Project 04: Scheme

Project 4: A Scheme Interpreter

Eval calls apply,
which just calls eval again!
When does it all end?

Change Log


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 the byproduct of implementation decisions in interpreters and compilers.

You will also implement some small programs in Scheme, including the count_change function that we studied in lecture. 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 the original 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.

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 of what you might create, 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 two: and tests.scm. You can download all of the project code as a zip archive. The Scheme interpreter A Tokenizer for scheme Primitive Scheme data structures and procedures A testing framework for Scheme
tests.scm A collection of test cases written in Scheme that are designed to test your Scheme interpreter. You will be adding additional test cases to this file, in addition to defining several Scheme procedures. Utility functions for 61A


This is a three-part project. As in the previous project, you'll work in a team of two people, person A and person B. In each part, you will do some of the work separately, but most questions can be completed as a pair. Both partners should understand the solutions to all questions.

After completing the first (short) part, you will be able to read and parse Scheme expressions. In the second (long) part, you will develop the interpreter in stages:

Finally, in the third part, you will implement Scheme procedures that are similar to some exercises that you previously completed in Python.

There are 30 possible points, along with 4 extra credit points. The extra credit problems are a bit involved; we recommend that you complete them all, but only after you have the regular stuff working.

The project is due on Tuesday, August 7th at 11:59:59 PM. You will must turn in both an electronic and a paper copy. Paper copies are to be turned in at the turn-in boxes in 283 Soda. On each file that you submit, make sure you indicate your TA's name, and the name and logins of both Person A and Person B.

The Scheme Language

Before you begin working on the project, review what you have learned in lecture about the Scheme language. If you would like to experiment with a working Scheme interpreter, look at Stk, which is installed on instructional machines as stk. You can also use the online wscheme REPL.

Read-Eval-Print. Run interactively, the interpreter reads Scheme expressions, evaluates them, and prints the results. The interpreter uses scm> as the prompt.

    scm> 2

The starter code for your Scheme interpreter in can successfully evaluate this simple expression, since it consists of a single literal numeral. The rest of the examples in this section will not work until you complete various portions of the project.

Non-standard Functions.

Load. Our load function differs from standard Scheme in that, since we don't have strings, we use a symbol for the file name. For example

    scm> (load 'problems.scm)

Turtle Graphics. Finally, to keep up the traditions of recent years, we've added some simple routines for turtle graphics, described later, that simply call functions in the Python turtle package (whose documentation we suggest you see; for one thing, it will let you try out turtle-graphics programs in Python).

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 test on your class account.


The tests for this project are largely taken from the Scheme textbook that 61A used for many years. Examples from relevant chapters (and a few more examples to test various corner cases) appear in tests.scm

You can also compare the output of your interpreter to the expected output by passing the file name to

    # python3 tests.scm

This is a rather useful script. As you'll see, tests.scm contains Scheme expressions interspersed with comments in the form:

    (+ 1 2)
    ; expect 3

Here, will evaluate (+ 1 2) using your code in, and output a test failure if 3 is not returned.

The scheme_test script collects these expected outputs and compares them with the actual output from the program, counting and reporting mismatches.

You can even test that your interpreter catches errors. The problem with error tests is that there is no "right" output. Our script, therefore, only requires that error messages start with "Error". Any such line will match

    ; expect Error

Finally, as always, you can run the doctests by doing:

 python3 -m doctest 

Don't forget to use the trace decorator from the ucb module to follow the path of execution in your interpreter.

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 crash. 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 caught and changed into SchemeError exceptions, which are caught and printed as error messages by the Scheme read-eval-print loop we've written for you. Python exceptions that "leak out" to the user in raw form are errors in your interpreter (tracebacks are for implementors, not civilians).

Preliminary: Read Some Code

This project is about modifying a complex piece of existing code. In such a situation, it's good to take time at the outset to read what's provided, try to understand what's there, and accumulate questions about parts you don't understand before trying to mess around with it. Indeed, a lot of what you take away from this project will simply be what you learn by reading all the code you don't have to write. In many cases, you'll be able to experiment with parts of it in isolation, simply by starting up an interactive Python session and using import to get access to the parts you'd like to play with.

Take a look over all the files provided with this project (with your partner, of course). Look particularly at, which defines the basic data structures that Scheme programs manipulate, and at the starter implementation in

In the file, look at read_eval_print, which is the top-level function that defines the interpreter's actions. Look also at the class Frame, which represents environment frames (just like those in the text and in lecture). Look at the run function and what it calls to see how the interpreter gets initialized and how the global environment comes to be.

You won't have to modify, but since you will be modifying the reader (and since we can ask you anything we want on tests), it might be a good idea to understand the routines it provides and how they are used. At any given time, the "current port" is a Buffer (see, which is used to provide a continuous stream of tokens from a token source. See if you can figure out how to look at the token stream produced for a small Scheme file.

Symbol Restrictions. For this project, your Scheme interpreter will only accept lower-case characters as valid characters in a symbol. For instance, the following is invalid in our Scheme:

scm> (define L '(1 2 3))
warning: invalid token: L
scm> (define myList '(1 2 3))
warning: invalid token: myList

Running your Scheme Interpreter

To run your Scheme interpreter in interactive mode, do:

# python3
Alternately, you can tell your Scheme interpreter to evaluate the lines of an input file by passing the file name as an argument to
    # python3 my_code.scm

Currently, your Scheme interpreter can handle a few simple expressions, such as:
scm> 1
scm> 42
scm> #t
scm> 'hi
To exit the Scheme interpreter, issue either Ctrl-c or Ctrl-d. Once you've implemented primitive functions, you can also invoke the exit primitive function to exit the interpreter:
scm> (exit)

Part 1: The Reader

The function scheme_read in is intended to read in tokens from the input source (such as a .scm file, or from the interactive prompt) and return Scheme expressions in our internal representation. In our Scheme interpreter, we will represent Scheme data types in the following way:

Scheme Data Type Our Internal Representation
Numbers Python's built-in int, float built-in 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 the file.

At the moment, the scheme_read can only handle atoms (like numbers, symbols, and boolean values) and the quote form. In particular, it can't handle pairs.

Problem 1 (2 pt). Your first task, with your partner, is to complete the scheme_read function. The scheme_read function should repeatedly pop tokens from the input_port (a Buffer instance) until it is able to return an expression. For instance:

>>> scheme_read(Buffer(tokenize_lines(["42"])))
>>> scheme_read(Buffer(tokenize_lines(["4 2 'hi"])))
4    # The Buffer object is now ["2 'hi"]
>>> scheme_read(Buffer(tokenize_lines(["(1 2 3) 'hi 4"])))
Pair(1, Pair(2, Pair(3, NULL)))
>> scheme_read(Buffer(tokenize_lines(["'bagel"])))
Pair("quote", Pair("bagel", NULL))        # 'bagel is treated as (quote bagel)
Note that, depending on the input, scheme_read may only have to pop one token from input_port, or it may have to pop many tokens.

In scheme_read, the first two cases (atoms, quotes) are already handled for you. Your task is to handle the third case, which deals with pairs.

The syntax for pairs and lists is a left parenthesis, followed by a "tail", defined as

The internal representation for Scheme's Null value that we will use is the global variable NULL, defined in

The provided (incomplete) nested function read_tail should read the tail and return its value. For example, the value that scheme_read should return for "(1 2 . 3)" consists of the value of the tail "1 2 . 3)", which is

Thus, the value that read_tail would return is: Pair(1, Pair(2, 3)).

As another example, the value returned for "(1 2)" is the value of the tail "1 2)", which is

so that the value denoted is Pair(1, Pair(2, NULL)).

If you've completed read_tail correctly, then the following expressions should evaluate correctly in your Scheme interpreter:

# python3
scm> 42
scm> '(1 2 3)
(1 2 3)
scm> '()
scm> '(1 (2 3) (4 (5)))
(1 (2 3) (4 (5)))
scm> '(1 (9 8) . 7)
(1 (9 8) . 7)
scm> '(hi there . (cs . (student)))
(hi there cs student)
The following expressions should throw a SchemeError:
scm> (lambda (x y . a b) (+ x y a b))
scm> (lambda (a b . c . d) (+ a b c d))
scm> '(1 2 . 3 4)
scm> '(a b (5 6 . doh doh))
Due to time limitations, here are a few cases that you do not have to throw errors for - your Scheme interpreter may behave in any way (i.e. throw mysterious errors, continue to operate strangely, etc.):
scm> (lambda (. x) (* x 2))
Add more tests to the beginning of the tests.scm to make sure that your reader is working correctly.

Note: Make sure that your scheme_read and read_tail functions are indeed working correctly before moving on. Because the subsequent exercises rely on scheme_read to correctly process lines, any bugs in your scheme_read function will cause strange and unexpected errors in the later exercises.

Part 2: The Evaluator

Now, we will implement an evaluator.

The main part of the evaluator is within scheme_eval and scheme_apply. Assuming Problem 1 has been done correctly, your evaluator should be able to correctly handle atoms (except for symbols), quoting, and lists.

However, function calls will not work yet -- the scheme_apply is incomplete. In the next exercise, we will implement primitive procedures.

Problem 2. Read scheme_apply. Notice that, for primitive procedures, scheme_apply calls the apply_primitive procedure, which should apply a primitive procedure to the provided arguments, with respect to some environment.

Scheme primitive procedures will be represented internally as instances of the PrimitiveProcedure class, defined in A PrimitiveProcedure instance has two instance attributes:

To see a list of all Scheme primitive functions, look in the file -- any function definition with the @primitive decorator will be added to the globally-defined _PRIMITIVES list.

The apply_primitive procedure is incomplete. With both you and your partner, complete the definition so that the primitive procedure is applied to its arguments. In particular, also make sure that your apply_primitive solution does the following:

Note that, even after you complete apply_primitive, your Scheme interpreter will still not be able to apply primitive procedures. This is because your Scheme interpreter still doesn't know how to look up the values for the primitive procedure symbols (such as +, *, and car). In the next two steps, you and your partner will implement symbol lookup and symbol definition.

Problem A2 (2 pt). In this step, you will get simple symbol evaluation and definition to work. Currently, your Scheme interpreter is unable to find the value of some defined symbol, even for the already-defined primitive procedures.

The main code that will deal with variable name lookups is located in the Frame class, in A Frame instance has two instance attributes:

Currently, the find method is incomplete -- at the moment, it always raises an "unknown identifier" error. The find method should, given a symbol sym, return the first Frame that contains a definition for sym. Your find procedure should behave in the following way:

After you complete this problem, you should be able to ask for the value of primitive procedures within the Scheme interpreter (such as +, *, and car). In particular, you should be able to call primitive procedures!

scm> +
<scheme_primitives.PrimitiveProcedure object at 0x2742d50>
scm> (+ 1 2)
scm> (* 3 4 (- 5 2) 1)

Problem B2 (2 pt). There are two missing parts in the method do_define_form, which handles the (define ...) construct. Here, we'll do just the first part: handling cases like

    (define pi2 (* 2 3.1415926))
where the defined symbol appears alone. Fill in the missing piece to make this work. First, you'll want to see how do_define_form is called (it's called within scheme_eval). Then, you'll want to take a look at the Frame.define method. Some questions that you might want to answer are: what type of object is vals? How do I get the symbol (i.e. variable name) from vals, and how do I compute the value that I should bind the symbol to in the current environment env?

When combined with your partner's work on A2, you should now be able to do things like

    scm> (define x 15) 
    scm> (define y (* 2 x))
    scm> y 
    scm> (+ y (* y 2) 1)
    scm> (define x 20)
    scm> x

Now that you can created symbols and given them simple values, it should be easy to come up with tests for A2 and B2. Add your tests to tests.scm.

At this point in the project, your Scheme interpreter should be be able to support the following features:

However, one notable missing feature is user-defined functions. In the next few steps, you will implement this very-useful feature.

In our interpreter, user-defined functions will be represented as instances of the LambdaProcedure class, defined in A LambdaProcedure instance consists of the following instance attributes:

Problem 3 (2 pt). Before we can call user-defined functions, we need to implement a piece of scheme syntax called begin. begin works as follows:

    scm> (begin (+ 2 3) (+ 5 6))
    scm> (begin (display 3) (newline) (+ 2 3))
begin executes all the statements given as arguments, and then returns what the last statement evaluates to.

Note: When scheme_eval evaluates one of the conditional constructs (if, and, or, cond, begin, case), notice that it calls scheme_eval on the return value of the relevant do_FORM procedures (do_if_form, do_cond_form, etc.). Take care that your Scheme interpreter doesn't inadvertantly call scheme_eval on the same value twice, or else you might get the following invalid behavior:

    scm> (begin 30 'hello)
    Error: unknown identifier: hello

Problem A4 (2 pt). Before we can call LambdaProcedures, we must be able to create them. At the moment, the do_lambda_form method, which creates LambdaProcedure values, is incomplete. Once you complete do_lambda_form, you can check your work by typing in lambda expressions to the interpreter. You should see something like this:

    scm> (lambda (x y) (+ x y))
    (lambda (x y) (+ x y))
Remember that, in Scheme, it is legal to have function bodies with more than one expression:
    STk> ((lambda (y) 42 (* y 2)) 5)
In order to more-easily implement this behavior, your do_lambda_form should do the following. If do_lambda_form detects that the current lambda body has multiple expressions, then the do_lambda_form function should place the expressions inside of an outer (begin ...) expression, and replace the lambda's body with the begin expression:
    scm> (lambda (y) 42 (* y 2))
    (lambda (y) (begin 42 (* y 2)))
For an explanation of why the begin is inserted, see the __init__ function for the LambdaProcedure class. This workaround will allow us to handle multi-expression function bodies in the same manner as single-expression function bodies.

Problem B4 (2 pt). Currently, your Scheme interpreter is able to define user-defined functions in the following manner (assuming that Problem A4 is done):

    scm> (define f (lambda (x) (* x 2)))
However, we'd like to be able to use the shorthand form of defining functions:
    scm> (define (f x) (* x 2))

Modify the do_define_form function so that it correctly handles the shorthand function definition form. Make sure that it can handle multi-expression bodies.

Here's a tip. You can think of the short-hand form of defining functions as doing the following:

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

After you complete Problem A4 and Problem B4, you should be able to define user-defined functions. However, your interpreter can't actually call them yet. Next, you and your partner will extend your interpreter to allow the invocation of user-defined functions.

Problem A5 (3 pt). The make_call_frame method of the Frame class is incomplete. Your task is to complete it.

The make_call_frame method should simulate the process of calling a user-defined function. Namely, this involves:

  1. Creating a new Frame.
  2. Binding formal parameters to its associated values (the arguments have already been evaluated with respect to the caller's environment).
  3. Extending the Frame that the user-defined function was defined in.

Don't forget the cases where the formal parameter list contains a trailing "varargs" entry, as in:

    (define (format port form . args) ...)
One unifying way to handle this case along with the simple lists-of-symbols is to consider the formals list as a kind of pattern that is matched against the list of argument values. That is, the formals list matches the argument list if you treat each symbol in the formals list as a pattern variable or wildcard that matches any expression. Thus, the list of values (1 2 3) has the internal structure
    Pair(number, Pair(number, Pair(number, NULL)))
while the formals list (a . b) has the structure
    Pair(symbol a, symbol b)
These have the same form if we match symbol a to the number 1 and symbol b to Pair(number, Pair(number, NULL)) Likewise, the ordinary formals list (a b c) has the structure
    Pair(symbol a, Pair(symbol b, Pair(symbol c, NULL)))
so it matches the argument list, too.

Problem B5 (3 pt). The function check_formals, which checks the formals-list argument to make_call_frame, currently does nothing at the moment. Fix it so that check_formals raises a SchemeError if the list of formals is invalid.

In particular, make sure that it supports the following argument syntax:

scm> (lambda (x y z) (+ x y z))
scm> (lambda (x . nums) (* x (reduce + nums)))
scm> (lambda nums (reduce * nums))
Make sure that your interpreter rejects the following. Where your interpeter rejects the following is not important (i.e. in scheme_read or check_formals), as long as you are correctly raising some SchemeError. It is an error for your interpreter to raise a Python exception.
scm> (lambda (x (y) z) (* x y z))
scm> (define (fn x 2) (+ x 2))

Problem 6 (3 pt). Finally, both you and your partner will modify scheme_apply to correctly handle user-defined functions. Complete the scheme_apply function so that it does the following:

  1. Create a new Frame, with all formal parameters bound to its associated evaulated arguments.
  2. Evaluate the body of procedure with respect to this new frame.
  3. Return the value of the call to procedure.

After you complete scheme_apply, user-defined functions (and lambda functions) should work in your Scheme interpreter. Be sure to add tests for Problems 3–6 to tests.scm

Problem 7. The basic Scheme conditional constructs are if, and, or, and cond. In the next section, you and your partner will implement these conditional special forms.

Remember that the logical forms (and, or) are short-circuiting. This means that not all arguments to an and/or should be evaluated, depending on the situation.

For and, your interpreter should evaluate each argument from left-to-right, and if any argument evaluates to a false value, then you stop evaluating the and and return False. In particular, if all arguments are a true value, then and should return True.

For or, your interpreter should evaluate each argument from left-to-right, and if any argument evaluates to a true value, then you stop evaluating the or and return True. If all arguments are false values, then or should return False.

Here are a few examples:

scm> (and 4 5 6)
True    ; all operands are true values
scm> (or 5 2 1)
True    ; 5 is a true value
scm> (and #t #f 42 (/ 1 0))
False    ; short-circuiting behavior of and
scm> (or 4 #t (/ 1 0))
True    ; short-circuiting behavior of or 

You might find the scheme_true function useful, defined in

Problem A7 (3 pt). Currently, if and and forms don't work correctly (if always evaluates to nil, and and always evaluates to True). Fill in the implementations of do_if_form and do_and_form. Make sure you correctly implement the short-circuiting behavior of and, as described above. Add test cases to tests.scm.

Note: For this project, we will only handle if expressions that contain three operands. The following expressions should be correctly supported by your interpreter:

scm> (if (= 4 2) 'true 'false)
scm> (if (= 4 4) (* 1 2) (+ 3 4))
And the following expression should be rejected by your interpreter:
scm> (if (= 4 2) 'true)
Error: too few operands in form

Problem B7 (3 pt). For the second half, fill in the implementations of do_cond_form and do_or_form. Make sure you correctly implement the short-circuiting behavior of or, as described above. Add test cases to tests.scm.

In particular, make sure that your do_cond_form correctly handles the following forms:

scm> (cond ((= 4 3) 'nope)
         ((= 4 4) 'hi)
         (else 'wait))
scm> (cond ((= 4 3) 'wat)
         ((= 4 4))
         (else 'hm))
scm> (cond ((= 4 4) 'here 42)
         (else 'wat 0))
For the last example, where the body of a cond has multiple expressions, you might find it helpful to 'convert' cond-bodies with multiple expression bodies into a single begin expression, i.e.:
scm> (cond ((= 4 4) 'here 42))
scm> (cond ((= 4 4) (begin 'here 42)))

A few clarifications:

Problem 8 (3 pt). The let special form introduces local variables, giving them their initial values. For example,

    scm> (define x 'hi)
    scm> (define y 'bye)
    scm> (let ((x 42)
             (y (* 5 10)))
          (list x y))
    (42 50)
    scm> (list x y)
    (hi bye)
Implement the do_let_form method to have this effect and (need we say it at this point?) test it, by adding test cases to tests.scm. Make sure your let correctly handles multi-expression bodies:
scm> (let ((x 42)) x 1 2)

As a reminder, you can think of the let form as doing the following:

scm> (let ((x 42) (y 16)) (+ x y))
scm> ((lambda (x y) (+ x y)) 42 16)
Thus, a let form implicitly creates a new Frame (containing the let bindings) which extends the current environment, and evaluates the body of the let with respect to this new Frame. This is very much exactly like a user-defined function call. Note that, in your project code, you don't have to actually create a LambdaProcedure and call it - instead, you can create a new Frame, add the necessary bindings, and evaluate the expressions of the let body with respect to the new Frame (indeed, the provided skeleton code points you towards this approach).

Reminder: the bindings created by a let are not able to refer back to previously-declared bindings from the same let:

STk> (let ((var1 42) (var2 (+ var1 2))) (+ var1 var2))
*** Error:
    unbound variable: x
Make sure your interpreter behaves in this way too.

Extra Credit 1. (2 pt). The let* construct is like let, except that each initialization expression "sees" the definitions that have gone before. Essentially,

    (let* ((x 10) 
          (y (+ x 5))) 
      (list x y))
is the same as
    (let ((x 10)) 
      (let ((y (+ x 5)))
        (list x y)))
and therefore has the value (10 15). Implement this special form (and, yes, test it).

Extra Credit 2 (2 pt). The case construct is a fancy conditional similar to the Java and C/C++ switch statement. Here are some examples from the Scheme reference manual:

    scm> (case (* 2 3) ((2 3 5 7) 'prime) ((1 4 6 8 9) 'composite)) 
    scm> (case (car '(c d)) ((a e i o u) 'vowel) ((w y) 'semivowel) (else 'consonant))
    scm> (define x 3) (define y 10)
    scm> (case (car '(+ * /)) ((+ add) (+ x y)) ((* mult) (* x y)) ((/ div) (/ x y))) 

The first operand is evaluated, but the first element of each of the subsequent operands is not evaluated (it's implicitly quoted). As for cond the remaining items in the matching operand are evaluated, and the value of the last is the value of the case. Implement and test this special form.

For this project, we will slightly differ from the behavior of STk's implementation of case. In our Scheme interpreter, assume that the correct syntax for the case is:

    (case <expr> (<lst_0> <expr_0>) ... (<lst_N> <expr_N>))

    (case <expr> (<lst_0> <expr_0>) ... (<lst_N> <expr_N>) (else <expr_else>))
There must be at least one (<lst_i> <expr_i>) operand. Each <lst_i> is a list of symbols or literals:
    <lst_i> := (sym_0 ... sym_N)
An empty <lst_i> should be accepted by your interpreter (even though that case will never be taken), such as:
    scm> (case 2 (() 'never-get-here) (else 'good))

If no cases of a case expression are taken, and an else is not present, then case should return False:

    scm> (case 42 ((1 2 3) 'case-a) (('a 'b 'c) (* 2 3)))

In particular, make sure that the all operands in the case are correctly formed. If any operands are ill-formed (according to our syntax), such as (case 2 (2 'a) (else 'b)), then raise a SchemeError with a meaningful error message.

We're There!

You should have been adding tests to tests.scm as you did each problem. In any case, make sure you have a reasonably complete set, since the readers will be looking at it. Your program should pass all your tests when you (or the autograder) run # python3 tests.scm

Your Scheme interpreter implementation is now complete.

Part 3: Write Some Scheme

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 at the bottom of tests.scm, along with some calls and expected results.

Problem A9 (3 pt). Implement the filter procedure, which takes two arguments, a procedure name and a list and returns a list that contains all elements of the input list for which applying the named procedure outputs a true value (i.e., something other than #f).

Problem B9 (3 pt). Implement the reverse procedure, which takes a list and returns the reverse of that list.

Problem 10 (2 pt). Implement the count-change procedure, which counts all of the ways to make change for a total amount, using coins with various denominations (denoms), but never uses more than max-coins in total. Write your implementation in tests.scm. The procedure definition line is provided, along with U.S. denominations.

Problem 11 (2 pt) Implement the count-partitions procedure, which counts all the ways to partition a positive integer total using only pieces less than or equal to another positive integer max-value. The number 5 has 5 partitions using pieces up to a max-value of 3:

3, 2 (two pieces) 3, 1, 1 (three pieces) 2, 2, 1 (three pieces) 2, 1, 1, 1 (four pieces) 1, 1, 1, 1, 1 (five pieces)

Problem 12 (3 pt). Implement the list-partitions procedure, which lists all of the ways to partition a positive integer total into at most max-pieces pieces that are all less than or equal to a positive integer max-value. Hint: Define a helper function to construct partitions.

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!

Contest: Recursive Art

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 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 (3 pt). Create a visualization of an iterative or recursive process of your choosing, using turtle graphics. Your implementation must be written entirely in Scheme, using STk (or the interpreter you have built (however, no extending the interpreter to do your work in Python)).

Prizes will be awarded for the winning entry in each of the following categories.

Entries (code and results) will be posted online, and winners will be selected by popular vote (in homework 14). 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.

Submission instructions for each category are included in the Homework 13 specifications.

Contest submissions are due next Saturday (August 4th, 2012). Note that this is before the project itself is due. In the following Tuesday's homework, you will vote on your favorite drawing.