Administration • HW1 due September 11 in class CS 611 – modify only interpretation.sml Advanced Programming Languages • New TA: James Cheney (jcheney@cs) Andrew Myers Cornell University Lecture 7: Lambda calculus 8 Sep 00 CS 611 Lecture 7 – Andrew Myers, Cornell University 2 Untyped Lambda Calculus Open vs. closed terms • IMP: no functions • term = expression denoting a value • Lambda calculus: all functions • Closed term: all identifiers bound by e ::= x | e 0 e 1 | λ x e 0 closest containing abstraction x Identifier. refers to variable defined by surrounding context. e 0 e 1 Application. Applies the function e 0 to the ( λ x … x ( λ y … y … )…) argument e 1 • Open term: some identifier(s) not λ x e 0 Abstraction/lambda term. Defines a new bound: ( λ x (y x)) function with argument variable x and body e 0 (ala ML’s fn x => e 0 ) • Legal lambda calculus programs: all closed terms • Universal, simple, core language (but not Lisp/Scheme ) CS 611 Lecture 7 – Andrew Myers, Cornell University 3 CS 611 Lecture 7 – Andrew Myers, Cornell University 4 Evaluation Higher-order functions • Application is evaluated by β -reduction: • Can express functions that return (or (( λ x e 1 ) e 2 ) → e 1 { e 2 / x } accept) other functions easily (all values are only functions) e 1 { e 2 / x } means “ e 1 with e 2 substituted for • A function that applies another function occurrences of x ” to 5: ( λ f (f 5)) ( note: defining “substituted” is tricky) • A function that returns a function that (( λ x x) e ) → x { e / x } = e applies another function to its (( λ x ( λ x x)) e ) → ( λ x x) { e / x } = ( λ x x) argument: ((( λ x ( λ y ( y x ))) 3) INC) → ( λ v ( λ f (f v))) (( λ y ( y 3)) INC) → (INC 3) → 4 CS 611 Lecture 7 – Andrew Myers, Cornell University 5 CS 611 Lecture 7 – Andrew Myers, Cornell University 6 1
Example Simulating multiple arguments • Don’t we need multiple arguments? ((( λ x ( λ y ( y x ))) 3) INC) → e ::= … | e 0 e 1 … e n | λ ( x 1 …x n ) e 0 (( λ y ( y 3)) INC) → (INC 3) → 4 • Can desugar (rewrite syntactically) into single-argument calculus: Shorthand: ( λ ( x 1 ... x n ) e ) � ( λ x 1 ( λ … ( λ x n e )…)) (( λ ( x y ) ( y x )) = ( λ x ( λ y ( y x ))) ( e 0 e 1 e 2 … e n ) � (...(( e 0 e 1 ) e 2 )… e n ) • Multi-argument functions are curried – applied one argument at a time (( λ ( x y ) ( y x )) 3 INC) → (INC 3) → 4 (+ 1 5) � ((+ 1) 5) CS 611 Lecture 7 – Andrew Myers, Cornell University 7 CS 611 Lecture 7 – Andrew Myers, Cornell University 8 Operational semantics Small-step semantics • Large-step semantics: configuration is ( β red’n) ( λ x e 1 ) e 2 → e 1 { e 2 / x } an expression of the language (no store) e 1 → e � 1 e 0 � λ x e 2 e 2 { e 1 / x } � v e 1 e 2 → e � 1 e 2 e 0 e 1 � v call-by-name • Call-by-name semantics: e 1 is not e 1 → e � 1 evaluated before substitution e 1 e 2 → e � 1 e 2 ( λ x e 1 ) v → e 1 { v / x } • Call-by-name + β -reduction: any e 2 → e � 2 v e 2 → v e � 2 lambda term is a value : v ::= λ x e call-by-value CS 611 Lecture 7 – Andrew Myers, Cornell University 9 CS 611 Lecture 7 – Andrew Myers, Cornell University 10 An infinite loop Simulating let • Lambda calculus has no “let” statement ala ML ( λ x (x x)) ( λ x (x x)) → ? This expression diverges (never stops taking ( λ x e 2 ) e 1 ( let x = e 1 in e 2 ) small steps) CS 611 Lecture 7 – Andrew Myers, Cornell University 11 CS 611 Lecture 7 – Andrew Myers, Cornell University 12 2
Definitions Representing booleans • Lambda calculus terms can become long; for • Lambda calculus is universal: no primitive compactness we will use names, multiple boolean type or “if” statement is needed arguments as shorthand (not part of language!) TRUE � ( λ x ( λ y x)) ~ ( λ (x y) x) IDENTITY � ( λ x x) FALSE � ( λ x ( λ y y)) ~ ( λ (x y) y) INC � (+ 1) if e 1 then e 2 else e 3 � ( IF e 1 e 2 e 3 ) APPLY - TO - FIVE � ( λ f (f 5)) IF � ( λ (x y z) (x y z)) COMPOSE � ( λ (f g) ( λ x (f (g x)))) ( IF TRUE e 2 e 3 ) → (( λ x ( λ y x)) e 2 ) e 3 ) TWICE � ( λ f ( COMPOSE f f )) → (( λ y e 2 ) e 3 ) → e 2 (( COMPOSE INC INC ) 2) → 4 (( TWICE ( TWICE INC )) 0) → 4 • Call-by-name important! e 2 and e 3 are not evaluated eagerly by IF CS 611 Lecture 7 – Andrew Myers, Cornell University 13 CS 611 Lecture 7 – Andrew Myers, Cornell University 14 Representing pairs (lists) Natural numbers • Pair/list operations: • Model the number n as a function that ( CONS x y) : construct list with head x, tail y composes an arbitrary function n times ( FIRST p) : return first item in list/first item in : Church numerals pair 0 � ( λ ( f a ) a ) (= FALSE ) ( REST p) : return remainder of list/second item in pair 1 � ( λ ( f a ) ( f a )) • One way to implement these operations: 2 � ( λ ( f a ) ( f ( f a ))) CONS � ( λ (x y) ( λ f (f x y))) 3 � ( λ ( f a ) ( f ( f ( f a )))) FIRST � ( λ p (p ( λ (x y) x))) (= ( λ p (p TRUE ))) n � ( λ ( f a ) ( f ( ... ( f a )...))) REST � ( λ p (p ( λ (x y) y))) (= ( λ p (p FALSE ))) n times CS 611 Lecture 7 – Andrew Myers, Cornell University 15 CS 611 Lecture 7 – Andrew Myers, Cornell University 16 Arithmetic • Can define INC function that adds one by writing a function that interposes an extra call to the function: n � ( λ ( f a ) ( f n a )) INC � ( λ n ( λ ( f a ) ( f (( n f ) a )))) Can define + and other arithmetic operators: + � ( λ ( n 1 n 2 ) ( λ ( f a ) (( n 1 f ) (( n 2 f ) a ))) or + � ( λ ( n 1 n 2 ) (( n 1 INC ) n 2 )) CS 611 Lecture 7 – Andrew Myers, Cornell University 17 3
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