Eval calls apply,
which just calls eval again!
When does it all end?
In this project, you will develop an interpreter for a subset of the Scheme language. As you proceed, think about the issues that arise in the design of a programming language; many quirks of languages are byproducts of implementation decisions in interpreters and compilers. The subset of the language used in this project is described in the functional programming section of Composing Programs.
You will also implement some small programs in Scheme. Scheme is a simple but powerful functional language. You should find that much of what you have learned about Python transfers cleanly to Scheme as well as to other programming languages. To learn more about Scheme, you can read Structure and Interpretation of Computer Programs online for free. Examples from Chapters 1 and 2 are included as test cases for this project. Language features from Chapters 3, 4, and 5 are not part of this project, but of course you are welcome to extend your interpreter to implement more of the language. Since we only include a subset of the language, your interpreter will not exactly match the behavior of other interpreters such as STk.
The project concludes with an open-ended graphics contest that challenges you to produce recursive images in only a few lines of Scheme. As an example, the picture above abstractly depicts all the ways of making change for $0.50 using U.S. currency. All flowers appear at the end of a branch with length 50. Small angles in a branch indicate an additional coin, while large angles indicate a new currency denomination. In the contest, you too will have the chance to unleash your inner recursive artist.
This project includes several files, but all of your changes will be made to
the first four: scheme.py
, scheme_reader.py
, questions.scm
, and
tests.scm
. You can download all of the project code as a zip
archive, which contains the following files:
scheme.py
: the Scheme evaluatorscheme_reader.py
: the Scheme syntactic analyzerquestions.scm
: a collection of functions written in Schemetests.scm
: a collection of test cases written in Schemescheme_tokens.py
: a tokenizer for Schemescheme_primitives.py
primitive Scheme proceduresscheme_test.py
: a testing framework for Schemebuffer.py
: a buffer implementationucb.py
: utility functions for 61Aok
: the autogradertests
: a directory of tests used by ok
You'll work in a team of two people, Partner A and Partner B. In each part, you will do some of the work separately and some together with your partner. For example, if a problem is marked 5A, then it is a solo problem for Partner A. Both partners should read, think about, and understand the solution to all questions. Feel free to help each other on the solo questions. If you choose to work on the whole project alone, you must complete all questions yourself.
In Parts I and II, you will develop the interpreter in several stages:
In Part III, you will implement Scheme procedures.
This is a 2-week project. You may work with one other partner. You should not share your code with students who are not your partner or copy from anyone else's solutions.
Start early! The amount of time it takes to complete a project (or any program) is unpredictable.
You are not alone! Ask for help early and often -- the TAs, readers, lab assistants, and your fellow students are here to help. Try attending office hours or posting on Piazza.
In the end, you will submit one project for both partners. The project is worth 30 points. 28 points are assigned for correctness, and 2 points for the overall composition of your program.
You will turn in the following files:
scheme_reader.py
scheme.py
questions.scm
tests.scm
You do not need to modify or turn in any other files to complete the project. To submit the project, run the following command. You will be able to view your submissions on the OK dashboard.
python3 ok --submit
For the functions that we ask you to complete, there may be some initial code that we provide. If you would rather not use that code, feel free to delete it and start from scratch. You may also add new function definitions as you see fit.
However, please do not modify any other functions. Doing so may result in your code failing our autograder tests. Also, please do not change any function signatures (names, argument order, or number of arguments).
Throughout this project, you should be testing the correctness of your code. It is good practice to test often, so that it is easy to isolate any problems.
We have provided an autograder called ok
to help you with
testing your code and tracking your progress. The first time you run
the autograder, you will be asked to log in with your @berkeley.edu
account using your web browser. Please do so. Each time you run
ok
, it will back up your work and progress on our
servers.
The primary purpose of ok
is to test your implementations, but there
is a catch. At first, the test cases are locked. To unlock tests,
run the following command from your terminal:
python3 ok -u
This command will start an interactive prompt that looks like:
===================================================================== Assignment: Project 4: A Scheme Interpreter OK, version ... ===================================================================== ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Unlocking tests At each "? ", type what you would expect the output to be. Type exit() to quit --------------------------------------------------------------------- Question 0 > Suite 1 > Case 1 (cases remaining: 1) >>> Code here ?
At the ?
, you can type what you expect the output to be.
If you are correct, then this test case will be available the next time
you run the autograder.
The idea is to understand conceptually what your program should do first, before you start writing any code.
Once you have unlocked some tests and written some code, you can check the correctness of your program using the tests that you have unlocked:
python3 ok
To help with debugging, ok
can also be run in
interactive mode:
python3 ok -i
If an error occurs, the autograder will start an interactive Python session in the environment used for the test, so that you can explore the state of the environment.
Most of the time, you will want to focus on a particular question.
Use the -q
option as directed in the problems below.
The tests
folder is used to store autograder tests, so make sure
not to modify it. You may lose all your unlocking progress if you
do. If you need to get a fresh copy, you can download the
zip archive and copy it over, but you
will need to start unlocking from scratch.
Read-Eval-Print. The interpreter reads Scheme expressions, evaluates them, and displays the results.
scm> 2
2
scm> (+ 2 3)
5
scm> (((lambda (f) (lambda (x) (f f x)))
(lambda (f k) (if (zero? k) 1 (* k (f f (- k 1)))))) 5)
120
The starter code for your Scheme interpreter in scheme.py
can successfully
evaluate the first expression above, since it consists of a single number. The
second (a primitive call) and the third (a computation of 5 factorial) will not
work just yet.
Load. Our load
procedure differs from standard Scheme in that we use a
symbol for the file name. For example, to load tests.scm
, evaluate the
following call expression.
scm> (load 'tests)
Symbols. Unlike some implementations of Scheme, in this project numbers and boolean values cannot be used as symbols. Also, symbols are always lowercased. This is illustrated in the following example, which won't work until a little bit later:
scm> (define 2 3)
Traceback (most recent call last):
0 (#define 2 3)
Error: bad argument to define
scm> 'Hello
hello
Turtle Graphics. In addition to standard Scheme procedures, we include
procedure calls to the Python turtle
package. This will come in handy in Part
IV, for the contest.
You can read the turtle module documentation online.
Note: The turtle
Python module may not be installed by default on your
personal computer. However, the turtle
module is installed on the
instructional machines. So, if you wish to create turtle graphics for this
project (i.e. for the contest), then you'll either need to setup turtle
on
your personal computer or use university computers.
Testing. The tests.scm
file contains a long list of example Scheme
expressions and their expected values.
(+ 1 2)
; expect 3
(/ 1 0)
; expect Error
You can compare the output of your interpreter to the expected output by running:
python3 scheme_test.py
For the example above, scheme_test.py
will evaluate (+ 1 2)
using your code
in scheme.py
, then output a test failure if 3
is not returned as the value.
The second example tests for an error (but not the specific error message.
Only a small subset of tests are designated to run by default because
tests.scm
contains an (exit)
call near the beginning, which halts testing.
As you complete more of the project, you should move or remove this call.
Note: your interpreter doesn't know how to exit
until Problems 3 and 4 are
completed; all tests will run until then.
Important: As you proceed in the project, add new tests to the top of
tests.scm
to verify the behavior of your implementation. Your composition
score for this project will depend on whether or not you have tested your
implementation in ways that are different from the ok
tests.
Exceptions. As you develop your Scheme interpreter, you may find that Python
raises various uncaught exceptions when evaluating Scheme expressions. As a
result, your Scheme interpreter will halt. Some of these may be the results of
bugs in your program, and some may be useful indications of errors in user
programs. The former should be fixed (of course!) and the latter should be
handled, usually by raising a SchemeError
. All SchemeError
exceptions are
handled and printed as error messages by the read_eval_print_loop
function in
scheme.py
. Ideally, there should never be unhandled Python exceptions for any
input to your interpreter.
To run your Scheme interpreter in an interactive session, type:
python3 scheme.py
You can use your Scheme interpreter to evaluate the expressions in an input file
by passing the file name as a command-line argument to scheme.py
:
python3 scheme.py tests.scm
Currently, your Scheme interpreter can handle a few simple expressions, such as:
scm> 1
1
scm> 42
42
scm> true
True
To exit the Scheme interpreter, press Ctrl-d
or evaluate the exit
procedure
(after completing problems 3 and 4):
scm> (exit)
The function scheme_read
in scheme_reader.py
parses a Buffer
(see
buffer.py
) instance that returns valid Scheme tokens when its current
and
pop
methods are invoked. This function returns the next full Scheme
expression in the src
buffer, using this representation:
Scheme Data Type | Our Internal Representation |
---|---|
Numbers | Python's built-in int and float data
types. |
Symbols | Python's built-in string data type. |
Booleans (#t , #f ) |
Python's built-in True , False values. |
Pairs | The Pair class, defined in
scheme_reader.py . |
nil |
The nil object, defined in
scheme_reader.py . |
Complete the scheme_read
function in scheme_reader.py
by adding support for
quotation. This function selects behavior based on the type of the next token:
src
is the string "nil"
, return the nil
object.
(provided)'bagel
), then return a quote special form (such as (quote bagel)
)."("
, return the result of
read_tail
. (provided)Test your understanding and implementation before moving on:
python3 ok -q 01 -u
python3 ok -q 01
Complete the read_tail
function in scheme_reader.py
by adding support for
dotted lists. A dotted list in Scheme is not necessarily a well-formed list, but
instead has an arbitrary second
attribute that may be any Scheme value.
The read_tail
function expects to read the rest of a list or dotted list,
assuming the open parenthesis of that list has already been popped by
scheme_read
.
Consider the case of calling scheme_read
on input "(1 2 . 3)
". The
read_tail
function will be called on the suffix "1 2 . 3)
", which is
1
and the value of the tail "2 .
3)
", which is2
and the Scheme value 3
.Thus, read_tail
would return Pair(1, Pair(2, 3))
.
Hint: In order to verify that only one element follows a dot, after
encountering a '.'
, read one additional expression and then
check to see that a closing parenthesis follows.
Test your understanding and implementation before moving on:
python3 ok -q 02 -u
python3 ok -q 02
You should also run the doctests for scheme_reader.py
(python3 -m doctest
scheme_reader.py
) and test your parser interactively by running python3
scheme_reader.py
. Every time you type in a value into the prompt, both the
str
and repr
values of the parsed expression are printed.
read> 42
str : 42
repr: 42
read> '(1 2 3)
str : (quote (1 2 3))
repr: Pair('quote', Pair(Pair(1, Pair(2, Pair(3, nil))), nil))
read> nil
str : ()
repr: nil
read> '()
str : (quote ())
repr: Pair('quote', Pair(nil, nil))
read> (1 (2 3) (4 (5)))
str : (1 (2 3) (4 (5)))
repr: Pair(1, Pair(Pair(2, Pair(3, nil)), Pair(Pair(4, Pair(Pair(5, nil), nil)), nil)))
read> (1 (9 8) . 7)
str : (1 (9 8) . 7)
repr: Pair(1, Pair(Pair(9, Pair(8, nil)), 7))
read> (hi there . (cs . (student)))
str : (hi there cs student)
repr: Pair('hi', Pair('there', Pair('cs', Pair('student', nil))))
All further changes to the interpreter will be made in scheme.py
. For each
question, add a few tests to the top of tests.scm
to verify the behavior of
your implementation.
In the implementation given to you, In fact, the
evaluator can only evaluate self-evaluating expressions: numbers, booleans, and
nil
.
Read the first two sections of scheme.py
, called Eval/Apply
and
Environemnts
. the scheme_eval
and scheme_apply
functions are complete,
but most of the functions or methods they use are not yet implemented. The
apply_primitive
and make_call_frame
functions assist in applying built-in
and user-define procedures, respectively. The Frame
class implements an
environment frame. The LambdaProcedure
class represents user-defined
procedures. These are all of the essential components of the interpreter; the
rest of scheme.py
defines special forms and input/output behaviour.
Test your understanding of how these components fit together by unlocking the
tests for eval_apply
.
python3 ok -q eval_apply -u
Implement the lookup
method of the Frame
class. It takes a symbol (Python
string) and returns the value bound to that name in the first Frame
of the
environment in which that name is found. A Frame
represents an environment
via two instance attributes:
bindings
is a dictionary that maps Scheme symbol keys (represented as
Python strings) to Scheme values.parent
is the parent Frame
instance. The parent of the Global Frame is
None
.Your lookup
implementation should:
self.bindings
if it exists.lookup
that symbol in the parent
if the parent
exists.SchemeError
. (provided)Test your understanding and implementation before moving on:
python3 ok -q 03 -u
python3 ok -q 03
After you complete this problem, you should be able to look up built-in procedure names.
scm> +
#[+]
scm> odd?
#[odd?]
scm> display
#[display]
However, your Scheme interpreter will still not be able to apply these procedures. Let's fix that.
Implement apply_primitive
, which is called by scheme_apply
. Primitive
procedures are applied by calling a corresponding Python function that
implements the procedure.
Scheme primitive procedures are represented as instances of the
PrimitiveProcedure
class, defined in scheme_primitives.py
. A
PrimitiveProcedure
has two instance attributes:
fn
is the Python function that implements the primitive Scheme
procedure.use_env
is a boolean flag that indicates whether or not this primitive
procedure will expect the current environment to be passed in as the last
argument. The environment is required, for instance, to implement the primitive
eval
procedure.To see a list of all Scheme primitive procedures used in the project, look in
the scheme_primitives.py
file. Any function decorated with @primitive
will
be added to the globally-defined _PRIMITIVES
list.
The apply_primitive
function takes a PrimitiveProcedure
instance, a Scheme
list of argument values, and the current environment. Your implementation
should:
procedure.use_env
is True
, then add the current environment env
as the last argument to this Python list.procedure.fn
on all of those arguments (Hint: use *args notation).TypeError
exception being thrown, then
handle it and raise a SchemeError
in the except
clause (raise
SchemeError
).Test your understanding and implementation before moving on:
python3 ok -q 04 -u
python3 ok -q 04
Your interpreter should now be able to evaluate primitive procedure calls, giving you the functionality of the Calculator language and more.
scm> (+ 1 2)
3
scm> (* 3 4 (- 5 2) 1)
36
scm> (odd? 31)
True
There are two missing parts in the do_define_form
function, which handles the
(defineĀ ...)
special forms. Implement just the first part, which binds
names to values but does not create new procedures. do_define_form
should
return the name after performing the binding.
scm> (define tau (* 2 3.1415926))
tau
Test your understanding and implementation before moving on:
python3 ok -q 05A -u
python3 ok -q 05A
You should now be able to give names to values and evaluate the resulting symbols.
scm> (define x 15)
x
scm> (define y (* 2 x))
y
scm> y
30
scm> (+ y (* y 2) 1)
91
scm> (define x 20)
x
scm> x
20
scm> (eval (define tau 6.28))
6.28
Implement the do_quote_form
function, which evaluates the quote
special
form.
Test your understanding and implementation before moving on:
python3 ok -q 06B -u
python3 ok -q 06B
You should now be able to evaluate quoted expressions.
scm> 'hello
hello
scm> '(1 . 2)
(1 . 2)
scm> '(1 (2 three . (4 . 5)))
(1 (2 three 4 . 5))
scm> (car '(a b))
a
scm> (eval (cons 'car '('(1 2))))
1
At this point in the project, your Scheme interpreter should support the following features:
quote
special form,(+ (- 4 2) 5)
.User-defined procedures are represented as instances of the LambdaProcedure
class. A LambdaProcedure
instance has three instance attributes:
formals
is a Scheme list of the formal parameters (symbols) that name the
arguments of the procedure.body
is a single Scheme expression; the body of the procedure.env
is the environment in which the procedure was defined.Implement the eval_all
function called from do_begin_form
, which will
complete the implementation of the begin
special form. A begin
expression
is evaluated by evaluating all sub-expressions in order. The value of the
begin
expression is the value of the final sub-expression.
scm> (begin (+ 2 3) (+ 5 6))
11
scm> (define x (begin (display 3) (newline) (+ 2 3)))
3
x
scm> (+ x 3)
8
scm> (begin (print 3) '(+ 2 3))
3
(+ 2 3)
If eval_all
is passed an empty list of expressions (nil
), then it should
return the special value okay
, which represents an undefined Scheme value.
Test your understanding and implementation before moving on:
python3 ok -q 07 -u
python3 ok -q 07
Implement the do_lambda_form
method, which creates LambdaProcedure
instances
by evaluating lambda
expressions. While you cannot call a user-defined
procedure yet, you can verify that you have created the procedure correctly by
evaluating a lambda expression:
scm> (lambda (x y) (+ x y))
(lambda (x y) (+ x y))
In Scheme, it is legal to place more than one expression in the body of a
procedure (although zero body expressions is disallowed). The body
attribute
of a LambdaProcedure
instance is a Scheme list of body expressions.
Test your implementation before moving on:
python3 ok -q 08
Currently, your Scheme interpreter is able to bind symbols to user-defined procedures in the following manner:
scm> (define f (lambda (x) (* x 2)))
f
However, we'd like to be able to use the shorthand form of defining named procedures:
scm> (define (f x) (* x 2))
f
Modify the do_define_form
function so that it correctly handles the shorthand
procedure definition form above. Make sure that it can handle multi-expression
bodies.
Test your understanding and implementation before moving on:
python3 ok -q 09A -u
python3 ok -q 09A
You should now find that defined procedures evaluate to lambda procedures.
scm> (define (square x) (* x x))
square
scm> square
(lambda (x) (* x x))
Implement the make_child_frame
method of the Frame
class, which:
Frame
instance, the parent of which is self
. (provided)Test your understanding and implementation before moving on:
python3 ok -q 10 -u
python3 ok -q 10
Implement the check_formals
function to raise an error whenever the Scheme
list of formal parameters passed to it is invalid. Raise a SchemeError
if
the list of formals
is not a well-formed list of symbols or if any symbol is
repeated.
Hint: The scheme_symbolp
function in scheme_primitives.py
returns whether
a value is a Scheme symbol.
Test your understanding and implementation before moving on:
python3 ok -q 11B -u
python3 ok -q 11B
Implement the make_call_frame
function called by scheme_apply
(at the end
of the Eval/Apply
section). It should create a new Frame
instance using the
make_child_frame
method of the appropriate parent frame, binding formal
parameters to argument values.
Test your understanding and implementation before moving on:
python3 ok -q 12 -u
python3 ok -q 12
At this point in the project, your Scheme interpreter should support the following features:
lambda
expressions,define
expressions, andNow is an excellent time to revisit the tests in tests.scm
and ensure that
you pass the tests that involve definition (Sections 1.1.2 and 1.1.4). You
should also add additional tests of your own at the top of tests.scm
to
verify that your interpreter is behaving as you expect.
Logical special forms include if
, and
, or
, and cond
. These expressions
are special because not all of their sub-expressions may be evaluated.
In Scheme, only False
is a false value. All other values are true values. You
can test whether a value is a true value or a false value using the provided
Python functions scheme_true
and scheme_false
, defined in
scheme_primitives.py
. (Note that Scheme traditionally uses #f
to indicate a
false value, which is equivalent to false
or False
. Similarly, true
and
True
and #t
are all equilvalent.)
Implement do_if_form
so that if
expressions are evaluated correctly. This
function should evaluate either the second (consequent) or third (alternative)
expression of the if
expression, depending on whether the value of the first
(predicate) expression is true.
scm> (if (= 4 2) 'a 'b)
b
scm> (if (= 4 4) (* 1 2) (+ 3 4))
2
It is legal to pass in just two expressions to the if
special form. In this
case, you should return the second expression if the first expression evaluates
to a true value. Otherwise, return the special okay
value, which represents
an undefined value. Hint: okay
is defined in scheme_primitives.py
and
imported to scheme.py
, so you can refer to it directly as okay
in your
Python code. The value okay
evaluates to itself.
scm> (if (= 4 2) 'a)
okay
Test your understanding and implementation before moving on:
python3 ok -q 13 -u
python3 ok -q 13
Implement do_and_form
and do_or_form
so that and
and or
expressions are
evaluated correctly.
The logical forms and
and or
are short-circuiting. For and
, your
interpreter should evaluate each sub-expression from left to right, and if any
of these evaluates to a false value, then False
is returned. Otherwise,
it should return the value of the last sub-expression. If there are no
sub-expressions in an and
expression, it evaluates to True
.
scm> (and)
True
scm> (and 4 5 6) ; all operands are true values
6
scm> (and 4 5 (+ 3 3))
6
scm> (and True False 42 (/ 1 0)) ; short-circuiting behavior of and
False
For or
, evaluate each sub-expression from left to right. If any evaluates to
a true value, return that value. Otherwise, return False
. If there are no
sub-expressions in an or
expression, it evaluates to False
.
scm> (or)
False
scm> (or 5 2 1) ; 5 is a true value
5
scm> (or False (- 1 1) 1) ; 0 is a true value in Scheme
0
scm> (or 4 True (/ 1 0)) ; short-circuiting behavior of or
4
Test your understanding and implementation before moving on:
python3 ok -q 14B -u
python3 ok -q 14B
Implement do_cond_form
so that it returns the value of the first result
sub-expression corresponding to a true predicate, or the sub-expression
corresponding to else
. Your implementation should match the following
examples and the additional tests in tests.scm
.
scm> (cond ((= 4 3) 'nope)
((= 4 4) 'hi)
(else 'wait))
hi
scm> (cond ((= 4 3) 'wat)
((= 4 4))
(else 'hm))
True
scm> (cond ((= 4 4) 'here (+ 40 2))
(else 'wat 0))
42
Hint: For the last example, where the body of a cond
case has multiple
expressions, use eval_all
.
If the body of a cond
case is empty for a true-valued predicate, then
do_cond_form
should return the value of the predicate.
scm> (cond (12))
12
scm> (cond ((= 4 3))
('hi))
hi
The value of a cond
is undefined if there are no true predicates and no
else
. In such a case, do_cond_form
should return okay
.
scm> (cond (False 1) (False 2))
okay
Test your understanding and implementation before moving on:
python3 ok -q 15A -u
python3 ok -q 15A
The let
special form binds symbols to values locally, giving them their
initial values. For example:
scm> (define x 'hi)
x
scm> (define y 'bye)
y
scm> (let ((x 42)
(y (* 5 10)))
(list x y))
(42 50)
scm> (list x y)
(hi bye)
Implement make_let_frame
, which returns a child frame of env
that binds
the symbol in each element of bindings
to the value of the corresponding
expression. The check_form
function can be used to check the structure
of each binding.
Test your understanding and implementation before moving on:
python3 ok -q 16 -u
python3 ok -q 16
Implement do_mu_form
to evaluate the mu
special form, a non-standard Scheme
expression type. A mu
expression is similar to a lambda
expression, but
evaluates to a MuProcedure
instance that is dynamically scoped. The
MuProcedure
class has been provided for you.
Additionally, update make_call_frame
so that the call frame used to evaluate
the body of a MuProcedure
is dynamically scoped. Calling a LambdaProcedure
uses lexical scoping: the parent of the new call frame is the environment in
which the procedure was defined. Calling a MuProcedure
created by a mu
expression uses dynamic scoping: the parent of the new call frame is the
environment in which the call expression was evaluated. As a result, a
MuProcedure
does not need to store an environment as an instance attribute.
It can refer to names in the environment from which it was called.
scm> (define f (mu (x) (+ x y)))
f
scm> (define g (lambda (x y) (f (+ x x))))
g
scm> (g 3 7)
13
Test your understanding and implementation before moving on:
python3 ok -q 17 -u
python3 ok -q 17
Your Scheme interpreter implementation is now complete. You should have been
adding tests to the top of tests.scm
as you did each problem. These tests
will be evaluated as part of your composition score for the project.
Not only is your Scheme interpreter itself a tree-recursive program, but it is
flexible enough to evaluate other recursive programs. Implement the following
procedures in Scheme in the questions.scm
file.
Implement the zip
procedure, which takes in a list of pairs and converts it
into a pair of lists, where the first list contains all of the first elements
of the original pairs, and the second list contains all of the second elements.
The "pairs" in the input are well-formed two-element lists, not Scheme pairs.
scm> (zip '((1 2)))
((1) (2))
scm> (zip '((1 2) (3 4) (5 6)))
((1 3 5) (2 4 6))
Test your understanding and implementation before moving on:
python3 ok -q 18 -u
python3 ok -q 18
Implement the list-partitions
procedure, which lists all of the ways to
partition a positive integer total
without using consecutive integers.
The contents of each partition must be listed in decreasing order.
Hint: Define a helper procedure to construct partitions. The built-in
append
procedure creates a list containing all the elements of two argument
lists. The cons-all
procedure in questions.scm
adds a first element to
each list in a list of lists.
The number 5 has 4 partitions that do not contain consecutive integers:
5
4, 1
3, 1, 1
1, 1, 1, 1, 1
The following partitions of 5 are not included because of consecutive integers:
3, 2
2, 2, 1
2, 1, 1, 1
Test your understanding and implementation before moving on:
python3 ok -q 19 -u
python3 ok -q 19
In Scheme, source code is data. Every non-primitive expression is a list, and we can write procedures that manipulate other programs just as we write procedures that manipulate lists.
Re-writing programs can be useful: we can write an interpreter that only
handles a small core of the language, and then write a procedure analyze
that
converts other special forms into the core language before a program is passed
to the interpreter.
For example, the let
special form is equivalent to a call expression that
begins with a lambda
expression. Both create a new frame extending the
current environment and evaluate a body within that new environment.
(let ((x 42) (y 16)) (+ x y))
;; Is equivalent to:
((lambda (x y) (+ x y)) 42 16)
We can use this rule to rewrite all let
special forms into lambda
expressions. We prevent evaluation of a program by quoting it, and then pass it
to analyze
:
scm> (analyze '(let ((a 1) (b 2)) (+ a b)))
((lambda (a b) (+ a b)) 1 2)
scm> (analyze '(let ((a 1)) (let ((b a)) b)))
((lambda (a) ((lambda (b) b) a)) 1)
In order to handle all programs, analyze
must be aware of Scheme syntax.
Since Scheme expressions are recursively nested, analyze
must also be
recursive. In fact, the structure of analyze
looks like that of
scheme_eval
:
(define (analyze expr)
(cond ((atom? expr) <Analyze atom>)
((quoted? expr) <Analyze quoted>)
((lambda? expr) <Analyze lambda>)
((define? expr) <Analyze define>)
((let? expr) <Analyze let>)
(else <Analyze other>)))
Implement the analyze
procedure, which takes in an expression and converts
all of the let
special forms in the expression into their equivalent lambda
expressions.
Hint: You may want to use the provided apply-to-all
procedure and the zip
procedure from Problem 18.
Test your understanding and implementation before moving on:
python3 ok -q 20 -u
python3 ok -q 20
Note: We used let
while defining analyze
. What if we want to run
analyze
on an interpreter that does not recognize let
? We can pass
analyze
to itself to compile itself into an equivalent program that does
not use let
:
;; The analyze procedure
(define (analyze expr)
...)
;; A list representing the analyze procedure
(define analyze-code
'(define (analyze expr)
...))
;; An analyze procedure that does not use 'let'
(define analyze-without-let
(analyze analyze-code))
Implement the hax
procedure that draws the following recursive illustration
when passed two arguments, a side length d
and recursive depth k
. The
example below is drawn from (hax 200 4)
.
To see how this illustration is constructed, consider this annotated version that gives the relative lengths of lines of the component shapes in the figure.
Complete the function scheme_optimized_eval
in scheme.py
. This alternative
to scheme_eval
is properly tail recursive. That is, the interpreter will
allow an unbounded number of active tail calls in constant space.
The Evaluate
class represents an expression that needs to be evaluated in an
environment. When scheme_optimized_eval
receives an expression in a tail
context, then it returns an Evaluate
instance. Otherwise, it repeatedly
evaluates expressions within the body of a while
statement, updating result
in each iteration.
A successful implementation will require changes to several other functions.
All tail calls should call scheme_eval
with True
as a third argument,
indicating a tail call.
Once you finish, uncomment the following line in scheme.py
to use your
implementation:
scheme_eval = scheme_optimized_eval
Test your understanding and implementation before moving on:
python3 ok -q EC -u
python3 ok -q EC
Congratulations! You have finished the final project for 61A! Assuming your tests are good and you've passed them all, consider yourself a proper computer scientist!
Now, get some sleep. You've earned it!
We've added a number of primitive drawing procedures that are collectively
called "turtle graphics". The turtle represents the state of the drawing
module, which has a position, an orientation, a pen state (up or down), and a
pen color. The tscheme__x_
functions in scheme_primitives.py
are the
implementations of these procedures, and show their parameters with a brief
description of each.
The Python documentation of the turtle module contains more detail.
Contest: Create a visualization of an iterative or recursive process of your
choosing, using turtle graphics. Your implementation must be written entirely in
Scheme using the interpreter you have built. However, you may add primitive
procedures to interface with Python's turtle
or math
modules. Other than
that all computation must be done in Scheme. If you do add new primitives,
then make sure to submit scheme_primitives.py
in addition to contest.scm
.
Prizes, as well as 3 extra credit points, will be awarded for the winning entry in each of the following categories:
You can check the number of tokens in a Scheme file called contest.scm
by
running the command
python3 scheme_tokens.py contest.scm
Entries (code and images) will be posted online, and winners will be selected by popular vote as part of a future homework. The voting instructions will read:
Please vote for your favorite entry in this semester's 61A Recursion Exposition contest. The winner should exemplify the principles of elegance, beauty, and abstraction that are prized in the Berkeley computer science curriculum. As an academic community, we should strive to recognize and reward merit and achievement (translation: please don't just vote for your friends).
To improve your chance of success, you are welcome to include a title and descriptive haiku in the comments of your entry, which will be included in the voting.
Entries that do not construct an image iteratively or recursively may be disqualified. Please don't just draw a preexisting image, even if the drawing function is iterative or recursive. If you're unsure, just ask.
Submission instructions will be posted closer to the deadline.
We have implemented a significant subset of Scheme in this project, but our interpreter can be extended with more features by following the extension instructions.