*Abstract:*
Joy is a functional programming language which is not based on the
application of functions to arguments but on the composition of
functions. The language makes extensive use of combinators which
perform the role of higher order functions. The algebra of Joy
programs can be used to express formal properties of many first and
second order functions without using variables ranging over values or
over functions. Some of these properties are idempotency, inverses,
converses, commutativity, symmetry, associativity, homomorphisms and
distribution. The paper also gives several analogues of concepts from
category theory.

*Keywords:* functional programming, function composition,
algebra of programs, monoids, categories, functors, natural
transformations, monads.

` Q1 `

and ```
Q2
```

are programs which denote the same function and ```
P
```

and ` R `

are other programs, then the two
concatenations ` P Q1 R `

and ` P Q2 R `

also
denote the same function. In other words, programs ` Q1 `

and ` Q2 `

can replace each other in concatenations. This
can serve as a rule of inference for As premises one needs axioms such as in the first three lines below, and definitions such as in the fourth line:

(+) 2 3 + == 5 (dup) 5 dup == 5 5 (*) 5 5 * == 25 (def square) square == dup *A derivation using the above axioms and the definition looks like this:

2 3 + square == 5 square (+) == 5 dup * (def square) == 5 5 * (dup) == 25 (*)The comments in the right margin explain how a line was obtained from the previous line. The derivation shows that the expressions in the first line and the last line denote the same function, or that the function in the first line is identical with the function in the last line.

Consider the following equations in infix notation: The first says
that multiplying a number `x`

by 2 gives the same result as
adding it to itself. The second says that the `size` of a
`reverse`d list is the same as the `size`

of the
original list.

2 * x = x + x size(reverse(x)) = size(x)In Joy the same equations would be written,

2 * == dup + reverse size == size

Other equivalences express algebraic properties of various operations.
For example, the predecessor `pred` of the successor
`succ` of a number is just the number itself. The
conjunction `and` of a truth value with itself gives just the
truth value. The less than relation `<`

is the converse of
the greater than relation `>`

. Inserting a number with
`cons` into a list of numbers and then taking the
`sum` of that gives the same result as first taking the sum
of the list and then adding the other number.

In conventional notation these are expressed by

pred(succ(x)) = x x and x = x x <y = y > x sum(cons(x,y)) = x + sum(y)In Joy these are expressed

succ pred == id dup and == id < == swap > cons sum == sum +Some properties of operations have to be expressed by combinators. One of these is the

In the first example below, the `dip`

combinator is used to
express the associativity of addition. Another combinator is the
`app2` combinator which expects a program on top of the stack
and below that two values. It applies the program to the two values.
In the second example below it expresses one of the De Morgan laws.
In the third example it expresses that the `size`

of two
lists `concat`enated is the sum of the `size`

s of
the two concatenands. The last example uses both combinators to
express that multiplication distributes (from the right) over
addition. (Note that the program parameter for `app2`

is
first constructed from the multiplicand and `*`

.)

[+] dip + == + + and not == [not] app2 or concat size == [size] app2 + [+] dip * == [*] cons app2 +

The remainder of this paper is organised as follows: The next five sections give detailed examples of algebraic laws which Joy operators satisfy. All of these laws are well known when expressed in familiar notation, what is new here is that they can be expressed in Joy notation without the use of implicitly or explicitly universally quantified variables. Then follow three sections using concepts from category theory, but no previous knowledge is assumed.

`f(x)`

is said to be `x`

f(f(x)) = f(x)For example, the function

abs(abs(n)) = abs(n) sort(sort(s)) = sort(s)Composition can be used to express that stack operations are

abs abs == abs sort sort == sortAnother idempotent Joy function, one which has no counterpart in conventional notation, is the

newstack newstack == newstack

The remainder of this section illustrates the use of the *identity
function* in Joy algebra. This function is denoted by the symbol
`id`. It has the property that for all programs
`P`

,

id P == P == P id

Let `f(x)`

be a unary function. Another unary function
`g(x)`

is said to be its *inverse* if for all
`x`

g(f(x)) = xFor example, the predecessor function on integers is the inverse of the successor function on integers: for all integers

`x`

pred(succ(x)) = xIt may or may not be that if one function is the inverse of a second then the second is the inverse of the first. This is true of the predecessor and successor functions when defined on integers, but not when defined on natural numbers. The identity function can be used to express that one function is the inverse of another and that certain values are unit elements.

The atomic program `succ` denotes a function which takes a
stack as argument and yields a stack as value. The argument stack has
to have an integer (or a character) on top. The value stack is like
the argument stack except that the integer (or character) has been
incremented by 1. The semantics for `pred` is analogous, it
decrements the integer (or character). The following express that the
functions denoted by the symbols `succ`

and
`pred`

are *inverse*s of each other:

pred succ == id succ pred == id

The `cons` function expects a list on top of the argument
stack. Below that it expects another value. The value that is
returned is another stack which is like the argument stack except that
the two top elements of the argument stack have been replaced by a new
list which has the value inserted into it at the front. The
`uncons` function undoes this. It expects a non-empty list
and leaves the first and the rest of the list. The two functions are
inverses of each other:

cons uncons == id uncons cons == idIt is worth pointing out this cannot be expressed in conventional notation because there the

`uncons`

operation makes no
sense. Actually, both functions are polymorphic in that instead of
lists they can operate on strings or on sets The two equations still
hold applied to strings. Only the right equation holds for sets.
The symbols `pairlist`, `pairstring` and
`pairset` denote functions which expect two potential members
of lists, strings or sets on top of the stack. They return a new
stack with the two members replaced by a single list, string or set.
The polymorphic `unpair` function is their inverse, but not
vice versa. Since `unpair`

leaves two results, the
following have no counterpart in conventional notation.

pairlist unpair == id pairstring unpair == id pairset unpair == id

Some functions are inverses of themselves. Examples of
*self-inverse* functions are the Boolean negation function and
the list reversal function: for all Boolean values `b`

and
for all lists `l`

not(not(b)) = b reverse(reverse(l)) = lIn Joy the two unary operators

`not`

operator expects a truth value
or a set on top of the stack and returns a stack which has the
complementary truth value or set on top of the stack. The
`reverse`

operator expects a list or a string on top of the
stack and returns a stack which has the reversal the list or string on
top of the stack. The two functions are self-inverses, and this is
expressed by
not not == id reverse reverse == id

Let `f(x,y)`

be a binary function. A constant
`c`

is called a left or right *unit element* if the
first or the second equation holds for all `x`

f(c,x) = x f(x,c) = xLeft and right unit elements often coincide, and then they are just called unit elements. In particular, this is true for the unit elements of commutative functions. For example, 0 is the unit element for addition, and 1 is the unit element for multiplication. In conventional infix notation:

n + 0 = n = 0 + n n * 1 = n = 1 * n

The identity function can also be used to express that certain
literals are *right unit* elements for binary operations: 0 for
addition, 1 for multiplication, the *empty string*
`""`

or the *empty list* `[]`

for
concatenation, the truth value `false` and the *empty
set* `{}`

for logical disjunction and set union, and
the truth value `true` for logical conjunction.

0 + == id 1 * == id "" concat == id [] concat == id false or == id {} or == id true and == id

`dup`

and `pop`

operators in Joy algebra. The Joy operator 42 dup == 42 42Both sides of the equation denote compositions of two functions. On the left the first function pushes a number and the second makes a duplicate of the top element. On the right the two functions are the same, each pushes a number. The equation says that the function on the left is identical to the one on the right. Both functions are defined for all stacks, and both return a stack which is like the argument stack except that two copies of the number 42 have been pushed.

A binary function `f(x,y)`

is called *idempotent* if
for all `x`

f(x,x) = xTwo examples are the Boolean conjunction and disjunction operations: for all

`b`

b and b = b b or b = bIn Joy the

`dup`

operator can express idempotency of the
Boolean operations dup and == id dup or == id dup min == id dip max == id

The Joy operator `pop` expects one value on top of the stack
and removes it. For example

17 42 pop == 17On the left the composition of three functions first pushes two numbers and then pops the second. On the right the function just pushes the first number. The two functions are identical since for all argument stacks they have the same result stack.

Let `f(x,y)`

be a binary function. A constant
`c`

is called a left or right *zero element* of
`f(x,y)`

if the first or second equation holds for all
`x`

:

f(c,x) = c f(x,c) = cFor example, the number zero is a left and right zero element for multiplication, and in conventional infix notation the laws looks like this:

0 * n = 0 n * 0 = 0The

`pop`

operator can also be used to express that
particular values are zero elements for binary operations: 0 for
multiplication, `false`

and the empty set `{}`

for logical conjunction and set intersection, and `true`

for
logical disjunction.
0 * == pop 0 false and == pop false {} and == pop {} true or == pop true

The two operators `dup`

and `pop`

are related by
the identity

dup pop == idThe

`pop`

operator can also be used to express the
42 pop == id succ pop == pop + pop == pop popSimilar laws express that some operators return two results on the stack:

uncons pop pop == pop dup pop pop == pop

`swap`

operator in
Joy algebra. The Joy operator swap swap == id

Let `f(x,y)`

be a binary function. Another binary function
`g(x,y)`

is its *converse* if for all `x`

and `y`

f(x,y) = g(y,x)For example, the numeric comparison relation

`<`

has as its
converse the relation `>`

:
(i <j) = (j > i)In Joy notation the

`swap`

operator can express that
comparison predicates `<`

and `<=`

have as their
converses the predicates `>`

and `>=`

by the
laws
swap > == < swap >= == <=The operator

`cons`

, it
expects an aggregate and a new value on top of the stack. It leaves a
new aggregate with the value inserted. But whereas `cons`

expects the aggregate on top and the value below, `swons`

expects them in the opposite order, the value on top and the aggregate
below. It follows that `swons`

is the converse of
`cons`

. In the same way, a binary string or list operation
`concat`

.
swap cons == swons swap concat == swoncat

One function is the converse of a second function if and only if the
second is the converse of the first. This says that converseness is a
symmetric relation. In Joy it is expressed by the following: for all
programs `P`

and `Q`

swap P == Q if and only if swap Q == PFrom this rule and the previous equalities it follows that

swap < == > swap <= == >= swap swons == cons swap swoncat == concat

A binary function `f(x,y)`

is *commutative* if it is
its own converse - if for all `x`

and `y`

f(x,y) = f(y,x)For example, addition of numbers is commutative, for all integers

`x`

and `y`

i + j = j + iIn Joy the

`swap`

operator can express that a function is
commutative.
swap + == + swap * == * swap and == and swap or == or swap max == max swap min == minTwo sorted sequences can be combined with the

swap merge == merge

A function which yields a truth value is often called a
*predicate*. Commutative predicates are often called
*symmetric*. For example, the *identity relation*
` = `

, a binary predicate, is commutative or symmetric.
Another is the `equal` predicate which tests lists for
identity, including sublists and their sublists. In conventional
notation, for all integers or lists `x`

and `y`

(i = j) = (j = i) equal(x,y) = equal(y,x)Turning these concepts on themselves, the converse relation is symmetric: for all functions

`f`

and `g`

(g is the converse of f) = (f is the converse of g)The same is not true for the inverse relation. The

`swap`

operator can express that a binary predicate is ` = `

and `equal`

are
symmetric:
swap = == = swap equal == equalWith

`swap`

one can express that elements are 0 swap + == id 1 swap * == id false swap or == id {} swap or == id true swap and == id "" swap concat == id [] swap concat == id "" swap merge == id [] swap merge == id

Some operators leave two elements on top of the stack, and two such
operator may be related in the sense that they just leave the elements
in a different order. This can also be expressed by
`swap`

:

uncons swap == unswons unswons swap == unconsThere is even one operator which is related to itself in this way, and that is

`dup`

:
dup swap == dup

Two operators related to `swap`

are `rollup` and
`rolldown`. The `rollup`

operator moves the third
and second element on the stack into second and first position, and it
moves the original first element into third position. The
`rolldown`

operator moves the second and first element on
the stack into third and second position, and it moves the original
third element into first position. They can express laws such as

rolldown concat concat == concat swoncat rollup swoncat concat == swoncat swoncat rollup merge merge == merge mergeTheir arities are expressed by

swap pop pop == pop pop rollup pop pop pop == pop pop pop rolldown pop pop pop == pop pop pop

This section illustrates the use of the `dip`

combinator in
Joy algebra.

The three previous sections have shown how a few Joy operations can
express a variety of well-known laws. In the sections to follow more
difficult Joy concepts will be needed. These resemble higher order
functions, but like everything else in Joy they really are just
functions from stacks to stacks. They differ from what are called the
operators in that they expect on top of the stack not just a passive
datum, but a quoted program which they execute. In accordance with an
older terminology they are here called *combinator*s.

One of these is the `dip` combinator. It expects a quoted
program on the top of the stack, and below at least one value of any
type. During execution it removes the program and the value from the
stack and saves them. Then it executes the program on the remainder
of the stack. Finally it restores the saved value to the top of the
stack. In most applications the program will be pushed just before
the combinator is to be applied. The combinator is useful for doing
something to the stack without disturbing the top value.

Here is an example:

1 2 3 4 + * 5 == 1 14 5 1 2 3 4 5 [+ *] dip == 1 14 5In the first line on the left the 3 and the 4 are immediately added, the result is multiplied by the 2 to give 14, and then the 5 is pushed on top. In the second line the 5 is pushed immediately after the 4, and consequently it is not possible to add the 3 and 4 without popping the 5 first. So, the program

`[+ *]`

is pushed and then
executed by `dip`

. The results are the same as those in
the two (identical) right sides.
A binary function `f(x,y)`

is said to be
*associative* if the result of applying it twice to three
values is independent of the order of application:

f(x,f(y,z)) = f(f(x,y),z)For example, addition of numbers is associative:

i + (j + k) = (i + j) + kIf

`g(x,y)`

is the converse of an associative
`f(x,y)`

, then `g(x,y)`

is also associative.
In Joy the `dip`

combinator can be used to express
associativity:

[+] dip + == + + [*] dip * == * * [and] dip and == and and [or] dip or == or or [max] dip max == max max [min] dip min == min min [concat] dip concat == concat concat [swoncat] dip swoncat == swoncat swoncat [merge] dip merge == merge merge

The following law expresses that the `dip`

combinator
leaves one value unchanged:

dip pop == [pop] dip i

`app2`

combinator
in Joy algebra.
The `app2` combinator expects a quoted program on top of the
stack, and below that two data parameters. As with all combinators,
the program will be executed, in this case twice. In case the program
computes a unary function, the effect is to replace the two data
parameters by two corresponding values of that function. The two
evaluations could be done in parallel. The more general case where
the program does not denote a unary function is described further
down.

Let `f(x1,x2)`

be a binary function defined on a type
`X`

, and let `g(y1,y2)`

be a binary function
defined on a type `Y`

. Let `h(x)`

be a function
from `X`

to `Y`

. Then `h(x)`

is a
*homomorphism* from `X`

and its binary function to
`Y`

and its binary function when the following holds for
all `x1`

and `x2`

:

h(f(x1,x2) = g(h(x1),h(x2))One example is the logarithm function which maps logarithms of products onto sums of logarithms. Two other examples are the doubling function which maps integers with addition into even integers with addition, and the squaring function which maps naturals with multiplication into square naturals with multiplication and the

log(x * y) = log(x) + log(y) double(x + y) = double(x) + double(y) square(x * y) = square(x) * square(y) size(concat(x,y)) = size(x) + size(y) not(p and q) = not p or not q not(p or q) = not p and not q

In Joy the `app2`

combinator can be used to express
*homomorphism* laws, and these all take the form:

f h == [h] app2 gSome such laws are

+ double == [double] app2 + * square == [square] app2 * max succ == [succ] app2 max concat size == [size] app2 + concat sum == [sum] app2 + concat product == [product] app2 * concat charset == [charset] app2 orIn the above,

`or`

operator computes
set union in this case. Another homomorphism is the concat sort == [sort] app2 merge

The `app2`

combinator can also be used to express the
familiar De Morgan laws for Boolean algebra and a (perhaps surprising)
isomorphic pair of laws for strings or lists:

and not == [not] app2 or or not == [not] app2 and concat reverse == [reverse] app2 swoncat swoncat reverse == [reverse] app2 concat

Laws like the above generalise to *distribution* laws. In
these the unary function is replaced by a new binary function, and for
each element in the domain a unary function can be defined from the
new binary function by letting one parameter be the given element. It
is useful to distinguish *right distribution* and *left
distribution*.

A binary function `f(x,y)`

distributes from the right over
another binary function `g(x,y)`

if the following holds:

f(g(x,y),z) = g(f(x,z),f(y,z))In arithmetic we have the familiar example of multiplication distributing from the right over addition. In Boolean algebra the conjunction and disjunction operators distribute from the right over

(i + j) * k = (i * k) + (j * k)

The `app2`

combinator can also express *right
distribution* laws. In each case there are three data parameters
on the stack, and the two ways of applying two functions are
equivalent. The one way is to apply the one function to the second
and third parameters (using the `dip`

combinator) and then
to apply the distributing function to the result and the first
parameters. The other way is to use the first parameter and the
distributing function to make a *constructed program* that
computes a unary function, use `app2`

to compute its values
for the second and third data parameters and to combine the two values
with the other function.

[+] dip * == [*] cons app2 + [and] dip or == [or] cons app2 and [or] dip and == [and] cons app2 or

A binary function `f(x,y)`

distributes from the left over
another binary function `g(x,y)`

if the following holds:

f(x,g(y,z)) = g(f(x,z),f(y,z))In arithmetic multiplication also distributes from the left over addition:

i * (j + k) = i * j + i * kThe

`app2`

combinator can also be applied to a quoted
program which does not compute a unary function, but accesses data
elements further down in the stack. In the examples below, these
elements have to be explicitly deleted later on, by ```
[pop]
dip
```

. It can be used to express + * == [*] app2 + [pop] dip or and == [and] app2 or [pop] dip and or == [or] app2 and [pop] dip

Apart from `app2`

there are similar combinators
`app1` and `app3`. Each expects a program
`[P]`

on top of the stack and below that 1, 2 or 3 further
parameters and produces 1, 2 or 3 values. Some pertinent laws are

[succ] app1 == succ [not] app1 == not [pop] dip app1 == app2 pop [swap] dip app2 == app2 swap [dup] dip app2 == app1 dupThe arities of these combinators are expressed by

app1 pop == pop pop app2 pop pop == pop pop pop app3 pop pop pop == pop pop pop pop

There is a sense in which one might say that an integer has two parts: a sign and an absolute value. When the two parts are multiplied, the result is the same as the original. In the same way, a non-empty list has two parts, its first and its rest. When the first is consed into the rest, the result is the same as the original list. In conventional notation this might be expressed as

sign(x) * abs(x) = x cons(first(x),rest(x)) = xThe same may be expressed in Joy notation using the

`dip`

combinator:
dup [sign] dip abs * == id dup [first] dip rest cons == id

The laws look somewhat cleaner when expressed in terms of another
combinator. The `cleave` combinator expects two programs and
below that another item. It applies both programs to produce two
results, for example

5 [pred] [dup *] cleave == 4 25The earlier laws about parts and wholes can then be expressed like this:

[sign] [abs] cleave * == id [first] [rest] cleave cons == id

The combinator `split` applied to a list and a test predicate
produces two lists, those members of the original list which pass the
test and those with fail. For any predicate, a list will have two
parts which can be merged to reconstitute the original list. In Joy
notation:

[sort] dip split merge == pop sortThe law cannot be expressed in conventional notation because

`split`

produces two results.
A *category* consists of a collection of *object*s and
for any two objects a collection of *morphism*s, each having
the one object as their *source* and the other object as their
*target*. In many categories the objects are just sets, or
they are sets with structure - algebras. Then the morphisms are unary
functions from sets to sets, or they are homomorphisms from algebras
to algebras. For any object the morphisms must include an
*identity morphism* with that object as source and target.
Often there will be other morphisms with that object as source and
target. For any object and two morphisms having a given object as
target and source respectively, there must be a composite morphism
having as source the source of the one component and as target the
target of the other.

This composition of morphisms must be associative, with identity
morphisms as left and right unit elements. These requirements are
satisfied for categories of sets and functions and for categories of
algebras and homomorphisms. But there are many categories that are
quite different. One kind of example are *monoid*s: an
associative binary operation over a set which includes a left and
right unit element. Here the category consists of just one object
(which is of no interest), but many morphisms, the elements of the
monoid. There are many other kinds of categories which are different
again.

Categories deal with two sorts of things, objects and morphisms. So
they are two-sorted algebras. Between categories there are morphisms
called *functor*s. These take objects and morphims of one
category into objects and morphisms of another category. In computer
science the most familiar functors are the *type constructor*s.
They take integers, characters, truth values and so on into
`LIST`

s of integers, `LIST`

s of characters,
`SET`

s of integers and so on. The functors must *
also* take integer functions such as squaring into corresponding
functions on `LIST`

s or `SET`

s of integers.

In computing circles the corresponding functions are usually written
`map(square,L)`

, for a list `L`

. In category
theory the same symbol is used for objects and morphisms, so the
examples are written `LIST(integer)`

and
`LIST(square)`

. In Joy there is no explicit type notation
at all, and `map` is just one of many combinators. Programs
to compute the list of squares of a given list can be written in
either of these two ways:

[square] map [dup *] map

Between any two functors `F`

and `G`

there can
be functions called *natural transformation*s. These take as
arguments the values of `F`

and `G`

at their
objects. A function `n`

is natural if for all morhisms
`m`

in the domains of `F`

and `G`

the
following holds for all `x`

:

n(F(m)(x)) = G(m)(n(x))Initially we shall only be concerned with the case where

`F`

and `G`

are the same functor
`LIST`

. Then an example of a natural transformation from
lists to lists is the `f`

and lists `L`

reverse(LIST(f)(L) = LIST(f)(reverse(L))or in conventional notation

reverse(map(f,L)) = map(f,reverse(L))In Joy algebra the naturality of

`reverse`

is expressed by
[reverse] dip map == map reverse

In computer science natural transformations are often called
*polymorphic* functions, in the case of lists they are
independent of the type of the elements of the lists. Four other
naturality laws, expressed in conventional notation:

map(f,rest(L)) = rest(map(f,L)) f(first(L)) = first(map(f,L)) map(f,concat(L1,L2)) = concat(map(f,L1),map(f,L2)) map(f,cons(x,[])) = cons(f(x),[]) map(f,unitlist(x)) = unitlist(f(x))The last two laws of course say the same thing. In Joy these would be expressed by

[rest] dip map == map rest [first] dip i == map first [concat] dip map == [map] cons app2 concat [[] cons] dip map == i [] cons [unitlist] dip map == i unitlistNote that in the third equation on the right the

`app2`

combinator has to use a
Somewhat more difficult is the naturality of `cons`. In
conventional notation this is expressed by

map(f,cons(x,L)) = cons(f(x),map(f,L))and in Joy notation by

[cons] dip map == dup [dip] dip map consThis is so complex that a step-by-step verification is called for. Let

`L`

and `x`

be the list and the additional
member. Let `[F]`

be a program which computes the function
`f`

. Let `x'`

be the result of applying
`f`

to `x`

, and let `L'`

be the
result of applying `f`

to all members of `L`

.
The proof of the equivalence of the LHS and the RHS consists in
showing that both reduce to the same program. For the LHS we have:
x L [F] [cons] dip map LHS == x L cons [F] map (dip) == [x L] [F] map (cons) == [x' L'] (map)For the RHS:

x L [F] dup [dip] dip map cons RHS == x L [F] [F] [dip] dip map cons (dup) == x L [F] dip [F] map cons (dip) == x' L [F] map cons (dip) == x' L' cons (map) == [x' L'] (cons)The two sides reduce to the same program, so they denote the same function.

A similar equation is the following:

map == [uncons] dip dup [dip] dip map consBut note that this is not suitable as a definition, since the RHS only applies to non-empty lists. The following is a suitable recursive definition:

map == [ pop null ] [ pop ] [ [uncons] dip dup [dip] dip map cons ] ifte

The fact that `map`

does not change the number of elements
in a list is expressed in conventional notation by

size(map(f,L)) = size(L)and in Joy notation by

map size == pop size

An important combinator for any aggregate is `filter`, which
expects an aggregate and below that a quotation which implememnts a
predicate. It returns an aggregate of the same type as the parameter
containing only those members for which the predicate yielded
`true`

. Given two aggregates, it does not matter whether
they are first combined and then filtered, or first filtered
separately and then combined. For sequences the law is this:

[concat] dip filter == [filter] cons app2 concatFor sets the combining operator has to be

`or`

instead of
`concat`

.
Another law concerns passing an aggregate first through one filter and
then passing the result through another filter. Passing the aggregate
through the conjunction of these filters produces the same result.
The `conjoin` operator takes two quoted predicates and
returns one quoted predicate which is their conjunction.

[filter] dip filter == conjoin filter

The following laws concern the `sum`

s and
`product`

s of lists of integers:

cons sum == sum + sum == uncons sum + cons product == product * product == uncons product *(Only the equations on the left could be expressed in conventional notation.)

This holds:

P == uncons Q iff cons P == Q

`LIST`

there are other functors such as `SET`

. So there is the
type `SET(integer)`

, the function `SET(square)`

which maps a set of integers into the set of their squares, and
similarly for other integer functions. Much of what was said about
lists and their natural transformations has counterparts for sets and
their natural transformations. In Joy there are several
implementations of `0`

.. `31`

,
such sets are written in curly braces, as in `{1 3 5}`

.
Values of this type can be manipulated by the combinator
`map`

and the operators `first`

,
`rest`

and `cons`

. Instead of the operator
`concat`

the set union operator [rest] dip map == map rest [first] dip map == map first [or] dip map == [map] cons app2 or [{} cons] dip map == i {} cons [cons] dip map == dup [dip] dip map cons

Now we have two functors, `LIST`

and `SET`

.
Lists have order and may have repetitions, sets have neither. A
useful function from lists to sets is the function `elements`
which removes order and repetitions. For example, in Joy notation

[ 3 1 5 1 ] elements == { 1 3 5 }It makes no difference whether the set of elements of a list is taken first and then the set is mapped through a function, or whether the list is first mapped through the same function and then the set of elements is taken. This is the naturality of the

`elements`

function, expressed by
[elements] dip map == map elementsFor example:

[ 3 1 5 1 ] [dup *] [elements] dip map == [ 3 1 5 1 ] elements [dup *] map == { 1 3 5 } [dup *] map == { 1 9 25 }and

[ 3 1 5 1 ] [dup *] map elements == [ 9 1 25 1 ] elements == { 1 9 25 }

Halfway between lists and sets are multisets or *bag*s; these
have no order but may have repetitions. A `BAG`

functor
would be similar to `LIST`

and `SET`

, and there
would be natural transformations from bags to bags, from lists to
bags, and from bags to sets. Currently Joy does not have an
implementation of bags.

A list can have as its members other lists, for example lists of
integers. Formally these are of type
`LIST(LIST(integer))`

. This uses the `LIST`

functor composed with itself: `LIST`

°
`LIST`

. Such a list can be mapped through a function by
mapping each sublist, for example

[[1 2 3][4 5]] [[dup *] map] map == [[1 4 9][16 25]]Here the second

`map`

is applied to the whole list, the
first or inner `map`

is applied to the sublists.
Alternatively a combinator mmap == [map] cons mapand then one can write

[[1 2 3][4 5]] [dup *] mmap == [[1 4 9][16 25]]

Whereas a list is one-dimensional, a *matrix* is
two-dimensional. Matrices can be implemented as lists of lists, and
the sublists can be interpreted either as the rows or the columns.
One important operation on matrices is the interchange of rows and
columns. The `transpose` operator does just that:

[[1 2][3 4]] transpose == [[1 3][2 4]]Transposition is another polymorphic function or natural transformation for matrices. It does not matter whether a matric is first transposed and then mapped elementwise through a function, or whether it is first mapped and then transposed.

[transpose] dip mmap == mmap transpose

The operator `zip` will transform two lists of the same
length into a list of pairs, for example

[1 2 3] [4 5 6] zip == [[1 4][2 5][3 6]]The

`zip`

operator can be defined by
zip == [] cons cons transposeThe

`zip`

function is natural, `zip`

ping the two
lists and then `mmap`

ing has the same effect as
`map`

ping and then `zip`

ping:
[zip] dip mmap == [map] app2 zip

A similar naturality law holds for the `cartproduct` operator
which produces the *cartesian product* of two aggregates which
do not have to be of the same type:

[cartproduct] dip mmap == [map] app2 cartproduct

Another useful datatype is that of *tree*s, also called
*recursive list*s. A tree of integers is either an integer or
a list of trees of integer. A *proper tree* is a list of
trees. The type gives rise to a functor `TREE`

, with the
data type `TREE(integer)`

and mapping functions such as
`TREE(square)`

. In Joy the combinator for tree mapping is
`treemap`. Most of the operations on lists also apply to
proper trees. Reversal can be done by `reverse`

just at
the top level, or by `treereverse` all the way down into all
sublists. Some naturality laws are:

[reverse] dip treemap == treemap reverse [treereverse] dip treemap == treemap treereverse [rest] dip treemap == treemap rest [first] dip i == treemap first [[] cons] dip treemap == treemap [] cons

Proper trees can be `treeflatten`ed to form a one-level list.
For example

[ [1 [2 3] [] 4] [5] ] treeflatten == [ 1 2 3 4 5 ]The

`treeflatten`

ing function is a natural transformation
between the `TREE`

and `LIST`

functors, the
order of treeflattening and mapping does not matter:
[treeflatten] dip map == treemap treeflattenThe following also holds:

treereverse treeflatten == treeflatten reverse

A *bare tree* is either the empty list `[]`

or it is
a list of bare trees. Formally there is a functor
`BARETREE`

, and for (degenerate) functions which can only
take `[]`

as parameters `BARETREE(f)`

maps bare
trees with contained `[]`

into trees. Proper trees can
also be `strip`ped of their leaves to form a bare tree:

[ [1 [2 3] [] 4] [5] ] strip == [ [ [] [] ] [] ]The

`strip`

function commutes with `reverse`

and
`treereverse`

:
reverse strip == strip reverse treereverse strip == strip treereverseOnce stripped, there is nothing for

`treemap`

to do:
[strip] dip treemap == treemap strip == pop strip

A *monad* `M`

over a category consists of a functor
from the category to itself, and two natural transformations. The
first transformation `joinM`

takes as argument an object in
the target of the square of the functor and gives as value an object
in the target of the functor. The second transformation
`unitM`

takes as argument an object in the category and
gives as value an object in the target of the functor. The two
transformations must satisfy two laws which are expressed in terms of
two variants obtained by applying the functor to the two
transformations:

From the first transformation one can define a variant by applying the functor to it. This variant is again a natural transformation, it takes as argument an object in the target of the cube of the functor and gives as value an object in the target of the square of the functor. The first transformation or its variant may be composed with the first transformation. The two compositions are again natural transformations, they take as argument an object in the cube of the functor and give as value an object in the target of the functor. The first defining law for monads is that these two compositions must be identical.

Similarly, from the second transformation one can define a variant by applying the functor to it. This variant is again a natural transformation, it takes as argument and value objects in the target of the functor. The second transformation or its variant may be composed with the first transformation. The two compositions take as argument an value objects in the target of the functor. The second defining law for monads is that these two compositions must both be equal to the identity function.

The above will now be illustrated with the *LIST monad*. Its
functor is the `LIST`

functor. Its first natural
transformation is usually called `flatten`, which
concatenates a two-level list to produce a single-level list. Its
second natural transformation is the unary `unitlist`
operation which takes any argument to produce its singleton list.
Here are two examples:

[[1 2 3] [peter paul]] flatten == [1 2 3 peter paul] [[1 2 3] [peter paul]] unitlist == [[[1 2 3] [peter paul]]]The two required variants are obtained by applying the

`LIST`

functor, as `map`

.
The variant of the `flatten`

operator is the polymorphic
operator

[flatten] mapwhich takes a list of (lists of lists) as argument and concatenates the (list of lists) but leaves the outer level list structure intact. This is an example:

[[[1 2] [3]] [[a] [b]]] [flatten] map == [[1 2 3] [a b]]The first monad law can now be written in Joy notation. It says that there are two equivalent ways of flattening a list of lists of lists to produce a list:

[flatten] map flatten == flatten flatten

The variant of `unitlist`

is the polymorphic operator

[unitlist] mapwhich takes a list of elements and produces the list of their unitlists. An example is

[1 2 [3 4]] [unitlist] map == [[1] [2] [[3 4]]]The second monad law can now be written in Joy as

[unitlist] map flatten == id == unitlist flatten

As natural transformations both `flatten`

and
`unitlist`

interact with the `LIST`

functor
operating (as `map`

) on arbitrary functions. There are two
further laws that arise. Because these two laws are more general than
the preceding ones, they are also more useful:

A list of lists can be mapped at the second level through an arbitrary
function using the `mmap`

combinator, producing another
list of lists. That can then be `flatten`

ed to produce a
single level list. The same original list of list can first be
`flatten`

ed to produce a single level list which can then
be mapped at the top level using `map`

. The two results
are the same, and in Joy this is expressed as

mmap flatten == [flatten] dip map

A function may be applied to an argument of any type, and then the
`unitlist`

can be taken. Alternatively the
`unitlist`

can be taken first and then the result can be
`map`

ped through the function. That the results are the
same can be written in Joy as

i unitlist == [unitlist] dip map

{Wadler92} shows that in any monad it is possible to define another
natural transformation, *monadic composition* which
simultaneously resembles function application and function
composition. For the `LIST`

monad it takes as one of its
arguments a list and as the other argument a function which yields a
list as value. The result is again a list. In Joy it might be
defined by

bind == map flattenIt satisfies the following laws:

[unitlist] dip bind == i [unitlist] bind == id [K [H] bind] bind == [K] bind [H] bindThe first two laws say that

`unitlist`

is a left and right
identity for `bind`

, the third says that `bind`

is associative. The third law is here expressed with program
variables `K`

and `H`

. Alternatively it is
expressed by
[bind] cons concat bind == [bind] dip bindWadler makes extensive use of many

`bind`

-like functions
for monads other than the `LIST`

monad.
A very general theory of lists, without the use of category theory, is
given in {Bird86}. A very readable introduction to the
`LIST`

functor can be found in {Spivey89}. The theory of
lists is generalised by {Malcolm89} to what have been called rose
trees. {Meijer-etal91} give a comprehensive collection of laws of
functional programming using very general functional forms for lists
and other data types. {Bird-deMoor92} use categories, homomorphisms
and algebraic techniques to solve sophisticated optimisation problems
in functional programming. It appears that most, and perhaps even
all, of the contributions in the above papers can be translated into
Joy notation.