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README.md

README.md

es6-lambda-calculus

implementing λ-calculus using EcmaScript 6 arrow notation (currently only on Chrome, FF, IE11 and Edge)

inspired by the study of "Functional Programming through Lambda Calculus" (G. Michaelson)

made possible only by the mind-blowing work of Alonzo Church

Alonzo Church

At each paragraph you can:

  • view the related ES6 code
  • run and exercise the ES6 code (open in new tab the HTML file and start the browser console - currently only on Chrome, FF, IE11 and Edge)

Table of Contents


definitions

ES6 code in 01-definitions-first-examples-es6.js
HTML is 01-definitions-first-examples.html

lambda-expression := name | function | function application

a "resolved" (or "evalued") λ-expression is called a value

name := character sequence
e.g. IDENTITY, pippo, C3PO

keyword def binds names to expressions

def IDENTITY = λx.x  // this is a comment (something that Alonzo Church actually did NOT foresee :-)
def TWELVE = (SUCCESSOR ELEVEN)
def id = IDENTITY  // that's a bit tricky, I admit...

functions

function := λ name . lambda-expression
name --> "variable"
lambda-expression --> "body" (contains one or more references to the variable)

e.g. λx.x, λpippo.pippo*2,

when the function body is again a function, we are actually watching a curried n-args function as a whole:

λx.(λy.x+y) = λx.λy.x+y

when a function is mentioned inside a def, we can shift variables to the left side of the definition:

def SUM = λx.λy.x+y --> SUM x = λy.x+y --> SUM x y = x+y // NB: "def" appears only in the first expression!!

NB: things like pippo*2 and x+y are used here just to give an idea of what function bodies are made for, but they are not legitimate. Actually, function bodies can host only precise λ-expressions. The most relevant function bodies contain function applications

why use EcmaScript 6?

λ-calculus is generally done with paper and pencil or in esoteric niche languages; ES6 is a good-enough trade-off, because it runs in (almost) every browser and it allows a good approximation of λ-functions using its brand new arrow function expressions

NB - as of 160322, arrow functions are implemented in released versions of Firefox, IE11 and Edge only, but widespread adoption seems underway

// NB: this is valid ES6 syntax
var IDENTITY = x => x;
var SUM = x => y => x + y;

alas, using ES6 we cannot shift variables to the left (use Haskell for that...)

throughout this README we will mix expressions in lambda-calculus notation with other ones in JavaScript/EcmaScript; you gotta pay attention whether the line begins with def or no keyword at all (λ-calculus notation), as opposed to starting with var (that's JavaScript)

function applications

function-application := (function lambda-expression)

when performing a function application, the λ-expression gets substituted to the function variable inside the function body;

after application the variable is bound, meaning that it has a given value inside the function body; bound variables cannot be further modified;

for example (still using illegitimate expressions +, 1 and 2 ;-):

(λx.x+1 1) = 2  // variable x gets bound to value 1; 1 plus 1 is 2

(λx.x λy.y) = λy.y  // variable x gets bound to lambda-expression λy.y and returned as-is in the body of the first function

(λx.x λx.x) = λx.x  // variable x in the FIRST function gets bound to lambda-expression λx.x (no relation whatsoever between the x's of each function)

((λx.λy.x+y 1) 2) = (λy.1+y 2) = 3  // variables x and y get bound stepwise to value 1 and value 2

the convention is that function application associates to the left:

(λx.λy.x+y 1 2) = ((λx.λy.x+y 1) 2)

the convention is also that function application may be omitted every time this does not bring ambiguities in the expression:

λx.λy.x+y 1 2 = (λx.λy.x+y 1 2)

two remarkable applicators

function APPLY

function APPLY is the abstraction of application; it applies whatever function to whatever argument:

def APPLY = λf.λx.(f x)
var APPLY = f => x => f(x)
function SELF_APPLY

function SELF_APPLY is a quirky thing; it applies whatever function to itself. It is customary to call s the argument of SELF_APPLY:

def SELF_APPLY = λs.(s s)
var SELF_APPLY = s => s(s)

this function has a very strange, rather unique characteristic: if you try with paper and pencil the next expression, you will verify that:

(SELF_APPLY SELF_APPLY) = (λs.(s s) λs.(s s)) = ... = (SELF_APPLY SELF_APPLY)

because of this peculiarity, SELF_APPLY takes a primary role in the λ-calculus definition and implementation of recursion

two ways of looking at function application

application inside function bodies means to bind every variable as soon as we know what it is (either a value or another λ-expression)

application to can be thought of in two ways:

  • call-by-value (or applicative order reduction): substitutions can be made only using values; if we are substituting a λ-expression, we must evaluate it first ⟹ eager languages

      var z = (x => x + 1)((y => 2*y)(6)) = (x => x + 1)(12) = 12 + 1 = 13
    

y gets bound straight away to 6; immediately after, x gets bound to 12 and z is immediately evaluated

  • call-by-name (or normal order reduction): substitutions are made without evaluating the expressions first; actual values will get computed only when actually needed ⟹ lazy languages

      var z = (x => x + 1)((y => 2*y)(6)) = (y => 2*y)(6) + 1
    

x gets literally bound to the operation (y => 2*y)(6); all actual values of y, x and z will get computed only when explicitly requested, for example by a console.log(z)

NB: ES6 is an eager language, so the call-by-name example ain't real running code, and it's here just for the show

trying to implement recursion using SELF_APPLY in an eager language is impossible: every useable recursive mechanism needs to have first thing a boolean check to decide whether to get out of the loop or recur once more; well, a pure call-by-value nature requests the interpreter to evaluate both left and right branch before deciding which path to take; but evaluating the right branch implies another turn of the wheel, a new right branch to interpret along with the left, and then another, and then another... ad libitum before anything useful can be done with the expression

our very first solution for implementing recursion will actually be a cheat: we will not use pure λ-expressions, we will mix JavaScript's if then else operator (which evaluates either branch depending on the boolean check) in our code

but, but, but... in a latter paragraph we will eventually manage to implement recursion in our ES6 pure λ-expressions, still using SELF_APPLY but feeding it with lazy function applications

pairs

a pair is a mean to freeze two expressions in order be able to operate on them later (i.e. to apply a function upon them):

def PAIR = λx.λy.λf.(f x y)  // PAIR a b = λf.(f a b)

f must be a function accepting two arguments

def f = λx.λy.whatever  // x will be bound to the original x's value; y will be bound to the original y's value

the two simplest operations on pairs are the getters FIRST and SECOND:

PAIR a b FIRST = a  --> def FIRST = λx.λy.x
PAIR a b SECOND = b --> def SECOND = λx.λy.y
back to the top

booleans

ES6 code in 02-booleans-es6.js
HTML is 02-booleans.html

Let's start our exposition from the ternary operator, one of the most basic concepts in computer science:

condition ? true_exp : false_exp

we want to define it in terms of λ-calculus, where it is customary to call the ternary operator COND; one possibility is the following function:

def COND = λtrue_exp.λfalse_exp.λcondition.(condition true_exp false_exp)

we realize that we've already encountered a function with exactly the same behavior as COND:

def COND = PAIR

we attempt now to define TRUE and FALSE as functions; whatever they'll come out to be, let's pass them as third variable to COND:

COND a b TRUE = a => def TRUE = λx.λy.x = FIRST

COND a b FALSE = b => def FALSE = λx.λy.y = SECOND

so we see that also TRUE and FALSE are functions that we'd encountered before

I hear your objection: "in language XYZ FIRST and SECOND are functions, while TRUE and FALSE are values"; well, as a matter of fact, in λ-calculus functions are values, so there's no contradiction

substituting in COND the function body as soon as all the variables have been bound, we come to a form that will be used a lot in the following paragraphs:

COND true_exp false_exp condition = condition true_exp false_exp

NOT operator

definition using the ternary operator is straightforward:

!x = x ? false : true

expressing NOT in terms of λ-calculus is also straightforward:

def NOT x = COND FALSE TRUE x = x FALSE TRUE   // if x is true, then NOT x is FALSE; otherwise it's TRUE

AND operator

definition using the ternary operator is the following:

x && y = x ? y : false

expressing AND in terms of λ-calculus derives immediately:

def AND x y = COND y FALSE x = x y FALSE   // if x is true, then AND x y depends on y; otherwise it's certainly FALSE

OR operator

definition using the ternary operator is the following:

x || y = x ? true : y

expressing OR in terms of λ-calculus derives immediately:

def OR x y = COND TRUE y x = x TRUE y   // if x is true, then OR x y is certainly TRUE; otherwise it depends on y
back to the top

natural numbers

ES6 code in 03-natural-numbers-es6.js
HTML is 03-natural-numbers.html

so far we have been using symbols like 1 or 2 to express numbers; unfortunately, we've been cheating: this practice is not allowed in λ-calculus

lambda-expressions are strictly limited to names, functions and applications; as we did in the previous paragraph with regards to booleans, we'll have to express numbers using only those three concepts; let's start...

we plainly decide to express number 0 with the identity function:

def ZERO = IDENTITY = λx.x

we express every natural number as the SUCCessor of its preceding one:

def ONE = SUCC ZERO
def TWO = SUCC ONE = SUCC (SUCC ZERO)

ok, now we need to define SUCC; we just decide to pick:

def SUCC = λn.(PAIR FALSE n)

or, in other words...

SUCC n = PAIR FALSE n

these picks for ZERO and SUCC are just two of the infinite possibilities; they just happen to be simple and powerful enough to start our conversation. Actually Church ended up with less simple yet (a lot) more powerful definitions.

def ONE = SUCC ZERO = PAIR FALSE IDENTITY  // rem ZERO = IDENTITY
def TWO = SUCC ONE = PAIR FALSE (PAIR FALSE IDENTITY)
def THREE = SUCC TWO = PAIR FALSE (PAIR FALSE (PAIR FALSE IDENTITY))

let's begin to slowly build something really useful; first step is to become able to tell whether a number is ZERO or not; here are the specs for the function ISZERO:

ISZERO = COND TRUE FALSE ZERO = ZERO TRUE FALSE
ISZERO IDENTITY = TRUE --> ISZERO λx.x = λx.λy.x
ISZERO ONE = FALSE
ISZERO (PAIR FALSE ZERO) = FALSE

a function that satisfies all these requirements is:

def ISZERO = λn.(n FIRST)

or, in other words...

ISZERO n = n FIRST

let's verify:

ISZERO ZERO = ZERO FIRST = IDENTITY FIRST = FIRST = TRUE
ISZERO ONE = PAIR FALSE ZERO FIRST = FALSE
ISZERO TWO = PAIR FALSE (PAIR FALSE ZERO) FIRST = FALSE

now we define the PREDecessor function, such that:

PRED ONE = PRED (PAIR FALSE ZERO) = ZERO
PRED TWO = PRED (PAIR FALSE (PAIR FALSE ZERO)) = ONE = PAIR FALSE ZERO 

therefore we could initially say that NAIVE_PRED n = n SECOND, but we need to guard against n = ZERO

NAIVE_PRED ZERO = ZERO SECOND = SECOND = FALSE // not a number anymore

we define then that PRED ZERO = ZERO and we get:

predecessor(n) = isZero(n) ? 0 : naivePredecessor(n)  // using the ternary operator

PRED n = COND ZERO (NAIVE_PRED n) (ISZERO n) = (ISZERO n) ZERO (NAIVE_PRED n) = (ISZERO n) ZERO (n SECOND)
back to the top

recursion and arithmetics

ES6 code in 04-recursion-arithmetics-es6.js
HTML is 04-recursion-arithmetics.html

addition

an intuitive definition of add is possible using recursion, first in ES6, then in λ-calculus notation:

var add = x => y => (y === 0) ? x : add(x+1)(y-1)
rec ADD x y = COND x (ADD (SUCC x) (PRED y)) (ISZERO y)

the big problem here is that we have the function name mentioned in its own body, and this normally speaking is not allowed by the rules of λ-calculus; keyword rec is used in λ-calculus (as well as in Scheme, by the way) to mean that a function is being defined recursively; it can be used freely to express the concept (and this is done a few times further down this text), but implementing rec is quite a complex matter, as we will see in the next paragraph

in order to implement recursion, we do some juggling remembering SELF_APPLY = λs.(s s), for which we know that an infinite loop (SELF_APPLY SELF_APPLY) = ... = (SELF_APPLY SELF_APPLY) exists; we create a helper function add2 that carries an additional function f in its signature and applies it to itself:

var add2 = x => y => (y === 0) ? x : f(f)(x+1)(y-1)
def ADD2 f x y = COND x (f f (SUCC x) (PRED y)) (ISZERO y) = (ISZERO y) x (f f (SUCC x) (PRED y))

you noticed that we wrote def ADD2 and not rec ADD2, because (contrary to ADD) the ADD2 function is not recursive, as it does not mention itself inside the body

this magic function ADD2 has the remarkable power that, when applied to itself, it behaves like we expected from ADD:

def ADD = ADD2 ADD2 = SELF_APPLY ADD2

let's repeat: the big gain here is that nobody mentions itself inside its own body anymore; you can see in the definition of ADD2 that the recursion is created implicitly by the self application of argument f and not by an explicit invocation

unfortunately, this truly interesting bit cannot be verified in ES6, because its eager interpreter tries to calculate both branches of the COND at all times, causing a stack overflow even if we know that only one of them should be evalued at a time

the codebase shows a little cheat about ADD2 (called ADD3) that makes use of the real JavaScript if then else (which evaluates the FALSE branch only when needed) to create a recursive addition helper that performs some crude and simple calculations

multiplication

an even intuitive, recursive definition exists for multiplication:

var mult = x => y => (y === 0) ? 0 : x + mult(x)(y-1)
rec MULT x y = COND ZERO (ADD x (MULT x (PRED y))) (ISZERO y)

which leads to a helper function which behaves like ADD2, and is also not runnable inside a browser:

MULT1 f x y = COND ZERO (ADD x (f f x (PRED y))) (ISZERO y)
def MULT = SELF_APPLY MULT1
var MULT = SELF_APPLY(MULT1) // 'Internal Error: too much recursion'

this second time we can try to abstract a bit the whole business and come up with a new way to define recursive functions like ADD and MULT:

def ADD = RECURSIVE ADD2
def MULT = RECURSIVE MULT1

where RECURSIVE is an abstraction upon which every helper function can be applied:

def RECURSIVE f = (λs.(f (s s)) λs.(f (s s)))

unfortunately RECURSIVE (also known as the fixed-point combinator or the Y-combinator) is practically a carbon copy of (SELF_APPLY SELF_APPLY) and causes an infinite loop at the very first moment we present it to the browser

the codebase shows a little cheat about MULT1 (called MULT2) that allows to run multiplication also in an eager interpreter

other operations

Alonzo Church managed to define also subtraction, power, absolute value and integer division; the numerals he designed eventually are different from the ones shown here

back to the top

introducing laziness

ES6 code in 05-laziness-es6.js
HTML is 05-laziness.html

at the previous paragraph we saw that recursion implemented through SELF_APPLY requires lazy evaluation of COND; this can be done putting a LAZY_COND in place, which operates on lazy expressions instead of evaluating them straight away

a lazy expression is a function that returns the actual expression when applied:

def LAZY_EXP = λ_.ACTUAL_EXP
var LAZY_EXP = _ => ACTUAL_EXP;  // example in JavaScript syntax

a LAZY_COND can be expressed in λ-calculus as:

def LAZY_COND true_lazy_exp false_lazy_exp condition = (condition LAZY_TRUE LAZY_FALSE) true_lazy_exp false_lazy_exp

in other words, a plain condition is evaluated; if it is TRUE, a LAZY_TRUE operator will be used to process the lazy expressions; if it is FALSE, then we'll use a LAZY_FALSE operator; here they go (also in JavaScript):

def LAZY_TRUE x y = (x)
def LAZY_FALSE x y = (y)

var LAZY_TRUE = x => y => x();
var LAZY_FALSE = x => y => y();

How to trick SELF_APPLY into using a LAZY_COND

as first simple example of how to put LAZY_COND at work, we prepare a BIGGER_X_THAN_Y evaluator; its very first naive recursive (and forbidden) expression is as follows:

rec BIGGER_X_THAN_Y x y = (AND (ISZERO y) (NOT (ISZERO x))) TRUE (BIGGER_X_THAN_Y (PRED x) (PRED y)) // forbidden syntax

this time we can express it through a helper function passed to SELF_APPLY; the helper function will exploit laziness. Here is its implementation in λ-calculus notation, as well as in JavaScript:

def BIGGER_X_THAN_Y1 f x y = LAZY_COND λ_.TRUE λ_.(f f (PRED x) (PRED y)) (AND (ISZERO y) (NOT (ISZERO x)))
def BIGGER_X_THAN_Y = SELF_APPLY BIGGER_X_THAN_Y1

var BIGGER_X_THAN_Y1 = f => x => y => LAZY_COND(_ => TRUE)(_ => f(f)(PRED(x))(PRED(y)))(AND(ISZERO(y))(NOT(ISZERO(x))));
var BIGGER_X_THAN_Y = SELF_APPLY(BIGGER_X_THAN_Y1);

this implementation is able to compute correctly only 1/3 of the possible cases:

BIGGER_X_THAN_Y(THREE)(TWO); // TRUE
BIGGER_X_THAN_Y(THREE)(FOUR); // Error - Too much recursion
BIGGER_X_THAN_Y(THREE)(THREE); // Error - Too much recursion

basically, this implementation is unable to respond FALSE; in order to allow input where the larger number is the second, we have to bring complexity in the NOT(TRUE) branch; this leads to the helper function BIGGER_X_THAN_Y2:

def BIGGER_X_THAN_Y2 f x y = LAZY_COND λ_.TRUE λ_.(LAZY_COND λ_.FALSE λ_.(f f (PRED x) (PRED y)) (AND (ISZERO x) (NOT (ISZERO y)))) (AND (ISZERO y) (NOT (ISZERO x)))
def BIGGER_X_THAN_Y = SELF_APPLY BIGGER_X_THAN_Y2

var BIGGER_X_THAN_Y2 = f => x => y => LAZY_COND(_ => TRUE)(_ => LAZY_COND(_ => FALSE)(_ => f(f)(PRED(x))(PRED(y)))(AND(ISZERO(x))(NOT(ISZERO(y)))))(AND(ISZERO(y))(NOT(ISZERO(x))));
var BIGGER_X_THAN_Y = SELF_APPLY(BIGGER_X_THAN_Y2);

this second implementation is able to compute correctly 2/3 of the possible cases:

BIGGER_X_THAN_Y(THREE)(TWO); // TRUE
BIGGER_X_THAN_Y(THREE)(FOUR); // FALSE
BIGGER_X_THAN_Y(THREE)(THREE); // Error - Too much recursion

in order to allow input where the arguments are equal, we OR one more check in the NOT(TRUE) branch; this leads to the final helper function BIGGER_X_THAN_Y3:

def BIGGER_X_THAN_Y3 f x y = LAZY_COND λ_.TRUE λ_.(LAZY_COND λ_.FALSE λ_.(f f (PRED x) (PRED y)) (OR (AND (ISZERO x) (ISZERO y)) (AND (ISZERO x) (NOT (ISZERO y))))) (AND (ISZERO y) (NOT (ISZERO x)))
def BIGGER_X_THAN_Y = SELF_APPLY BIGGER_X_THAN_Y3

// the good one!
var BIGGER_X_THAN_Y3 = f => x => y => LAZY_COND(_ => TRUE)(_ => LAZY_COND(_ => FALSE)(_ => f(f)(PRED(x))(PRED(y)))(OR(AND(ISZERO(x))(ISZERO(y)))(AND(ISZERO(x))(NOT(ISZERO(y))))))(AND(ISZERO(y))(NOT(ISZERO(x))));
BIGGER_X_THAN_Y = SELF_APPLY(BIGGER_X_THAN_Y3);

BIGGER_X_THAN_Y(THREE)(TWO); // TRUE
BIGGER_X_THAN_Y(THREE)(FOUR); // FALSE
BIGGER_X_THAN_Y(THREE)(THREE); // FALSE

a variation on the same lazy stuff can be used to define a much useful EQUAL N M function:

def EQUAL1 f x y = LAZY_COND λ_.TRUE λ_.(LAZY_COND λ_.FALSE λ_.(f f (PRED x) (PRED y)) (OR (AND (NOT (ISZERO x)) (ISZERO y)) (AND (ISZERO x) (NOT (ISZERO y))))) (AND (ISZERO y) (ISZERO x))
def EQUAL = SELF_APPLY EQUAL1

var EQUAL1 = f => x => y => LAZY_COND(_ => TRUE)(_ => LAZY_COND(_ => FALSE)(_ => f(f)(PRED(x))(PRED(y)))(OR(AND(NOT(ISZERO(x)))(ISZERO(y)))(AND(ISZERO(x))(NOT(ISZERO(y))))))(AND(ISZERO(y))(ISZERO(x)));
var EQUAL = SELF_APPLY(EQUAL1);

EQUAL(THREE)(TWO); // FALSE
EQUAL(THREE)(FOUR); // FALSE
EQUAL(THREE)(THREE); // TRUE

all the recursive functions (ADD3 and MULT2) that in the previous paragraph have been created by cheating with JavaScript's lazy if then else operator can now be implemented in pure λ-calculus notation.

a truly insightful function of this family is jsinteger, which takes a numeral and returns its value as a plain JavaScript integer; it gives a clear demonstration of the power given by the faculty of mixing JS with λ-notation:

var jsinteger1 = n => f => N => LAZY_COND(_ => n)(_ => f(n + 1)(f)(PRED(N)))(ISZERO(N)) // first n will be the initial value
var jsinteger = jsinteger1(0)(jsinteger1); // sandwiched self-application
NB: this paragraph about laziness is not coming from the Michaelson book; it is instead an original contribution (with a big credit due to @MirkoBonadei)
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types

ES6 code in 06-types-es6.js
HTML is 06-types.html

types ensure that operators are applied on the objects they are supposed to handle;

we want to be able to pass only TRUE and FALSE to functions like NOT, AND, OR; we want to be able to pass only ZERO, ONE, etc to functions like PRED and SUCC

we represent a typed object with a pair (type, value):

def MAKE_OBJ type value = PAIR type value = λs.(s type value)

we get to know the type by applying the whole object to FIRST, we get to know the value by doing the same with SECOND:

def TYPE obj = obj FIRST
def VALUE obj = obj SECOND

we use natural numbers to represent types and numeric comparison to test the type

def ISTYPE t obj = EQUAL (TYPE obj) t

where EQUAL is graciously provided by the previous paragraph about laziness; please note that this EQUAL can only compare numerals, and therefore types. Nothing else.

zeroeth object type: errors

we decide that errors are object type ZERO; we define errors by means of a set of useful functions

def error_type = ZERO
def MAKE_ERROR e = MAKE_OBJ error_type e
def ISERROR e = ISTYPE error_type e

the values for errors will be defined based upon the function where the error took place; but we define also the ultimate error:

def ERROR = MAKE_ERROR error_type // = PAIR error_type error_type

we verify that ERROR satisfied the functions mentioned before:

ISERROR ERROR = ISTYPE ZERO ERROR = EQUAL (TYPE ERROR) ZERO
TYPE ERROR = ERROR FIRST = error_type = ZERO

first object type: booleans

we decide that booleans are object of type ONE; we define booleans by means of a set of useful functions

def bool_type = ONE
def MAKE_BOOL b = MAKE_OBJ bool_type b
def ISBOOL b = ISTYPE bool_type b

we define now the typed versions of TRUE and FALSE:

def TRUE_OBJ = MAKE_BOOL TRUE
def FALSE_OBJ = MAKE_BOOL FALSE

plus a specific error for boolean troubles:

def BOOL_ERROR = MAKE_ERROR bool_type

then we make sure that TYPED_NOT operates only on booleans:

def NOT x = COND FALSE TRUE x // this was untyped NOT

ISBOOL(X) ? MAKE_BOOL (NOT (VALUE X) : BOOL_ERROR // using the ternary operator

def TYPED_NOT X = COND (MAKE_BOOL (NOT (VALUE X))) (BOOL_ERROR) (ISBOOL(X)) = (ISBOOL(X)) (MAKE_BOOL (NOT (VALUE X))) (BOOL_ERROR)

TYPED_AND and TYPED_OR follow along the same lines:

def TYPED_AND X Y = (AND (ISBOOL X) (ISBOOL Y)) (MAKE_BOOL (AND (VALUE X) (VALUE Y))) (BOOL_ERROR)
def TYPED_OR X Y = (OR (ISBOOL X) (ISBOOL Y)) (MAKE_BOOL (OR (VALUE X) (VALUE Y))) (BOOL_ERROR)

second object type: numerals

we decide that natural numbers are object of type TWO; we define numerals by means of a set of useful functions

def num_type = TWO
def MAKE_NUM = MAKE_OBJ num_type
def ISNUM = ISTYPE num_type

we define now the typed versions of a few integers:

def ONE_OBJ = MAKE_NUM ONE
def TWO_OBJ = MAKE_NUM TWO
def THREE_OBJ = MAKE_NUM THREE

plus a specific error that will be returned in case of troubles with computations involving numerals:

def NUM_ERROR = MAKE_ERROR num_type

now we make sure that TYPED_ADD and TYPED_MULT will operate only on numerals:

def TYPED_ADD X Y = (AND (ISNUM X) (ISNUM Y)) (MAKE_NUM (ADD (VALUE X) (VALUE Y))) (NUM_ERROR)
def TYPED_MULT X Y = (AND (ISNUM X) (ISNUM Y)) (MAKE_NUM (MULT (VALUE X) (VALUE Y))) (NUM_ERROR)
before talking about chars and strings (the last object types presented) here is a little detour about lists; after talking about that, we will be able to represent strings as lists of characters
back to the top

lists

ES6 code in 07-lists-es6.js
HTML is 07-lists.html

a list in λ-calculus is implemented as a binary tree by chaining pairs one inside the other (guess where LISP got the idea from, in the first place...)

first of all we define the type for lists and the basic type error and operators:

def list_type = THREE
def ISLIST = ISTYPE list_type
def LIST_ERROR = MAKE_ERROR list_type
def MAKE_LIST = MAKE_OBJ list_type

the building blocks

we start building lists from the empty list NIL, which will need to have a few characteristics:

  • it is a list; in other words: EQUAL(TYPE(NIL))(list_type)
  • it has value such that whoever tries to read it, will get a LIST ERROR
  • it's gotta be a PAIR, so to fit inside the binary tree

all this is satisfied by the following λ-expression

def NIL = MAKE_LIST (PAIR LIST_ERROR LIST_ERROR)

once we have NIL, we may start building the lists; the atomic list builder is called the CONStructor and its working is shown here:

[1, 2, 5] --> CONS ONE (CONS TWO (CONS FIVE NIL)) // from JS array to λ-calculus LIST

we seem to have almost everything already at hand to start managing this stuff:

def CONS = PAIR
def HEAD list = list FIRST  // also known as CAR
def TAIL list = list SECOND  // als known as CDR
def ISEMPTY list = COND TRUE FALSE (ISZERO list) = (ISZERO list) TRUE FALSE

but we want to keep track of type, so this time we need the CONStructor and the getters to be a bit more careful:

def CONS H T = (ISLIST T) (MAKE_LIST (PAIR H T)) LIST_ERROR
def HEAD list = (ISLIST list) (VALUE list FIRST) LIST_ERROR
def TAIL list = (ISLIST list) (VALUE list SECOND) LIST_ERROR

basic list manipulation

then we need some further basic operations on lists, all based on recursion:

rec LENGTH list = (ISEMPTY list) ZERO (SUCC (LENGTH TAIL list)) // recursive definition, no way...
rec APPEND element list = (ISEMPTY list) (CONS element NIL) (CONS (HEAD list) (APPEND element (TAIL list))) // also recursive

we are now able to obtain recursion implicitly by using the lazy version of the SELF_APPLY trick:

def LEN1 f list = LAZY_COND λ_.ZERO λ_.(SUCC (f f (TAIL list))) (ISEMPTY list)
def LENGTH SELF_APPLY LEN1

def APP1 f element list = LAZY_COND λ_.(CONS element NIL) λ_.(CONS (HEAD list) (f f element (TAIL list))) (ISEMPTY list)
def APPEND = SELF_APPLY APP1

in the codebase you will find the EcmaScript 6 version of the helper functions LEN1 and APP1

advanced list manipulation: MAP and REDUCE

we end up thinking about some more sophisticated stuff:

rec MAP list mapper = COND NIL (CONS (mapper (HEAD list))(MAP (TAIL list) mapper) (ISEMPTY list) // recursive, no way!

we realize MAP once more with the implicit recursion of the lazy SELF_APPLY method:

def MAP1 f list mapper = LAZY_COND λ_.NIL λ_.(CONS (mapper (HEAD list))(f f (TAIL list) mapper)) (ISEMPTY list)
def MAP = SELF_APPLY MAP1 // the good version

the recursive definition for REDUCE (fold left) is:

rec REDUCE fun acc list = (ISEMPTY list) acc (REDUCE fun (fun acc HEAD(list)) (TAIL list)) // explicit recursion, no way!

the version with implicit recursion is:

def RED1 f fun acc list = LAZY_COND λ_.acc λ_.(f f fun (fun acc HEAD(list)) (TAIL list)) (ISEMPTY list)
def REDUCE = SELF_APPLY RED1

the codebase contains an ES6 implementation for both MAP1 and RED1

helper functions for managing JavaScript arrays

due to the striking similarity between λ-calculus lists and JavaScript arrays, we can benefit from having a few helper functions to mediate between the two worlds. They are relatively straightforward to implement by mixing idioms from both realms:

var LIST2array = REDUCE(array => head => [head].concat(array))([]);
var array2LIST = array => array.reverse().reduce((acc, curr) => CONS(curr)(acc), NIL);

the codebase contains a few examples of usage of these functions.

functions for analyzing the content of lists

the two most typical applications are ALL/EVERY (a chain of AND's) and ANY/SOME (a chain of OR's):

ALL predicate list = REDUCE TYPED_AND TRUE_OBJ (MAP predicate list)
ANY predicate list = REDUCE TYPED_OR FALSE_OBJ (MAP predicate list)

from ALL we derive helper functions that check whether a list contains only homogeneous elements, for example a STRING may be built only using a list of CHAR's:

ALL_CHARS charlist = ALL (λx.MAKE_BOOL (ISCHAR x)) charlist // NB - produces TYPED booleans
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characters and strings

ES6 code in 08-chars-strings-es6.js
HTML is 08-chars-strings.html

characters (type FOUR)

we start by creating the usual basic definitions:

def char_type = FOUR
def MAKE_CHAR = MAKE_OBJ char_type
def ISCHAR = ISTYPE char_type
def CHAR_ERROR = MAKE_ERROR char_type

then we need a few numerals that we will use to indicate ASCII codes:

def forty_eight = MULT SIX (MULT FOUR TWO);
def sixty_five = MULT FIVE (ADD FIVE (ADD FIVE THREE));
def ninety_seven = ADD ONE (MULT TWO forty_eight);

in λ-calculus we may easily use a function name as its own representation:

def 'a' = MAKE_CHAR(ninety_seven)
def 'b' = MAKE_CHAR(SUCC(VALUE 'a'))
VALUE('a') // --> 'a'
TYPE('a') // --> FOUR

in JavaScript we can use the relevant character as var name (for numbers we need to prepend an _), but if we want to receive its representation as a value, we need to store it first by defining VALUE(char) as a PAIR; for example:

var A = MAKE_CHAR(PAIR(sixty_five)('A'));
var B = MAKE_CHAR(PAIR(SUCC(VALUE(A)(FIRST)))('B'));
var _0 = MAKE_CHAR(PAIR(forty_eight)('0'));
var _1 = MAKE_CHAR(PAIR(SUCC(VALUE(_0)(FIRST)))('1'));

we may make good use of a JavaScript mapper e.g. from "a" to var a:

function char2CHAR(char) {
  return {
    '0': _0,
    ...
    'a': a,
    ...
    'A': A,
    ...
    'Z': Z
  }[char];
}

now the whole basic alphanumeric charset is at our disposal, with everything numerically ordered

strings (type FIVE)

strings are lists of CHAR's:

def string_type = FIVE
def ISSTRING = ISTYPE string_type
def STRING_ERROR = MAKE_ERROR string_type

in this case MAKE_STRING, the basic operation for strings, needs to check both the type of the list and all of its elements:

def MAKE_STRING string = (AND (ISLIST string) (ALL_CHARS string)) (MAKE_OBJ string_type string) STRING_ERROR;

a conversion function from JavaScript strings to λ-calculus STRINGs builds upon the conversion functions for JS arrays and LISTs:

var string2STRING = string => MAKE_STRING(array2LIST(string.split('').map(char2CHAR)));

the codebase shows various examples of utilisation of all these functions

NB: again, this paragraph about chars and strings is mostly not coming from the Michaelson book; it is instead an original contribution.
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