Contrasts

Comparing things – Semantics, Arithmetic expressions

Semantics
Semantics is the study of meaning. In language, it deals with how words, symbols, and sentences connect to concepts or truth. In computer science, semantics describes what a program or expression does rather than just how it looks (its syntax).

Example:

  • Syntax: 2 + 2 is a valid arrangement of symbols.
  • Semantics: its meaning is the number 4.

Reverse Polish Notation (RPN)
RPN, also called postfix notation, is a way of writing arithmetic expressions without parentheses. Instead of writing operators between numbers (infix), operators come after the operands.

Example:

  • Infix: (3 + 4) * 5
  • RPN: 3 4 + 5 *

RPN is especially useful for computers and calculators because it is evaluated step by step with no ambiguity.

Semantics of RPN (how RPN gains meaning)
The rules that define RPN’s meaning are:

  1. Read tokens left to right.
  2. Numbers: push onto the stack.
  3. Operators: pop the required number of operands from the stack, apply the operation, and push the result.
  4. The meaning of the whole expression is the final value left on the stack.
  5. If the stack empties too soon or has extra values at the end, the expression is semantically invalid.

Example: 5 1 2 + 4 * + 3 -

  • Push 5 → [5]
  • Push 1 → [5, 1]
  • Push 2 → [5, 1, 2]
  • + → pop 1 and 2 → 3 → [5, 3]
  • Push 4 → [5, 3, 4]
  • * → pop 3 and 4 → 12 → [5, 12]
  • + → pop 5 and 12 → 17 → [17]
  • Push 3 → [17, 3]
  • - → pop 17 and 3 → 14 → [14]
    Result = 14

Invalid example: 2 +

  • Push 2 → [2]
  • + requires two operands, but only one exists.
  • Expression is semantically invalid.

Similarities Between Semantics and RPN

  1. Purpose
  • Semantics: assigns meaning to words, symbols, or sentences.
  • RPN: assigns meaning to tokens (numbers and operators) through stack rules.
  1. Rule of interpretation
  • Semantics: words and grammar map to concepts or truth values.
  • RPN: numbers push, operators pop/compute/push.
  1. Order of elements
  • Semantics: word order changes meaning (man bites dog vs dog bites man).
  • RPN: token order changes result (2 3 + vs 3 2 +).
  1. Handling ambiguity
  • Semantics: resolves multiple meanings (bank as river or money).
  • RPN: eliminates ambiguity in arithmetic (no need for parentheses).
  1. Compositionality
  • Semantics: meaning of a sentence = meaning of parts + rules of combination.
  • RPN: value of an expression = combination of stack operations step by step.
  1. Final product
  • Semantics: a clear, unambiguous meaning of a statement.
  • RPN: a single, unambiguous numeric result on the stack.

In short:
Semantics and RPN both describe how symbols gain meaning through rules. Semantics is the broad theory of meaning in language and logic, while RPN is a specific example of a semantic system for arithmetic expressions. Both rely on order, structure, and unambiguous rules to ensure that symbols are understood the same way every time.