Untyped Lambda Calculus Principles of Programming Languages CSE 526 - PDF document

Untyped Lambda Calculus Principles of Programming Languages CSE 526 Syntax 1 Variables and Substitution 2 Reductions 3 Nameless Representation 4 Compiled at 13:40 on 2018/02/15 Programming Languages The Untyped Lambda Calculus CSE 526 1 / 30 Lambda Calculus A formal notation to study computability and programming. Can be considered as the smallest universal programming language. Universal: Can be used to express any computation that can be performed on a Turing Machine Small: Has only two constructs: abstraction and application. Brief History: Introduced by Church and Kleene in 1930s. Used by Church to study problems in computability. Concepts have heavily influenced functional programming . Used to study types and type systems in programming languages Programming Languages The Untyped Lambda Calculus CSE 526 2 / 30

Syntax Lambda Terms Syntax of the λ -calculus t ::= Terms Variable x | Abstraction λ x . t | t t Application Textual Representation: Use parentheses to represent trees using linear text Programming Languages The Untyped Lambda Calculus CSE 526 3 / 30 Syntax Informal Semantics λ -expressions can be considered as expressions in a functional language Abstraction: ( λ x . t ) is a “function” with formal parameter x that returns (the value of) term t . Example 1: λ x . x is the identity function: one that returns the argument value itself. Example 2: λ x .λ y . x is a function that takes two arguments x and y and returns the first argument. Application: ( t 1 t 2 ) is a “function call” where t 1 is a function and t 2 is the supplied argument. Example: (( λ x . x ) y ) supplies y as the argument to the identity function. Programming Languages The Untyped Lambda Calculus CSE 526 4 / 30

Syntax Syntactic Conventions and Syntactic Sugar Parentheses can be dropped using the following conventions: application is left associative e.g. (( f f ) x ) is same as f f x a λ binds as much as possible to its right. e.g λ f . λ x . f ( f x ) is same as ( λ f . ( λ x . f ( f x ))) Multiple consecutive abstractions can be combined: e.g. λ f .λ x . f ( f x ) is same as λ f x . f ( f x ) Programming Languages The Untyped Lambda Calculus CSE 526 5 / 30 Variables and Substitution The Meaning of Lambda Expressions Recall: λ x . t stands for a function with x as the parameter and (the value of) t as the return value. ( t 1 t 2 ) stands for “calling” the function t 1 with t 2 as the parameter. Example: Consider the expression (( λ wyx . y ( w y x )) ( λ sz . z )) This is an instance of an application. The expression in blue is passed as an argument to the function in red. The meaning of an application: replace every occurrence of the formal parameter in the body of the function with the given argument. In the above example λ yx . y (( λ sz . z ) y x ) 1 λ yx . y (( λ z . z ) x ) 2 λ yx . y x 3 Programming Languages The Untyped Lambda Calculus CSE 526 6 / 30

Variables and Substitution Encoding Booleans in the λ -Calculus Example: B λ -calculus ( true && false ) true λ x . λ y . x ≡ ( λ x . λ y . (( x y ) false ))) false λ x . λ y . y ( λ x . λ y . x ) ( λ x . λ y . y ) λ x . λ y . (( x y ) false ) && → ( λ y . ((( λ x . λ y . x ) y ) false ))) λ x . λ y . (( x true ) y ) || ( λ x . λ y . y ) λ x . (( x false ) true ) ! → ( (( λ x . λ y . x ) ( λ x . λ y . y )) false ) if λ c . λ t . λ e . (( c t ) e ) → ( ( λ y . ( λ x . λ y . y )) false ) This is known as the → ( λ x . λ y . y ) Church encoding of Booleans , or simply Church Booleans . ≡ false Programming Languages The Untyped Lambda Calculus CSE 526 7 / 30 Variables and Substitution Encoding Natural Numbers in the λ -Calculus N λ -calculus 0 λ s . λ z . z λ s . λ z . ( s z ) 1 λ s . λ z . ( s ( s z )) 2 λ s . λ z . ( s ( s ( s z ))) 3 . . . λ n . λ s . λ z . ( s (( n s ) z )) inc plus λ m . λ n . λ s . λ z . (( m s ) (( n s ) z )) λ m . λ n . (( m ( plus n )) 0 ) times iszero λ m . (( m ( λ x . false )) true ) . . . This is known as the Church encoding of Naturals , or simply Church Numerals . Programming Languages The Untyped Lambda Calculus CSE 526 8 / 30

Variables and Substitution Encoding Data Structures in the λ -Calculus fst ( pair ϕ 1 ϕ 2 ) λ f . λ s . λ c . (( c f ) s ) pair ≡ ( λ p . ( p true )) ( pair ϕ 1 ϕ 2 ) fst λ p . ( p true ) → ( pair ϕ 1 ϕ 2 ) true snd λ p . ( p false ) → ∗ ( λ c . (( c ϕ 1 ) ϕ 2 )) true → (( true ϕ 1 ) ϕ 2 ) Example: Let ϕ 1 and ϕ 2 be two arbitrary → ϕ 1 expressions. snd ( pair ϕ 1 ϕ 2 ) pair ϕ 1 ϕ 2 ≡ ( λ p . ( p false )) ( pair ϕ 1 ϕ 2 ) ≡ ( ( λ f . λ s . λ c . (( c f ) s ) ϕ 1 ) ϕ 2 ) → ∗ (( false ϕ 1 ) ϕ 2 ) → ∗ λ c . (( c ϕ 1 ) ϕ 2 ) → ϕ 2 Programming Languages The Untyped Lambda Calculus CSE 526 9 / 30 Variables and Substitution Evaluating Lambda Expressions: An Informal Intro. t ✔ ❚ ✔ ❚ Basic reduction: ( λ x . t 1 ) t 2 → [ x �→ t 2 ] t 1 , ✔ apply ❚ ✔ ❚ � ❅ where ✔ ❚ t 2 λ x ✔ ❚ ✄ ❈ [ x �→ t 2 ] t 1 be the term obtained by replacing all ✔ ❚ t 1 ✄ ❈ ✔ ❚ ✂ ❇ ✄ ❈ ✔ ✂ ❇ ❚ ✄ ❈ “free” occurrences of x in t 1 by t 2 . A sub-term of t of the form ( λ x . t 1 ) t 2 is called a redex of t . apply One step in evaluating a λ -term t is � ❅ apply λ x replacing some redex in t according to the � ❅ above reduction schema. x λ x λ z In general, there may be many redexes in x apply � ❅ a term. z λ x Example: Let id = ( λ x . x ) in term x id ( id ( λ z . id z )) Programming Languages The Untyped Lambda Calculus CSE 526 10 / 30

Variables and Substitution Reduction Strategies A reduction strategy is used to choose a redex where the basic reduction step will be done. Full β -reduction: Pick a redex non-deterministically apply � ❅ Normal Order: choose the left-most, apply λ x outer-most redex. � ❅ x λ x λ z Call-By-Name: like normal-order, but ignore redexes inside abstractions. x apply � ❅ Call-By-Value: choose the right-most, z λ x inner-most redex that is not inside an x abstraction. Programming Languages The Untyped Lambda Calculus CSE 526 11 / 30 Variables and Substitution Evaluating Lambda Expressions The key step in evaluating an application then is: replace every occurrence of a formal parameter with the actual argument . Example: (( λ x . ( λ z . x z )) y ) → ( λ z . y z ) We can formalize the meaning of application by introducing a function, called substitution that maps terms to terms: ( λ x . t 1 ) t 2 → [ x �→ t 2 ] t 1 The central problem now is how we define this substitution function. Programming Languages The Untyped Lambda Calculus CSE 526 12 / 30

Variables and Substitution Substitutions (1 st attempt) [ x �→ s ] x = s [ x �→ s ] y = if y � = x y [ x �→ s ]( λ y . t ) = λ y . [ x �→ s ] t [ x �→ s ]( t 1 t 2 ) = ([ x �→ s ] t 1 ) ([ x �→ s ] t 2 ) Appears to be correct. Example: [ x �→ y ]( λ z . x z ) = ( λ z . y z ) Use: ( λ x . ( λ z . x z )) y ) → ( λ z . y z ) But is incorrect! Example: [ x �→ y ]( λ x . x ) = ( λ x . y ) Use: (( λ x . ( λ x . x )) y ) → ( λ x . y ) Programming Languages The Untyped Lambda Calculus CSE 526 13 / 30 Variables and Substitution Substitutions (2 nd attempt) [ x �→ s ] x = s [ x �→ s ] y = if y � = x y � λ y . t if x = y [ x �→ s ]( λ y . t ) = λ y . [ x �→ s ] t if x � = y [ x �→ s ]( t 1 t 2 ) = ([ x �→ s ] t 1 ) ([ x �→ s ] t 2 ) [ x �→ y ]( λ x . x ) = ( λ x . x ) But is still incorrect! e.g. [ x �→ y ]( λ y . x y ) = ( λ y . y y ) In the result of the above example, one y is local to the function while the other y is not local. But going by our definition, there is no way to distinguish between the two y ’s! Solution: We should get ( λ w . y w ) instead (by suitably renaming “local” variables). Programming Languages The Untyped Lambda Calculus CSE 526 14 / 30

Variables and Substitution Bound and Free Variables: An Informal Intro. Variable x in λ -expression λ x . t is said to be bound . Example 1: x in λ x . x is a bound variable. Example 2: in λ x . ( x y ), x is bound but y is not bound. Rough meaning: parameters are local to a function definition. A variable that is not bound is said to be free . Example 2: in λ x . ( x y ), y is free. Rough meaning: free variables in a function definition are analogous to non-local variables. Programming Languages The Untyped Lambda Calculus CSE 526 15 / 30 Variables and Substitution Bound and Binding Occurrences ❄ ✎☞ ✎☞ λ x . x ✍✌ ✍✌ Bound Occurrence (use) Binding Occurrence (declaration) ❄ ❄ ✎☞ ✎☞ ✎☞ ✎☞ ( λ x . x )( λ z . ( x x z )) ‘ ✍✌ ✍✌ ✍✌ ✍✌ Free Occurrence ❄ ( λ z . ( λ x . z ( x x )) ( λ x . z ( x x ))) ✻ ✻ Programming Languages The Untyped Lambda Calculus CSE 526 16 / 30