PLT: Lecture 3

1
PLT Lecture 3

• Mathematical Notation Continued

2
Last Lecture

• Outlined the need for compilation
• Outlined the requirements for compilation
• Sought a domain where they have experience with
  these requirements
• Started exploring this domain mathematical
  notation
• Its parts
• Their properties

3
Expression Trees

• Advantages
• Ease of reading
• Spaced out
• No need for decoding delimiters
• Spacing means space for annotations
• But they suffer from the word-processor
  1-dimension problem
• Often addressed by using images to portray
  expressions.
• This can lead to problems too

4
Maths as Images Problems

• Searching problems
• Re-use problems
• Display Problems
• Font size variations
• Resolution, emphasized when printed
• The halo effect (admittedly small)
• Horizontal alignment with text
• Bandwidth problems

5
1-D Notations

• Demarking first and last symbols in a
  sub-expression
• ( )
• lt gt
• Free to use which-ever you like in Mathematics
  but not so in computer languages (C for e.g.)
• Meanings of ( ) 4
• Meanings of 2
• Meanings of 5.5

6
Communication Issues

• In reading expressions in 1-D, identical
  delimiters can cause problems
• ( ((bb)-((4a)c)))/(2a)
• ( bb-lt4agtc)/(2a)
• See how easy it is when using stretchable
  operators
• (ab)/(cd)
• ( (ab))/(cd)
• ( ((ab)/(cd))
• ( ((ab)/(cd)) (d(d1))

7
Reducing the number of delimiter expressions

• The scoping issue
• Stretchable operators
• Piggy in the middle
• Prefix and postfix unary operator
• Binary infix operators
• 243
• What happens the 4
• If its the right operand of then 24 is the
  left operand of 18
• If its the left operand of then 43 is the
  right operant of 14
• More formally
• Priority
• Associativity

8
Priority

• If o1 has a greater priority than o2 then (for
  binary infix ops)
• a o1 b o2 c (a o1 b) o2 c
• a o2 b o1 c a o2 (b o1 c)
• This can only help so much in our effort to
  minimize delimiters

9
Extending Priority for unary operators

• Scenario
• Unary prefix operator / unary postfix operator
• Unary prefix operator / binary infix operator
• Binary infix operator / unary prefix operator
• Latter 2 assume the unary operators have higher
  precedence than binary ones
• !altb (a1, b0)

10
Priorities / Associativity in C

• Associativity (a16, b4, c2)
• int a3
• int a-b-3
• (a5)(b8)(c2)
• a / b / c
• sizeof(sizeof(8))
• Priority (w1, x0, y6, z5)
• if(x!0)
• --j // illegal
• Data ptr
• ty / z w
• dataptry

11
Keywords in Programming Language Expressions

• Delimiters
• Algol 68 / Modula-2
• if..fi
• dood
• case.esac
• Operators
• ADD X TO Y GIVING Z

12
A Scheme for Generating Definitions of
Expressions

• Generic Rules
• Starter Rule
• Provide a starter set of expressions
• Complex Rules
• Build on the starter rules to provide a more
  complex set of expressions
• Uses 3 sets
• Literal symbols
• Variable symbols
• Punctuation symbols ( )
• Operator symbols with their arity, and fixity

13
Informal Structure

• Every literal symbol is an expression
• Every variable symbol is an expression
• Is E1 is an expression, then (E1) is an
  expression
• If o is a unary prefix operator and E1 is an
  expression then oE1 is an expression
• If o is a binary infix operator and E1, E2 are
  expressions then E1 o E2 is an expression
• If o is a unary postfix operator and E1 is an
  expression then E1o is an expression
• If o is a n-ary prefix operator and E1, E2En are
  expressions then o(E1, E2En) is an expression
• If o is a n-ary postfix operator and E1, E2En
  are expressions then (E1, E2En)o is an
  expression

14
mOn Notation

• Succinctly specifying arity and fixity
• 0O1
• 1O0
• 1O1
• 0On
• nO0
• Now we are in a better position to formalize the
  scheme
• We will do this in the next lecture