Operator precedence parsing - The Logic Tutorial

Operator precedence parsing is a technique used in computer science to analyze expressions and determine their order of evaluation. It's a specific type of bottom-up parsing, meaning it starts from the individual tokens (numbers, variables, operators) and builds its way up to a parse tree representing the entire expression.

Here's a breakdown of the key concepts:

1. Operator Precedence Grammar:

This is a special kind of grammar used to define the rules for expressions. It has two key properties:

• No empty string (ε) on the right-hand side (RHS) of any production rule. This ensures that the parser doesn't encounter empty elements while building the parse tree.
• No two non-terminals (variables representing parts of the expression) appearing consecutively on the RHS. This prevents ambiguity in interpreting the order of operations.

2. Precedence Table:

This table is the heart of operator precedence parsing. It defines the relative order of operations between different operators. The table typically has rows and columns corresponding to the operators, and the cell at each intersection indicates the relationship between the corresponding operators. Common relationships include:

• a > b: Operator a has higher precedence than b.
• a < b: Operator a has lower precedence than b.
• a = b: Operator a and b have the same precedence.

3. Parsing Algorithm:

The most common algorithm for operator precedence parsing is the Shunting-yard algorithm. It works by iterating through the expression's tokens and performing two main actions:

• Shift: If the current token is an operand (number or variable), it's pushed onto a stack.
• Reduce: If the current token is an operator, the parser checks the precedence table and the stack. Based on the relationships, it might:
  • Pop operands and the previous operator from the stack and combine them according to the operator's meaning, forming a sub-expression. This sub-expression is then pushed back onto the stack.
  • Simply push the operator onto the stack if it has lower precedence than the top element on the stack.

4. Applications:

Operator precedence parsing is widely used in various applications, including:

• Compilers and interpreters: These programs rely on parsing expressions to understand and evaluate code written in programming languages.
• Calculators: Most calculators use operator precedence to determine the order of operations when evaluating user input.
• Spreadsheet software: Similar to calculators, spreadsheets use precedence rules to calculate formulas entered by users.

In summary, operator precedence parsing is a powerful technique for analyzing expressions and ensuring they are evaluated correctly based on the defined order of operations.