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2: A Recursive-Descent Expression Parser
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Assume this precedence for each operator: Highest
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Operators of equal precedence evaluate from left to right. The parser developed here will be subject to a few constraints. First, all variables are single letters (in other words, 26 variables, A through Z, are available). The variables are not case sensitive (a and A are treated as the same variable). Second, all numeric values are assumed to be double, although you could easily modify the parser to handle other types of values. Finally, to keep the logic clear and easy to understand, only rudimentary error checking is included.
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If you have not thought much about the problem of expression parsing, you might assume that it is a simple task, but it isn't. To better understand the problem, try to evaluate this sample expression: 10 2 * 3 You know that this expression is equal to the value 4. Although you could easily create a program that would compute that specific expression, the problem is how to create a program that gives the correct answer for any arbitrary expression. At first you might think of an algorithm something like this: a = get first operand while(operands present) { op = get operator b = get second operand a = a op b } This approach gets the first operand, the operator, and the second operand to perform the first operation, and then gets the next operator and operand to perform the next operation, and so on. However, if you try this basic approach, the expression 10 2 * 3 evaluates to 24 (that is, 8 * 3) instead of 4 because this procedure neglects the precedence of the operators. You cannot just take the operands and operators in order from left to right because the rules of algebra dictate that multiplication must be done before subtraction. Some beginners think that this problem can be easily overcome, and sometimes, in very restricted cases, it can. But the problem only gets worse when you add parentheses, exponentiation, variables, unary operators, and the like.
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Although there are a number of ways to write the code that processes an expression, the one developed here is the approach most easily written by a person. It is called a recursive-descent parser, and in the course of this chapter you will see how it got its name. (Some of the other methods used to write parsers employ complex tables that are usually generated by another computer program. These are sometimes called table-driven parsers.)
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There are a number of ways to parse and evaluate an expression. For use with a recursivedescent parser, think of expressions as recursive data structures that is, expressions that are defined in terms of themselves. If, for the moment, we assume that expressions can only use +, , *, /, and parentheses, all expressions can be defined with the following rules: expression term [+ term] [ term] term factor [* factor] [/ factor] factor variable, number, or (expression) The square brackets designate an optional element, and the means produces. In fact, the rules are usually called the production rules of the expression. Therefore, for the definition of term you could say: Term produces factor times factor or factor divided by factor. Notice that the precedence of the operators is implicit in the way an expression is defined. Here is an example. The expression 10 + 5 * B has two terms: 10 and 5 * B. The second term contains two factors: 5 and B. These factors consist of one number and one variable. On the other hand, the expression 14 * (7 C) has two factors: 14 and (7 C). The factors consist of one number and one parenthesized expression. The parenthesized expression contains two terms: one number and one variable. This process forms the basis for a recursive-descent parser, which is a set of mutually recursive methods that work in a chainlike fashion and implement the production rules. At each appropriate step, the parser performs the specified operations in the algebraically correct sequence. To see how the production rules are used to parse an expression, let s work through an example using the following expression: 9/3 (100 + 56) Here is the sequence that you will follow:
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