basics of programming languages (pre lecture) · basics of programming languages (pre lecture) dr....

Basics of Programming Languages (Pre Lecture)

Dr. Neil T. Dantam

CSCI-400, Colorado School of Mines

Spring 2021

Dantam (Mines CSCI-400) Basics Spring 2021 1 / 34

Introduction

PL BasicsI Common features across

programming languages:I ExpressionsI VariablesI FunctionsI Etc.

I Mathematical basis:Lambda Calculus

I A simple/clean/elegant abstraction

OutcomesI Know definitions of expressions,

functions, common constructs

I Understand (basic) evaluation andtype checking

I Understand how to define anonymousfunctions (lambdas)


Expressions

Outline

Expressions

Functions


Expressions

Expressions

Definition (Expression)

An expression is a finite combination of symbols which may bereduced (rewritten) to obtain a simpler expression or a result.

Example

1 + 2︸︷︷︸expression

3︸︷︷︸also an expression

Rewrite


Expressions

Expressions in Imperative and Functional Programming

Use of Expressions

I Primary building block of functional programs

I Imperative languages: some expressions and somestatements/commands (not expressions)

I (Purely) Functional Languages: Everything is an expression

Example (C Expressions)

I 1+2

I 3

I x ? 1 : 2

Counterexample (C Statements)

I i f ( x ) { y =1; } e l s e { y =2; }I whi le ( i ) { i −−;}


Expressions

Syntax and Semantics

Expression Syntax

I Is the expression a valid combinationof symbols?

C Syntax

I Valid: (1 + 2) * 3

I Invalid: (1 + 2 * 3

Expression Semantics

I Type-checking rules(static semantics):Produce a type or fail with an error

I Evaluation rules(dynamic semantics):Produce a value, exception, or infiniteloop

We will precisely define syntax and semantics over this course.


Expressions

Values

Definition (Value)

A value is an expression that does not need further evaluation.

Illustration

Values Expressions

Example

1 + 2︸︷︷︸expression

3︸︷︷︸value, expression

Rewrite


Expressions

Notation: Evaluation and Typeing

Evaluation

“α evaluates to β”or “α rewrites to β”

α β

Example

I 1 + 2 3

I ¬true false

I ¬¬true ¬false true

Typing

“α has type τ”or “α is in τ”

α : τ or α ∈ τ

Example

I 3 : int

I false : bool

I ¬true : bool


Expressions

If Expressionsaka Conditionals

If Expression

if e1︸︷︷︸test

then e2︸︷︷︸then clause

else e3︸︷︷︸else clause

EvaluationI When e1 true and e2 v2,

if e1 then e2 else e2 v2

I When e1 false and e3 v3,if e1 then e2 else e2 v3

Type checking

When

I e1 : bool

I e2 : τ

I e3 : τ

(if e1 then e2 else e3) : τ


Expressions

Type inference and annotation

Type Inference

I We can infer the type of an if expression fromthe types of its constituent expressions.

I type inference is generally possible in functionallanguages and is performed by compilers

I If types cannot be inferred, compilation failswith a type error

I May add annotations to test/diagnose:Replace e with e:t

If Expression Type

(if e1 then e2 else e3) : τ

where

I e1 : bool

I e2 : τ

I e3 : τ

Capability: “If it compiles, it (probably) works.”


Expressions

Example: If Evaluation and Type-checking

if true then 1 else 2

1

if

Evaluation


if (true : bool) then (1 : int) else (2 : int)

if (true : bool) then (1 : int) else (2 : int) : int

subexpressions

if

Type Checking


Expressions

Example: Type Errors

if 1 then false else true

if (1 : int) then (false : bool) else (true : bool)

ERROR

if true then 1 else false

if (true : bool) then (1 : int) else (false : bool)

ERROR


Expressions

Definitions

Definition (definition)

A definition gives a name to a value.

IllustrationI Definitions are disjoint from expressions

I But syntactically, definitions contain expressions

Values Expressions Definitions


Expressions

Variables

DefinitionI A Variable is a symbol representing an expression.

I Bound variables refer to some expression.

I Free variables DO NOT not refer to some expression (areunbound).

I Immutable variables CANNOT be changed (are constant).

I Mutable variables CAN be changed (reassigned).


Expressions

Let ExpressionLet Expression

let x︸︷︷︸identifier

← e1︸︷︷︸binding exp.

in e2︸︷︷︸body exp.

EvaluationI Evaluate e1 v1

I Substitute v1 for x in e2, yielding newexpression e ′2

I Implementation: there is a memory locationnamed x that contains/references v

I Evaluate: e ′2 v2

I Result:let x ← e1 in e2 v2

Type checkingI e1 : τ1

I x : τ1

I e2 : τ2

I let x ← e1 in e2 : τ2


Expressions

Example: Expression Evaluation

let x ← true in if x then 1 else 2


1

let

if

let x ← if false then 1 else 2 in x

let x ← 2 in x

2

if

let


Expressions

Exercise: Expression Evaluation

if let x ← true in ¬x then false else true

if ¬true then false else true

if false then false else true

true

let

¬

if

if if true then false else true then 1 else 2

if false then 1 else 2

2

if

if


Expressions

Example: Simple Type Propagation (0)

let x ← true in if x then 1 else 2

let x ← (true : bool) in if x then (1 : int) else (2 : int)

let (x : bool)← (true : bool) in if (x : bool) then (1 : int) else (2 : int)

let (x : bool)← (true : bool) in (if (x : bool) then (1 : int) else (2 : int) : int)

let (x : bool)← (true : bool) in (if (x : bool) then (1 : int) else (2 : int) : int) : int

literals

let binding

if

let body


Expressions

Example: Simple Type Propagation (1)

let x ← 1 in if x then false else true

let x ← (1 : int) in if x then (false : bool) else (true : bool)

let (x : int)← (1 : int) in if (x : int) then (false : bool) else (true : bool)

ERROR!

literals

let binding

if


Expressions

Exercise: Simple Type Propagation (0)

if let x ← true in ¬x then false else true

if let x ← (true : bool) in ¬x then (false : bool) else (true : bool)

if let (x : bool)← (true : bool) in ¬(x : bool) then (false : bool) else (true : bool)

if let (x : bool)← (true : bool) in (¬(x : bool) : bool) then (false : bool) else (true : bool)

if (let (x : bool)← (true : bool) in (¬(x : bool) : bool) : bool) then (false : bool) else (true : bool)

if (let (x : bool)← (true : bool) in (¬(x : bool) : bool) : bool) then (false : bool) else (true : bool) : bool

literal

let binding

not (¬)

let body

if


Expressions

Exercise: Simple Type Propagation (1)

if if true then false else true then 1 else 2

if if (true : bool) then (false : bool) else (true : bool) then (1 : int) else (2 : int)

if (if (true : bool) then (false : bool) else (true : bool) : bool) then (1 : int) else (2 : int)

if (if (true : bool) then (false : bool) else (true : bool) : bool) then (1 : int) else (2 : int) : int

literal

if

if


Functions

Outline

Expressions

Functions


Functions

Lambda

Function Definition

λ

formal parameter︷︸︸︷x . α︸︷︷︸

body exp.

I Create an anonymous (unnamed) function.

I Do not evaluate the body until applied (called)

I The function is itself a value.

Function Application

function︷︸︸︷f z︸︷︷︸

actual parameter

I Bind actual parameter z tothe formal parameter of f .

I Evaluate the body of f .

Example

(λa . 1 + a) 2 1 + 2 3a = 2 +


Functions

Notation Reorientation

Grade-school Algebra: Implicit Multiply

a b

“multiply a times b”

readas

Lambda Calculus: Function Call

a b

“apply function a to parameter b”

readas

Importance of abstract syntax:What operation is specified?


Functions

Contrasting Notations

Function Definition

Other notation

function(x) {return 1 + x}

Lambda Calculus

λx . 1 + x

Function Application

Other notation

f (2)

Lambda Calculus

f 2

Minimalist.


Functions

Lambda: Very Important!

I Lambda can performany possible computation!

I Trivial with Lambda:I N-ary functionsI Variable binding (let)I Statements (;)I Named functionsI OOP

I Also possible with Lambda:I RecursionI Conditionals (if)I ListsI StructsI Arithmetic (+, −, etc.)

λ“I’m kind of a big deal.”


Functions

Function Call AssociativityEvaluate Left to Right

(a0a1a2a3) = ((((a0a1)a2)a3)

call

call

call

a0

fun.

a1arg.

fun.

a2

arg.

fun.

a3

arg.


Functions

Function Call ParenthesizationUse parenthesis to change evaluation order

(a0 a1) (a2 a3)

call

call

a0

fun.

a1

arg.

fun.

call

a2

fun.

a3

arg.

arg.


Functions

Functions are values

Bind variables to functions

Pass one function as the parameter to another

Return one function as the result of another function


Functions

Example: Binding and Passing Functions

Example (Binding)

let f ← (λa . 1 + a) in f 2

(λa . 1 + a) 2

1+2

3

let

a=2

+

Example (Passing)

(λf . f 2) (λa . 1 + a)

(λa . 1 + a) 2

1+2

3

f = λa . 1+a

a=2

+

Notice the similarity?Dantam (Mines CSCI-400) Basics Spring 2021 30 / 34

Functions

Example: Returning Functions

Example (Returning)

(λb . λa . b + a) 1 2

(λa . 1 + a) 2

1+2

3

b = 1

a=2

+


Functions

Exercise: Lambda Evaluation

I(λx . x) y

y

I(λx . x) (λy . y)

λy . y

I(λx . (λy . xy)) z

λy . zy

I(λx . x x) (λx . x x)

(λx . x x) (λx . x x) (λx . x x) (λx . x x)

. . .


Functions

Summary

I Everything is an expressionI Evaluate itI Type Check it

I lambda defines anonymous functions(which are also expressions) λ

“I’ll be back.”


Functions

References

I Clarkson. Functional Programming in OCaml.I Ch 2 The basics of OCaml

I Pierce. Type Systems and Programming Languages.I Ch 5 The Untyped Lambda-Calculus

I Hennessy. The Semantics of Programming Languages.I Ch 1.1 Preliminaries: IntroductionI Ch 2.1-2.3 Arithmetic Expressions: Concrete Operational Semantics, Evaluation Semantics,

Computation SemanticI Ch 3: A Simple Functional Language


https://www.cs.cornell.edu/courses/cs3110/2020fa/textbook/

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