Slide 12.4: Syntax of the lambda calculus (cont.)

Slide 12.3: Syntax of the lambda calculus
Slide 12.5: Syntax of the lambda calculus (cont.)
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Syntax of the Lambda Calculus (Cont.)

Some examples of lambda expressions are

The lambda expression λx.x denotes the identity function in the sense that ((λx.x) E) = E for any lambda expression E.

The expression λn.(add n 1) denotes the successor function on the integers so that ((λn.(add n 1)) 5) = 6.

The abstraction (λf.(λx.(f (f x)))) describes a function with two arguments, a function and a value, that applies the function to the value twice. If sqr is the (predefined) integer function that squares its argument, then
```
     ( ( ( λf . ( λx . ( f ( f x ) ) ) ) sqr ) 3 )
   = ( ( λx . ( sqr ( sqr x ) ) ) 3 )
   = ( sqr ( sqr 3 ) )
   = ( sqr 9 )
   = 81
```
Here f is replaced by sqr and then x by 3.

These examples show that the number of parentheses in lambda expressions can become quite large. The following notational conventions allow abbreviations that reduce the number of parentheses:

Uppercase letters and identifiers beginning with capital letters will be used as metavariables ranging over lambda expressions.

Function application associates to the left.
```
     E₁ E₂ E₃  means  ( ( E₁ E₂ ) E₃ )
```
The scope of “λ<variable>“ in an abstraction extends as far to the right as possible.
```
   λx.E₁ E₂ E₃ means (λx.(E₁ E₂ E₃)) and not ((λx.E₁ E₂) E₃)
```