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The vertical lines mean or. To simplify the discussion so that we need not also supply rules for creating identifiers and numbers, assume that identifiers are valid variable names and numbers are valid numbers. We will treat them as terminals. Production 1 says that an expression can consist either of a term, or of an expression plus an add_op (addition operator) plus a term. Production 2 says that a term can be a factor, or it can be another term plus a mult_op (multiplication operator) plus a factor. For example, we can parse the following expression according to the grammar: X * 3 + 4 We can, by rule 1, replace the original expression with another expression (X * 3), an add_op (+), and a term (4). By rule 2 the single-token term (4) can be replaced by a factor, which can, by rule 3 be replaced by a number (4), which is a terminal for us. It remains for us to parse the expression (X * 3). At this point, by rule 1 the only legal substitution for (X * 3) is a term. By rule 2 the term (X * 3) can be replaced by another term (X), a mult_op (*), and a factor (3). Again, rule 3 says the factor (3) can be replaced by a number (3), which is a terminal. By rule 2 the term (X) can be replaced by a factor (X), which by rule 3 can be replaced by an identifier (X), which we said was a terminal for us.

CHAP. 4]

The information shown in Table Name ISO Language Area Population 4.3 provides a good example. If Austria AT German 83,870 8,192,880 these data were stored in a map Poland PL Polish 312,685 38,536,869 data structure, then each row France FR French 547,030 60,876,136 would be stored in a separate keyGermany DE German 357,021 82,422,299 value pair. The keys would be the Italy IT Italian 301,230 58,133,509 strings in the ISO column. The Portugal PT Portuguese 92,391 10,605,870 value for each key would be the Sweden SE Swedish 449,964 9,016,596 rest of the data for the country that Table 4.3 A table of seven records is uniquely identified by that key. These data could be the fields of a Country object, using the Country class defined in Example 4.14. The Map interface is defined like this:

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Such decomposition of a more complex expression into its terminals according to the rules of the grammar is called a derivation The result of a successful derivation is a parse tree or syntax tree Here is the parse tree for the derivation we just completed: ( X * 3 + 4 ) expression / | \ / | \ (X * 3) + 4 expression add_op term / | \ 4 factor / | \ 4 number / | \ / | \ X * 3 term mult_op factor X factor 3 number X identifier To compute the meaning of the expression, the parse tree can be traversed from the bottom up, computing the multiplication first and then performing the addition If an expression can be parsed according to the grammar of the language, the expression conforms to the syntax of the language.

The harmonic mean of two numbers x and y is the number h defined by the formula h = 2/(1/x + 1/y). In pseudocode for Fraction types, this could be expressed as:

harmonicMean(x: Fraction, y: Fraction) returns Fraction return x.reciprocal().plus(y.reciprocal()).reciprocal().times(2);

Once the parser creates the parse tree, the compiler can work from the bottom of the tree to the top, creating the machine instructions to implement the expression This last phase is called code generation Today most descriptions of language syntax use a version (there are several) of EBNF Some notational changes simplify the representations of productions In particular, EBNF uses curly brackets to denote zero or more occurrences of, and it uses square brackets to denote optional parts of a production EBNF uses parentheses and vertical or separators to denote multiple-choice options for a single element.

JAVA INTERFACES In Java, an ADT can be represented by an interface. Recall that a Java interface is just like a Java class, except that it contains no executable code. EXAMPLE 1.3 A Fraction Interface

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We can rewrite the grammar above using this EBNF notation: expression -> term { (+ | -) term } term -> factor { (* | /) factor } factor -> identifier | number | - factor | ( expression ) If it is not obvious that these rules agree with our earlier grammar, consider our earlier first rule for expressions: expression -> term | expression add_op term From this rule, we can generate: expression expression expression expression .. expression -> -> -> -> term expression + term expression + term + term expression + term + term + term.

public interface Fraction { Fraction plus(Fraction x); Fraction times(int n); Fraction times(Fraction x); Fraction reciprocal(); double value(); }

If an ADT is translated into Java as an interface, then we can implement algorithms that use it as Java methods.

So, the EBNF notation says more simply: expression -> term { (+ | -) term }

CHAP. 1]

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