lcalc - Lambda-Calculus Overview Susan Older Version...

Lambda-Calculus Overview
Susan Older
Version : February 21, 2007
Abstract
These notes provide a review of some of the terminology from lectures that do not explicitly
appear in the textbook but that I expect you to know for assignments and exam.
1 Syntactic Properties
Definition 1:
The set of free variables in E , written fv [[ E ]] , is defined inductively as follows:
fv [[ X ]] = { X }
fv [[( E
1
E
2
)]] = fv [[ E
1
]] ∪ fv [[ E
2
]]
fv [[( lambda ( X ) E )]] = fv [[ E ]] - { X }
Definition 2:
The set of bound variables in E , written bv [[ E ]] , is defined inductively as follows:
bv [[ X ]] = ∅
bv [[( E
1
E
2
)]] = bv [[ E
1
]] ∪ bv [[ E
2
]]
bv [[( lambda ( X ) E )]] =
bv [[ E ]] ∪ { X } if X ∈ fv [[ E ]]
bv [[ E ]] otherwise
Definition 3:
In any abstraction ( lambda ( X ) E ) :
• E is the body of the abstraction
• The occurrence of X in ( lambda ( X ) · · · ) is called a binding occurrence .
• The extent of the binding (or declaration) of X is the entire body E .
• The scope of the binding of X is that portion of the body in which occurrences of X are (or
would be) free. Thus, the scope of the binding of X is the extent of the binding, minus any
subsequent declarations of X that occur within the body E .
1

2 Substitution
Definition 4:
Let M and N be λ -calculus expressions, and let X be a variable. The substitution of N for X
in M , denoted M [ N / X ] , is defined inductively as follows:
• If M is a variable reference Y :
Y [ N / X ] =
N if X and Y are the same variable (i.e., X = Y )
Y if X and Y are not the same variable (i.e.,