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Backus-Naur Form
Recursive Descent Parsing
Building an Expression Tree

Section 9.5

A Simple Recursive Descent Parser

I have always been fascinated by language -- both natural languages like English and the artificial languages that are used by computers. There are many difficult questions about how languages can convey information, how they are structured, and how they can be processed. Natural and artificial languages are similar enough that the study of programming languages, which are pretty well understood, can give some insight into the much more complex and difficult natural languages. And programming languages raise more than enough interesting issues to make them worth studying in their own right. How can it be, after all, that computers can be made to "understand" even the relatively simple languages that are used to write programs? Computers, after all, can only directly use instructions expressed in very simple machine language. Higher level languages must be translated into machine language. But the translation is done by a compiler, which is just a program. How could such a translation program be written?

9.5.1  Backus-Naur Form

Natural and artificial languages are similar in that they have a structure known as grammar or syntax. Syntax can be expressed by a set of rules that describe what it means to be a legal sentence or program. For programming languages, syntax rules are often expressed in BNF (Backus-Naur Form), a system that was developed by computer scientists John Backus and Peter Naur in the late 1950s. Interestingly, an equivalent system was developed independently at about the same time by linguist Noam Chomsky to describe the grammar of natural language. BNF cannot express all possible syntax rules. For example, it can't express the fact that a variable must be defined before it is used. Furthermore, it says nothing about the meaning or semantics of the langauge. The problem of specifying the semantics of a language -- even of an artificial programming langauge -- is one that is still far from being completely solved. However, BNF does express the basic structure of the language, and it plays a central role in the design of translation programs.

In English, terms such as "noun", "transitive verb," and "prepositional phrase" are syntactic categories that describe building blocks of sentences. Similarly, "statement", "number," and "while loop" are syntactic categories that describe building blocks of Java programs. In BNF, a syntactic category is written as a word enclosed between "<" and ">". For example: <noun>, <verb-phrase>, or <while-loop>. A rule in BNF specifies the structure of an item in a given syntactic category, in terms of other syntactic categories and/or basic symbols of the language. For example, one BNF rule for the English language might be

<sentence>  ::=  <noun-phrase> <verb-phrase>

The symbol "::=" is read "can be", so this rule says that a <sentence> can be a <noun-phrase> followed by a <verb-phrase>. (The term is "can be" rather than "is" because there might be other rules that specify other possible forms for a sentence.) This rule can be thought of as a recipe for a sentence: If you want to make a sentence, make a noun-phrase and follow it by a verb-phrase. Noun-phrase and verb-phrase must, in turn, be defined by other BNF rules.

In BNF, a choice between alternatives is represented by the symbol "|", which is read "or". For example, the rule

<verb-phrase>  ::=  <intransitive-verb>  |
                    ( <transitive-verb> <noun-phrase> )

says that a <verb-phrase> can be an <intransitive-verb>, or a <transitive-verb> followed by a <noun-phrase>. Note also that parentheses can be used for grouping. To express the fact that an item is optional, it can be enclosed between "[" and "]". An optional item that can be repeated one or more times is enclosed between "[" and "]...". And a symbol that is an actual part of the language that is being described is enclosed in quotes. For example,

<noun-phrase>  ::=  <common-noun> [ "that" <verb-phrase> ]  |
                    <common-noun> [ <prepositional-phrase> ]...

says that a <noun-phrase> can be a <common-noun>, optionally followed by the literal word "that" and a <verb-phrase>, or it can be a <common-noun> followed by zero or more <prepositional-phrase>'s. Obviously, we can describe very complex structures in this way. The real power comes from the fact that BNF rules can be recursive. In fact, the two preceding rules, taken together, are recursive. A <noun-phrase> is defined partly in terms of <verb-phrase>, while <verb-phrase> is defined partly in terms of <noun-phrase>. For example, a <noun-phrase> might be "the rat that ate the cheese", since "ate the cheese" is a <verb-phrase>. But then we can, recursively, make the more complex <noun-phrase> "the cat that caught the rat that ate the cheese" out of the <common-noun> "the cat", the word "that" and the <verb-phrase> "caught the rat that ate the cheese". Building from there, we can make the <noun-phrase> "the dog that chased the cat that caught the rat that ate the cheese". The recursive structure of language is one of the most fundamental properties of language, and the ability of BNF to express this recursive structure is what makes it so useful.

BNF can be used to describe the syntax of a programming language such as Java in a formal and precise way. For example, a <while-loop> can be defined as

<while-loop>  ::=  "while" "(" <condition> ")" <statement>

This says that a <while-loop> consists of the word "while", followed by a left parenthesis, followed by a <condition>, followed by a right parenthesis, followed by a <statement>. Of course, it still remains to define what is meant by a condition and by a statement. Since a statement can be, among other things, a while loop, we can already see the recursive structure of the Java language. The exact specification of an if statement, which is hard to express clearly in words, can be given as

<if-statement>  ::=  
             "if" "(" <condition> ")" <statement>
             [ "else" "if" "(" <condition> ")" <statement> ]...
             [ "else" <statement> ]

This rule makes it clear that the "else" part is optional and that there can be, optionally, one or more "else if" parts.

9.5.2  Recursive Descent Parsing

In the rest of this section, I will show how a BNF grammar for a language can be used as a guide for constructing a parser. A parser is a program that determines the grammatical structure of a phrase in the language. This is the first step to determining the meaning of the phrase -- which for a programming language means translating it into machine language. Although we will look at only a simple example, I hope it will be enough to convince you that compilers can in fact be written and understood by mortals and to give you some idea of how that can be done.

The parsing method that we will use is called recursive descent parsing. It is not the only possible parsing method, or the most efficient, but it is the one most suited for writing compilers by hand (rather than with the help of so called "parser generator" programs). In a recursive descent parser, every rule of the BNF grammar is the model for a subroutine. Not every BNF grammar is suitable for recursive descent parsing. The grammar must satisfy a certain property. Essentially, while parsing a phrase, it must be possible to tell what syntactic category is coming up next just by looking at the next item in the input. Many grammars are designed with this property in mind.

I should also mention that many variations of BNF are in use. The one that I've described here is one that is well-suited for recursive descent parsing.

When we try to parse a phrase that contains a syntax error, we need some way to respond to the error. A convenient way of doing this is to throw an exception. I'll use an exception class called ParseError, defined as follows:

 * An object of type ParseError represents a syntax error found in 
 * the user's input.
private static class ParseError extends Exception {
   ParseError(String message) {
} // end nested class ParseError

Another general point is that our BNF rules don't say anything about spaces between items, but in reality we want to be able to insert spaces between items at will. To allow for this, I'll always call the routine TextIO.skipBlanks() before trying to look ahead to see what's coming up next in input. TextIO.skipBlanks() skips past any whitespace, such as spaces and tabs, in the input, and stops when the next character in the input is either a non-blank character or the end-of-line character.

Let's start with a very simple example. A "fully parenthesized expression" can be specified in BNF by the rules

<expression>  ::=  <number>  |
                   "(" <expression> <operator> <expression> ")"
<operator>  ::=  "+" | "-" | "*" | "/"

where <number> refers to any non-negative real number. An example of a fully parenthesized expression is "(((34-17)*8)+(2*7))". Since every operator corresponds to a pair of parentheses, there is no ambiguity about the order in which the operators are to be applied. Suppose we want a program that will read and evaluate such expressions. We'll read the expressions from standard input, using TextIO. To apply recursive descent parsing, we need a subroutine for each rule in the grammar. Corresponding to the rule for <operator>, we get a subroutine that reads an operator. The operator can be a choice of any of four things. Any other input will be an error.

 * If the next character in input is one of the legal operators,
 * read it and return it.  Otherwise, throw a ParseError.
static char getOperator() throws ParseError {
   char op = TextIO.peek(); 
   if ( op == '+' || op == '-' || op == '*' || op == '/' ) {
      return op;
   else if (op == '\n')
      throw new ParseError("Missing operator at end of line.");
      throw new ParseError("Missing operator.  Found \"" +
            op + "\" instead of +, -, *, or /.");
} // end getOperator()

I've tried to give a reasonable error message, depending on whether the next character is an end-of-line or something else. I use TextIO.peek() to look ahead at the next character before I read it, and I call TextIO.skipBlanks() before testing TextIO.peek() in order to ignore any blanks that separate items. I will follow this same pattern in every case.

When we come to the subroutine for <expression>, things are a little more interesting. The rule says that an expression can be either a number or an expression enclosed in parentheses. We can tell which it is by looking ahead at the next character. If the character is a digit, we have to read a number. If the character is a "(", we have to read the "(", followed by an expression, followed by an operator, followed by another expression, followed by a ")". If the next character is anything else, there is an error. Note that we need recursion to read the nested expressions. The routine doesn't just read the expression. It also computes and returns its value. This requires semantical information that is not specified in the BNF rule.

 * Read an expression from the current line of input and return its value.
 * @throws ParseError if the input contains a syntax error
private static double expressionValue() throws ParseError {
   if ( Character.isDigit(TextIO.peek()) ) {
          // The next item in input is a number, so the expression
          // must consist of just that number.  Read and return
          // the number.
      return TextIO.getDouble();
   else if ( TextIO.peek() == '(' ) {
          // The expression must be of the form 
          //         "(" <expression> <operator> <expression> ")"
          // Read all these items, perform the operation, and
          // return the result.
      TextIO.getAnyChar();  // Read the "("
      double leftVal = expressionValue();  // Read and evaluate first operand.
      char op = getOperator();             // Read the operator.
      double rightVal = expressionValue(); // Read and evaluate second operand.
      if ( TextIO.peek() != ')' ) {
             // According to the rule, there must be a ")" here.
             // Since it's missing, throw a ParseError.
         throw new ParseError("Missing right parenthesis.");
      TextIO.getAnyChar();  // Read the ")"
      switch (op) {   //  Apply the operator and return the result. 
      case '+':  return leftVal + rightVal;
      case '-':  return leftVal - rightVal;
      case '*':  return leftVal * rightVal;
      case '/':  return leftVal / rightVal;
      default:   return 0;  // Can't occur since op is one of the above.
                            // (But Java syntax requires a return value.)
   else {  // No other character can legally start an expression.
      throw new ParseError("Encountered unexpected character, \"" + 
            TextIO.peek() + "\" in input.");
} // end expressionValue()

I hope that you can see how this routine corresponds to the BNF rule. Where the rule uses "|" to give a choice between alternatives, there is an if statement in the routine to determine which choice to take. Where the rule contains a sequence of items, "(" <expression> <operator> <expression> ")", there is a sequence of statements in the subroutine to read each item in turn.

When expressionValue() is called to evaluate the expression (((34-17)*8)+(2*7)), it sees the "(" at the beginning of the input, so the else part of the if statement is executed. The "(" is read. Then the first recursive call to expressionValue() reads and evaluates the subexpression ((34-17)*8), the call to getOperator() reads the "+" operator, and the second recursive call to expressionValue() reads and evaluates the second subexpression (2*7). Finally, the ")" at the end of the expression is read. Of course, reading the first subexpression, ((34-17)*8), involves further recursive calls to the expressionValue() routine, but it's better not to think too deeply about that! Rely on the recursion to handle the details.

You'll find a complete program that uses these routines in the file SimpleParser1.java.

Fully parenthesized expressions aren't very natural for people to use. But with ordinary expressions, we have to worry about the question of operator precedence, which tells us, for example, that the "*" in the expression "5+3*7" is applied before the "+". The complex expression "3*6+8*(7+1)/4-24" should be seen as made up of three "terms", 3*6, 8*(7+1)/4, and 24, combined with "+" and "-" operators. A term, on the other hand, can be made up of several factors combined with "*" and "/" operators. For example, 8*(7+1)/4 contains the factors 8, (7+1) and 4. This example also shows that a factor can be either a number or an expression in parentheses. To complicate things a bit more, we allow for leading minus signs in expressions, as in "-(3+4)" or "-7". (Since a <number> is a positive number, this is the only way we can get negative numbers. It's done this way to avoid "3 * -7", for example.) This structure can be expressed by the BNF rules

<expression>  ::=  [ "-" ] <term> [ ( "+" | "-" ) <term> ]...
<term>  ::=  <factor> [ ( "*" | "/" ) <factor> ]...
<factor>  ::=  <number>  |  "(" <expression> ")"

The first rule uses the "[ ]..." notation, which says that the items that it encloses can occur zero, one, two, or more times. This means that an <expression> can begin, optionally, with a "-". Then there must be a <term> which can optionally be followed by one of the operators "+" or "-" and another <term>, optionally followed by another operator and <term>, and so on. In a subroutine that reads and evaluates expressions, this repetition is handled by a while loop. An if statement is used at the beginning of the loop to test whether a leading minus sign is present:

 * Read an expression from the current line of input and return its value.
 * @throws ParseError if the input contains a syntax error
private static double expressionValue() throws ParseError {
   boolean negative;  // True if there is a leading minus sign.
   negative = false;
   if (TextIO.peek() == '-') {
      TextIO.getAnyChar();  // Read the minus sign.
      negative = true;
   double val;  // Value of the expression.
   val = termValue();
   if (negative)
      val = -val;
   while ( TextIO.peek() == '+' || TextIO.peek() == '-' ) {
          // Read the next term and add it to or subtract it from
          // the value of previous terms in the expression.
      char op = TextIO.getAnyChar();  // Read the operator.
      double nextVal = termValue(); 
      if (op == '+')
         val += nextVal;
         val -= nextVal;
   return val;
} // end expressionValue()

The subroutine for <term> is very similar to this, and the subroutine for <factor> is similar to the example given above for fully parenthesized expressions. A complete program that reads and evaluates expressions based on the above BNF rules can be found in the file SimpleParser2.java.

9.5.3  Building an Expression Tree

Now, so far, we've only evaluated expressions. What does that have to do with translating programs into machine language? Well, instead of actually evaluating the expression, it would be almost as easy to generate the machine language instructions that are needed to evaluate the expression. If we are working with a "stack machine", these instructions would be stack operations such as "push a number" or "apply a + operation". The program SimpleParser3.java can both evaluate the expression and print a list of stack machine operations for evaluating the expression. Here is an applet that simulates the program:

It's quite a jump from this program to a recursive descent parser that can read a program written in Java and generate the equivalent machine language code -- but the conceptual leap is not huge.

The SimpleParser3 program doesn't actually generate the stack operations directly as it parses an expression. Instead, it builds an expression tree, as discussed in Section 9.4, to represent the expression. The expression tree is then used to find the value and to generate the stack operations. The tree is made up of nodes belonging to classes ConstNode and BinOpNode that are similar to those given in Section 9.4. Another class, UnaryMinusNode, has been introduced to represent the unary minus operation. I've added a method, printStackCommands(), to each class. This method is responsible for printing out the stack operations that are necessary to evaluate an expression. Here for example is the new BinOpNode class from SimpleParser3.java:

private static class BinOpNode extends ExpNode {
   char op;        // The operator.
   ExpNode left;   // The expression for its left operand.
   ExpNode right;  // The expression for its right operand.
   BinOpNode(char op, ExpNode left, ExpNode right) {
          // Construct a BinOpNode containing the specified data.
      assert op == '+' || op == '-' || op == '*' || op == '/';
      assert left != null && right != null;
      this.op = op;
      this.left = left;
      this.right = right;
   double value() {
          // The value is obtained by evaluating the left and right
          // operands and combining the values with the operator.
      double x = left.value();
      double y = right.value();
      switch (op) {
      case '+':  
         return x + y;
      case '-':  
         return x - y;
      case '*':  
         return x * y;
      case '/':  
         return x / y;
         return Double.NaN;  // Bad operator!
   void  printStackCommands() {
          // To evaluate the expression on a stack machine, first do
          // whatever is necessary to evaluate the left operand, leaving
          // the answer on the stack.  Then do the same thing for the
          // second operand.  Then apply the operator (which means popping
          // the operands, applying the operator, and pushing the result).
      TextIO.putln("  Operator " + op);

It's also interesting to look at the new parsing subroutines. Instead of computing a value, each subroutine builds an expression tree. For example, the subroutine corresponding to the rule for <expression> becomes

    static ExpNode expressionTree() throws ParseError {
           // Read an expression from the current line of input and
           // return an expression tree representing the expression.
       boolean negative;  // True if there is a leading minus sign.
       negative = false;
       if (TextIO.peek() == '-') {
          negative = true;
       ExpNode exp;   // The expression tree for the expression.
       exp = termTree();  // Start with a tree for first term.
       if (negative) {
              // Build the tree that corresponds to applying a
              // unary minus operator to the term we've
              // just read.
          exp = new UnaryMinusNode(exp);
       while ( TextIO.peek() == '+' || TextIO.peek() == '-' ) {
                // Read the next term and combine it with the
                // previous terms into a bigger expression tree.
           char op = TextIO.getAnyChar();
           ExpNode nextTerm = termTree();
                // Create a tree that applies the binary operator
                // to the previous tree and the term we just read.
           exp = new BinOpNode(op, exp, nextTerm);
       return exp;
    } // end expressionTree()

In some real compilers, the parser creates a tree to represent the program that is being parsed. This tree is called a parse tree. Parse trees are somewhat different in form from expression trees, but the purpose is the same. Once you have the tree, there are a number of things you can do with it. For one thing, it can be used to generate machine language code. But there are also techniques for examining the tree and detecting certain types of programming errors, such as an attempt to reference a local variable before it has been assigned a value. (The Java compiler, of course, will reject the program if it contains such an error.) It's also possible to manipulate the tree to optimize the program. In optimization, the tree is transformed to make the program more efficient before the code is generated.

And so we are back where we started in Chapter 1, looking at programming languages, compilers, and machine language. But looking at them, I hope, with a lot more understanding and a much wider perspective.

End of Chapter 9

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