Infix to Postfix converter

Infix to Postfix Converter: A Practical Guide and Calculator

An infix to postfix converter transforms mathematical and logical expressions written in infix notation (operators between operands, e.g., A + B) into postfix notation (Reverse Polish Notation, RPN, e.g., AB+). This conversion enables simple stack-based evaluation without parentheses, making it ideal for calculators, compilers, and expression evaluators. In this article you will learn the infix to postfix algorithm, step-by-step conversion examples with numbers, postfix evaluation, the Shunting Yard approach, and real-world applications.

Infix to Postfix Converter Formula

Core variables and definitions

– Expression tokens: sequence T = t1, t2, …, tn where each token ti is an operand, operator, left parenthesis “(” or right parenthesis “)”. – S: stack used for operators and parentheses. – O: output list (postfix sequence). – prec(op): precedence value of operator op (higher number = higher precedence). – assoc(op): associativity of operator op, either ‘left’ or ‘right’.

All variables are defined so formulas below are unambiguous.

Conversion rules (the formula)

Given tokens t1..tn, process each token using the following rules:

• If ti is an operand: O ← O + ti.
• If ti is a left parenthesis “(” : push “(” onto S.
• If ti is a right parenthesis “)” : pop operators from S and append to O until “(” is popped from S (do not append “(“).
• If ti is an operator o1:
• After all tokens processed: pop all remaining operators from S to O.

– While S is not empty and top(S) is an operator o2 and: – ((assoc(o1) = left) AND (prec(o1) ≤ prec(o2))) OR – ((assoc(o1) = right) AND (prec(o1) < prec(o2))), then pop top(S) and append to O. – Push o1 onto S. This concise set of rules is the working formula for any infix to postfix converter implementation.

Operator precedence and associativity table

| Operator | Symbol(s) | Precedence (prec) | Associativity (assoc) | |———-|———–|——————-:|———————-:| | Exponentiation | ^ | 4 | right | | Multiplication/Division | * , / | 3 | left | | Addition/Subtraction | + , – | 2 | left | | Unary operators | unary -, unary + | 5 (or handled separately) | right |

Use this table to evaluate comparisons in the formula above. Precedence values are integers; higher means evaluated earlier.

How to calculate Conversion of expressions from infix to postfix

The Shunting Yard vs. basic stack-based algorithm

The Shunting Yard algorithm, created by Dijkstra, is a specific implementation of the conversion rules above. It uses a stack (S) for operators and a queue or list (O) for output. The Shunting Yard formalizes handling of functions, separators (commas), and unary operators, while the basic stack-based variant implements the simple operator/parentheses rules.

Both methods share the same core idea: use a stack to manage operators and output operands immediately.

Step-by-step conversion example (symbolic)

Convert: (A+B)*C – D/E

• Tokens: (, A, +, B, ), *, C, -, D, /, E
• Start with S = empty, O = empty.
• Process each token with the rules:

Step-by-step states (short form S=stack, O=output):

• ‘(‘ → S: ( ; O:
• ‘A’ → S: ( ; O: A
• ‘+’ → S: ( + ; O: A
• ‘B’ → S: ( + ; O: A B
• ‘)’ → pop until ‘(‘ → pop + to O then pop ‘(‘ → S: empty ; O: A B +
• ‘‘ → S: ; O: A B +
• ‘C’ → S: * ; O: A B + C
• ‘-‘ → compare ‘-‘ with top ‘‘ : prec(-)=2 ≤ prec()=3 and ‘-‘ left assoc → pop ‘‘ to O. Push ‘-‘ → S: – ; O: A B + C
• ‘D’ → S: – ; O: A B + C * D
• ‘/’ → compare ‘/’ with top ‘-‘ : prec(/)=3 > prec(-)=2 → push ‘/’ → S: – / ; O: A B + C * D
• ‘E’ → S: – / ; O: A B + C * D E
• End: pop all → pop ‘/’ then ‘-‘ → O: A B + C * D E / –

Resulting postfix: AB+C*DE/-

This is the expected result and matches the example often used for teaching.

Step-by-step conversion example (numeric)

Convert: (3+4)*5 – 6/2

• Tokens: (, 3, +, 4, ), *, 5, -, 6, /, 2
• Apply algorithm like above:

Detailed steps (numbered):

• ‘(‘ → S: ( ; O:
• 3 → O: 3
• ‘+’ → S: ( +
• 4 → O: 3 4
• ‘)’ → pop ‘+’ → O: 3 4 + ; S: empty
• ‘‘ → S: ; O: 3 4 +
• 5 → O: 3 4 + 5
• ‘-‘ → pop ‘‘ → O: 3 4 + 5 ; push ‘-‘ → S: –
• 6 → O: 3 4 + 5 * 6
• ‘/’ → push ‘/’ because prec(/) > prec(-) → S: – /
• 2 → O: 3 4 + 5 * 6 2
• End → pop ‘/’, then ‘-‘ → O: 3 4 + 5 * 6 2 / –

Final postfix: 34+5*62/-

This numeric postfix expression will be used in the evaluation section.

Evaluation of postfix expressions and stack techniques

Postfix evaluation formula

To compute a numeric result from a postfix expression:

• Let E be an evaluation stack (initially empty).
• For each token t in postfix sequence:
• After all tokens processed, the final result is result = pop(E). If E is not empty or underflows, the expression was invalid.

– If t is an operand (numeric value): push t onto E. – If t is a binary operator op: pop b ← pop(E), a ← pop(E), compute r = apply(op, a, b), push r back on E. – If t is unary operator: pop a ← pop(E), r = apply(op, a), push r. Define variables: – E: evaluation stack – a, b: operand values popped from E – op: operator symbol – r: result of apply(op, a, b)

Example numeric evaluation (from earlier)

Postfix: 34+5*62/-

Step-by-step evaluation (stack E shown after each action):

• Read 3 → E: [3]
• Read 4 → E: [3, 4]
• Read ‘+’ → pop 4, 3 → compute 3 + 4 = 7 → push 7 → E: [7]
• Read 5 → E: [7, 5]
• Read ‘‘ → pop 5, 7 → compute 7 5 = 35 → push 35 → E: [35]
• Read 6 → E: [35, 6]
• Read 2 → E: [35, 6, 2]
• Read ‘/’ → pop 2, 6 → compute 6 / 2 = 3 → push 3 → E: [35, 3]
• Read ‘-‘ → pop 3, 35 → compute 35 – 3 = 32 → push 32 → E: [32]

Result = 32.

Tabular summary of stack operations for evaluation

| Step | Token | Operation | Stack after operation | |——|——-|———–|————————| | 1 | 3 | push 3 | [3] | | 2 | 4 | push 4 | [3,4] | | 3 | + | pop 4,3; push 7 | [7] | | 4 | 5 | push 5 | [7,5] | | 5 | * | pop 5,7; push 35 | [35] | | 6 | 6 | push 6 | [35,6] | | 7 | 2 | push 2 | [35,6,2] | | 8 | / | pop 2,6; push 3 | [35,3] | | 9 | – | pop 3,35; push 32 | [32] |

This method is straightforward and efficient: evaluation is linear in the number of tokens.

Shunting Yard algorithm/stack techniques used for parsing and conversion

Shunting Yard algorithm overview

The Shunting Yard algorithm is a reliable infix to postfix algorithm that handles: – Operators with different precedence and associativity. – Parentheses for grouping. – Function calls and argument separators. – Optional handling for unary operators and identifiers.

The algorithm operates in O(n) time where n is the number of tokens, because each token is pushed or popped at most a constant number of times.

Pseudocode (compact)

• For each token t in input:
• After reading input, pop remaining operators to output.

– If t is operand → output it. – If t is function → push to S. – If t is operator o1 → while top of S is operator o2 and ((assoc(o1) = left and prec(o1) ≤ prec(o2)) or (assoc(o1) = right and prec(o1) < prec(o2))) pop o2 to output. Push o1. – If t is “(” → push to S. – If t is “)” → pop operators from S to output until “(” popped. This concise pseudocode is how most compilers implement parsing for expressions when converting to postfix (or abstract syntax trees).

Handling special cases

– Unary operators: treat unary minus as a different operator (e.g., u-) with higher precedence and right associativity. – Functions: push function token, pop when encountering closing parenthesis, then push function into output. – Errors: mismatched parentheses or insufficient operands during evaluation indicate invalid input.

Practical applications and benefits

Real-world applications

– Compilers and interpreters: convert infix expressions into postfix or ASTs for efficient code generation. – Embedded systems: postfix evaluation avoids recursion and complex parsing, ideal for constrained devices. – Scientific and RPN calculators: many calculators (e.g., HP series) use RPN internally for quick evaluation. – Expression evaluators in spreadsheets and databases: use conversion to avoid repeated parsing and to cache compiled forms. – Math and symbolic manipulation systems: convert between representations for algebraic simplification and code generation.

Features and benefits (bullet list)

– Fast, linear-time conversion using stack techniques. – Eliminates parentheses for evaluation, simplifying runtime logic. – Unambiguous order of operations using precedence and associativity. – Easy to implement and debug in many languages. – Suited for streaming tokenized input (no need to store full parse tree).

Implementation tips and pitfalls

Implementation tips

– Tokenize input carefully: distinguish multi-digit numbers, decimals, variable names, and functions. – Handle whitespace and separators (commas) consistently. – Implement precise precedence and associativity rules; document any non-standard behavior. – Treat unary plus/minus as special operators to avoid ambiguity. – Validate parentheses and operator counts to catch malformed expressions early.

Common pitfalls

– Treating ‘-‘ as always binary leads to errors with expressions like -3*2. – Using incorrect precedence table (e.g., swapping * and +) will produce wrong results. – Forgetting to flush the stack at the end will yield incomplete postfix output. – Using floating-point division vs integer division inconsistently during evaluation.

Examples and test cases

Symbolic example summary

| Infix Expression | Postfix Result | |——————|—————-| | (A+B)C – D/E | AB+CDE/- | | A^B^C (right assoc) | ABC^^ (or AB C ^ ^) | | A + B C | ABC+ |

Numeric example summary

| Infix | Postfix | Evaluation | |——-|———|———–| | (3+4)5 – 6/2 | 34+562/- | 32 | | 2^3^2 | 2 3 2 ^ ^ (232^^) | 512 (because 3^2=9, 2^9=512) | | 10 + 2 6 | 10 2 6 + (1026*+) | 22 |

These test cases are useful for validating your infix to postfix converter implementation.

FAQ

What does “Infix to Postfix Converter” mean?

An infix to postfix converter is a tool or algorithm that transforms expressions written in standard infix notation (operators between operands) into postfix notation (operators after operands). This conversion simplifies evaluation because it removes the need for parentheses and respects operator precedence explicitly via ordering.