PL: Lecture #4  Tuesday, January 15th
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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))

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.