Download Functional Programming and Lambda Calculus

Functional Programming
and Lambda Calculus
CSCI 3136
Principles of Programming Languages
Faculty of Computer Science
Dalhousie University
Winter 2012
Reading: Chapter 10
Review: Programming Models
Imperative programming: Computation is a sequence of side effects.
Functional Programming: Computation by function evalution. There are no
side effects.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Review: Programming Models
Imperative programming: Computation is a sequence of side effects.
Functional Programming: Computation by function evalution. There are no
side effects.
This requires functions to be first-class objects: they can be passed as function
arguments and returned as the results of computations.
Functions that operate on functions and/or return functions are called
higher-order functions.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Review: Programming Models
Imperative programming: Computation is a sequence of side effects.
Functional Programming: Computation by function evalution. There are no
side effects.
This requires functions to be first-class objects: they can be passed as function
arguments and returned as the results of computations.
Functions that operate on functions and/or return functions are called
higher-order functions.
Iteration is impossible in functional languages because iteration inherently
requires side effects. Iteration can be simulated using recursion.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Review: Programming Models
Imperative programming: Computation is a sequence of side effects.
Functional Programming: Computation by function evalution. There are no
side effects.
This requires functions to be first-class objects: they can be passed as function
arguments and returned as the results of computations.
Functions that operate on functions and/or return functions are called
higher-order functions.
Iteration is impossible in functional languages because iteration inherently
requires side effects. Iteration can be simulated using recursion.
Other common features of functional programming languages
•
•
•
•
Strong reliance on list types and strong support for manipulating them
Structured function returns
Structured data constructors
Garbage collection
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Functional Programming and the Real World
Advantages of functional programming
• The lack of side effects makes it easier to reason about program:
– Formal proofs of program correctness become easier.
– Tests always produce the same result.
• Higher level of abstraction → more expressive
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Functional Programming and the Real World
Advantages of functional programming
• The lack of side effects makes it easier to reason about program:
– Formal proofs of program correctness become easier.
– Tests always produce the same result.
• Higher level of abstraction → more expressive
Real-world concerns
• The interaction with the real world (file I/O, user I/O, . . . ) happens
through side effects.
• Most functional programming languages have support for side effects to
accommodate this.
– Scheme: very flexible: supports mixing of different programming styles
– Haskell: Controlled side effects through monads
Lambda Calculus
CSCI 3136: Principles of Programming Languages
λ-Calculus (Alonzo Church 1941)
. . . is the theoretical foundation of functional programming. It expresses
computation entirely using function application.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
λ-Calculus (Alonzo Church 1941)
. . . is the theoretical foundation of functional programming. It expresses
computation entirely using function application.
A context-free grammar for lambda expressions
Expr → name | number | λ name . Expr | Func Arg
Func → name | (λ name . Expr) | Func Arg
Arg → name | number | (λ name . Expr) | (Func Arg)
Numbers are not really part of this, but we introduce them here to make the
exposition easier.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
λ-Calculus (Alonzo Church 1941)
. . . is the theoretical foundation of functional programming. It expresses
computation entirely using function application.
A context-free grammar for lambda expressions
Expr → name | number | λ name . Expr | Func Arg
Func → name | (λ name . Expr) | Func Arg
Arg → name | number | (λ name . Expr) | (Func Arg)
Numbers are not really part of this, but we introduce them here to make the
exposition easier.
Examples
•
•
•
•
•
•
x, times, plus, 7
λx.x
λ y . times
λx.x yz
λ y . times y y
λ x . (times x λ y. y)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Interpretation
A variable is bound if it is in the scope of a λ and free otherwise.
Examples
• x is bound in λ x . x y but free in the subexpression x y
• x is free in λ y . x y
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Interpretation
A variable is bound if it is in the scope of a λ and free otherwise.
Examples
• x is bound in λ x . x y but free in the subexpression x y
• x is free in λ y . x y
λ x . Expr (function definition)
• is a function with formal parameter x, which becomes a bound variable in
Expr
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Interpretation
A variable is bound if it is in the scope of a λ and free otherwise.
Examples
• x is bound in λ x . x y but free in the subexpression x y
• x is free in λ y . x y
λ x . Expr (function definition)
• is a function with formal parameter x, which becomes a bound variable in
Expr
(λ x . Expr) A ≡ Expr[A \ x] (function application)
• replaces every free occurrence of x in Expr with A
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Interpretation
A variable is bound if it is in the scope of a λ and free otherwise.
Examples
• x is bound in λ x . x y but free in the subexpression x y
• x is free in λ y . x y
λ x . Expr (function definition)
• is a function with formal parameter x, which becomes a bound variable in
Expr
(λ x . Expr) A ≡ Expr[A \ x] (function application)
• replaces every free occurrence of x in Expr with A
(λ x . λ y . Expr) A B ≡ (λ y . Expr[A \ x]) B ≡ Expr[A \ x and B \ y]
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Computing Using λ-Calculus
Computation is done using three transformations:
β-reduction
• (λx . E) M →β E[M \ x]
• Permissible only if no free variable of M becomes bound in E[M \ x]
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Computing Using λ-Calculus
Computation is done using three transformations:
β-reduction
• (λx . E) M →β E[M \ x]
• Permissible only if no free variable of M becomes bound in E[M \ x]
α-conversion
• λ x . E →α λ y . E[ y \ x], where y is a variable not occurring in E
• Needed to make β-reduction possible
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Computing Using λ-Calculus
Computation is done using three transformations:
β-reduction
• (λx . E) M →β E[M \ x]
• Permissible only if no free variable of M becomes bound in E[M \ x]
α-conversion
• λ x . E →α λ y . E[ y \ x], where y is a variable not occurring in E
• Needed to make β-reduction possible
η-reduction
• λ x . F x →η F , where F contains no free occurrences of x
• Eliminates “surplus” lambda abstractions
Lambda Calculus
CSCI 3136: Principles of Programming Languages
The Need for α-Conversion
(λ x . λ y . x y)(λ z . y) c
Lambda Calculus
CSCI 3136: Principles of Programming Languages
The Need for α-Conversion
(λ x . λ y . x y)(λ z . y) c
Substitution without α-conversion
(λ x . λ y . x y)(λ z . y) c →β (λ y . (λ z . y) y) c
→β (λ z . c) c
→β c
Lambda Calculus
CSCI 3136: Principles of Programming Languages
The Need for α-Conversion
(λ x . λ y . x y)(λ z . y) c
Substitution without α-conversion
(λ x . λ y . x y)(λ z . y) c →β (λ y . (λ z . y) y) c
→β (λ z . c) c
→β c
Substitution with α-conversion
(λ x . λ y . x y)(λ z . y) c →α (λ x . λ w . x w)(λ z . y) c
→β (λ w . (λ z . y) w) c
→η (λ z . y) c
→β y
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Associativity in λ-Expressions
Function definition is right-associative:
λ x . λ y . λ z . E ≡ λ x . (λ y . (λ z . E)))
Function application is left-associative:
A B C D ≡ ((A B) C) D
λ extends as far to the right as possible (to the end of the expression or to the
next parenthesis):
λ x . A B C D ≡ λ x . (A B C D)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Labels
Labels can be created as shortcuts for long expressions. This is kind of like a
global function definition in a programming language.
Examples
square ≡ λ x . times x x
id ≡ λ x . x
const ≡ λ x . λ y . x
hypot ≡ λ x . λ y . sqrt (plus (square x) (square y))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Example 1
not true ≡ (λ t . t (λx . λ y . y) (λ x . λ y . x)) (λ x . λ y . x)
→β (λ x . λ y . x) (λx . λ y . y) (λ x . λ y . x)
→β (λ y . (λ x . λ y . y)) (λ x . λ y . x)
→β (λ x . λ y . y)
≡ false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Example 1
not true ≡ (λ t . t (λx . λ y . y) (λ x . λ y . x)) (λ x . λ y . x)
→β (λ x . λ y . x) (λx . λ y . y) (λ x . λ y . x)
→β (λ y . (λ x . λ y . y)) (λ x . λ y . x)
→β (λ x . λ y . y)
≡ false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Example 1
not true ≡ (λ t . t (λx . λ y . y) (λ x . λ y . x)) (λ x . λ y . x)
→β (λ x . λ y . x) (λx . λ y . y) (λ x . λ y . x)
→β (λ y . (λ x . λ y . y)) (λ x . λ y . x)
→β (λ x . λ y . y)
≡ false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Example 1
not true ≡ (λ t . t (λx . λ y . y) (λ x . λ y . x)) (λ x . λ y . x)
→β (λ x . λ y . x) (λx . λ y . y) (λ x . λ y . x)
→β (λ y . (λ x . λ y . y)) (λ x . λ y . x)
→β (λ x . λ y . y)
≡ false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (1)
true ≡ λ x . λ y . x
false ≡ λ x . λ y . y
not ≡ λ t . t false true
if ≡ λ c . λ t . λ e . c t e
Example 1
not true ≡ (λ t . t (λx . λ y . y) (λ x . λ y . x)) (λ x . λ y . x)
→β (λ x . λ y . x) (λx . λ y . y) (λ x . λ y . x)
→β (λ y . (λ x . λ y . y)) (λ x . λ y . x)
→β (λ x . λ y . y)
≡ false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Logic and Control Flow (2)
Example 2
if true 3 4 ≡ (λ c . λ t . λ e . c t e) (λ x . λ y . x) 3 4
→β (λ t . λ e . (λ x . λ y . x) t e) 3 4
→β (λ e . (λ x . λ y . x) 3 e) 4
→β (λ x . λ y . x) 3 4
→β (λ y . 3) 4
→β 3
What about “and” and “or”?
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Representing Natural Numbers Using Functions
An inductive definition of
numbers
• 0 ≡ const id ≡ λ f . λ x . x
• succ ≡ λ n. λ f . λ x . f (n f x)
• For n > 0, n ≡ succ (n − 1)
–
–
–
–
1 ≡ λ f .λ x . f x
2 ≡ λ f . λ x . f ( f x)
3 ≡ λ f . λ x . f ( f ( f x))
...
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Representing Natural Numbers Using Functions
An inductive definition of
numbers
• 0 ≡ const id ≡ λ f . λ x . x
• succ ≡ λ n. λ f . λ x . f (n f x)
• For n > 0, n ≡ succ (n − 1)
–
–
–
–
1 ≡ λ f .λ x . f x
2 ≡ λ f . λ x . f ( f x)
3 ≡ λ f . λ x . f ( f ( f x))
...
Arithmetic operations
• Addition: + ≡
λ a . λ b . λ f . λ x . a f (b f x)
• Multiplication:
∗ ≡ λ a . λ b . λ f . a (b f )
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Representing Natural Numbers Using Functions
An inductive definition of
numbers
• 0 ≡ const id ≡ λ f . λ x . x
• succ ≡ λ n. λ f . λ x . f (n f x)
• For n > 0, n ≡ succ (n − 1)
–
–
–
–
1 ≡ λ f .λ x . f x
2 ≡ λ f . λ x . f ( f x)
3 ≡ λ f . λ x . f ( f ( f x))
...
Arithmetic operations
• Addition: + ≡
λ a . λ b . λ f . λ x . a f (b f x)
• Multiplication:
∗ ≡ λ a . λ b . λ f . a (b f )
Example in Haskell
type Number a = (a -> a) -> a -> a
zero :: Number a
zero = \f -> \x -> x
succ :: Number a -> Number a
succ = \n -> \f -> \x -> f (n f x)
plus, times :: Number a -> Number a -> Number a
plus = \a -> \b -> \f -> \x -> a f (b f x)
times = \a -> \b -> \f -> \x -> a (b f) x
one, two,
one
=
two
=
three
=
five
=
fifteen =
three, five, fifteen :: Number a
succ zero
succ one
succ two
plus two three
times five three
> three (+1) 0
3
> five (+1) 0
5
> fifteen (+1) 0
15
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (1)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (1)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 1
null? nil ≡ (λ l . l (λ x . λ y . false)) (λ x . true)
→β (λ x . true) (λ x . λ y . false)
→β true
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (2)
cons ≡ λ h . λ t . λ x . x h t
(prepend single element h to list t)
car ≡ λ l . l (λ x . λ y . x)
(return head of list l)
cdr ≡ λ l . l (λ x . λ y . y)
(return tail of list l)
nil ≡ λ x . true
(empty list)
null? ≡ λ l . l (λ x . λ y . false)
(is list l empty?)
Example 2
null? (cons 1 nil) ≡ (λ l . l (λ x . λ y . false)) ((λ h . λ t . λ x . x h t) 1 (λ x . true))
→β (λ h . λ t . λ x . x h t) 1 (λ x . true) (λ x . λ y . false)
→β (λ t . λ x . x 1 t) (λ x . true) (λ x . λ y . false)
→β (λ x . x 1 (λ x . true)) (λ x . λ y . false)
→β (λ x . λ y . false) 1 (λ x . true)
→β (λ y . false) (λ x . true)
→β false
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Building Data Structures: Lists (3)
Example 3
car (cdr (cons 1 (cons 2 (cons 3 nil))))
≡ (λ l . l (λ x . λ y . x)) (cdr (cons 1 (cons 2 (cons 3 nil))))
→β cdr (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
≡ (λ l . l (λ x . λ y . y)) (cons 1 (cons 2 (cons 3 nil))) (λ x . λ y . x)
→β cons 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 1 (cons 2 (cons 3 nil)) (λ x . λ y . y) (λ x . λ y . x)
→∗β (λ x . λ y . y) 1 (cons 2 (cons 3 nil)) (λ x . λ y . x)
→∗β (cons 2 (cons 3 nil)) (λ x . λ y . x)
≡ (λ h . λ t . λ x . x h t) 2 (cons 3 nil) (λ x . λ y . x)
→∗β (λ x . λ y . x) 2 (cons 3 nil)
→∗β 2
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying
(Haskell Curry, logician)
Currying: No need for multi-argument functions. Instead, represent such
functions as functions returning functions returning functions . . .
Currying makes “partial function application” possible:
• times ≡ λ x . λ y . x ∗ y (a function with two arguments)
• Full application: times 2 3 ≡ 6 (a number)
• Partial application: times 2 ≡ λ y . 2 ∗ y (a function doubling its argument)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (1)
Example: Depending on a condition cond, multiply every element in a
sequence by 10 or add 2.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (1)
Example: Depending on a condition cond, multiply every element in a
sequence by 10 or add 2.
C
Attempt 1
if( cond ) {
for( i = 0; i < n; i++ ) A[i] *= 10;
} else {
for( i = 0; i < n; i++ ) A[i] += 2;
}
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (1)
Example: Depending on a condition cond, multiply every element in a
sequence by 10 or add 2.
C
Attempt 1
if( cond ) {
for( i = 0; i < n; i++ ) A[i] *= 10;
} else {
for( i = 0; i < n; i++ ) A[i] += 2;
}
Code
duplication
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (1)
Example: Depending on a condition cond, multiply every element in a
sequence by 10 or add 2.
C
Attempt 1
if( cond ) {
for( i = 0; i < n; i++ ) A[i] *= 10;
} else {
for( i = 0; i < n; i++ ) A[i] += 2;
}
Code
duplication
Attempt 2
for( i = 0; i < n; i++ )
if( cond )
A[i] *= 10;
else
A[i] += 2;
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (1)
Example: Depending on a condition cond, multiply every element in a
sequence by 10 or add 2.
C
Attempt 1
if( cond ) {
for( i = 0; i < n; i++ ) A[i] *= 10;
} else {
for( i = 0; i < n; i++ ) A[i] += 2;
}
Code
duplication
Attempt 2
for( i = 0; i < n; i++ )
if( cond )
A[i] *= 10;
else
A[i] += 2;
Inefficient if cond
is expensive to evaluate
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (2)
Attempt 3
int plus2( int x ) { return 2+x; }
int times10( int x ) { return 10*x; }
void map( int *A, int n, int *f( int ) ) {
for( i = 0; i < n; i++ ) A[ i ] = f( A[ i ] );
}
if( cond )
map( A, n, times10 );
else
map( A, n, plus2 );
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (2)
Attempt 3
int plus2( int x ) { return 2+x; }
int times10( int x ) { return 10*x; }
void map( int *A, int n, int *f( int ) ) {
for( i = 0; i < n; i++ ) A[ i ] = f( A[ i ] );
}
if( cond )
map( A, n, times10 );
else
map( A, n, plus2 );
Way too much
boilerplate
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (3)
Scheme (Anonymous functions and higher-order functions for list processing,
but no currying)
(if cond
(map (lambda (x) (* 10 x)) A)
(map (lambda (x) (+ 2 x)) A))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Currying Is Cool (3)
Scheme (Anonymous functions and higher-order functions for list processing,
but no currying)
(if cond
(map (lambda (x) (* 10 x)) A)
(map (lambda (x) (+ 2 x)) A))
Haskell (Anonymous functions + currying)
if cond then map (* 10) A
else map (+ 2) A
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (1)
Example: Computing factorials
Attempt 1 (Does not work)
fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (1)
Example: Computing factorials
Attempt 1 (Does not work)
fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
The reason this is not correct is because “fac” is just a shorthand for the
λ-expression on its right-hand side. In order to obtain a valid λ-expression, all
such shorthands need to be replaced with the λ-expressions they represent.
This leads to an infinite process here.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (1)
Example: Computing factorials
Attempt 1 (Does not work)
fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
The reason this is not correct is because “fac” is just a shorthand for the
λ-expression on its right-hand side. In order to obtain a valid λ-expression, all
such shorthands need to be replaced with the λ-expressions they represent.
This leads to an infinite process here.
Towards a correct definition
fac ≡ (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))) fac
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (1)
Example: Computing factorials
Attempt 1 (Does not work)
fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
The reason this is not correct is because “fac” is just a shorthand for the
λ-expression on its right-hand side. In order to obtain a valid λ-expression, all
such shorthands need to be replaced with the λ-expressions they represent.
This leads to an infinite process here.
Towards a correct definition
fac ≡ (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))) fac
In other words,
fac ≡ F fac,
where
F ≡ (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (1)
Example: Computing factorials
Attempt 1 (Does not work)
fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
The reason this is not correct is because “fac” is just a shorthand for the
λ-expression on its right-hand side. In order to obtain a valid λ-expression, all
such shorthands need to be replaced with the λ-expressions they represent.
This leads to an infinite process here.
Towards a correct definition
fac ≡ (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))) fac
In other words,
fac ≡ F fac,
where
F ≡ (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))
“fac” is a fixed point of F .
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (2)
The Y-combinator (fixed-point operator)
Y ≡ λ h . (λ x . h (x x)) (λ x . h (x x))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (2)
The Y-combinator (fixed-point operator)
Y ≡ λ h . (λ x . h (x x)) (λ x . h (x x))
Whenever we would normally write a recursive function
f ≡ λ x . ... f ...,
we now write
F ≡ Y (λ f . λ x . . . . f . . . )
and then evaluate F x instead of f x.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (2)
The Y-combinator (fixed-point operator)
Y ≡ λ h . (λ x . h (x x)) (λ x . h (x x))
Whenever we would normally write a recursive function
f ≡ λ x . ... f ...,
we now write
F ≡ Y (λ f . λ x . . . . f . . . )
and then evaluate F x instead of f x.
Back to factorials:
• Incorrect: fac ≡ λ n . if (= n 0) 1 (∗ n (fac (− n 1)))
• Correct: Fac ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (3)
Fac 3 ≡ Y (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
≡ (λ h . (λ x . h (x x)) (λ x . h (x x)))
(λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) 3
→β (λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x)) 3
→β (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1))))
((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 3
→∗β if (= 3 0) 1
(∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 3 1)))
→∗β ∗ 3 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 2)
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (4)
...
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
→∗β ∗ 3 (∗ 2 (∗ 1 (if (= 0 0) 1
(∗ 0 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 0 1))))))
→∗β ∗ 3 (∗ 2 (∗ 1 1))
→∗β 6
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (4)
...
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
→∗β ∗ 3 (∗ 2 (∗ 1 (if (= 0 0) 1
(∗ 0 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 0 1))))))
→∗β ∗ 3 (∗ 2 (∗ 1 1))
→∗β 6
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (4)
...
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
→∗β ∗ 3 (∗ 2 (∗ 1 (if (= 0 0) 1
(∗ 0 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 0 1))))))
→∗β ∗ 3 (∗ 2 (∗ 1 1))
→∗β 6
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Recursion in λ-Calculus (4)
...
→∗β ∗ 3 (∗ 2 (∗ 1 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) 0)))
→∗β ∗ 3 (∗ 2 (∗ 1 (if (= 0 0) 1
(∗ 0 (((λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))
(λ x . (λ f . λ n . if (= n 0) 1 (∗ n ( f (− n 1)))) (x x))) (− 0 1))))))
→∗β ∗ 3 (∗ 2 (∗ 1 1))
→∗β 6
Lambda Calculus
CSCI 3136: Principles of Programming Languages
(Non-)Termination of Reductions
Some reductions never terminate:
(λ x . x x) (λ x . x x) →β (λ x . x x) (λ x . x x)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
(Non-)Termination of Reductions
Some reductions never terminate:
(λ x . x x) (λ x . x x) →β (λ x . x x) (λ x . x x)
Others may produce longer and longer expressions:
(λ x . x x x) (λ x . x x x) →β (λ x . x x x) (λ x . x x x) (λ x . x x x)
→β (λ x . x x x) (λ x . x x x) (λ x . x x x) (λ x . x x x)
→β . . .
Lambda Calculus
CSCI 3136: Principles of Programming Languages
(Non-)Termination of Reductions
Some reductions never terminate:
(λ x . x x) (λ x . x x) →β (λ x . x x) (λ x . x x)
Others may produce longer and longer expressions:
(λ x . x x x) (λ x . x x x) →β (λ x . x x x) (λ x . x x x) (λ x . x x x)
→β (λ x . x x x) (λ x . x x x) (λ x . x x x) (λ x . x x x)
→β . . .
This is the same as infinite loops or infinite recursion in incorrect programs.
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Normal-Order vs Applicative-Order Reduction (1)
Normal-order reduction: Subtitute the argument into the function
unevaluated and continue reducing the resulting expression.
(λ x . h)((λ x . x x) (λ x . x x)) →β h
Applicative-order reduction: Reduce both the function and the argument
expressions to the simplest possible form before substituting the argument into
the function. Then continue to reduce the resulting expression.
(λ x . h)((λ x . x x) (λ x . x x)) →∗β . . . (infinite expansion)
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Normal-Order vs Applicative-Order Reduction (2)
Why does this matter?
Normal-order reduction allows us to construct infinite objects and use them, as
long as we only ever use a finite part of them.
Example: Fibonacci numbers in Haskell
fibonacci = 1:1:(zipWith (+) fibonacci (tail fibonacci))
> take 10 fibonacci
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
Lambda Calculus
CSCI 3136: Principles of Programming Languages
Church-Rosser Theorem
If two reductions of the same λ-expression terminate, they terminate with the
same result.
If some reduction of a λ-expression terminates, the normal-order reduction
terminates.
Lambda Calculus
CSCI 3136: Principles of Programming Languages