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 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. Finally, you can receive extra credit for various enhancements to your project.

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

If you add extensions, you may need to modify and turn in some of the other files as well.

Initial Advice

Most of the work in the Python part of this project will be in reading and understanding the code. Don't allow the portions that don't say "YOUR CODE HERE" to remain a mystery to you: you will have to understand much of it. As usual, ask us and your fellow students for help understanding anything. A large part of what you get from this project will come from actually figuring out what's already there!

In contrast, the amount of Python code you have to write is not large. Our solution, including the optional extra credit, and a few pieces of the Extra for Experts added 126 lines to and Don't let that mislead into thinking you'll be able to toss this off in an hour! Figuring out which lines to add will take a great deal more time.

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. The load procedure reads a file of Scheme code, as if typed into the terminal. For example, to load tests.scm, evaluate either of the following call expression.

    scm> (load 'tests)
    ;; or
    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 lower-cased.

    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 sample 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. Your interpreter doesn't know how to exit until you complete Problems 3 and 4; 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

The Autograder

We've included an autograder that includes tests for each question. For this project, you do not have to unlock anyt of the tests. You can invoke autograder for a particular question as follows:

python3 -q <question number>

To help with debugging, you can also start an interactive prompt if an error occurs by adding the -t flag at the end:

python3 -q <question number> -i

You can also invoke the autograder for all problems at once using:


One last note: you might have noticed a file called tests.pkl that came with the project. This file is used to store autograder tests, so make sure not to modify it. If you need to get a fresh copy, you can follow this link for tests.pkl.

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
If your interpreter goes into an infinite loop, you should be able to stop it with Ctrl-c (or Ctrl-c Ctrl-c in an Emacs shell buffer). To exit the Scheme interpreter, issue Ctrl-d (Ctrl-c Ctrl-d in Emacs) or (after finishing problems 3 and 4) evaluate the exit procedure:
    scm> (exit)

Scheme Values

The interpreter represents Scheme values using subtypes of the class SchemeValue, which is defined in You will find definitions of all the methods used in these values in SchemeValue, where they are given default definitions. Many of the these definitions simply cause an error, since many methods work only on particular types. For example, the length method (for determining the length of a list) is defined only on Pairs and nil, so its default definition is to raise a SchemeError. That definition, in turn, is inherited by default by all the other subtypes of SchemeValue, but overridden in Pair and nil.

It is characteristic of this object-oriented approach that it avoids the use of if statements in most cases. Instead of writing something like

    def apply(proc, args, env):
	if type(proc) is PrimitiveProcedure:
	    # Code to apply primitive function
	elif type(proc) is LambdaProcedure:
	    # Code to apply lambda function
	elif ...
	    raise SchemeError("attempt to call something other than a function")
the programmer instead writes
    def apply(self, args, env):
        # Code to apply a primitive method
in the class PrimitiveProcedure and
    def apply(self, args, env):
        # Code to apply a lambda function
in the class LambdaProcedure, and so forth, with a default definition in SchemeValue that contains the final raise statement. SchemeValue and its subclasses are defined in and Here are these types and examples of expressions you can write in your Python code to produce them:
Scheme Data Type Our Internal Representation Classes Python Code
Types defined in
Numbers (0, -3.2) SchemeInt and SchemeFloat. scnum(0), scnum(3.2)
Symbols (merge, define) SchemeSymbol intern('merge'), intern('define')
Strings ("foo") SchemeStr scstr('foo')
Booleans (#t, #f) scheme_true and scheme_false scheme_true, scheme_false
Pairs ((a . b) Pair Pair(intern('a'), intern('b'))
nil, () nil nil
Lists: (a b) Pair and nil Pair(intern('a'), Pair(intern('b'), nil))
okay okay okay
Types defined in
Functions PrimitiveProcedure, LambdaProcedure, MuProcedure PrimitiveProcedure(...), etc.


The intern function returns a symbol, given its name (as a Python string or a Scheme symbol). It always returns the same symbol for equal strings, so that all instances of the same symbol return true when compared using the Python is operator (or the Scheme eq? function).

The okay class represents the undefined value returned by, for example, the Scheme function load. It's sole interesting property is that it prints as "okay".

Classes whose names are uncapitailized (nil, scheme_true, etc.) are immediately replaced by instances of those types. For example:

  class nil(SchemeValue):

  nil = nil()
so that at from this point on, nil is the sole object of (what was) the class nil, which we no longer need to be able to name, since we'll never create another instance of it. As a convenience, the Pair constructor will also accept Python numbers, converting them into SchemeInts and SchemeFloats, and Python strings, converting them into SchemeSymbols with intern.

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, converted into some kind of SchemeValue. Ths scheme_read functions does not return values of any of the procedure types, nor the special type okay.

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:

To test your code, use the following command:

  python3 -q 1

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 test your code, use the following command:

  python3 -q 2

You can also verify interactively that your solutions to Problem 1 and 2 work correctly 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.

There are a number of if statements in the code that test a variable proper_tail_recursion. Until you get the first extra-credit problem (#23), this variable is the constant False, which you should bear in mind with reading the code. 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. In this course, we've seen two approaches to data-directed programming. In one, illustrated by scheme_eval, the data contain some identifying tag that indicates what the various operations on that data are supposed to do. In the case of scheme_eval, this tag is a combination of the type of the expr argument and (when that argument is non-atomic) the first item in the list. We use a few if clauses to handle base cases (symbols, numbers, etc.). Then, for pairs that start with a symbol, we first use a dispatch table (see Lecture #17) to handle the special forms. The remaining cases are calls, handled in the rest of scheme_eval. The other approach is to have the data in effect contain the implementations of the operations upon it. We did this, for example, in Lecture #10, where one version of make_rat returned a function that contained implementations of the accessor functions for the numerator and denominator of a rational number. The more common way, and the one used in this project, is to use object-oriented programming, where methods (functions) for manipulating and object are in effect attached to that object. The class SchemeValue and its subtypes are an example.

Problem 3 (2 pt). Implement the apply method in the class PrimitiveProcedure. Primitive procedures are applied by calling a corresponding Python function that implements the procedure. The apply method overrides the default definition in SchemeValue, of which PrimitiveProcedure is a subtype. The default definition simply raises an exception, since most Python types do not implement function application. All definitions of apply that do return a value are defined to return a pair, (V, E), where V is a Scheme value, and E is an environment frame (type Frame) or None. The apply method in PrimitiveProcedure always returns None as the value of E, which indicates that "V is the final value resulting from applying this function to its arguments." As you'll see, overridings of apply for other classes of function have the option to return an environment frame for E, which means "V is a Scheme expression that has to be further evaluated in environment E to get the actual result of applying this function to its arguments."

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 for later processing into a PrimitiveProcedure and assignment to the global environment in

The apply method for a PrimitiveProcedure instance takes a Scheme list of argument values and the current environment. As for all overridings of the apply method, it returns a pair consisting of a Scheme value (the one returned by the fn function when it is called) and an environment, which in the Python code for the primitive function) and (value, env) Your implementation should:

To test your code, run the following command:

  python3 -q 3

The tests for PrimitiveProcedure.apply 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,

To test your code, run the following command:

  python3 -q 4

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)

Handling special forms

The do_*_form functions implement the "special forms" in Scheme---expressions represented as lists that are not evaluated according to the general rules for evaluating function calls. The scheme_eval function uses a dispatch table, SPECIAL_FORMS, to detect special forms and select the proper do_*_form function to handle them. All these functions take the rest of the special form (other than the first symbol) and an environment and return a Scheme value and an environment. The environment is None if there is nothing further to do to evaluate the form and the returned Scheme value is the final result. A non-nil value for the environment indicates that to get the actual final result of evaluating the form, the interpreter must evaluate (in the sense of scheme_eval) the returned expression in the returned environment.

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))

To test your code, run the following command:

  python3 -q A5

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))))

You can test your code by running the following command:

  python3 -q B6

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)

You can test your code by running the following command:

  python3 -q 7

Problem 8 (2 pt). Implement the do_lambda_form method, which evaluates lambda expressions by returning LambdaProcedure instances. 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)))

You can test your code by running the following command:

  python3 -q 8

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.

You can test your code by running the following command:

  python3 -q A9

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:

You can test your code by running the following command:

  python3 -q 10

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.)

You can test your code by running the following command:

  python3 -q B11

Problem 12 (2 pt). Implement the apply method in the LambdaProcedure class. It should:

You can test your code by running the following command:

  python3 -q 12

After you complete LambdaProcedure.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.

Logical 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. The __bool__ method causes true and false SchemeValues to be true and false Python values as well, so you can easily test whether a value is a true value or a false value.

It makes sense for the do_*_form functions for the logical forms to take advantage of their freedom to return an expression and environment for further evaluation rather than a value, rather than a value and None. For example, the expression (if (zero? x) (f x) (g x)) means "if x is 0, evaluate (f x) in the current environment and otherwise evaluate (g x) in the current enviornment. By returning one or the other of these expressions and its environment argument, do_if_form fulfills its contract. Doing it this way will turn out to be useful when you get to Problem 22.

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)

You can test your code by running the following command:

  python3 -q A13

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 return it. Otherwise, return the last sub-expression from do_or_form.

    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))
    #f   ; short-circuiting behavior of and
    scm> (or 4 #t (/ 1 0))
    4    ; short-circuiting behavior of or

You can test your code by running the following command:

  python3 -q B14

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.

You can test your code by running the following command:

  python3 -q A15

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.

You can test your code by running the following command:

  python3 -q A16

Problem B17 (2 pt). Implement 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.

To do this, complete MuProcedure.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)

You can test your code by running the following command:

  python3 -q B17

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)

You can test your code by running the following command:

  python3 -q 18

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)

You can test your code by running the following command:

  python3 -q 19

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):


You can test your code by running the following command:

  python3 -q 20

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). So far, we have paid no attention to handling tail recursion correctly. For example, to evaluate this call on the tail-recursive sum-series function:

    (define (sum-series f low high init) 
       (if (> low high) init (sum-series f (+ low 1) high (+ init (f low))))
    (sum-series (lambda x: (* x x)) 0 1000000)
Your implementation will end up recursively calling scheme_eval to evaluate the recursive call to sum-series, with the result that the Python interpreter would have to recurse about 1000000 levels deep to perform the computation (which it generally will refuse to do). For this problem, we'll make the Scheme interpreter properly tail recursive so that it will allow an unbounded number of active tail calls in constant space.

We have arranged that proper_tail_recursion is initially false, with the result that only the false branches of several if statements will get executed. Complete the other branches of these statements (and possibly make other modifications, depending on what you've done in implementing the special forms) to handle tail calls properly. 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 and take advantage of the while loop in scheme_eval to use iteration in the interpreter in place of recursive calls in these cases. For call expressions, this change only applies to calling user-defined procedures.

Once you finish, uncomment the line proper_tail_recursion = True in

You can test your code by running the following command:

  python3 -q 22

Problem 23 (3 pt). The year 1960 saw the publication of the Revised Report on the Algorithmic Language Algol 60 as well as an implementation of that language. The Report was a model of clear and concise language description that served as a model for many later language reference manuals (including that of Scheme). In at least one instance, however, it is possible that the desire for a simple and elegant description led to a language feature that proved a bit problematic to implement, however much fun it was to use.

Specifically, to describe parameter passing to functions, the Report used a form of the substitution model (see Lecture #2). That is, they described calls to functions as if the effect were to replace the call (at execution time) with a copy of the function's body with the actual parameters substituted for all occurrences of the formal parameters, first changing the names of local variables so as to avoid clashes with names used in the actual parameters. It's a simple explanation, but there is a subtle, very significant difference from our substitution model: it is the actual parameter expressions, not their values, that are substituted. The result is known as call-by-name parameter passing (whereas Python and standard Scheme use call-by-value parameter passing).

Consider a call in Scheme such as (f (/ x y) y). In standard Scheme, this requires that the interpreter evaluate f, (/ x y), and y and then call f with the argument values bound to f's formal parameters. If y should happen to be 0, the argument evaluation will fail and f will never be called. Suppose, however, that we change the semantics of Scheme so that we evaluate (/ x y) only if and when the body of f actually uses its value? If f is something like this:

  (define (f a b)
       (if (= b 0) 0 a))
then f's first parameter would never be evaluated during the call (f (/ x y) y) and the function would return 0.

Back in the 1960s, getting this to work turned out to be "interesting", especially if one was trying to get fast executable code. However, if speed is not of the essence, the implementation is fairly easy: we just convert call-by-name parameters into ordinary call-by-value parameters by means of a trick. For the definition above, we can write instead:

    (define (f a b)
         (if (= (b) 0) 0 (a)))
and then change the sample call to
    (f (lambda () (/ x y)) (lambda () y))  

This approach is an illustration on the old CS saying "Any difficulty in computer science can be overcome by adding a level of indirection." Here, we pass in a parameterless function that yields the parameter's value only when it is called. Such functions, when implicitly introduced by a compiler, interpreter, or runtime system, are known as thunks (for obscure reasons). The usual scoping rules for Scheme automatically avoid name clashes, so no renaming of parameters is needed.

Implementing the feature exactly this way is trickier than it looks (trust us), so we suggest an alternative. First, we'll introduce a new special form with the same syntax as lambda: (nu (formals) body), which produces call-by-name functions represented in our interpreter by NuProcedures. Although we could represent thunks with LambdaProcedures, we instead suggest subtyping LambdaProcedure and modifying your code so that whenever scheme_eval fetches a thunk from a symbol, the value actually obtained is the result of calling the thunk. You'll also have to fill in NuProcedure so that when such functions are called, they "thunkify" all their parameters rather than simply evaluating them.

You can test your code by running the following command:

  python3 -q 23

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.