Natural languages

  • We use natural languages to communicate with each other.
    • Syntax and grammar rules determine valid word order.
    • Semantics determine what a group of words ‘means’.
  • Syntax can be fine, but the semantics can be bad.
    • The petrol ate the car
  • One statement could be ambiguous—have multiple meanings
    • Bath the dog wearing an apron.

Formal Languages

  • Formal languages are much more carefully defined than a natural language.

  • As a result, ambiguity is not possible.

  • Some formal languages include:

    • Chemical formulae
    • Maths
    • Programming languages

Reverse Polish Notation (RPN)

Infix Notation

  • When we look at mathematical expression the operands are placed on either side of the operator.
    • 3+4
    • The operand being the data we want to use and the operator being what we want to do with it
  • This is infix notation
  • The problem is that we often need brackets to alter the order of operations.
    • (5+7) x 3


  • Polish mathematician Jan Łukasiewicz devised a way to represent mathematical expressions so that we don’t need to use brackets

  • This uses postfix notation so the operator is shown after the items we need to operate on

    • 3 4 +
  • The result of an operation replaces the items that caused it to happen in the expression.

    • 5 7 + 3 x => 12 3 x => 36
  • Representing calculations in reverse polish makes it easier for machines to interpret.

  • In particular, calculations express in RPN take less steps to process on a computer.

  • To perform an RPN calculation, we use a stack.

RPN and the Stack

  • Here’s an infix notation:

    • 5 x ( 1 + 2)
  • Represented as RPN it looks like

    • 1 2 + 5 x
  • In operation

    • Push 1 and 2
    • Pop 2, 1 and add.
    • Push 3 (result of add)
    • Pop 3,5 and multiply.
    • Push 15 (result of multiply)
  • Stack now contains the result

RPN as a Tree

  • You can also represent RPN as a tree.

  • The tree can be reversed post order with each calculation replacing the operator.

Regular Expressions

  • A regular expression provides a way to define sequences of characters
  • The regular expression can then be used to pattern match
    • To find examples in a bigger body of characters
    • To determine if an input matches an allowed pattern
  • The regular expression defines the whole set