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Scope,
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local
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,
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and
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Abstract
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Functions
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Instructions
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for
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students
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&
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labbies:
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Students
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use
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DrScheme,
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following
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the
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design
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recipe,
...
on
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the
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exercises
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at
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their
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own
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pace,
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while
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labbies
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wander
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among
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the
...
students,
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answering
...
questions,
...
bringing
...
the
...
more
...
important
...
ones
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to
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the
...
lab's
...
attention.
Quick Review of Scope
A function definition introduces a new local scope - the parameters are defined within the body.
(define (
parameters...)
body)
A local
expression introduces a new local scope--the names in the definitions are visible both within the bodies of the definitions and within the body. If a local name collides with (matches) a name in the surrounding program, the local name shadows the enclosing name; i.e., the matching name in the enclosing program is invisible within the local
expression; only the matching local name can be accessed. The same shadowing phenomenon happens when a parameter name in a function definition collides with a name defined in the surrounding program.
(local [
definitions...]
body)
Note that the use of square brackets [ ] here is equivalent to parentheses( ). The brackets help set off the definitions a little more clearly for readability.
In order to use local and the other features about to be introduced in class, you need to set the DrScheme language level to Intermediate Student.
Exercises
Finding the maximum element of a list.
Let's consider the problem of finding the maximum number in a list which is used an example in Lecture 7.
- Develop the function
max-num
without usinglocal
exactly as in lecture.
- Develop the optimized version of this function (call it
opt-max-num
) usinglocal
exactly as in lecture.
Try running each version on the following input (or longer ones):
(list 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
If it is convenient, use your up
function from last lab to avoid typing long input lists like the preceding one.
What difference do you see?
For each version, determine approximately how many calls to your function is made for the input
(list 1 2 3 ... n)
for various values of n
.
Which version is more efficient?
Note that you can write an efficient solution without using local
; you can use an auxiliary function instead. The auxiliary function takes the expression appearing on the right hand side of the local definition as an argument. Quickly write this version of the function.
In general, you can take any program using local
, and turn it into an equivalent program without local
. Using local
doesn't let us write programs which were impossible before, but it does let us write them more cleanly and concisely.
Generating lists of ascending numbers
Retrieve your code for the up
function and its helper upfrom
from last lab.
Rewrite this program so that the helper function upfrom
is hidden inside a local
expression and takes only one argument since the upper bound argument is available as the parameter to the enclosing definition of up
.
Don't forget to revise the contract and purpose for the improved upFrom
function.
Advice on when to use local
These examples point out the reasons why to use local
:
- Avoid code duplication. I.e., increase code reuse.
- Avoid recomputation.
This sometimes provides dramatic benefits. Avoid re-mentioning an unchanging argument. - To hide the name of helper functions. An aside: many programming languages (including Java) provide alternate mechanisms (such as
public
/private
attributes) for hiding helper functions and data which scale better to large programs and accomodate the unit testing of hidden functions (whichlocal
does not!). - Name individual parts of a computation, for clarity.On the other hand, don't get carried away.
Here are two easy ways to misuse local:
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;; h3. Quick Review of Scope A function definition introduces a new local scope - the parameters are defined within the body. * {{(define (}} _parameters_ {{...)}} _body_ {{)}} A {{local}} expression introduces a new local scope--the names in the definitions are visible both within the bodies of the definitions and within the body. If a local name collides with (matches) a name in the surrounding program, the local name _shadows_ the enclosing name; _i.e._, the matching name in the enclosing program is invisible within the {{local}} expression; only the matching local name can be accessed. The same shadowing phenomenon happens when a parameter name in a function definition collides with a name defined in the surrounding program. * {{(local \[}} definitions {{...\]}} body {{)}} Note that the use of square brackets \[ \] here is equivalent to parentheses( ). The brackets help set off the definitions a little more clearly for readability. In order to use _local_ and the other features about to be introduced in class, you need to set the DrScheme language level to *Intermediate Student*. h3. Exercises h5. Finding the maximum element of a list. Let's consider the problem of finding the maximum number in a list which is used an example in Lecture 7. * Develop the function {{max-num}} without using local exactly as in lecture. * Develop the optimized version of this function (call it {{opt-max-num}}) using local exactly as in lecture. Try running each version on the following input (or longer ones): {{(list 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)}} If it is convenient, use your {{up}} function from last lab to avoid typing long input lists like the preceding one. What difference do you see? For each version, determine approximately how many calls to your function is made for the input {{(list 1 2 3 ... _n_)}} for various values of {{_n_}}. Which version is more efficient? Note that you can write an efficient solution without using {{local}}; you can use an auxiliary function instead. The auxiliary function takes the expression appearing on the right hand side of the local definition as an argument. Quickly write this version of ma function including a contra In general, you can take any program using local, and turn it into an equivalent program without local. Using local doesn't let us write programs which were impossible before, but it does let us write them more cleanly and concisely. h5. Generating lists of ascending numbers Retrieve your code for the {{up}} function and its helper {{upfrom}} from last lab. Rewrite this program so that the helper function {{upfrom}} is hidden inside a {{local} expression and takes only one argument since the upper bound argument is available as the parameter to the enclosing definition of {{up}}. Don't forget to revise the contract and purpose for the improved {{upFrom}} function. h3. Advice on when to use local These examples point out the reasons why to uselocal: # Avoid code duplication. I.e., increase code reuse. # Avoid recomputation. This sometimes provides dramatic benefits. Avoid re-mentioning an unchanging argument. # To hide the name of helper functions.An AsideWhile important conceptually, most programming languages (including Scheme) provide alternate mechanisms for this which scale better to large programs. These mechanisms typically have names like modules orpackages. As this one example shows, even small programs can get a bit less clear when hiding lots of helpers. # Name individual parts of a computation, for clarity.On the other hand, don't get carried away. Here are two easy ways to misuse local: {code} max-num: non-empty-list-of-number -> number (define (max-num a-nelonalon) (local [(define max-rest (max-num (rest a-nelonalon)))] (cond [(empty? (rest a-nelonalon)) (first a-nelonalon)] [else (if (<= (first a-nelonalon) max-rest) max-rest (first a-nelonalon))]))) {code} _Question_: What's wrong with this? Make sure you don't put a local too early in your code. {code} |
Try running this example on any legal input (according to the contract). It blows up!
Make sure you don't put a local too early in your code.
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;; max-num: non-empty-list ->-of-number -> number (define (max-num a-nelonalon) (cond [(empty? (rest a-nelonalon)) (first a-nelonalon)] [else (local [(define max-rest (max-num (rest a-nelonalon))) (define first-smaller? (<=(first a-nelonalon) max-rest))] (if first-smaller? max-rest (first a-nelonalon)))])) {code} |
This
...
isn't
...
wrong,
...
but
...
the
...
local
...
definition
...
of
...
first-smaller?
...
is
...
unnecessary.
...
Since
...
the
...
comparison
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is
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only
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used
...
once,
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this
...
refactoring
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is
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not
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a
...
case
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of
...
code
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reuse
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or
...
recomputation.
...
It
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provides
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a
...
name
...
to
...
a
...
part
...
of
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the
...
computation,
...
but
...
whether
...
that
...
improves
...
clarity
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in
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this
...
case
...
is
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a
...
judgement
...
call.
...
Scope
...
and
...
the
...
semantics
...
of
...
local
...
How
...
are
...
local
...
expressions
...
evaluated?
...
The
...
following
...
reduction rule
...
describes
...
how
...
to
...
evaluate
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a
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local
...
expression
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when
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it
...
is
...
the
...
leftmost
...
un-evaluated
...
expression.
...
1.
...
Rename
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each
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defined
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name
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in
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the
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local
...
expression
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by
...
appending
...
_
...
i
...
to
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its
...
name
...
for
...
a
...
suitable
...
choice
...
of i
. A good choice for i
is simply the index (starting from 0) of this particular local
evaluation in the course of the evaluation. The DrScheme stepper follows this convention. When renaming local variables,
make sure that you consistently change each defined name in the entire local
expression.
2. Lift the modified define
statements from the local
expression to the top level (at the end of top level program), and replace the local
expression by it body (which has been transformed by renaming).
This two-part process is technically a single step in an evaluation. But you can make each part of the rule a separate step if you choose. See Example 1 below.
Evaluating Program Text
Now that we have extended our hand-evaluation model to include the definitions forming a program, we can describe how to handle define
statements that bind ordinary variables rather than functions. The right hand sides of define
statements are not necessarily values. When evaluating a program, you must always reduce the leftmost expression that is not a value; in some cases this expression may be part of the right hand size of a define
.
Sample evaluations involving local
and explicit program text
Example 1
Observe how the rewriting makes it clear that there are two separate variables initially named x
.
Code Block |
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number {{_i_}} to the name. A good choice for {{_i_}} is simply the index of this {{local}} evaluation starting from 0. make sure that you consistently change each defined name in the entire {{local} expression. 2. Lift the modifed {{define} statements from the {{local}} expression to the top level (at the end of top level program), and replace the {{local}} expression by it body (which has been transformed by renaming). This two-part process is technically a single step in an evaluation. But you can make each part of the rule a separate step if you choose. h3. Evaluating Program Text Now that we have extended our hand-evaluation model to include the definitions forming a program, we can describe how to handle {{define}} statements that bind ordinary variables rather than functions. The right hand sides of {{define} statements are not necessarily values. When evaluating a program, you must always reduce the leftmost expression that is not a value; in some cases this expression may be part of the right hand size of a {{define}}. h3. Sample evaluations involving {{local}} and bexplicit program text h5. Example 1 Observe how the rewriting makes it clear that there are two separate variables initially named {{x}}. {code} (define x 5) (local [(define x 7)] x) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define x_0 7) x_0 {code} {{ |
=>
Code Block |
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}} {code} 7 {code} |
This
...
evaluation
...
can
...
also
...
be
...
written
...
in
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a
...
slightly
...
longer
...
form
...
where
...
the
...
evaluation
...
of
...
a
...
local
...
expression
...
takes
...
two
...
reduction
...
steps
...
rather
...
than
...
one.
...
(The
...
one-step
...
form
...
is
...
better
...
technically,
...
but
...
the
...
two-step
...
form
...
is
...
acceptable
...
in
...
the
...
context
...
of
...
an
...
introductory
...
course.)
Code Block |
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} (define x 5) (local (define x 7) x) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (local (define x_0 7) x_0) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define x_0 7) x_0 {code} |
=>
Code Block |
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7
|
Example 2.
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{{=>}} {code} 7 {code} h5. Example 2. {code} (define x 5) (define y x) (define z (+ x y)) (local [(define x 7) (define y x)] (+ x y z)) {code} {{ |
=>
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}} {code} (define x 5) (define y 5) (define z (+ x y)) (local [(define x 7) (define y x)] (+ x y z)) {code} {{ |
=>
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}} {code} (define x 5) (define y 5) (define z (+ 5 y)) (local [(define x 7) (define y x)] (+ x y z)) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z (+ 5 5)) (local [(define x 7) (define y x)] (+ x y z)) {code} {{=> |
=>
Code Block |
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(}} {code} (define x 5) (define y 5) (define z 10) (local [(define x 7) (define y x)] (+ x y z)) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z 10) (local [(define x\_0 7) (define y\_0 x\_0) ] (+ x\_0 y\_0 z) {code} {{) |
=>
Code Block |
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=>}} {code} (define x 5) (define y 5) (define z 10) (define x\_0 7) (define y\_0 7x_0) (+ x\_0 y\_0 z) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z 10) (define x\_0 7) (define y\_0 7) (+ 7x_0 y\_0 z) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z 10) (define x\_0 7) (define y\_0 7) (+ 7 7y_0 z) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z 10) (define x\_0 7) (define y\_0 7) (+ 7 7 10z) {code} {{ |
=>
Code Block |
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}} {code} (define x 5) (define y 5) (define z 10) (define x\_0 7) (define y\_0 7) 24 {code} Note: {{+}} is a function of an arbitrary number of numbers in Scheme. Hence {{(+ 7 7 10)}} reduces to {{24}} in a single step. The longer form of this evaluation adds an extra step in lifiting the {{local}}. If any aspect of hand evaluation confuses you, try running examples in DrScheme using the stepper. DrScheme provides an easy way to look at this: the Check Syntax button. Clicking on this does two things. First, it checks for syntactic correctness of your program in the definitions window. If there are errors, it reports the first one. But, if there aren't any, it then annotates your code with various colors and, when you move your cursor on top of a placeholder, draws arrows between placeholder definitions and uses. h5. Scoping Exercise What does the following code do? Explain it in terms of the evaluation rule above. {code} (define growth-rate 1.1) (define (grow n) (* n growth-rate)) (local (define growth-rate 1.3) (grow 10)) (grow 10) {code} Confirm your understanding with DrScheme, with the Check Syntax button, and by evaluating the code. Is the following code essentially different from the previous example? Why or why not? {code} (define growth-rate 1.3) (define (grow n) (local (define growth-rate 1.1) (* n growth-rate))) (grow 10) {code} Once again, confirm your understanding with DrScheme. In each case, grow "knows" which placeholder named growth-rate was in scope when it was defined, because that knowledge is part of grow's closure. h3. More scoping exercises In each of the following, determine which definition corresponds to each placeholder usage. As a result, first figure out without DrSchemewhat each example produces. To do so, you may want to use local's hand-evaluation rules above. Then confirm your results by using DrScheme. (Note - Some examples give errors about unbound placeholders.) {code} (define w 1) (define x 3) (define y 5) {code} {code} (local [(define w 2) (define x 4) (define z 6)] (+ w x y z)) (+ w x y) {code} {code} (define w 1) (define x 3) (define (test x y) (local [(define w 2) (define x 4) (define z 6)] (+ w x y z)))) {code} {code} (test 8 3) (test x w) (test w x) {code} {code} (define x 1) (define y 1) (local [(define x 2) (define y x)] (+ x y)) (+ x y) {code} Here is an example of mutually recursive functions following the template for natural numbers. It is motivated by the observation "a positive number n is even if n-1 is odd, a positive number n is odd if n-1 is even, and 0 is even." {code} (local [; my-odd?: natnum --> boolean (define (my-odd? n) (cond (zero? n) false (positive? n) (my-even? (sub1 n)))) {code} {code} ; my-even?: natnum --> boolean (define (my-even? n) (cond (zero? n) true (positive? n) (my-odd? (sub1 n))))] ; Now, the body of the local: (my-odd? 5)) ; Outside the local: (my-odd? 5) {code} Note that the two local functions are in each others' closure; they will always call each other, no matter how you define other placeholders with the same names my-even?. Also note that this is an example of mutually recursive functions, where the mutual recursion does not result from a mutually recursive data structure. {code} (define x 1) (define y 2) (define (f x) (local [(define (g x) (+ x y)) (define y 100)] (g (g x)))) (local [(define x 30) (define y 31) (define (g x) (* x y))] (g 20)) (g 20) (f 20) {code} h3. Review As a reminder, we've discussedfilter. Starting from several examples using specific predicates, we noticed a great deal of similarity. By abstracting away that similarity into a function parameter, we obtained the following definition: ; filter : (X \->boolean) [X] \-> [X] ; Purpose: Returns a list like the one given, except with those ; elements not passing the given predicate test removed. (define (filter keep? l) (cond [(empty? l) empty|(empty? l) empty] \[(cons? l) (cond [(keep? (first l)) (cons (first l) (filter keep? (restl)))|(keep? (first l)) (cons (first l) (filter keep? (restl)))] [else(filter keep? (restl))|else(filter keep? (restl))])\])) As an alternative, we could note that the above definition repeats the code (filter keep? ...) . I.e.,keep? is an invariant argument. We could choose to avoid that, as follows: ; filter : (X \->boolean) [X] \-> [X] ; Returns a list like the one given, except with those ; elements not passing the given predicate test removed. (define (filter keep? l) (local \[(define (filter-helper l) (cond [(empty? l) empty|(empty? l) empty] \[(cons? l) (cond [(keep? (first l)) (cons (first l) (filter-helper (restl)))|(keep? (first l)) (cons (first l) (filter-helper (restl)))] [else(filter-helper (restl))|else(filter-helper (restl))])\]))\] (filter-helper l))) filterexercises UsingDrScheme's "Check Syntax" feature to identify where each variable is bound in the second definition. Recall that when you put your cursor above a binding instance, it will draw arrows to the uses, and when you put your cursor above a use, it will draw an arrow to its binding instance. Using either definition of filter , define a function that takes a list of numbers and returns a list of all the positive numbers in that list. {anchor:map_Review}mapReview We are familiar with writing functions such as the following: ;double-nums : [number] \-> [number] ; Given a list of numbers,returna list of ; numbers that are twice the value of the corresponding items in ; the original list. (define (double-nums a-lon) (cond [(empty? a-lon) empty|(empty? a-lon) empty] [(cons? a-lon) (cons (\* 2 (first a-lon)) (double-nums (resta-lon)))|(cons? a-lon) (cons (* 2 (first a-lon)) (double-nums (resta-lon)))])) ; lessthan3-nums : [number] \-> [boolean] ; Given a list of numbers,returna list of ; booleans that indicate whether the corresponding items in ; the original list are less than 3. (define (lessthan3-nums a-lon) (cond [(empty? a-lon) empty|(empty? a-lon) empty] [(cons? a-lon) (cons (< (first a-lon) 3) (lessthan3-nums (resta-lon)))])) Since these functions are so similar, we'd like to package together the similar parts and separate out the different parts. We'll "package" the similar parts as a new function that can take either of the "different" parts as an argument that tells us what to do: ; map : (X \-> Y) [X] \-> [Y] (define (map f l) (cond [(empty? l) empty|(empty? l) empty] [(cons? l) (cons (f (first l)) (map f (restl)))|(cons? l) (cons (f (first l)) (map f (restl)))])) ;double-num : number \-> number (define (double-num n) (\* 2 n)) ;double-nums : [number] \-> [number] (define (double-nums a-lon) (mapdouble-num a-lon)) ; lessthan-3-nums : [number] \-> [boolean] (define (lessthan3-nums a-lon) (map (lambda (n) (< n 3)) a-lon)) mapabstracts the general idea of "computing a new element from each old element and building a list of the new elements" away from what specifically you are computing from each element. Another way to understandmapis by the following equation. It describeswhatmapdoes without the usually extraneous details ofhowit does it. (map f (list x<sub>1</sub> ... x<sub>n</sub>)) = (list (f x<sub>1</sub>) ... (f x<sub>n</sub>)) mapexercises Writedouble-numsusingmapand local, without aglobalfunction double-num. Writedouble-numsusingmapand lambda, without any named function like double-num. {anchor:The_Big_Picture}The Big Picture Abstract functions are both powerful and convenient. By using abstract functions to group all the common similar elements of many functions, we can concentrate on what's different. This allows us to write shorter code that is also clearer and more understandable. The examples in this lab are certainly not the only abstract functions, but they are among those that are particularly useful because they correspond to common list computations. Because Scheme has lists built in and since these functions are so useful, Scheme has them pre-defined. Usually, using abstract functions takes the place of following our standard templates. So, what happens to our design recipes? In the short term, while you are still getting used to abstract functions, we strongly recommend that you first follow the design recipe, and then go back and edit your code to use abstract functions where applicable. In the long term, you will learn to identify common patterns before you actually write code and be able to go directly to writing the shorter versions using abstract functions. You will probably findfilterandmapamong the easier ones to use at first. On the other hand,foldrandfoldlare more general and take more practice to get comfortable with. You will get practice on homeworks and the next two exams. {anchor:foldr}foldr We have written many functions to combine elements of a list, e.g., to sum the numbers in a list and to find the maximum element in a list. In each, we have some valuebaseas the result for the empty list and a functionfto combine the first element and the result on all the rest of the elements. Combining all the elements means satisfying the following equation: (foldr f base (list x<sub>1</sub> x<sub>2</sub> ... x<sub>n</sub>)) = (f x<sub>1</sub> (f x<sub>2</sub> ... (f x<sub>n</sub> base))) Many functions we've written fit this pattern, although this might not be obvious at first. This function was discussed in class, but with the name fun-for-l and its arguments in a different order. There we saw this is also equivalent to turning our list template into an abstract function. foldrexercises Based upon this equation, what should the following evaluate to? Think about them first, then try them inDrScheme (where foldr is pre-defined). (foldr + 0 (list \-1 5 \-3 4 2)) (foldr - 0 (list \-1 5 \-3 4 2)) (foldr cons empty (list \-1 5 \-3 4 2)) (foldr append empty (list (list 1 2) (list 4) empty (list 8 1 2))) What is the contract forfoldr? You should be able to determine this from the equation and examples above. First ask yourself the question assuming the input list is a list of numbers, then generalize. Usingfoldr, define a function to compute the product of a list of numbers. Usingfoldr, definemap. Define the functionmy-foldrto satisfy the above equation. As you might expect, it follows the template for a function consuming a list. Test your function against Scheme's built-in foldrto make sure they give the same results for the same inputs. If you have time... Usingfoldr, define a function to compute whether all elements in a list of numbers are greater than 6. Then generalize this to usingfoldr to define filter . {anchor:More_examples}More examples Here's a few more pre-defined abstract functions: (andmap f (list x<sub>1</sub> ... x<sub>n</sub>)) = (and (f x<sub>1</sub>) ... (f x<sub>n</sub>)) (ormap f (list x<sub>1</sub> ... x<sub>n</sub>)) = (or (f x<sub>1</sub>) ... (f x<sub>n</sub>)) (build-list n f) = (list (f 0) ... (f (sub1 n))) The following is a quick summary of the abstract functions mentioned in this lab: filter \- selects some elements from a list map \- applies a function to each list element andmap /ormap \- applies a function to each list element and combines the resulting booleans build-list \- constructs a list of the given length according to the given function foldr /foldl \- combines all elements of a list Exercises The following can be defined using some combination of the pre-defined abstract functions. Define a function that, given a list of numbers, returns the sum of all the positive numbers in the list. Define a function that, given a list of anything, determines whether all of the numbers in the list are even. If you have time... (This is from the last lab, but use abstract functions this time, and then compare the function to the one you wrote last week.) Develop a function that, given n and i normally returns the list of length i of numbers up to n : (list <VAR>n</VAR>-(<VAR>i</VAR>-1) ... <VAR>n</VAR>-1 <VAR>n</VAR>)More concretely, e.g., (nums-upto-help 5 0) ? empty (nums-upto-help 5 2) ? (list 4 5) (nums-upto-help 5 5) ? (list 1 2 3 4 5) (nums-upto-help 6 5) ? (list 2 3 4 5 6) (nums-upto-help 7 5) ? (list 3 4 5 6 7) For simplicity, you may assume that i = n . This should just follow the template for natural numbers on i . If you have time... Defineandmap, which computes whether all elements in a list of booleans are true: (andmap f (list x<sub>1</sub> ... x<sub>n</sub>)) = (and (f x<sub>1</sub>) ... (f x<sub>n</sub>))First, define it recursively following the list template, then define it using foldr andmap , then using onlyfoldr . {anchor:foldl}foldl The mathematically inclined might have noticed thatfoldr groups the binary operations right-associatively. Thus the "r" in the name. What if we want to group left-associatively? Well, we also have the following: (foldl f base (list x<sub>1</sub> x<sub>2</sub> ... x<sub>n</sub>)) = (f x<sub>n</sub> ... (f x<sub>2</sub> (f x<sub>1</sub> base))...) foldlexercises Based upon this equation, what should the following evaluate to? Think about them, then try them inDrScheme (where foldl is pre-defined). (foldl + 0 (list \-1 5 \-3 4 2)) (foldl - 0 (list \-1 5 \-3 4 2)) (foldl cons empty (list \-1 5 \-3 4 2)) (foldl append empty (list (list 1 2) (list 4) empty (list 8 1 2)))Compare the results to those for foldr . Do (foldr + 0 <em>numlist</em>)and (foldl + 0 <em>numlist</em>)always give the same results? Hint: Think back a couple labs. Define the function my-foldl to satisfy the above equation, testing it against Scheme's built-in foldl . Hint: Usereverse . If you've read ahead a few chapters: Definemy-foldlwithout using reverse . Hint: Use an accumulator. {anchor:Other_types}Other types Abstract functions are also quite useful. Scheme has lists and their abstract functions built-in, so they get more attention, but the ideas transfer to trees perfectly well. For example, one definition of binary tree is the following: (define-struct node (n left right)) ; A binary tree of numbers (btn) is ; - empty ; - (make-node n l r) ; where n is a number and l,r are btns Tree exercises If you have time... Define a map \-like function on btns. Semi-challenge: Define afoldr \-like function on btns. Hint: Whereasfoldr on lists took a binary function argument, this one needs a ternary function argument. h2. \!\! Access Permissions * Set ALLOWTOPICCHANGE = Main.TeachersComp211Group(+ 7 7 10) |
=>
Code Block |
---|
(define x 5)
(define y 5)
(define z 10)
(define x_0 7)
(define y_0 7)
24
|
Note: + is a function of an arbitrary number of numbers in Scheme. Hence (+ 7 7 10)
reduces to 24
in a single step. The longer form of this evaluation adds an extra step in lifiting the local
.
If any aspect of hand evaluation confuses you, try running examples in DrScheme using the stepper.
DrScheme provides an easy way to look at this: the Check Syntax button. Clicking on this does two things. First, it checks for syntactic correctness of your program in the definitions window. If there are errors, it reports the first one. But, if there aren't any, it then annotates your code with various colors and, when you move your cursor on top of a placeholder, draws arrows between placeholder definitions and uses.
Review
In Lectures 8-9, we developed the Scheme filter
function, a functional that takes a unary predicate and a list as arguments. Starting from several examples using specific predicates, we noticed a great deal of similarity. By abstracting away that similarity into a function parameter, we obtained the following definition:
Code Block |
---|
;; filter : (X -> boolean) (list-of X) -> (list-of X)
;; Purpose: (filer p alox) returns the elements e of alox for which (p e) is true
(define (filter keep? alox)
(cond [(empty? alox) empty]
[(cons? alox) (if (keep? (first alox))
(cons (first alox) (filter keep? (rest alox)))
(filter keep? (rest alox)))]))
|
We can putatively improve the preceding definition by eliminating the repetition of the code (filter keep? (rest alox))
:
Code Block |
---|
;; filter : (X -> boolean) (list-of X) -> (list-of X)
;; (filer p alox) returns the elements e of alox for which (p e) is true
(define (filter keep? alox)
(cond [(empty? alox) empty]
[(cons? alox)
(local [(define kept-rest (filter keep? (rest alox)))]
(if (keep? (first alox))
(cons (first alox) kept-rest)
kept-rest))]))
|
The refactoring of the first definition of filter
given above is borderline. It does not improve the running time of the program since the duplicate calls appear in disjoint control paths. Which of the two preceding definitions is easier to understand? The answer is debatable; it is not clear that the "improved" definition is better.
Some functional programmers advocate performing yet another "optimization": introducing a recursive helper function in the body of filter
because the keep?
parameter is unnecessary in filter
except as single a top-level binding. Since keep?
never changes in any recursive call, it can be eliminated as a parameter in a helper function nested inside filter
where the binding of keep?
is visible. We are not persuaded that this revision is a good idea for two reasons. First, eliminating constant parameters from function calls is a low-level transformation that is best performed by a compiler. In many cases, it may degrade rather than improve efficiency. Second, the program with the helper function is harder to understand than either program above.
Note the preceding definitions of filter
are not acceptable to DrScheme because the name filter
collides with the Scheme library function named filter
(which performs the same operation). If you want to run either of the preceding programs in DrScheme, you will have to rename filter
(as say Filter
).
Optional Exercise: rewrite filter
in the following "optimized" form:
Code Block |
---|
(define (Filter keep? alox)
(local [(define (filter-help alox) ...]
(filter-help alox)))
|
Do not bother writing a contract, purpose statement, or template instantiation for this function since we recommend that you throw this code away after the lab is over. The name has been changed to Filter
to avoid colliding with the Scheme library function named filter
.
Exercises
- Load one of the more interesing Scheme programs you have written (such as HW2) into DrScheme. (If you did the preceding optional exercise, you can use this program.) Perform the "Check Syntax" command to identify where each variable is bound. Recall that when you put your cursor above a binding instance, it will draw arrows to the uses, and when you put your cursor above a use, it will draw an arrow to its binding instance.
- X Using the Scheme library function
filter
, develop a function that takes a list of numbers and returns a list of all the positive numbers in that list.
The map
functional
We are familiar with writing functions such as the following:
Code Block |
---|
;; double-nums : (list-of number) -> (list-of number)
;; (double-num (list e1 ... ek)) returns (list d1 ... dk) where each element di is twice ei.
(define (double-nums alon)
(cond [(empty? alon) empty]
[(cons? alon) (cons (* 2 (first alon)) (double-nums (rest alon)))]))
;; <3-nums : (list-of number) -> (list-of boolean)
;; (<3-nums (list e1 ... en)) returns (list b1 ... bn) where bi has the value (< ei 3)
(define (<3-nums alon)
(cond [(empty? alon) empty]
[(cons? alon) (cons (< (first alon) 3) (<3-nums (rest alon))))]))
|
These functions are very similar and can be written trivially using the Scheme library function map
discussed in Lecture 9.
Exercises.
- Write
double-nums and <3-nums
usingmap
.
The Big Picture
Abstract functions are both powerful and convenient. By using abstract functions to group all the common similar elements of many functions, we can concentrate on what's different. This practice allows us to write shorter code that is also clearer and more understandable
We have written many functions to combine elements of a list, e.g., to sum the numbers in a list and to find the maximum element in a list. In each, we have some value base
as the result for the empty list and a function f
to combine the elements of the list. The elements can be combined in two different ways: one associating to the right (foldr
) and one associating to the left foldl
. The following equations describe how these two form of combining work:
Code Block |
---|
(foldr f base (list x1 x2 ... xn)) = (f x1 (f x2 ... (f xn base))) [1]
(foldl f base (list x1 x2 ... xn)) = (f xn ... (f x2 (f x1 base))...) [2]
|
Many functions we've written fit this pattern, although this fact might not be obvious at first.
Exercises
- X Based upon the preceding equation ([1]), what should the following evaluate to? Think about them first, then try them in DrScheme (where
foldr
is pre-defined).- (foldr + 5 (list -1 5 -3 4 2))
- (foldl + 5 (list -1 5 -3 4 2))
- (foldr - 5 (list -1 5 -3 4 2))
- (foldl - 5 (list -1 5 -3 4 2))
- (foldr cons empty (list -1 5 -3 4 2))
- (foldl cons empty (list -1 5 -3 4 2))
- (foldr append empty (list (list 1 2) (list 4) empty (list 8 1 2)))
- (foldr append empty (list (list 1 2) (list 4) empty (list 8 1 2)))
- X What is the contract for
foldr
? Forfoldl
? You should be able to determine this from the equations and the examples above.
(We also covered the typing offoldr
in lecture.) Hint: First, determine the type offoldr
andfoldl
assuming the input list is a list of numbers, and then generalize. - Using
foldr
, define a function to compute the product of a list of numbers. Do the same forfoldl
. - Using
foldr
, definemap
. (Also done in lecture.) Do the same forfoldl
. - Define the function
Foldr
to satisfy equation (1) above. As you might expect, it follows the template for a function consuming a list. (It was also done in lecture.) Test your function against Scheme's built-infoldr
to make sure they give the same results for the same inputs. - Define the function
Foldl
to satisfy equation (2) above. As you might expect, it follows the template for a function consuming a list. Test your function against Scheme's built-infoldl
to make sure they give the same results for the same inputs. - Define a function to compute whether all elements in a list of numbers are greater than 6. Write two version versions, one using
foldr
and one usingfoldl
, choosing suitable names (distinct fromfilter
for each.
Then generalize both definitions to definefilter
. Are your twofilter
functions identical? Hint: look at computations that generate errors. - Define a function that, given a list of numbers, returns the sum of all the positive numbers in the list. Write two versions, one using
foldr
and one usingfoldl
. - Without using explicit recursion, develop a function
upfrom
that, giveni
andn
returns the list of lengthi
of numbers up ton
.
More examples
Here are a few more pre-defined abstract functions:
(andmap f (list x1 ... xn)) = (and (f x1) ... (f xn))
(ormap f (list x1 ... xn)) = (or (f x1) ... (f xn))
(build-list n f) = (list (f 0) ... (f (sub1 n)))
Exercises
The following can be defined using some combination of the pre-defined abstract functions.
- X Define a function that, given a list of numbers, determines whether all of the numbers in the list are even. Write two versions one using
foldr
and one usingfoldl
. - Define
andmap
andormap
usingfoldr
rather than recursion. Do the same usingfoldl
.
Summary
The following table presents a quick summary of the abstract functions mentioned
in this lab:
| selects some elements from a list |
---|---|
| applies a function to each list element |
| applies a function to each list element and reduces this list using |
| applies a function to each list element and reduces this list using |
| given a unary function |
| associating to the right, reduces a list using the specified binary function and base case |
| associating to the left, reduces a list using the specified binary function and base case |