A Theoreticians Programming Language Do we need... Function - - PowerPoint PPT Presentation

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a theoretician s programming language

A Theoreticians Programming Language Do we need... Function - - PowerPoint PPT Presentation

Oct 29, 2022 10 likes •443 views

A Theoreticians Programming Language Do we need... Function definition? Martin Henz From the Lambda Calculus to JavaScript A Theoreticians Programming Language Do we need... Function definition? granted! Martin Henz From the Lambda

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A Theoretician’s Programming Language

Do we need... Function definition?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no! Conditionals?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no! Conditionals? no!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no! Conditionals? no! Recursive functions?

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no! Conditionals? no! Recursive functions? no!

Martin Henz From the Lambda Calculus to JavaScript

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A Theoretician’s Programming Language

Do we need... Function definition? granted! Function application? granted! Functions with multiple parameters? no! Numbers? no! Conditionals? no! Recursive functions? no!

Martin Henz From the Lambda Calculus to JavaScript

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Some Examples

function square(x) { return x * x; } square(13);

Martin Henz From the Lambda Calculus to JavaScript

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Do we need multiple arguments?

function plus(x,y) { return x + y; } plus(5,7);

Martin Henz From the Lambda Calculus to JavaScript

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Do we need multiple arguments?

function plus(x,y) { return x + y; } plus(5,7); becomes function plus(x) { function plusx(y) { return x + y; } return plusx; } var plusfive = plus(5); plusfive(7);

Martin Henz From the Lambda Calculus to JavaScript

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Another Example

function power(x,y) { if (y === 0) return 1; return x * power(x,y-1); } power(2,4);

Martin Henz From the Lambda Calculus to JavaScript

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Do we need multiple arguments?

function power(x,y) { if (y === 0) return 1; return x * power(x,y-1); } power(2,4); translates to: function power(x) { return function(y) { if (y === 0) return 1; return x * power(x)(y-1); }; } power(2)(4);

Martin Henz From the Lambda Calculus to JavaScript

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Do we need numbers?

Representing 0 using Church numerals: function zero(f) { return function(x) { return x; } } zero(’something’)(’somethingelse’)

Martin Henz From the Lambda Calculus to JavaScript

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Do we need numbers?

Representing 1 using Church numerals: function one(f) { return function(x) { return f(x); } }

ne(function(x) { return x*2; })(4)

Martin Henz From the Lambda Calculus to JavaScript

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Do we need numbers?

Representing 2 using church numerals: function two(f) { return function(x) { return f(f(x)); } } two(function(x) { return x*2; })(4)

Martin Henz From the Lambda Calculus to JavaScript

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Getting the number back

function two(f) { return function(x) { return f(f(x)); } } function church2js(c) { return c(function(x) { return x+1; })(0); } church2js(two);

Martin Henz From the Lambda Calculus to JavaScript

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Can you do the reverse?

Wanted: Define a function js2church that takes a JediScript number as argument and returns its Church numeral?

Martin Henz From the Lambda Calculus to JavaScript

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Multiplication

function times(x) { return function(y) { return function(f) { return x(y(f)); } } } function two(f) { return function(x) { return f(f(x)); } }

Martin Henz From the Lambda Calculus to JavaScript

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Multiplication

function three(f) { return function(x) { return f(f(f(x))); } } function church2js(c) { return c(function(x) { return x+1; })(0); } church2js(times(two)(three));

Martin Henz From the Lambda Calculus to JavaScript

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Conditionals

Conditional statements if (20 < 10) { return 5; } else { return 7; } Conditional expressions (20 < 10) ? 5 : 7

Martin Henz From the Lambda Calculus to JavaScript

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Do we need conditionals?

Idea Represent booleans with functions The function “true” function True(x) { return function(y) { return x; } }

Martin Henz From the Lambda Calculus to JavaScript

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Do we need conditionals?

Idea Represent booleans with functions The function “false” function False(x) { return function(y) { return y; } }

Martin Henz From the Lambda Calculus to JavaScript

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Do we need conditionals?

Conditional in JediScript True ? 5 : 7; Conditional using Encoding True(5)(7);

Martin Henz From the Lambda Calculus to JavaScript

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Factorial using Conditional Expressions

function factorial(x) { return (x === 0) ? 1 : x * factorial(x - 1); } factorial(5);

Martin Henz From the Lambda Calculus to JavaScript

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Step 1: Eliminate Recursive Call

function F(f) { return function(x) { return (x === 0) ? 1 : x * f(x - 1); }; }

Martin Henz From the Lambda Calculus to JavaScript

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Step 2: Find a Fix-Point Function (aka Y-Combinator)

We need a function Y with the following properties: Y(F) ≡ F(Y(F))

Martin Henz From the Lambda Calculus to JavaScript

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Step 2: A Y-Combinator

function (f) { return (function (x) { return f(function(y) { return x(x)(y); }); }) (function (x) { return f(function(y) { return x(x)(y); }); }); }

Martin Henz From the Lambda Calculus to JavaScript

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The Pure (Untyped) Lambda Calculus

As a sublanguage of JediScript, the Lambda Calculus looks like this: L ::= id | (L)(L); | function(id) { return L; }

Martin Henz From the Lambda Calculus to JavaScript

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So: Why don’t we program using the Lambda Calculus?

Answer Other design goals are equally important!

Martin Henz From the Lambda Calculus to JavaScript

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So: Why don’t we program using the Lambda Calculus?

Answer Other design goals are equally important! Some design goals for full JavaScript Expressive Easy to learn Convenient to use

Martin Henz From the Lambda Calculus to JavaScript

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So: Why don’t we program using the Lambda Calculus?

Answer Other design goals are equally important! Some design goals for full JavaScript Expressive Easy to learn Convenient to use At the expense of...

Martin Henz From the Lambda Calculus to JavaScript

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So: Why don’t we program using the Lambda Calculus?

Answer Other design goals are equally important! Some design goals for full JavaScript Expressive Easy to learn Convenient to use At the expense of... simplicity!

Martin Henz From the Lambda Calculus to JavaScript

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Lambda Calculus: Some History

Introduced by Alonzo Church in 1930s as a minimal formal system for recursion theory Later found to be equivalent to other computing frameworks (Church-Turing thesis) Used extensively in programming language theory and theoretical computer science

Martin Henz From the Lambda Calculus to JavaScript

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Summary

Simplicity is an important and highly useful driving force behind science and engineering

Martin Henz From the Lambda Calculus to JavaScript

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Summary

Simplicity is an important and highly useful driving force behind science and engineering Enables insights that would otherwise remain lost in a thicket of details

Martin Henz From the Lambda Calculus to JavaScript

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Summary

Simplicity is an important and highly useful driving force behind science and engineering Enables insights that would otherwise remain lost in a thicket of details In practice, simplicity competes with other goals; keep it in mind when thinking about complex systems

Martin Henz From the Lambda Calculus to JavaScript

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Homework

Write a lambda expression EXP such that lambda2jediscript((EXP)(jediscript2lambda(6))); will result in the factorial of 6 when entered in the Web Console

f Firefox.

You will need: conditionals using the shown encoding your implementation of lambda2jediscript and jediscript2lambda the Y combinator shown above addition, multiplication predecessor “n - 1” is the hardest

Martin Henz From the Lambda Calculus to JavaScript