PL: Lecture #4  Tuesday, September 22nd

BNF, Grammars, the AE Language

Getting back to the theme of the course: we want to investigate programming languages, and we want to do that using a programming language.

The first thing when we design a language is to specify the language. For this we use BNF (Backus-Naur Form). For example, here is the definition of a simple arithmetic language:

<AE> ::= <num>
      | <AE> + <AE>
      | <AE> - <AE>

Explain the different parts. Specifically, this is a mixture of low-level (concrete) syntax definition with parsing.

We use this to derive expressions in some language. We start with <AE>, which should be one of these:

<num> is a terminal: when we reach it in the derivation, we’re done. <AE> is a non-terminal: when we reach it, we have to continue with one of the options. It should be clear that the + and the - are things we expect to find in the input — because they are not wrapped in <>s.

We could specify what <num> is (turning it into a <NUM> non-terminal):

<AE> ::= <NUM>
      | <AE> + <AE>
      | <AE> - <AE>

<NUM> ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
        | <NUM> <NUM>

But we don’t — why? Because in Racket we have numbers as primitives and we want to use Racket to implement our languages. This makes life a lot easier, and we get free stuff like floats, rationals etc.

To use a BNF formally, for example, to prove that 1-2+3 is a valid <AE> expression, we first label the rules:

<AE> ::= <num>        (1)
      | <AE> + <AE>  (2)
      | <AE> - <AE>  (3)

and then we can use them as formal justifications for each derivation step:

<AE> + <AE>        ; (2)
<AE> + <num>        ; (1)
<AE> - <AE> + <num> ; (3)
<AE> - <AE> + 3    ; (num)
<num> - <AE> + 3    ; (1)
<num> - <num> + 3  ; (1)
1 - <num> + 3      ; (num)
1 - 2 + 3          ; (num)

This would be one way of doing this. Alternatively, we can can visualize the derivation using a tree, with the rules used at the nodes.

These specifications suffer from being ambiguous: an expression can be derived in multiple ways. Even the little syntax for a number is ambiguous — a number like 123 can be derived in two ways that result in trees that look different. This ambiguity is not a “real” problem now, but it will become one very soon. We want to get rid of this ambiguity, so that there is a single (= deterministic) way to derive all expressions.

There is a standard way to resolve that — we add another non-terminal to the definition, and make it so that each rule can continue to exactly one of its alternatives. For example, this is what we can do with numbers:

<NUM>  ::= <DIGIT> | <DIGIT> <NUM>

<DIGIT> ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Similar solutions can be applied to the <AE> BNF — we either restrict the way derivations can happen or we come up with new non-terminals to force a deterministic derivation trees.

As an example of restricting derivations, we look at the current grammar:

<AE> ::= <num>
      | <AE> + <AE>
      | <AE> - <AE>

and instead of allowing an <AE> on both sides of the operation, we force one to be a number:

<AE> ::= <num>
      | <num> + <AE>
      | <num> - <AE>

Now there is a single way to derive any expression, and it is always associating operations to the right: an expression like 1+2+3 can only be derived as 1+(2+3). To change this to left-association, we would use this:

<AE> ::= <num>
      | <AE> + <num>
      | <AE> - <num>

But what if we want to force precedence? Say that our AE syntax has addition and multiplication:

<AE> ::= <num>
      | <AE> + <AE>
      | <AE> * <AE>

We can do that same thing as above and add new non-terminals — say one for “products”:

<AE>  ::= <num>
        | <AE> + <AE>
        | <PROD>

<PROD> ::= <num>
        | <PROD> * <PROD>

Now we must parse any AE expression as additions of multiplications (or numbers). First, note that if <AE> goes to <PROD> and that goes to <num>, then there is no need for an <AE> to go to a <num>, so this is the same syntax:

<AE>  ::= <AE> + <AE>
        | <PROD>

<PROD> ::= <num>
        | <PROD> * <PROD>

Now, if we want to still be able to multiply additions, we can force them to appear in parentheses:

<AE>  ::= <AE> + <AE>
        | <PROD>

<PROD> ::= <num>
        | <PROD> * <PROD>
        | ( <AE> )

Next, note that <AE> is still ambiguous about additions, which can be fixed by forcing the left hand side of an addition to be a factor:

<AE>  ::= <PROD> + <AE>
        | <PROD>

<PROD> ::= <num>
        | <PROD> * <PROD>
        | ( <AE> )

We still have an ambiguity for multiplications, so we do the same thing and add another non-terminal for “atoms”:

<AE>  ::= <PROD> + <AE>
        | <PROD>

<PROD> ::= <ATOM> * <PROD>
        | <ATOM>

<ATOM> ::= <num>
        | ( <AE> )

And you can try to derive several expressions to be convinced that derivation is always deterministic now.

But as you can see, this is exactly the cosmetics that we want to avoid — it will lead us to things that might be interesting, but unrelated to the principles behind programming languages. It will also become much much worse when we have a real language rather such a tiny one.

Is there a good solution? — It is right in our face: do what Racket does — always use fully parenthesized expressions:

<AE> ::= <num>
      | ( <AE> + <AE> )
      | ( <AE> - <AE> )

To prevent confusing Racket code with code in our language(s), we also change the parentheses to curly ones:

<AE> ::= <num>
      | { <AE> + <AE> }
      | { <AE> - <AE> }

But in Racket everything has a value — including those +s and -s, which makes this extremely convenient with future operations that might have either more or less arguments than 2 as well as treating these arithmetic operators as plain functions. In our toy language we will not do this initially (that is, + and - are second order operators: they cannot be used as values). But since we will get to it later, we’ll adopt the Racket solution and use a fully-parenthesized prefix notation:

<AE> ::= <num>
      | { + <AE> <AE> }
      | { - <AE> <AE> }

(Remember that in a sense, Racket code is written in a form of already-parsed syntax…)

Simple Parsing

On to an implementation of a “parser”:

Unrelated to what the syntax actually looks like, we want to parse it as soon as possible — converting the concrete syntax to an abstract syntax tree.

No matter how we write our syntax:

we always mean the same abstract thing — adding the number 3 and the number 4. The essence of this is basically a tree structure with an addition operation as the root and two leaves holding the two numerals.

With the right data definition, we can describe this in Racket as the expression (Add (Num 3) (Num 4)) where Add and Num are constructors of a tree type for syntax, or in a C-like language, it could be something like Add(Num(3),Num(4)).

Similarly, the expression (3-4)+7 will be described in Racket as the expression:

(Add (Sub (Num 3) (Num 4)) (Num 7))

Important note: “expression” was used in two different ways in the above — each way corresponds to a different language, and the result of evaluating the second “expression” is a Racket value that represents the first expression.

To define the data type and the necessary constructors we will use this:

(define-type AE
  [Num Number]
  [Add AE AE]
  [Sub AE AE])

To make things very simple, we will use the above fact through a double-level approach:

This is achieved by the following simple recursive function:

(: parse-sexpr : Sexpr -> AE)
;; parses s-expressions into AEs
(define (parse-sexpr sexpr)
  (cond [(number? sexpr) (Num sexpr)]
        [(and (list? sexpr) (= 3 (length sexpr)))
        (let ([make-node
                (match (first sexpr)
                  ['+ Add]
                  ['- Sub]
                  [else (error 'parse-sexpr "unknown op: ~s"
                              (first sexpr))])
                #| the above is the same as:
                (cond [(equal? '+ (first sexpr)) Add]
                      [(equal? '- (first sexpr)) Sub]
                      [else (error 'parse-sexpr "unknown op: ~s"
                                  (first sexpr))])
          (make-node (parse-sexpr (second sexpr))
                      (parse-sexpr (third sexpr))))]
        [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

This function is pretty simple, but as our languages grow, they will become more verbose and more difficult to write. So, instead, we use a new special form: match, which is matching a value and binds new identifiers to different parts (try it with “Check Syntax”). Re-writing the above code using match:

(: parse-sexpr : Sexpr -> AE)
;; parses s-expressions into AEs
(define (parse-sexpr sexpr)
  (match sexpr
    [(number: n) (Num n)]
    [(list '+ left right)
    (Add (parse-sexpr left) (parse-sexpr right))]
    [(list '- left right)
    (Sub (parse-sexpr left) (parse-sexpr right))]
    [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

And finally, to make it more uniform, we will combine this with the function that parses a string into a sexpr so we can use strings to represent our programs:

(: parse : String -> AE)
;; parses a string containing an AE expression to an AE
(define (parse str)
  (parse-sexpr (string->sexpr str)))

The match Form

The syntax for match is

(match value
  [pattern result-expr]

The value is matched against each pattern, possibly binding names in the process, and if a pattern matches it evaluates the result expression. The simplest form of a pattern is simply an identifier — it always matches and binds that identifier to the value:

(match (list 1 2 3)
  [x x]) ; evaluates to the list

Another simple pattern is a quoted symbol, which matches that symbol. For example:

(match foo
  ['x "yes"]
  [else "no"])

will evaluate to "yes" if foo is the symbol x, and to "no" otherwise. Note that else is not a keyword here — it happens to be a pattern that always succeeds, so it behaves like an else clause except that it binds else to the unmatched-so-far value.

Many patterns look like function application — but don’t confuse them with applications. A (list x y z) pattern matches a list of exactly three items and binds the three identifiers; or if the “arguments” are themselves patterns, match will descend into the values and match them too. More specifically, this means that patterns can be nested:

(match (list 1 2 3)
  [(list x y z) (+ x y z)]) ; evaluates to 6
(match '((1) (2) 3)
  [(list (list x) (list y) z) (+ x y z)]) ; also 6

There is also a cons pattern that matches a non-empty list and then matches the first part against the head for the list and the second part against the tail of the list.

In a list pattern, you can use ... to specify that the previous pattern is repeated zero or more times, and bound names get bound to the list of respective matching. One simple consequent is that the (list hd tl ...) pattern is exactly the same as (cons hd tl), but being able to repeat an arbitrary pattern is very useful:

> (match '((1 2) (3 4) (5 6) (7 8))
    [(list (list x y) ...) (list x y)])
'((1 3 5 7) (2 4 6 8))

A few more useful patterns:

id              -- matches anything, binds `id' to it
_              -- matches anything, but does not bind
(number: n)    -- matches any number and binds it to `n'
(symbol: s)    -- same for symbols
(string: s)    -- strings
(sexpr: s)      -- S-expressions (needed sometimes for Typed Racket)
(and pat1 pat2) -- matches both patterns
(or pat1 pat2)  -- matches either pattern (careful with bindings)

Note that the foo: patterns are all specific to our #lang pl, they are not part of #lang racket or #lang typed/racket

The patterns are tried one by one in-order, and if no pattern matches the value, an error is raised.

Note that ... in a list pattern can follow any pattern, including all of the above, and including nested list patterns.

Here are a few examples — you can try them out with #lang pl untyped at the top of the definitions window. This:

(match x
  [(list (symbol: syms) ...) syms])

matches x against a pattern that accepts only a list of symbols, and binds syms to those symbols. And here’s an example that matches a list of any number of lists, where each of the sub-lists begins with a symbol and then has any number of numbers. Note how the n and s bindings get values for a list of all symbols and a list of lists of the numbers:

> (define (foo x)
    (match x
      [(list (list (symbol: s) (number: n) ...) ...)
      (list 'symbols: s 'numbers: n)]))
> (foo (list (list 'x 1 2 3) (list 'y 4 5)))
'(symbols: (x y) numbers: ((1 2 3) (4 5)))

Here is a quick example for how or is used with two literal alternatives, how and is used to name a specific piece of data, and how or is used with a binding:

> (define (foo x)
    (match x
      [(list (or 1 2 3)) 'single]
      [(list (and x (list 1 _)) 2) x]
      [(or (list 1 x) (list 2 x)) x]))
> (foo (list 3))
> (foo (list (list 1 99) 2))
'(1 99)
> (foo (list 1 10))
> (foo (list 2 10))

The define-type Form

The class language that we’re using, #lang pl, is based on Typed Racket: a statically-typed dialect of Racket. It is not exactly the same as Typed Racket — it is restricted in many ways, and extended in a few ways. (You should therefore try to avoid looking at the Typed Racket documentation and expect things to be the same in #lang pl.)

The most important extension is define-type, which is the construct we will be using to create new user-defined types. In general, such definitions looks like what we just used:

(define-type AE
  [Num Number]
  [Add AE AE]
  [Sub AE AE])

This defines a new type called AE, an AE? predicate for this type, and a few variants for this type: Num, Add, and Sub in this case. Each of these variant names is a constructor, taking in arguments with the listed types, where these types can include the newly defined type itself in (the very common) case we’re defining a recursive type. The return type is always the newly defined type, AE here.

To summarize, this definition gives us a new AE type, and three constructors, as if we wrote the following type declarations:

The newly defined types are known as “disjoint unions”, since values in these types are disjoint — there is no overlap between the different variants. As we will see, this is what makes this such a useful construct for our needs: the compiler knows about the variants of each newly defined type, which will make it possible for it to complain if we extend a type with more variants but not update all uses of the type.

Furthermore, since the return types of these constructors are all the new type itself, there is no way for us to write code that expects just one of these variants. We will use a second form, cases, to handle these values.

The cases Form

A define-type declaration defines only what was described above: one new type name and a matching predicate, and a few variants as constructor functions. Unlike HtDP, we don’t get predicates for each of the variants, and we don’t get accessor functions for the fields of the variants.

The way that we handle the new kind of values is with cases: this is a form that is very similar to match, but is specific to instances of the user-defined type.

Many students find it confusing to distinguish match and cases since they are so similar. Try to remember that match is for primitive Racket values (we’ll mainly use them for S-expression values), while cases is for user-defined values. The distinction between the two forms is unfortunate, and doesn’t serve any purpose. It is just technically difficult to unify the two.

For example, code that handles AE values (as defined above) can look as follows:

(cases some-ae-value
  [(Num n)  "a number"]
  [(Add l r) "an addition"]
  [(Sub l r) "a subtraction"])

As you can see, we need to have patterns for each of the listed variants (and the compiler will throw an error if some are missing), and each of these patterns specifies bindings that will get the field values contained in a given variant object.

We can also use nested patterns:

(cases some-ae-value
  [(Num n)              "a number"]
  [(Add (Num m) (Num n)) "a simple addition"]
  [(Add l r)            "an addition"]
  [(Sub (Num m) (Num n)) "a simple subtraction"]
  [(Sub l r)            "a subtraction"])

but this is a feature that we will not use too often.

The final clause in a cases form can be an else clause, which serves as a fallback in case none of the previous clauses matched the input value. However, using an else like this is strongly discouraged! The problem with using it is that it effectively eliminates the advantage in getting the type-checker to complain when a type definition is extended with new variants. Using these else clauses, we can actually mimic all of the functionality that you expect in HtDP-style code, which demonstrates that this is equivalent to HtDP-style definitions. For example:

(: Add? : AE -> Boolean)
;; identifies instances of the `Add` variant
(define (Add? ae)
  (cases ae
    [(Add l r) #t]
    [else #f]))

(: Add-left : AE -> AE)
;; get the left-hand subexpression of an addition
(define (Add-left ae)
  (cases ae
    [(Add l r) l]
    [else (error 'Add-left "expecting an Add value, got ~s" ae)]))


Important reminder: this is code that you should not write! Doing so will lead to code that is more fragile than just using cases, since you’d be losing the protection the compiler gives you in the form of type errors on occurrences of cases that need to be updated when a type is extended with new variants. You would therefore end up writing a bunch of boiler-plate code only to end up with lower-quality code. The core of the problem is in the prevalent use of else which gives up that protection.

In these examples the else clause is justified because even if AE is extended with new variants, functions like Add? and Add-left should not be affected and treat the new variants as they treat all other non-Add instances. (And since else is inherent to these functions, using them in our code is inherently a bad idea.) We will, however, have a few (very few!) places where we’ll need to use else — but this will always be done only on some specific functionality rather than a wholesale approach of defining a different interface for user-defined types.

Semantics (= Evaluation)


Back to BNF — now, meaning.

An important feature of these BNF specifications: we can use the derivations to specify meaning (and meaning in our context is “running” a program (or “interpreting”, “compiling”, but we will use “evaluating”)). For example:

<AE> ::= <num>        ; <AE> evaluates to the number
      | <AE1> + <AE2> ; <AE> evaluates to the sum of evaluating
                      ;      <AE1> and <AE2>
      | <AE1> - <AE2> ; ... the subtraction of <AE2> from <AE1>
                              (... roughly!)

To do this a little more formally:

a. eval(<num>) = <num> ;*** special rule: translate syntax to value
b. eval(<AE1> + <AE2>) = eval(<AE1>) + eval(<AE2>)
c. eval(<AE1> - <AE2>) = eval(<AE1>) - eval(<AE2>)

Note the completely different roles of the two +s and -s. In fact, it might have been more correct to write:

a. eval("<num>") = <num>
b. eval("<AE1> + <AE2>") = eval("<AE1>") + eval("<AE2>")
c. eval("<AE1> - <AE2>") = eval("<AE1>") - eval("<AE2>")

or even using a marker to denote meta-holes in these strings:

a. eval("$<num>") = <num>
b. eval("$<AE1> + $<AE2>") = eval("$<AE1>") + eval("$<AE2>")
c. eval("$<AE1> - $<AE2>") = eval("$<AE1>") - eval("$<AE2>")

but we will avoid pretending that we’re doing that kind of string manipulation. (For example, it will require specifying what does it mean to return <num> for $<num> (involves string->number), and the fragments on the right side mean that we need to specify these as substring operations.)

Note that there’s a similar kind of informality in our BNF specifications, where we assume that <foo> refers to some terminal or non-terminal. In texts that require more formal specifications (for example, in RFC specifications), each literal part of the BNF is usually double-quoted, so we’d get

<AE> ::= <num> | <AE1> "+" <AE2> | <AE1> "-" <AE2>

An alternative popular notation for eval(X) is [[X]]:

a. [[<num>]] = <num>
b. [[<AE1> + <AE2>]] = [[<AE1>]] + [[<AE2>]]
c. [[<AE1> - <AE2>]] = [[<AE1>]] - [[<AE2>]]

Is there a problem with this definition? Ambiguity:

eval(1 - 2 + 3) = ?

Depending on the way the expression is parsed, we can get either a result of 2 or -4:

eval(1 - 2 + 3) = eval(1 - 2) + eval(3)          [b]
                = eval(1) - eval(2) + eval(3)    [c]
                = 1 - 2 + 3                      [a,a,a]
                = 2

eval(1 - 2 + 3) = eval(1) - eval(2 + 3)          [c]
                = eval(1) - (eval(2) + eval(3))  [a]
                = 1 - (2 + 3)                    [a,a,a]
                = -4

Again, be very aware of confusing subtleties which are extremely important: We need parens around a sub-expression only in one side, why? — When we write:

eval(1 - 2 + 3) = ... = 1 - 2 + 3

we have two expressions, but one stands for an input syntax, and one stands for a real mathematical expression.

In a case of a computer implementation, the syntax on the left is (as always) an AE syntax, and the real expression on the right is an expression in whatever language we use to implement our AE language.

Like we said earlier, ambiguity is not a real problem until the actual parse tree matters. With eval it definitely matters, so we must not make it possible to derive any syntax in multiple ways or our evaluation will be non-deterministic.

Quick exercise:

We can define a meaning for <digit>s and then <num>s in a similar way:

<NUM> ::= <digit> | <digit> <NUM>

eval(0) = 0
eval(1) = 1
eval(2) = 2
eval(9) = 9

eval(<digit>) = <digit>
eval(<digit> <NUM>) = 10*eval(<digit>) + eval(<NUM>)

Is this exactly what we want? — Depends on what we actually want…

Side-note: Compositionality

The example of

<NUM> ::= <digit> | <NUM> <digit>

being a language that is easier to write an evaluator for leads us to an important concept — compositionality. This definition is easier to write an evaluator for, since the resulting language is compositional: the meaning of an expression — for example 123 — is composed out of the meaning of its two parts, which in this BNF are 12 and 3. Specifically, the evaluation of <NUM> <digit> is 10 * the evaluation of the first, plus the evaluation of the second. In the <digit> <NUM> case this is more difficult — the meaning of such a number depends not only on the meaning of the two parts, but also on the <NUM> syntax:

eval(<digit> <NUM>) =
  eval(<digit>) * 10^length(<NUM>) + eval(<NUM>)

This this case this can be tolerable, since the meaning of the expression is still made out of its parts — but imperative programming (when you use side effects) is much more problematic since it is not compositional (at least not in the obvious sense). This is compared to functional programming, where the meaning of an expression is a combination of the meanings of its subexpressions. For example, every sub-expression in a functional program has some known meaning, and these all make up the meaning of the expression that contains them — but in an imperative program we can have a part of the code be x++ — and that doesn’t have a meaning by itself, at least not one that contributes to the meaning of the whole program in a direct way.

(Actually, we can have a well-defined meaning for such an expression: the meaning is going from a world where x is a container of some value N, to a world where the same container has a different value N+1. You can probably see now how this can make things more complicated. On an intuitive level — if we look at a random part of a functional program we can tell its meaning, so building up the meaning of the whole code is easy, but in an imperative program, the meaning of a random part is pretty much useless.)

Implementing an Evaluator

Now continue to implement the semantics of our syntax — we express that through an eval function that evaluates an expression.

We use a basic programming principle — splitting the code into two layers, one for parsing the input, and one for doing the evaluation. Doing this avoids the mess we’d get into otherwise, for example:

(define (eval sexpr)
  (match sexpr
    [(number: n) n]
    [(list '+ left right) (+ (eval left) (eval right))]
    [(list '- left right) (- (eval left) (eval right))]
    [else (error 'eval "bad syntax in ~s" sexpr)]))

This is messy because it combines two very different things — syntax and semantics — into a single lump of code. For this particular kind of evaluator it looks simple enough, but this is only because it’s simple enough that all we do is replace constructors by arithmetic operations. Later on things will get more complex, and bundling the evaluator with the parser will be more problematic. (Note: the fact that we can replace constructors with the run-time operators mean that we have a very simple, calculator-like language, and that we can, in fact, “compile” all programs down to a number.)

If we split the code, we can easily include decisions like making

{+ 1 {- 3 "a"}}

syntactically invalid. (Which is not, BTW, what Racket does…) (Also, this is like the distinction between XML syntax and well-formed XML syntax.)

An additional advantage is that by using two separate components, it is simple to replace each one, making it possible to change the input syntax, and the semantics independently — we only need to keep the same interface data (the AST) and things will work fine.

Our parse function converts an input syntax to an abstract syntax tree (AST). It is abstract exactly because it is independent of any actual concrete syntax that you type in, print out etc.

Implementing The AE Language

Back to our eval — this will be its (obvious) type:

(: eval : AE -> Number)
;; consumes an AE and computes
;; the corresponding number

which leads to some obvious test cases:

(equal? 3 (eval (parse "3")))
(equal? 7 (eval (parse "{+ 3 4}")))
(equal? 6 (eval (parse "{+ {- 3 4} 7}")))

which from now on we will write using the new test form that the #lang pl language provides:

(test (eval (parse "3"))            => 3)
(test (eval (parse "{+ 3 4}"))      => 7)
(test (eval (parse "{+ {- 3 4} 7}")) => 6)

Note that we’re testing only at the interface level — only running whole functions. For example, you could think about a test like:

(test (parse "{+ {- 3 4} 7}")
      => (Add (Sub (Num 3) (Num 4)) (Num 7)))

but the details of parsing and of the constructor names are things that nobody outside of our evaluator cares about — so we’re not testing them. In fact, we shouldn’t even mention parse in these tests, since it is not part of the public interface of our users; they only care about using it as a compiler-like black box. (This is sometimes called “integration tests”.) We’ll address this shortly.

Like everything else, the structure of the recursive eval code follows the recursive structure of its input. In HtDP terms, our template is:

(: eval : AE -> Number)
(define (eval expr)
  (cases expr
    [(Num n)  ... n ...]
    [(Add l r) ... (eval l) ... (eval r) ...]
    [(Sub l r) ... (eval l) ... (eval r) ...]))

In this case, filling in the gaps is very simple

(: eval : AE -> Number)
(define (eval expr)
  (cases expr
    [(Num n)  n]
    [(Add l r) (+ (eval l) (eval r))]
    [(Sub l r) (- (eval l) (eval r))]))

We now further combine eval and parse into a single run function that evaluates an AE string.

(: run : String -> Number)
;; evaluate an AE program contained in a string
(define (run str)
  (eval (parse str)))

This function becomes the single public entry point into our code, and the only thing that should be used in tests that verify our interface:

(test (run "3")            => 3)
(test (run "{+ 3 4}")      => 7)
(test (run "{+ {- 3 4} 7}") => 6)

The resulting full code is:

#lang pl

#| BNF for the AE language:
  <AE> ::= <num>
          | { + <AE> <AE> }
          | { - <AE> <AE> }
          | { * <AE> <AE> }
          | { / <AE> <AE> }

;; AE abstract syntax trees
(define-type AE
  [Num Number]
  [Add AE AE]
  [Sub AE AE]
  [Mul AE AE]
  [Div AE AE])

(: parse-sexpr : Sexpr -> AE)
;; parses s-expressions into AEs
(define (parse-sexpr sexpr)
  (match sexpr
    [(number: n) (Num n)]
    [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
    [(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
    [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

(: parse : String -> AE)
;; parses a string containing an AE expression to an AE AST
(define (parse str)
  (parse-sexpr (string->sexpr str)))

(: eval : AE -> Number)
;; consumes an AE and computes the corresponding number
(define (eval expr)
  (cases expr
    [(Num n)  n]
    [(Add l r) (+ (eval l) (eval r))]
    [(Sub l r) (- (eval l) (eval r))]
    [(Mul l r) (* (eval l) (eval r))]
    [(Div l r) (/ (eval l) (eval r))]))

(: run : String -> Number)
;; evaluate an AE program contained in a string
(define (run str)
  (eval (parse str)))

;; tests
(test (run "3") => 3)
(test (run "{+ 3 4}") => 7)
(test (run "{+ {- 3 4} 7}") => 6)

(Note that the tests are done with a test form, which we mentioned above.)

For anyone who thinks that Racket is a bad choice, this is a good point to think how much code would be needed in some other language to do the same as above.