Download The λ – Calculus

Introduction
• A λ – expression specifies the:
The λ – Calculus
– Parameters of a function
– Definition of a function
• A λ – expression is of the form:
Joseph Spring
7COM1023 Programming Paradigms
(A. B. Tucker & R. E. Noonan, Programming Languages: Principles and Paradigms, 2nd Ed, McGraw Hill, 2007)
( x  x  x )
(parameter/identifier  definition)
• Note: A λ – expression does NOT specify
the name of a function
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Discussion Points
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Expressions
• So given a function
Introduction
Expressions
Bound and Free Variables
Substitution
Beta Reduction
Eager and Lazy Evaluation
Square :    by x  x  x
So Square( x)  x  x the corresponding
lambda expression would be ( x  x  x)
• An instance of a lambda expression to a
particular value ‘a’ takes the form:
(( x  x  x) a)  a a  a 2
• So, for example: (( x  x  x)3)  3  3  9
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Introduction
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Pure / Uninterpreted Lambda Calculus
Definition (Church)
1. An identifier is a lambda expression
e.g. x
(λx  x)
2. Given two lambda expressions M and N the application
of M to N is also a lambda expression and is denoted by
(M N)
• The λ – Calculus is said to be:
– Developed by Alonzo Church in 1941
– ‘The foundation of functional programming’
which was developed in the early 1960’s
• A λ – expression specifies the:
– Parameters of a function
– Definition of a function
e.g. ((λx  x) (λy  y))
• A λ – expression does NOT specify the
name of a function
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Pure / Uninterpreted Lambda Calculus
Substitution
Substituting an expression N for a variable x in M is
denoted by M[x  N] and defined as follows:
Definition (Church)
3. An abstraction, written as ( x  M) with x an
identifier and M a lambda expression is also a lambda
expression
1. If the free variables of N have no bound occurrences
in M, then the term M[x  N] is formed by replacing
all free occurrences of x in M by N. Otherwise:
In terms of the BNF grammar rules:
LambdaExpression  variable| (M N) | ( x  M)
M  LambdaExpression
N  LambdaExpression
2. Assume that the variable y is free in N and bound in
M. Then consistently replace the binding and
corresponding bound occurrences of y in M, by a
new variable, say u. Repeat until the condition in 1.
above applies. Then proceed as in step 1.
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Bound and Free Variables
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Substitution - Examples
Definition (Church)
1. Given a lambda expression ( x  M) the variable x is
said to be bound in the sub-expression M
2. A bound variable is one whose name is the same as
the parameter. Otherwise the variable is said to be free
3. Any variable not bound in M is said to be free
Note: bound variables are placeholders just like function
parameters in the imperative and OOP paradigms
x[ x  y ]  y
(xx)[ x  y ]  ( yy )
(zw)[ x  y ]  ( zw)
(zx)[ x  y ]  ( zy )
( x  ( z x))[ x  y]  (u  ( zu))[ x  y]  (u  ( zu))
( x  ( z x))[y  x]  (u  ( zu))[y  x]  (u  ( zu))
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Renaming Variables
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Reference
Any such variable can be renamed in a consistent
manner with any variable free in M, without changing
the meaning of the lambda expression
Free variables can be defined in a lambda expression
as:
free( x)  x
1.
A. B. Tucker & R. E. Noonan, Programming Languages: Principles and
Paradigms, 2nd Ed, McGraw Hill, 2007
2.
M. Gabrielli & S. Martini, Programming Languages: Principles and
Paradigms, Springer, 2010
3.
H. E. Bal & D. Grune, Programming Language Essentials, Addison
Wesley, 1994
free( MN )  free( M )  free( N )
free( x  M )  free( M ) \{x}  free( M )  {x}
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