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Recall that the parser must produce output (e.g., an abstract-syntax tree) for the next phase of the compiler. This involves doing a syntax-directed translation -- translating from a sequence of tokens to some other form, based on the underlying syntax.

A syntax-directed translation is defined by augmenting the CFG: a translation rule is defined for each production. A translation rule defines the translation of the left-hand side nonterminal as a function of:

constants
the right-hand-side nonterminals' translations
the right-hand-side tokens' values (e.g., the integer value associated with an INTLITERAL token, or the String value associated with an ID token)

To translate an input string:

Build the parse tree.
Use the translation rules to compute the translation of each nonterminal in the tree, working bottom up (since a nonterminal's value may depend on the value of the symbols on the right-hand side, you need to work bottom-up so that those values are available).

The translation of the string is the translation of the parse tree's root nonterminal.

Example 1

Below is the definition of a syntax-directed translation that translates an arithmetic expression to its integer value. When a nonterminal occurs more than once in a grammar rule, the corresponding translation rule uses subscripts to identify a particular instance of that nonterminal. For example, the rule exp $\rightarrow$ exp PLUS term has two exp nonterminals; exp₁ means the left-hand-side exp, and exp₂ means the right-hand-side exp. Also, the notation xxx.value is used to mean the value associated with token xxx.

CFG Production Translation rules

exp $\longrightarrow$ exp PLUS term exp₁.trans = exp₂.trans + term.trans

exp $\longrightarrow$ term exp.trans = term.trans

term $\longrightarrow$ term TIMES factor term₁.trans = term₂.trans * factor.trans

term $\longrightarrow$ factor term.trans = factor.trans

factor $\longrightarrow$ INTLITERAL factor.trans = INTLITERAL.value

factor $\longrightarrow$ LPARENS exp RPARENS factor.trans = exp.trans

consider evaluating these rules on the input 2 * (4 + 5). The result is the following annotated parse tree:

Example 2

Consider a language of expressions that use the three operators: +, &&, == using the terminal symbols PLUS, AND , EQUALS, respectively. Integer literals are represented by the same INTLITERAL token we've used before, and TRUE and FALSE represent the literals true and false (note that we could have just as well defined a single BOOLLITERAL token that the scanner would populate with either true or false).

Let's define a syntax-directed translation that type checks these expressions; i.e., for type-correct expressions, the translation will be the type of the expression (either int or bool), and for expressions that involve type errors, the translation will be the special value error. We'll use the following type rules:

Both operands of the + operator must be of type int.
Both operands of the && operator must be of type bool.
Both operands of the == operator must have the same (non-error) type.

Here is the CFG and the translation rules:

CFG Production			Translation rules
exp	$\longrightarrow$	exp `PLUS` term	if (exp₂.trans == int and (term.trans == int) then exp₁.trans = int else exp₁.trans = error
exp	$\longrightarrow$	exp `AND` term	if (exp₂.trans == bool and (term.trans == bool) then exp₁.trans = bool else exp₁.trans = error
exp	$\longrightarrow$	exp `EQUALS` term	if (exp₂.trans == term.trans) and (term.trans $\neq$ error) then exp₁.trans = bool else exp₁.trans = error
exp	$\longrightarrow$	term	exp.trans = term.trans
term	$\longrightarrow$	`TRUE`	term.trans = bool
term	$\longrightarrow$	`FALSE`	term.trans = bool
term	$\longrightarrow$	`INTLITERAL`	term.trans = int
term	$\longrightarrow$	`LPARENS exp RPARENS`	term.trans = exp.trans

Here's an annotated parse tree for the input (2 + 2) == 4

TEST YOURSELF #1

The following grammar defines the language of base-2 numbers:

B -> 0
  -> 1
  -> B 0
  -> B 1

Define a syntax-directed translation so that the translation of a binary number is its base 10 value. Illustrate your translation scheme by drawing the parse tree for 1001 and annotating each nonterminal in the tree with its translation.

solution

Building an Abstract-Syntax Tree

So far, our example syntax-directed translations have produced simple values (an int or a type) as the translation of an input. In practice however, we want the parser to build an abstract-syntax tree as the translation of an input program. But that is not really so different from what we've seen so far; we just need to use tree-building operations in the translation rules instead of, e.g., arithmetic operations.

The AST vs the Parse Tree

First, let's consider how an abstract-syntax tree (AST) differs from a parse tree. An AST can be thought of as a condensed form of the parse tree:

Operators appear at internal nodes instead of at leaves.
"Chains" of single productions are collapsed.
Lists are "flattened".
Syntactic details (e.g., parentheses, commas, semi-colons) are omitted.

In general, the AST is a better structure for later stages of the compiler because it omits details having to do with the source language, and just contains information about the essential structure of the program.

Below is an example of the parse tree and the AST for the expression 3 * (4 + 2) (using the usual arithmetic-expression grammar that reflects the precedences and associativities of the operators). Note that the parentheses are not needed in the AST because the structure of the AST defines how the subexpressions are grouped.

For constructs other than expressions, the compiler writer has some choices when defining the AST -- but remember that lists (e.g., lists of declarations lists of statements, lists of parameters) should be flattened, that operators (e.g., "assign", "while", "if") go at internal nodes, not at leaves, and that syntactic details are omitted.

Input
=====

{               
   x = 0;        
   while (x<10) { 
      x = x+1;     
   }      
   y = x*2;
}

Note that in the AST there is just one stmtList node, with a list of three children (the list of statements has been "flattened"). Also, the "operators" for the statements (assign and while) have been "moved up" to internal nodes (instead of appearing as tokens at the leaves). And finally, syntactic details (curly braces and semi-colons) have been omitted.

Translation Rules That Build an AST

To define a syntax-directed translation so that the translation of an input is the corresponding AST, we first need operations that create AST nodes. Let's use java code, and assume that we have the following class hierarchy:

class ExpNode { }

class IntLitNode extends ExpNode {
    public IntLitNode(int val) {...}
}

class PlusNode extends ExpNode {
    public PlusNode( ExpNode e1, ExpNode e2 ) { ... }
}

class TimesNode extends ExpNode {
    public TimesNode( ExpNode e1, ExpNode e2 ) { ... }
}

Now we can define a syntax-directed translation for simple arithmetic expressions, so that the translation of an expression is its AST:

CFG Production			Translation rules
exp	$\longrightarrow$	exp `PLUS` term	exp₁.trans = new PlusNode(exp₂, term.trans)
exp	$\longrightarrow$	term	exp.trans = term.trans
term	$\longrightarrow$	term `TIMES` factor	term₁.trans = new TimesNode(term₂.trans, factor.trans)
term	$\longrightarrow$	factor	term.trans = factor.trans
factor	$\longrightarrow$	`INTLITERAL`	factor.trans = new IntLitNode(`INTLITERAL`.value)
factor	$\longrightarrow$	`LPARENS` exp `RPARENS`	factor.trans = exp.trans

TEST YOURSELF #2

Illustrate the syntax-directed translation defined above by drawing the parse tree for the expression 2 + 3 * 4, and annotating the parse tree with its translation (i.e., each nonterminal in the tree will have a pointer to the AST node that is the root of the subtree of the AST that is the nonterminal's translation).

solution