val (const unknown-procedure-type-error))
(goto (label signal-error))
signal-error
(perform (op user-print) (reg val))
(goto (label read-eval-print-loop))
For the purposes of the simulation, we initialize the stack each time through the driver loop, since it might not be empty after an error (such as an undefined variable) interrupts an evaluation.31
If we combine all the code fragments presented in sections 5.4.1-5.4.4, we can create an evaluator machine model that we can run using the register-machine simulator of section 5.2.
(define eceval
(make-machine
'(exp env val proc argl continue unev)
eceval-operations
'(
read-eval-print-loop
<entire machine controller as given above>
)))
We must define Scheme procedures to simulate the
operations used as primitives by the evaluator. These are
the same procedures we used for the metacircular evaluator in
section 4.1, together with the few additional ones
defined in footnotes throughout section 5.4.
(define eceval-operations
(list (list 'self-evaluating? self-evaluating)
<complete list of operations for eceval machine>))
Finally, we can initialize the global environment and run the evaluator:
(define the-global-environment (setup-environment))
(start eceval)
;;; EC-Eval input:
(define (append x y)
(if (null? x)
y
(cons (car x)
(append (cdr x) y))))
;;; EC-Eval value:
ok
;;; EC-Eval input:
(append '(a b c) '(d e f))
;;; EC-Eval value:
(a b c d e f)
Of course, evaluating expressions in this way will take much longer than if we had directly typed them into Scheme, because of the multiple levels of simulation involved. Our expressions are evaluated by the explicit-control-evaluator machine, which is being simulated by a Scheme program, which is itself being evaluated by the Scheme interpreter.
Simulation can be a powerful tool to guide the implementation of
evaluators. Simulations make it easy not only to explore variations
of the register-machine design but also to monitor the performance of
the simulated evaluator. For example, one important factor in
performance is how efficiently the evaluator uses the stack. We can
observe the number of stack operations required to evaluate various
expressions by defining the evaluator register machine with the
version of the simulator that collects statistics on stack use
(section 5.2.4), and adding an instruction at the
evaluator's print-result entry point to print the
statistics:
print-result
(perform (op print-stack-statistics)); added instruction
(perform
(op announce-output) (const ";;; EC-Eval value:"))
... ; same as before
Interactions with the evaluator now look like this:
;;; EC-Eval input:
(define (factorial n)
(if (= n 1)
1
(* (factorial (- n 1)) n)))
(total-pushes = 3 maximum-depth = 3)
;;; EC-Eval value:
ok
;;; EC-Eval input:
(factorial 5)
(total-pushes = 144 maximum-depth = 28)
;;; EC-Eval value:
120
Note that the driver loop of the evaluator reinitializes the stack at the start of each interaction, so that the statistics printed will refer only to stack operations used to evaluate the previous expression.
Exercise 5.26. Use the monitored stack to explore the tail-recursive property of the
evaluator (section 5.4.2). Start the
evaluator and define the iterative factorial procedure from
section 1.2.1:
(define (factorial n)
(define (iter product counter)
(if (> counter n)
product
(iter (* counter product)
(+ counter 1))))
(iter 1 1))
Run the procedure with some small values of n. Record the maximum stack depth and the number of pushes required to compute n! for each of these values.
a. You will find that the maximum depth required to evaluate n! is independent of n. What is that depth?
b. Determine from your data a formula in terms of n for the total number of push operations used in evaluating n! for any n > 1. Note that the number of operations used is a linear function of n and is thus determined by 2 constants.
Exercise 5.27. For comparison with exercise 5.26, explore the
behavior of the following procedure for computing factorials
recursively:
(define (factorial n)
(if (= n 1)
1
(* (factorial (- n 1)) n)))
By running this procedure with the monitored stack, determine, as a function of n, the maximum depth of the stack and the total number of pushes used in evaluating n! for n > 1. (Again, these functions will be linear.) Summarize your experiments by filling in the following table with the appropriate expressions in terms of n:
| Maximum depth | Number of pushes | |
| Recursive | ||
| factorial | ||
| Iterative | ||
| factorial | ||
The maximum depth is a measure of the amount of space used by the evaluator in carrying out the computation, and the number of pushes correlates well with the time required.
Exercise 5.28. Modify the definition of the evaluator by changing
eval-sequence as described in
section 5.4.2 so that the evaluator is no
longer tail-recursive. Rerun your experiments from
exercises 5.26 and 5.27 to demonstrate
that both versions of the factorial procedure now require space
that grows linearly with their input.
Exercise 5.29. Monitor the stack operations in the tree-recursive Fibonacci computation:
(define (fib n)
(if (< n 2)
n
(+ (fib (- n 1)) (fib (- n 2)))))
a. Give a formula in terms of n for the maximum depth of the stack required to compute Fib(n) for n > 2. Hint: In section 1.2.2 we argued that the space used by this process grows linearly with n.
b. Give a formula for the total number of pushes used to compute
Fib(n) for n > 2. You should find that the number of
pushes (which correlates well with the time used) grows exponentially
with n. Hint: Let S(n) be the number of pushes used in computing
Fib(n). You should be able to argue that there is a formula
that expresses S(n) in terms of S(n - 1), S(n - 2), and some fixed
overhead
constant k that is independent of n. Give the
formula, and say what k is. Then show that S(n) can be expressed
as a Fib(n + 1) + b and give the values of a and b.
Exercise 5.30. Our evaluator currently catches and signals only 2 kinds of errors -- unknown expression types and unknown procedure types. Other errors will take us out of the evaluator read-eval-print loop. When we run the evaluator using the register-machine simulator, these errors are caught by the underlying Scheme system. This is analogous to the computer crashing when a user program makes an error.32 It is a large project to make a real error system work, but it is well worth the effort to understand what is involved here.
a. Errors that occur in the evaluation process, such as an attempt to
access an unbound variable, could be caught by changing the lookup
operation to make it return a distinguished condition code, which cannot
be a possible value of any user variable. The evaluator can test
for this condition code and then do what is necessary to go to signal-error. Find all of the places in the evaluator where such a
change is necessary and fix them. This is lots of work.
b. Much worse is the problem of handling errors that are signaled by
applying primitive procedures, such as an attempt to divide by zero or
an attempt to extract the car of a symbol. In a professionally
written high-quality system, each primitive application is checked for
safety as part of the primitive. For example, every call to car
could first check that the argument is a pair. If the argument is not
a pair, the application would return a distinguished condition code to
the evaluator, which would then report the failure. We could arrange
for this in our register-machine simulator by making each primitive
procedure
check for applicability and returning an appropriate distinguished
condition code on failure. Then the primitive-apply code in the
evaluator can check for the condition code and go to signal-error if necessary. Build this structure and make it work.
This is a major project.
19 See Batali et al. 1982 for more information on the chip and the method by which it was designed.
20 In our controller, the dispatch is written as a
sequence of test and branch instructions. Alternatively,
it could have been written in a data-directed style (and in a real
system it probably would have been) to avoid the need to perform
sequential tests and to facilitate the definition of new expression
types. A machine designed to run Lisp would probably include a dispatch-on-type instruction that would efficiently execute such
data-directed dispatches.
21 This is an important but subtle point in translating
algorithms from a procedural language, such as Lisp, to a
register-machine language. As an alternative to saving only what is
needed, we could save all the registers (except val) before each
recursive call. This is called a framed-stack discipline. This
would work but might save more registers than necessary; this could be
an important consideration in a system where stack operations are
expensive. Saving registers whose contents will not be needed later
may also hold onto useless data that could otherwise be
garbage-collected, freeing space to be reused.
22 We add to the evaluator data-structure procedures in
section 4.1.3 the following 2 procedures
for manipulating argument lists:
(define (empty-arglist) '())
(define (adjoin-arg arg arglist)
(append arglist (list arg)))
We also use an additional syntax procedure to test for the
last operand in a combination:
(define (last-operand? ops)
(null? (cdr ops)))
23 The optimization of treating the last operand
specially is known as evlis tail recursion (see Wand 1980).
We could be somewhat more efficient
in the argument evaluation loop if we made evaluation of the first
operand a special case too. This would permit us to postpone
initializing argl until after evaluating the first operand, so
as to avoid saving argl in this case. The compiler in
section 5.5 performs this optimization. (Compare
the construct-arglist procedure of
section 5.5.3.)
24 The
order of operand evaluation in the metacircular evaluator is
determined by the order of evaluation of the arguments to cons
in the procedure list-of-values of section 4.1.1
(see exercise 4.1).
25 We saw in section 5.1 how to implement such a process with a register machine that had no stack; the state of the process was stored in a fixed set of registers.
26 This implementation of tail recursion in
ev-sequence is one variety of a well-known optimization technique used
by many compilers. In compiling a procedure that ends with a procedure call,
one can replace the call by a jump to the called procedure's entry point.
Building this strategy into the interpreter, as we have done in this section,
provides the optimization uniformly throughout the language.
27 We can define no-more-exps? as follows:
(define (no-more-exps? seq) (null? seq))
28 This isn't really cheating. In an actual
implementation built from scratch, we would use our explicit-control
evaluator to interpret a Scheme program that performs source-level
transformations like cond->if in a syntax phase that runs before
execution.
29 We assume here that read and the
various printing operations are
available as primitive machine operations, which is useful for our
simulation, but completely unrealistic in practice. These
are actually extremely complex operations. In practice, they would be
implemented using low-level input-output operations
such as transferring single characters to and from a device.
To support the get-global-environment operation we define
(define the-global-environment (setup-environment))
(define (get-global-environment)
the-global-environment)
30 There are other errors that we would like the interpreter to handle, but these are not so simple. See exercise 5.30.
31 We could perform the stack initialization only after errors, but doing it in the driver loop will be convenient for monitoring the evaluator's performance, as described below.
32 Regrettably, this is the normal state of affairs in
conventional compiler-based language systems such as C.
In UNIX TM the system dumps core,
and in
DOS/Windows TM it becomes catatonic.
The Macintosh TM displays a
picture of an exploding bomb and offers you the opportunity to reboot
the computer -- if you're lucky.