A grammar that is generated to define the mathematical operators is called **operator grammar** with some restrictions on grammar. An **operator precedence grammar** is a context-free grammar that has the property that no production has either an empty right-hand side (null productions) or two adjacent non-terminals in its right-hand side.

**Examples –**

This is the example of operator grammar:

E->E+E/E*E/id

However, grammar that is given below is not an operator grammar because two non-terminals are adjacent to each other:

S->SAS/a A->bSb/b

Although, we can convert it into an operator grammar:

S->SbSbS/SbS/a A->bSb/b

**Operator precedence parser –**

An operator precedence parser is a one of the bottom-up parser that interprets an operator-precedence grammar. This parser is only used for operator grammars. *Ambiguous grammars are not allowed* in case of any parser except operator precedence parser.

There are two methods for determining what precedence relations should hold between a pair of terminals:

- Use the conventional associativity and precedence of operator.
- The second method of selecting operator-precedence relations is first to construct an unambiguous grammar for the language, a grammar that reflects the correct associativity and precedence in its parse trees.

This parser relies on the following three precedence relations: **⋖, ≐, ⋗**

**a ⋖ b ** This means a “yields precedence to” b.

**a ⋗ b ** This means a “takes precedence over” b.

**a ≐ b ** This means a “has precedence as” b.

**Figure –** Operator precedence relation table for grammar E->E+E/E*E/id

There is not given any relation between id and id as id will not be compared and two variables can not come side by side. There is also a disadvantage of this table as if we have n operators than size of table will be n*n and complexity will be 0(n^{2}). In order to increase the size of table, use **operator function table**.

The operator precedence parsers usually do not store the precedence table with the relations; rather they are implemented in a special way. Operator precedence parsers use **precedence functions** that map terminal symbols to integers, and so the precedence relations between the symbols are implemented by numerical comparison. The parsing table can be encoded by two precedence functions **f** and **g** that map terminal symbols to integers. We select f and g such that:

- f(a) < g(b) whenever a is precedence to b
- f(a) = g(b) whenever a and b having precedence
- f(a) > g(b) whenever a takes precedence over b

**Example –** Consider the following grammar:

E -> E + E/E * E/( E )/id

The directed graph representing the precedence function:

Since there is not any cycle in the graph so we can make function table:

fid -> g* -> f+ ->g+ -> f$ gid -> f* -> g* ->f+ -> g+ ->f$

Size of the table is **2n**.

One disadvantage of function table is that evev though we have blank entries in relation we got non-blank entries in function table. Blank entries are also called error. Hence error detection capability of relational table is greater than function table.

`#include <stdio.h> ` `#include <string.h> ` ` ` `// function f to exit from the loop ` `// if given condition is not true ` `void` `f() ` `{ ` ` ` `printf` `(` `"Not operator grammar"` `); ` ` ` `exit` `(0); ` `} ` ` ` `void` `main() ` `{ ` ` ` `char` `grm[20][20], c; ` ` ` ` ` `// Here using flag variable, ` ` ` `// considering grammar is not operator grammar ` ` ` `int` `i, n, j = 2, flag = 0; ` ` ` ` ` `// taking number of productions from user ` ` ` `scanf` `(` `"%d"` `, &n); ` ` ` `for` `(i = 0; i < n; i++) ` ` ` `scanf` `(` `"%s"` `, grm[i]); ` ` ` ` ` `for` `(i = 0; i < n; i++) { ` ` ` `c = grm[i][2]; ` ` ` ` ` `while` `(c != ` `''` `) { ` ` ` ` ` `if` `(grm[i][3] == ` `'+'` `|| grm[i][3] == ` `'-'` ` ` `|| grm[i][3] == ` `'*'` `|| grm[i][3] == ` `'/'` `) ` ` ` ` ` `flag = 1; ` ` ` ` ` `else` `{ ` ` ` ` ` `flag = 0; ` ` ` `f(); ` ` ` `} ` ` ` ` ` `if` `(c == ` `'$'` `) { ` ` ` `flag = 0; ` ` ` `f(); ` ` ` `} ` ` ` ` ` `c = grm[i][++j]; ` ` ` `} ` ` ` `} ` ` ` ` ` `if` `(flag == 1) ` ` ` `printf` `(` `"Operator grammar"` `); ` `} ` |

Input :3 A=A*A B=AA A=$ Output : Not operator grammar Input :2 A=A/A B=A+A Output : Operator grammar

$ is a null production here which are also not allowed in operator grammars.

**Advantages –**

- It can easily be constructed by hand
- It is simple to implement this type of parsing

**Disadvantages –**

- It is hard to handle tokens like the minus sign (-), which has two different precedence (depending on whether it is unary or binary)
- It is applicable only to small class of grammars

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