Functional Programming Languagescs3161/19t3/Week 05/1Mon/Slides.pdfThe Functional Paradigm MinHS...

The Functional Paradigm MinHS

Functional Programming Languages

Liam O’ConnorCSE, UNSW (and data61)

Term3 2019

Functional ProgrammingMany languages have been called functional over the years:

Lisp(define (max-of lst)

(cond[(= (length lst) 1) (first lst)][else (max (first lst) (max-of (rest lst)))]))

HaskellmaxOf :: [Int] → IntmaxOf = foldr1 max

JavaScript?function maxOf(arr) {

var max = arr .reduce(function(a, b) {return Math.max(a, b);

JavaScript?function maxOf(arr) {

var max = arr .reduce(function(a, b) {return Math.max(a, b);

What do theyhave in common?

Definitions

Unlike imperative languages, functional programming languagesare not very crisply defined.

Attempt at a Definition

A functional programming language is a programming languagederived from or inspired by the λ-calculus, or derived from orinspired by another functional programming language.

The result? If it has λ in it, you can call it functional.

In this course, we’ll consider purely functional languages, whichhave a much better definition.

Definitions

Unlike imperative languages, functional programming languagesare not very crisply defined.

Attempt at a Definition

A functional programming language is a programming languagederived from or inspired by the λ-calculus, or derived from orinspired by another functional programming language.

The result? If it has λ in it, you can call it functional.

In this course, we’ll consider purely functional languages, whichhave a much better definition.

Why Study FP Languages?

Think of a major innovation in the area of programming languages.

Garbage Collection?

Lisp, 1958

Functions as Values?Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

Monads?Haskell, 1991

Software Transactional Memory?

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Functions as Values?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

Monads?

Haskell, 1991

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Garbage Collection?

Lisp, 1958

Polymorphism?

ML, 1973

Type Inference?

ML, 1973

Metaprogramming?

Lisp, 1958

Lazy Evaluation?

Miranda, 1985

GHC Haskell, 2005

Purely Functional Programming LanguagesThe term purely functional has a very crisp definition.

Definition

A programming language is purely functional if β-reduction (orevaluation in general) is actually a confluence.In other words, functions have to be mathematical functions, andfree of side effects.

Consider what would happen if we allowed effects in a functionallanguage:

count = 0;f x = {count := count + x ; return count};m = (λy . y + y) (f 3)

If we evaluate f 3 first, we will get m = 6, but if we β-reduce mfirst, we will get m = 9. ⇒ not confluent.

Purely Functional Programming LanguagesThe term purely functional has a very crisp definition.

Definition

A programming language is purely functional if β-reduction (orevaluation in general) is actually a confluence.In other words, functions have to be mathematical functions, andfree of side effects.

Consider what would happen if we allowed effects in a functionallanguage:

count = 0;f x = {count := count + x ; return count};m = (λy . y + y) (f 3)

If we evaluate f 3 first, we will get m = 6, but if we β-reduce mfirst, we will get m = 9. ⇒ not confluent.

Making a Functional Language

We’re going to make a language called MinHS.

1 Three types of values: integers, booleans, and functions.

2 Static type checking (not inference)

3 Purely functional (no effects)

4 Call-by-value (strict evaluation)

In your Assignment 1, you will be implementing an evaluator for aslightly less minimal dialect of MinHS.

Syntax

Integers n ::= · · ·Identifiers f , x ::= · · ·Literals b ::= True | FalseTypes τ ::= Bool | Int | τ1 → τ2Infix Operators ~ ::= * | + | == | · · ·Expressions e ::= x | n | b | (e) | e1 ~ e2

| if e1 then e2 else e3

| e1 e2| recfun f :: (τ1 → τ2) x = e↑ Like λ, but with recursion.

As usual, this is ambiguous concrete syntax. But all theprecedence and associativity rule apply as in Haskell. We assume asuitable parser.

Syntax

| if e1 then e2 else e3| e1 e2

| recfun f :: (τ1 → τ2) x = e↑ Like λ, but with recursion.

Syntax

| if e1 then e2 else e3| e1 e2| recfun f :: (τ1 → τ2) x = e↑ Like λ, but with recursion.

Examples

Example (Stupid division by 5)

recfun divBy5 :: (Int→ Int) x =if x < 5

then 0else 1 + divBy5 (x − 5)

Example (Average Function)

recfun average :: (Int→ (Int→ Int)) x =recfun avX :: (Int→ Int) y =

(x + y) / 2

As in Haskell, (average 15 5) = ((average 15) 5).

We don’t need no let

This language is so minimal, it doesn’t even need let expressions.How can we do without them?

let x :: τ1 = e1 in e2 :: τ2 ≡ (recfun f :: (τ1 → τ2) x = e2) e1

We don’t need no let

This language is so minimal, it doesn’t even need let expressions.How can we do without them?

let x :: τ1 = e1 in e2 :: τ2 ≡ (recfun f :: (τ1 → τ2) x = e2) e1

Abstract Syntax

Moving to first order abstract syntax, we get:

1 Things like numbers and boolean literals are wrapped in terms(Num, Lit)

2 Operators like a + b become (Plus a b).

3 if c then t else e becomes (If c t e).

4 Function applications e1 e2 become explicit (Apply e1 e2).

5 recfun f :: (τ1 → τ2) x = e becomes (Recfun τ1 τ2 f x e).

6 Variable usages are wrapped in a term (Var x).

What changes when we move to higher order abstract syntax?

1 Var terms go away – we use the meta-language’s variables.

2 (Recfun τ1 τ2 f x e) now uses meta-language abstraction:(Recfun τ1 τ2 (f . x . e)).

Abstract Syntax

Working Statically with HOAS

To Code

We’re going to write code for an AST and pretty-printer for MinHSwith HOAS.Seeing as this requires us to look under abstractions withoutevaluating the term, we have to extend the AST with special “tag”values.

Static SemanticsTo check if a MinHS program is well-formed, we need to check:

1 Scoping – all variables used must be well defined2 Typing – all operations must be used on compatible types.

Our judgement is an extension of the scoping rules to include types:

Γ ` e : τ

Under this context of assumptions

The expression is assigned this type

The context Γ includes typing assumptions for the variables:

x : Int, y : Int ` (Plus x y) : Int

Static SemanticsTo check if a MinHS program is well-formed, we need to check:

1 Scoping – all variables used must be well defined2 Typing – all operations must be used on compatible types.

Our judgement is an extension of the scoping rules to include types:

Γ ` e : τ

Under this context of assumptions

The expression is assigned this type

The context Γ includes typing assumptions for the variables:

x : Int, y : Int ` (Plus x y) : Int

Static Semantics

Γ ` (Num n) : Int Γ ` (Lit b) : Bool

Γ ` e1 : Int Γ ` e2 : Int

Γ ` (Plus e1 e2) : Int

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) :

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Γ ` (Apply e1 e2) :

Let’s implement a type checker.

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) :

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool

Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) :

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

x : τ1, f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1,

f : (τ1 → τ2)

` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1, f : (τ1 → τ2) ` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) :

τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1, f : (τ1 → τ2) ` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) : τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1, f : (τ1 → τ2) ` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) : τ1 → τ2

Γ ` e1 : τ1 → τ2

Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1, f : (τ1 → τ2) ` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) : τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Static Semantics

Γ ` e1 : Bool Γ ` e2 : τ Γ ` e3 : τ

Γ ` (If e1 e2 e3) : τ

(x : τ) ∈ Γ

Γ ` x : τ

Γ, x : τ1, f : (τ1 → τ2) ` e : τ2

Γ ` (Recfun τ1 τ2 (f . x . e)) : τ1 → τ2

Γ ` e1 : τ1 → τ2 Γ ` e2 : τ1

Γ ` (Apply e1 e2) : τ2

Dynamic Semantics

Structural Operational Semantics (Small-Step)

Initial states:

All well typed expressions.Final states: (Num n), (Lit b), Recfun too!

Evaluation of built-in operations:

e1 7→ e ′1

(Plus e1 e2) 7→ (Plus e ′1 e2)

(and so on as per arithmetic expressions)

Dynamic Semantics

Initial states: All well typed expressions.Final states:

(Num n), (Lit b), Recfun too!

e1 7→ e ′1

Dynamic Semantics

Initial states: All well typed expressions.Final states: (Num n), (Lit b),

Recfun too!

e1 7→ e ′1

Dynamic Semantics

Initial states: All well typed expressions.Final states: (Num n), (Lit b), Recfun too!

e1 7→ e ′1

Specifying If

e1 7→ e ′1

(If e1 e2 e3) 7→ (If e ′1 e2 e3)

(If (Lit True) e2 e3) 7→ e2

(If (Lit False) e2 e3) 7→ e3

How about Functions?Recall that Recfun is a final state – we don’t need to evaluate itwhen it’s alone.

Evaluating function application requires us to:

1 Evaluate the left expression to get the function being applied

2 Evaluate the right expression to get the argument value

3 Evaluate the function’s body, after supplying substitutions forthe abstracted variables.

e1 7→ e′1(Apply e1 e2) 7→ (Apply e′1 e2)

e2 7→ e′2(Apply (Recfun . . . ) e2) 7→ (Apply (Recfun . . . ) e′2)

v ∈ F

(Apply (Recfun τ1 τ2 (f .x . e)) v) 7→ e[x := v , f := (Recfun τ1 τ2 (f .x . e))]

Functional Programming Languagescs3161/19t3/Week 05/1Mon/Slides.pdfThe Functional Paradigm MinHS...

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