Project 04: Scheme

Project 4: A Scheme Interpreter

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 the byproduct of implementation decisions in interpreters and compilers.

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 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. Since we only include a subset of the language, your interpreter will not match exactly 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 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 four:,, questions.scm, and tests.scm. You can download all of the project code as a zip archive. The Scheme evaluator The Scheme syntactic analyzer
questions.scm A collection of test cases written in Scheme
tests.scm A collection of test cases written in Scheme A tokenizer for scheme Primitive Scheme procedures A testing framework for Scheme A suite of tests for the project Utility functions for 61A Utility functions for grading


This is a two-part, two-person project. All questions are labeled sequentially, but some are designated for certain people by a prefix of their letter (A or B). Both partners should understand the solutions to all questions.

In the first part, you will develop the interpreter in stages:

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

There are 27 possible correctness points and 3 composition points. The composition score in this project will evaluate the clarity of your code and your ability to write tests that verify the behavior of your interpreter.

Submit the project using submit proj4. The only files you are required to submit are,, questions.scm, and tests.scm.

The Scheme Language

Before you begin working on the project, review what you have learned in lecture about the Scheme language in Section 3.2 of Composing Programs.

Read-Eval-Print. The interpreter reads Scheme expressions, evaluates them, and prints the results.

    scm> 2
    scm> (((lambda (f) (lambda (x) (f f x)))
           (lambda (f k) (if (zero? k) 1 (* k (f f (- k 1)))))) 5)

The starter code for your Scheme interpreter in can successfully evaluate the first expression above, since it consists of a single number. The second (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.

    scm> (define 2 3)
    Traceback (most recent call last):
      0 (#define 2 3)
    Error: bad argument to define
    scm> 'Hello

Turtle Graphics. In addition to standard Scheme procedures, we include procedure calls to the Python turtle package. 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.


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


For the example above, will evaluate (+ 1 2) using your code in, 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 that 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 autograder.

As always, you can run the doctests for the project.

    python3 -m doctest

You can also run the autograder tests.

    python3 -q 1

Debugging. Try using the trace decorator from the ucb module to follow the path of execution in your interpreter.

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 Ideally, there should never be unhandled Python exceptions for any input to your interpreter.

Running Your Scheme Interpreter

To run your Scheme interpreter in an interactive mode, type:

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
    python3 tests.scm
Currently, your Scheme interpreter can handle a few simple expressions, such as:
    scm> 1
    scm> 42
    scm> #t
To exit the Scheme interpreter, issue either Ctrl-c or Ctrl-d or evaluate the exit procedure:
    scm> (exit)

The Reader

The function scheme_read in parses a Buffer ( instance that returns valid Scheme tokens on invocations of current and pop methods. 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
nil The nil object, defined in

Problem 1 (1 pt). Complete the scheme_read function in by adding support for quotation. This function dispatches on the type of the next token:

Problem 2 (2 pt). Complete the read_tail function in 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

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.

To verify that your solutions to Problem 1 and 2 work correctly, run the doctests for and test your parser interactively by running,

    # python3
    read> 42
    read> '(1 2 3)
    (quote (1 2 3))
    read> nil
    read> '()
    (quote ())
    read> (1 (2 3) (4 (5)))
    (1 (2 3) (4 (5)))
    read> (1 (9 8) . 7)
    (1 (9 8) . 7)
    read> (hi there . (cs . (student)))
    (hi there cs student)

The Evaluator

All further changes to the interpreter will be made in 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, the scheme_eval function is complete, but few of the functions or methods it uses are implemented. In fact, the evaluator can only evaluate self-evaluating expressions: numbers, booleans, and nil.

Problem 3 (2 pt). 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 A PrimitiveProcedure has two instance attributes:

To see a list of all Scheme primitive procedures used in the project, look in the 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:

The doctest for apply_primitive should now pass. However, your Scheme interpreter will still not be able to apply primitive procedures, because your Scheme interpreter still doesn't know how to look up the values for the primitive procedure symbols (such as +, *, and car).

Problem 4 (2 pt) 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 it is found. A Frame represents an environment via two instance attributes:

Your lookup implementation should,

After you complete this problem, you should be able to evaluate primitive procedure calls, giving you the functionality of the Calculator language and more.

    scm> +
    scm> (+ 1 2)
    scm> (* 3 4 (- 5 2) 1)
    scm> (odd? 31)

Problem A5 (1 pt). There are two missing parts in the do_define_form function, which handles the (define ...) 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))

You should now be able to give names to values and evaluate symbols to those values.

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

Problem B6 (1 pt). Implement the do_quote_form function, which evaluates the quote special form. Once you have done so, you can evaluate quoted expressions.

    scm> 'hello
    scm> '(1 . 2)
    (1 . 2)
    scm> '(1 (2 three . (4 . 5)))
    (1 (2 three 4 . 5))
    scm> (car '(a b))
    scm> (eval (cons 'car '('(1 2))))

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

User-Defined Procedures

User-defined procedures are represented as instances of the LambdaProcedure class, defined in A LambdaProcedure instance has three instance attributes:

Problem 7 (2 pt). First, implement the begin special form, which includes a list of one or more sub-expressions that are each evaluated in order. The value of the final sub-expression is the value of the begin expression.

    scm> (begin (+ 2 3) (+ 5 6))
    scm> (begin (display 3) (newline) (+ 2 3))
    scm> (begin (print 3) '(+ 2 3))
    (+ 2 3)

Hint: When scheme_eval evaluates one of the LOGICAL_FORMS in, it calls scheme_eval on the returned value. Take care that your Scheme interpreter doesn't inadvertently call scheme_eval on the same value twice, or else you might have the following incorrect behavior:

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

Problem 8 (2 pt). 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 read 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 have function bodies with more than one expression. In order to implement this feature, your do_lambda_form should detect when the body of a lambda expression contains multiple expressions. If so, then do_lambda_form should place those expressions inside of a (begin ...) form, and use that begin expression as the body:
    scm> (lambda (y) (print y) (* y 2))
    (lambda (y) (begin (print y) (* y 2)))

Problem A9 (1 pt). Currently, your Scheme interpreter is able to define user-defined procedures in the following manner:

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

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. Hint: construct a lambda expression and evaluate it with do_lambda_form.

Once you have completed this problem, you should find that defined procedures evaluate to lambda procedures.

    scm> (define (square x) (* x x))
    scm> square
    (lambda (x) (* x x))

Problem 10 (2 pt). Implement the make_call_frame method of the Frame class, which:

Problem B11 (1 pt). 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 symbol? procedure in returns whether a value is a Scheme symbol.)

Problem 12 (2 pt). Implement scheme_apply to correctly apply user-defined LambdaProcedure instances. (The case of MuProcedures is handled later in the project). It should:

After you complete scheme_apply, user-defined functions (and lambda functions) should work in your Scheme interpreter. Now is an excellent time to revisit the tests in tests.scm and ensure that you pass the ones 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.

Special Forms

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 #f (also known as false or 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

Problem A13 (1 pt). Implement do_if_form so that if expressions are evaluated correctly. This function should return either the second (consequent) or third (alternative) expression of the if expression, depending on the value of the first (predicate) expression.

    scm> (if (= 4 2) true false)
    scm> (if (= 4 4) (* 1 2) (+ 3 4))

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.

    scm> (if (= 4 2) true)

Problem B14 (2 pt). 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 False, then False is returned. If all but the last sub-expressions evaluate to true values, return the last sub-expression from do_and_form.

For or, evaluate each sub-expression from left to right. If any evaluates to a true value, then quote that value and return it. These return values must be quoted because they are evaluated in scheme_eval. If all but the last sub-expression evaluate to false, return the last sub-expression from do_or_form without quoting it.

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

Problem A15 (1 pt). Implement do_cond_form so that it returns the first result sub-expression corresponding to a true predicate (or 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))
    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 case has multiple expressions, you might find it helpful to replace cond-bodies with multiple expression bodies into a single begin expression, i.e., the following two expressions are equivalent.
    (cond ((= 4 4) 'here 42))
    (cond ((= 4 4) (begin 'here 42)))

If the body of a cond case is empty, then do_cond_form should quote the value of the predicate and return it, if the predicate evaluates to a true value.

    scm> (cond (12))
    scm> (cond ((= 4 3))

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.

Problem A16 (2 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 test it, by adding test cases to the top of tests.scm. Make sure your let correctly handles multi-expression bodies:
    scm> (let ((x 42)) x 1 2)

The let special form is equivalent to creating and then calling a lambda procedure. That is, the following two expressions are equivalent:

    (let ((x 42) (y 16)) (+ x y))
    ((lambda (x y) (+ x y)) 42 16)
Thus, a let form 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. 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 in this new environment.

Problem B17 (2 pt). 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, complete scheme_apply to call MuProcedure procedures using dynamic scoping. 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)))
    scm> (define g (lambda (x y) (f (+ x x))))
    scm> (g 3 7)

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.

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 in questions.scm.

Problem 18 (2 pt). Implement the merge procedure, which takes in a comparator and two sorted list arguments and combines them into one sorted list. A comparator is a function that compares two values. For example:

    scm> (merge < '(1 4 6) '(2 5 8))
    (1 2 4 5 6 8)
    scm> (merge > '(6 4 1) '(8 5 2))
    (8 6 5 4 2 1)

Problem 19 (2 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.

The number 5 has 4 partitions using pieces up to a max-value of 3 and a max-pieces of 4:

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

Problem 20 (2 pt). You have been given the definition to an abstract implementation of trees. Use it to implement tree-sums, which is a function that returns a list of all possible sums of nodes, when traversing from root to leaf. For example, the following tree when passed through tree-sums will return (20 19 13 16 11):


Problem 21 (0 pt). 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.


Extra Credit

Problem 22 (3 pt). Complete the function scheme_optimized_eval in 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.

Instead of recursively calling scheme_eval for tail calls and logical special forms, and let, replace the current expr and env with different expressions and environments. For call expressions, this change only applies to calling user-defined procedures.

Once you finish, uncomment the line scheme_eval = scheme_optimized_eval in

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. 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 in addition to contest.scm.

Prizes will be awarded for the winning entry in each of the following categories, as well as 3 extra credit points.

Entries (code and results) 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. This includes just drawing a preexisting image, even if the drawing function is iterative or recursive.

Submission instructions will be posted on the course website.

Extra for Experts

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.