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The following 4 exercises involve the data type arith-expr
. If you are asked to write a function(s), follow the design recipe: contract, purpose, examples/tests, template instantiation, code, testing (which happens automatically if the examples are given in (check-expect ...)
form). Follow the same recipe for any help function that you introduce.
- Write an evaluator for arithmetic expressions as follows:
- Write the (function) template for
arith-expr
- Write a function
to-list
that maps an arith-expr
to the corresponding "list" representation in Scheme. Numbers are unchanged. Some other examples include: Code Block |
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(to-list (make-sum (make-prod 4 7) 25)) => '(+ (* 4 7) 25)
(to-list (make-quot (make-diff 4 7) 25)) => '(/ (- 4 7) 25)
|
Note: you need to define the output type (which I call gen-expr) for this function, but you can omit the template because this assignment does not include any functions that process this type. - Write a function
eval
that evaluates an arith-expr
. Your evaluator should produce
exactly the same result for an arith-expr E
that Scheme evaluation would produce for
the list representation (to-list E)
.
- Extend the definition of
<arith-expr>
} by adding a clause for variables (represented as Scheme
symbols). Write the (function) template for this definition. - Modify your definition of
to-list
to support the new definition of arith-expr. - Given the Scheme structure defintion:
Code Block |
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(define-structure binding (var val))
|
a binding
is (make-binding s n)
where s
is a symbol and n
is a number and an environment
is a (listOf binding)
. Write a (function) template for processing an environment
. - Define a top-level variable (constant)
empty-env
that is bound to the empty environment. - Write a function
extend
that takes environment env
, a symbol s
, and a number n
,
and returns an extended environment identical to env
except that it adds the additional binding of s
to n
.
The definition of extend
trivial; it requires no recursion. As a result, extend
satisfies the invariant Code Block |
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(check-expect (extend empty-env _s_ _env_ _n_) (list (make-binding _s_ _n_)))
|
In the remainder of the problem, use empty-env
and extend
to define example environments for test cases. - Write a function
lookup
that takes a symbol s
and an environment env
and returns the first binding in env
with a var
component that equals s
. If no match is found, lookup
returns empty. Note that the return type of lookup
is not simply binding
. Define the a new union type called option-binding
for the the return type. - Write a new
eval
function for the new definition of arith-expr
. The new eval
takes two arguments: an arith-expr E
to evaluate and an environment env}} specifying the values of free variables in {{E
. For example, Code Block |
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(eval 'x (extend empty-env 'x 17)) => 17
(eval (make-prod 4 7) (extend empty-env 'x 17)) = 28
(eval 'y (extend empty-env 'x 17)) => some form of run-time error
|
If an arith-expr E
contains a free variable that is not bound in the environment env
, then (eval E env)
will naturally produce some form of run-time error if you have correctly coded eval
. Do NOT explicitly test for this form of error.
- An
environment
is really a finite function. It is finite in the sense that it can be completely defined by a finite table, which is not true of nearly all the primitive and library functions in Scheme (and other programming languages). Even the identity function is not finite. For the purpose of this exercise, we redefine the type environment
.
as (symbol -> option-binding)
. - Rewrite
eval
to use environment
defined as a finite function in (symbol -> option-binding)
instead of as a (listOf option-binding)
. If you cleanly coded your definition of eval
in the preceding problem using lookup
, make-binding
, and extend
, all that you have to do to your solution to the previous problem is
redefine the bindings of lookup
, empty-env
, and extend
, and revise your test cases for extend
. You can literally copy the entire text of your solution to problem 2; change the definitions of lookup
, empty-env
, and extend
; update your documentation (annotations) concerning
the environment
type; and revise your tests for extend
. Note that extend
cannot be tested (since the result is a function!) without using lookup
to examine it. (If you wrote a correct solution to problem 2, you can do this problem is less than 15 minutes!)
- Extra Credit (50%)
Extend the definition of <arith-expr>
by adding a clause for unary lambda
-expressions and a clause for unary applications of an arith-expr
to an arith-expr
. Use the name lam
for the structure representing a lambda
-expression and the names var
and body
for the accessors of this structure. Use the name app
for the structure representing an application and the names head
and arg
for the accessors of this structure. Note that the head of an app
is an arith-expr
not a lam
. - Write a (function) template for the newest definition of
arith-expr
. - Extend the definition of
to-list
to support the newest definition of arith-expr
. - Extend the definition of
eval
to support the newest definition of arith-expr
. Note that eval
can now return functions as well as numbers. Your biggest challenge is determining a good representation for function values. What does eval
return for a lam
input? That input may contain free variables. In principle, you could represent the value of the lam
input by a revised lam
(with no free variables) obtained by substituting the values for free variables from the environment input (just like we do in hand-evaluation). But this approach is tedious and computationally expensive. A better strategy is to define a structure type (called a closure) to represent a function value. The structure type must contain the original lam
and a description of what substitution would have been made, deferring the actual substitution just as eval
defers substitutions by maintaining an environment.