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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. 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. 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
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-> term + term + term + term ...+ term
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So, the EBNF notation says more simply: expression -> term { (+ | -) term }
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An expression is a term followed by zero, one, or many additive terms. Here is an example of EBNF used to represent an optional element in a production: if-statement -> if( expression ) statement [else statement] This production says that an if-statement consists of the key word if, followed by an open parenthesis, followed by an expression, followed by a closed parenthesis, followed by a program statement, optionally followed by the key word else and another program statement.
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A very important requirement for a programming language grammar is that it be unambiguous. Given an expression in the language, there must be one and only one valid derivation in the language. To illustrate an ambiguous grammar, consider this simplification of the grammar for mathematical expressions: 1 expression -> expression operator expression | identifier | number | - expression | ( expression ) 2 operator -> + | - | * | / We can again parse the expression (X * 3 + 4) proceeding from the left to the right, and the result will be the same parse tree we derived from the more complex grammar. However, this simpler grammar would also allow a rightmost approach, with the following result: ( X * 3 + 4 ) expression / | \ / | \ X * (3 + 4) expression operator expression | \ X Identifier \ (3 + 4) expression operator expression / \ / | \ / | \ / | \ 3 number + 4 number The meaning of the second parsing is very different from the first, because in the rightmost parsing the addition occurs before the multiplication. That is not the customary hierarchy of operations, and the second parse tree will, in general, produce a different value for the expression than the first. Because the simpler grammar can produce two different and valid parse trees for the same expression, the grammar is ambiguous. Programming language grammars must be unambiguous. Look again at the first grammar, the more complex example, and notice how the grammar enforces a hierarchy of operations; multiplication and division occur before addition or subtraction. Correct grammars place higher precedence operations lower in the cascade of productions. Another key to a correctly specified grammar is the associativity of language elements. Does a mathematical operator associate left to right, or right to left This makes a difference with expressions like (9 - 4 - 2). Left associativity of operators yields 3, while right associativity yields 7. How do the grammar rules express associativity A production like this is left-associative: expression -> term | expression add_op term A production like this is right-associative: expression -> term | term add_op expression The significant difference is that the recursion (where an expression is part of an expression) is on the left in the first case, and on the right in the second case. Using the left-associative production to parse (9 - 4 - 2) results in this parse tree: ( 9 - 4 - 2 ) expression / | \ / | \ (9 - 4) 2 expression add_op term
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