A little Jupyter notebook illustrating some basic lambda calculus with JavaScript :)

lambda-calculus.ipynb

{
 "cells": [
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   "cell_type": "markdown",
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   "source": [
    "# f(x)\n",
    "\n",
    "or\n",
    "\n",
    "# lambda calculus in 3 minutes, lol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Lambda Calculus\n",
    "\n",
    "The idea is that we can describe all computation with functions of one argument."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With these fundamental building blocks, how can higher concepts be encoded?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "## Booleans\n",
    "\n",
    "What are they? A binary choice!\n",
    "\n",
    "How might that be expressed as a function? Make a choice between two inputs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Function: falz]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "troo = x => y => x\n",
    "falz = x => y => y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "'wat'"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "// use the \"booleans\" to make a choice\n",
    "troo('lol')('wat')\n",
    "falz('lol')('wat')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "## Boolean logic?"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
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     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Function: falz]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "not = b => b(falz)(troo)\n",
    "not(troo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Function: falz]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "and = b1 => b2 => b1(b2)(b1)\n",
    "and(troo)(falz)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Function: troo]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "or = b1 => b2 => b1(b1)(b2)\n",
    "or(troo)(falz)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "'lol'"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iff = c => thn => els => c(thn)(els)\n",
    "iff(troo)('lol')('wat')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Numbers (well, natural numbers)\n",
    "\n",
    "What are they? Good for counting!\n",
    "\n",
    "How might that be expressed as a function? Counting function applications."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Function: two]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero = f => x => x\n",
    "one = f => x => f(zero(f)(x)) // f(x)\n",
    "two = f => x => f(one(f)(x)) // f(f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_int = n => n(i => i+1)(0)\n",
    "to_int(two)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Each successor _can_ be implementated in terms of its predecessor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "succ = n => f => x => f(n(f)(x))\n",
    "three = succ(two)\n",
    "four = succ(three)\n",
    "to_int(four)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Arithmetic?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add = m => n => f => x => n(f)(m(f)(x))\n",
    "to_int(add(two)(three))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mul = m => n => f => x => n(m(f))(x) // simplifies to: m => n => f => n(m(f))\n",
    "to_int(mul(two)(three))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp = b => e => f => x => e(b)(f)(x) // simplifies to: b => e => e(b)\n",
    "to_int(exp(two)(three))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "h/t Anjana Vakil! 🙌\n",
    "- https://www.youtube.com/watch?v=7BsfMMYvGaU&t=2s\n",
    "- https://gist.github.com/vakila/3d5cebaebf01c4c77b289b9a0388e3c8\n",
    "\n",
    "It me! ❤️\n",
    "- twitter/iamvery\n",
    "- slack/jay"
   ]
  }
 ],
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