Programming with Lambda (Pre Lecture)

Dr. Neil T. Dantam

CSCI-400, Colorado School of Mines

Spring 2021

Dantam (Mines CSCI-400) Lambda Spring 2021 1/29 Origin of Lambda: What’s an “algorithm”? Entscheidungsproblem: mathematical true algorithm statement false

λ-calculus Turing machines

David Hilbert Alonzo Church

Alan Turing

Dantam (Mines CSCI-400) Lambda Spring 2021 2/29 Introduction

Lambda Outcomes ◮ Lambda: the ultimate operator ◮ Implement common language ◮ Everything programmable can be constructs using Lambda programmed with lambda ◮ Practical: if a language lacks a desired feature, but has lambda, you can implement that feature with lambda (see also: JavaScript)

Dantam (Mines CSCI-400) Lambda Spring 2021 3/29 Lambda Expression Review Function Deﬁnition Function Application formal parameter function λ . z}|{x α z}|{f y body actual parameter |{z} |{z} ◮ Create an unnamed function. ◮ Bind actual parameter y to ◮ The function is itself a value. the formal parameter of f . ◮ Do not evaluate the body until applied (called) ◮ Evaluate the body of f . Example a = 2 + (λa . 1+ a) 2 1 + 2 3

Dantam (Mines CSCI-400) Lambda Spring 2021 4/29 Lambda: Very Important! Review

◮ Lambda can perform any possible computation! ◮ Trivial with Lambda: ◮ N-ary functions ◮ Variable binding (let) “I’m kind of a big deal.” ◮ Statements (;) ◮ Named functions ◮ OOP ◮ Also possible with Lambda: ◮ Recursion λ ◮ Conditionals (if) ◮ Lists ◮ Structs ◮ Arithmetic (+, −, etc.)

Dantam (Mines CSCI-400) Lambda Spring 2021 5/29 Lambda: Very Important! Review

Dantam (Mines CSCI-400) Lambda Spring 2021 6/29 Binary and N-ary Functions Outline

Binary and N-ary Functions

Bindings

Dantam (Mines CSCI-400) Lambda Spring 2021 7/29 Binary and N-ary Functions Binary Functions Reduction of Binary to Unary Functions Example λ . Binary Unary ( ab a + b) 1 2 formal params. body λx .λy . e λ x y . e z}|{ z}|{ (λa .λb . a + b) 1 2 ◮ Convert to two lambdas a = 1 ◮ Outer lambda binds ﬁrst param (x) and (λb . 1+ b) 2 returns inner lambda ◮ Inner lambda binds second param (y) and b = 2 evaluates 1 + 2

Each function call creates new bindings of its arguments.

Dantam (Mines CSCI-400) Lambda Spring 2021 8/29 Binary and N-ary Functions N-ary Functions

N-ary functions Example arity: n + 1 arity: n (λabc . a + b + c) 1 2 3

λx0 x1 ... xn . e λx0 . (λx1 ... xn . α)

(λa .λbc . a + b + c) 1 2 3 ◮ Recursively convert to unary functions.

(λa .λb .λc . a + b + c) 1 2 3

Dantam (Mines CSCI-400) Lambda Spring 2021 9/29 Binary and N-ary Functions N-ary Functions example, continued

Example (λabc . a + b + c) 1 2 3 (λb .λc . 1+ b + c) 2 3 b = 2 λ .λ . λ . ( a bc a + b + c) 1 2 3 = 1 ( c 1+2+ c) 3

a c = 3 (λa .λb .λc . a + b + c) 1 2 3 1 + 2 + 3 + 6

Dantam (Mines CSCI-400) Lambda Spring 2021 10/29 Binary and N-ary Functions Currying: Specializing Functions

Binary Curry

curry ≡ λf a . (λb . f ab)

Function curry(f, a) 1 function g(b) is f(a,b) 2 g

Example curry (λab . a + b) 42 Haskell B. Curry λb . (λab . a + b) 42 b λb . 42 + b

Dantam (Mines CSCI-400) Lambda Spring 2021 11/29 Binary and N-ary Functions Historical Interlude 0: Mathematics of Lambda Curry, Hilbert, Church, and Turing

“decision problem” via λ-calculus “decision problem” via Turing machines David Hilbert Alonzo Church

Haskell Curry Alan Turing Dantam (Mines CSCI-400) Lambda Spring 2021 12/29 Binary and N-ary Functions Exercise: Currying

Reduce the n-ary functions to unary functions:

◮ λxy . xy

◮ λxyz . (xy)+ z

Dantam (Mines CSCI-400) Lambda Spring 2021 13/29 Bindings Outline

Binary and N-ary Functions

Bindings

Dantam (Mines CSCI-400) Lambda Spring 2021 14/29 Bindings Let Expression Review

Let Expression Evaluation ◮ let x ← e1 in e2 Evaluate e1 v1 ◮ identiﬁer binding exp. body exp. Substitute v1 for x in e2, |{z} |{z} |{z} ′ yielding new e2 ◮ ′ Evaluate e2 v2

Dantam (Mines CSCI-400) Lambda Spring 2021 15/29 Bindings Example: Let and Lambda

Example (Variable Binding) Example (Function Application)

let a ← 2 in 1+ a (λa . 1+ a) 2 a = 2 a=2 1 + 2 1 + 2 + + 3 3

Notice the similarity?

Dantam (Mines CSCI-400) Lambda Spring 2021 16/29 Bindings Let to Lambda Reduction of Let to Lambda Let Lambda identiﬁer binding exp. body exp. identiﬁer body exp. binding exp. λ . let z}|{x ← z}|{e1 in z}|{e2 ( z}|{x z}|{e2 ) z}|{e1

◮ Identiﬁer x becomes lambda parameter ◮ Body exp. e2 becomes the lambda body ◮ Binding exp. e1 becomes the call argument

Example let λ a = 2 + let a ← 2 in 1+ a (λa . 1+ a) 2 1 + 2 3

Dantam (Mines CSCI-400) Lambda Spring 2021 17/29 Bindings Sequential Deﬁnition

Sequential Deﬁnition identiﬁer binding exp. Example var x ← e1 ; z}|{ z}|{ e2 var a ← 2; 1+ a body |{z} a = 2 Evaluation ◮ 1 + 2 Evaluate e1 v1 ◮ + Substitute v1 for x in e2, ′ yielding new e2 3 ◮ ′ Evaluate e2 v2

Dantam (Mines CSCI-400) Lambda Spring 2021 18/29 Bindings Example: Sequential Deﬁnitions and Let

Example (Sequential Deﬁnition) Example (Let)

var a ← 2; let a ← 2 in 1+ a 1+ a a = 2 a = 2 1 + 2 1 + 2 + + 3 3

Dantam (Mines CSCI-400) Lambda Spring 2021 19/29 Bindings Sequential Deﬁnition to Let

Reduction of Sequential Definition to Let Sequential Def. identifier binding exp. Let var x ← e1 ; identifier binding exp. body z}|{ z}|{ e2 let x ← e1 in e2 z}|{ z}|{ z}|{ body |{z}

Example var a ← 2; let a ← 2 in 1+ a (λa . 1+ a) 2 1 + 2 3 1+ a

Dantam (Mines CSCI-400) Lambda Spring 2021 20/29 Bindings Named Local Functions

Example ﬂet f (a) → 1+ a in f 2 Local Functions

name a = 2 formal param. fun. body ﬂet body   1 + 2 ﬂet f x → e1 in e2 z}|{ z}|{ z}|{ z}|{   + 3

Dantam (Mines CSCI-400) Lambda Spring 2021 21/29 Bindings Example: Local Functions and Let/Lambda

Example (Local Function) Example (Let and Lambda)

ﬂet f (a) → 1+ a in f 2 let f ← λa . 1+ a in f 2 a = 2 f = λa . 1+ a 1 + 2 (λa . 1+ a) 2 + a = 2 3 1 + 2 + 3

Dantam (Mines CSCI-400) Lambda Spring 2021 22/29 Bindings Local Functions to Let and Lambda

Reduction of Local Functions to Let and Lambda Local Function name Let / Lambda formal param. fun. body ﬂet body   let f ← λx . e1 in e2 ﬂet f x → e1 in e2 z}|{ z}|{ z}|{ z}|{  

Example ... ﬂet f (a) → 1+ a in f 2 let f ← λa . 1+ a in f 2 (λa . 1+ a) 2

Dantam (Mines CSCI-400) Lambda Spring 2021 23/29 Bindings Global Functions to Local Functions

Reduction Global Functions to Sequential Assignment Global Function name formal param. fun body   defun f x → e1 ; Local Function z}|{ z}|{ z}|{   ﬂet f (x) → e1 in e2 e2 more expressions |{z}

Example defun f (a) → 1+ a; ... ﬂet f (a) → 1+ a in f 2 let f ← λa . 1+ a in f 2 f 2

Dantam (Mines CSCI-400) Lambda Spring 2021 24/29 Bindings Example: Reduction to Lambda

name binding exp body name body binding exp ◮   let    a=2 let a ← 2 in 1+ a  λ a . 1+ a 2  1 + 2  z}|{ z}|{ z }| {  z}|{ z }| { z}|{             +    3 ◮ let x ← 1 in (let y ← 2 in x + y) let x let y (λx . let y ← 2 in x + y) 1 (λx . λy . x + y) 1 2 x=1 y=1 + (λy . 1+ y) 2 1 + 2 3 ◮ ﬂet f (x) → if x then 1 else 0 in f true ﬂet let f ← λx . if x then 1 else 0 in f true let (λf . f true)(λx . if x then 1 else 0) f =λx . ... (λx . if x then 1 else 0) true x=true if if true then 1 else 0 1

Dantam (Mines CSCI-400) Lambda Spring 2021 25/29 Bindings Exercise: Reduction to Lambda ◮ let x ← 1 in λa . x + a ◮ (let x ← 1 in λa . x + a) 2

◮ ﬂet f (a b) → a + b in f 1 2

◮ ﬂet f (n) → (if n = 0 then 1 else n ∗ f (n − 1)) in f 2

Dantam (Mines CSCI-400) Lambda Spring 2021 26/29 Bindings Summary We can program with lambda?

◮ Everything is an expression var n-ary ◮ Everything is a lambda ◮ We’ve seen: let, var, ﬂet, defun ◮ Also possible: Recursion, Conditionals, Lists, Arithmetic ◮ More to come: defun let ◮ How do we actually implement lambda? λ ◮ How do we detect errors at compile time and avoid errors at runtime? ◮ What other programming “tricks” can we now use? if {+, −, /, ∗}

Dantam (Mines CSCI-400) Lambda Spring 2021 27/29 Bindings Historical Interlude 1: From Lambda to Functional Programming

Alonzo Church. A set of postulates for the foundation of logic (Lambda Calculus). 1932. John McCarthy. Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I. (Lisp). 1960.

Guy Steele, Jr. Lambda: The ultimate {imperative, declarative}. (Scheme). 1976.

Dantam (Mines CSCI-400) Lambda Spring 2021 28/29 Bindings References

◮ Pierce. Type Systems and Programming Languages. ◮ Ch 5.2 Programming in the Lambda-Calculus

Dantam (Mines CSCI-400) Lambda Spring 2021 29/29