Operator Precedence Parser

What is Operator Precedence Grammar?

A grammar is said to be an operator grammar if it follows these two properties:

1. There should be no ε (epsilon) on the right-hand side of any production.
2. There should be no two non-terminals adjacent to each other.

Examples of operator grammar are A + B, A - B, A x B, etc.

What is Operator Precedence Relations?

There are three operator precedence relations. These are-

1. a ⋗ b: This relation implies that terminal “a” has higher precedence than terminal “b”.
2. a ⋖ b: This relation implies that terminal “a” has lower precedence than terminal “b”.
3. a ≐ b: This relation implies that terminal “a” has equal precedence to terminal “b”.

Now, let’s see the steps to perform the operator precedence parsing.

What is an Operator Precedence Table?

A precedence table is used in operator precedence parsing to establish the relative precedence of operators and to resolve shift-reduce conflicts during the parsing process. The table instructs the parser when to shift (consume the input and proceed to the next token) and when to reduce (apply a production rule to reduce a set of tokens to a non-terminal symbol). It is an essential Data Structure for building a shift-reduce parser.

A precedence table is commonly expressed as a two-dimensional matrix, with rows and columns corresponding to grammatical operators. The table's elements define the order of precedence between the operators. 

Parsing Action

• Add the $ sign to both ends of the provided input text
   
• Now, scan the input string from left to right until you find the
   
• Scan to the leftover all the equal precedence until you reach the first leftmost
   
• Everything between the left and rightmost corners is a handle
   
• $ on $ indicates that parsing was successful

Examples

Example 1: 

Operator grammar includes operators with different levels of precedence and associativity. The grammar specifies the syntactic structure of expressions, which can be used to derive parse trees for expressions.

In this example, you will learn how to parse a string step by step using operator precedence parsing.

Consider the following grammar 

T → T+T/TxT/id

With the help of the above grammar, parse the input string “id+idxid”.

Solution:

Step 1: 

Step 2: The given grammar T → T+T/TxT/id is an operator grammar. So we don’t need to modify it.

Step 3: Operator Precedence Table

For drawing the operator precedence table, we will take the terminals present in the grammar on the left-hand side and right-hand side. 

• If the symbol on the left-hand side has higher precedence than the right-hand side symbol, insert the ⋗ symbol in the table.
• If the symbol on the left-hand side has lower precedence than the right-hand side symbol, insert the ⋖ symbol in the table.  
• If the symbol on the left-hand side has equal precedence to the right-hand side symbol, insert nothing in the case of terminals, ⋗ symbol in the operators, and A in the case of the $ symbol in the table.  

Thus, the operator precedence table for the given grammar will be-

Step 4: Parsing the input string.

We will use the operator precedence table to parse the given input string. We will compare the symbol at the top of the stack and the current input symbol. Based on the precedence relation, we will perform either the shift operation or the reduced operation. 

Thus, the parsing will be as follows.

The given string is parsed and accepted.

Step 5: Generating the parse tree.

We will start from the bottom of the stack and generate the parse tree.

Example 2: 

Operator grammar defines the precedence and associativity of operators, which is necessary for successfully parsing phrases. It is simple to read and understand, making it a common choice for constructing programming language syntax.

In this example, you'll see how efficiently we can parse a string using the operational parsing method.

Consider the following grammar: 

E → E+T/T

T → TxV/V

V → a/b/c/d

With the help of the above grammar, parse the input string “a+bxcxd”.

Solution:

Step 1: 

Step 2: The grammar E → E+T/T, T → TxV/V, V → a/b/c/d is an operator grammar. So we don’t need to modify it.

Step 3: Operator Precedence Table

The operator precedence table for the given grammar will be-

Step 4: Parsing the input string.

Since no production contains E on its right-hand side, this parsing will lead to an error.

Thus, the given input string cannot be parsed using the operator precedence operator by the given grammar. So, we cannot generate a parse tree.

Advantages of Operator Precedence Parsing

• Simplicity: Operator precedence parsing is straightforward to implement, making it accessible for simple grammar parsing.
• Efficiency: It offers efficient parsing for expressions involving operators, with minimal overhead.
• No Backtracking: This method does not require backtracking, leading to predictable parsing performance.
• Handle Operator Precedence: It naturally handles different levels of operator precedence and associativity rules.
• Deterministic: Provides a deterministic parsing method for a subset of grammars, ensuring consistent results.

Disadvantages of Operator Precedence Parsing

• Limited Applicability: Effective primarily for arithmetic and similar expressions; not suitable for more complex grammatical structures.
• Ambiguity Handling: Struggles with handling ambiguities in grammar without additional rules or modifications.
• Restrictive Grammars: Requires grammars to be free of left recursion and to have a unique precedence relationship between every pair of terminals.
• Error Recovery: Offers limited error recovery options, making it challenging to diagnose and correct syntax errors.
• Complexity with Non-Operators: Incorporating non-operator tokens or constructs increases complexity and may require additional parsing mechanisms.

Frequently Asked Questions

Which binary operators have the same precedence in parsing?

There are three operator precedence relations. The relation a < b means that a has lower precedence than b. Similarly, a > b means that a has higher precedence than b. And the relations a = b, here a "has equal precedence as" b. So, = have the same precedence in parsing.

What are the common errors in operator precedence parsing?

Character pair errors and reducibility errors are the two types of operator precedence parsing mistakes. A character pair mistake arises when there is no operator precedence relationship in the grammar between two symbols. A reducibility error happens when you are unable to lower the handle to the left side of a production.

How are handles found in operator precedence parsing?

Having precedence relations allows to identify handles as follows: - Scan the string from the left until seeing •> - Scan backward the string from right to left until seeing <• - Everything between the two relations <• and •> forms the handle. Note that not the entire sentential form is scanned to find the handle.

Which class of operator has the highest precedence?

The operator precedence is in charge of evaluating expressions. In Java, the parenthesis() and Array subscript[] take the highest precedence. For example, addition and subtraction take higher precedence over left and right shift operators.

Which operator has the lowest precedence?

The operator with the lowest precedence is the assignment operator ("="). This means that in an expression, assignment is evaluated last, ensuring that all other operations are completed before assigning the result to a variable. This helps in maintaining the correct order of operations in programming.

Conclusion

In this article, we have studied operator precedence in compiler design. We went through the concept thoroughly, discussing operator grammar, operator precedence relations, steps, and examples.

Recommended Reading:

• Lexical Analysis in Compiler Design
• Input Buffering in Compiler Design
• Top Down Parsing and Bottom Up Parsing
• Intermediate Code Generation in Compiler Design