You are here: The Algebra of Joy
TITLE>The Algebra of JoyAbstract: 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 rewriting.
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
reversed 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, without
variables, like this:
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 without variables
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 dip combinator which expects a program
on top of the stack and below that another value. It saves the value,
executes the program on the remainder of the stack and then restores
the saved value.
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 concatenated is the sum of the sizes 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 idempotent
if applying it once is "as good as" applying it twice. In
conventional notation this means that for all x
f(f(x)) = f(x)
For example, the function abs which returns the absolute
value of a number is idempotent. Another one is the function defined
on lists or strings which returns a sorted version - sorting an
already sorted sequence makes no difference.
abs(abs(n)) = abs(n) sort(sort(s)) = sort(s)
Composition can be used to express that stack operations are
idempotent. The following express the idempotency of
abs and sort:
abs abs == abs sort sort == sort
Another idempotent Joy function, one which has no counterpart in
conventional notation, is the newstack function, it throws
away anything that is on the stack. Doing it twice in succession
gives the same result as doing it once:
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)) = x
For example, the predecessor function on integers is the inverse of
the successor function on integers: for all integers x
pred(succ(x)) = x
It 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 inverses 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 == id
It 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)) = l
In Joy the two unary operators not and reverse
are polymorphic. The 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) = x
Left 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 dup expects one
value on top of the stack and pushes a duplicate on top. For example,
42 dup == 42 42
Both 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) = x
Two examples are the Boolean conjunction and disjunction operations:
for all b
b and b = b b or b = b
In Joy the dup operator can express idempotency of the
Boolean operations and and or which are defined
for truth values and for sets. It can also express the idempotency of
the numeric binary min and max operators:
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 == 17
On 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) = c
For 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 = 0
The 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 == id
The pop operator can also be used to express the
arity of a function, the number of parameters which it
expects. For example, numbers are nullary, the successor function is
unary, and addition is binary. There is no way to express this in
conventional notation. In Joy it is expressed by:
42 pop == id succ pop == pop + pop == pop pop
Similar 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 expects two values of
any type on top of the stack; its effect is to interchange them. The
operator is its own inverse:
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 swons is similar to 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
swoncat is defined to be the converse of
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 == P
From 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 + i
In Joy the swap operator can express that a function is
commutative.
swap + == + swap * == *
swap and == and swap or == or
swap max == max swap min == min
Two sorted sequences can be combined with the merge
operator to form one new sorted sequence. Unlike
concatenation, merging is commutative:
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 symmetric.
The following express that = and equal are
symmetric:
swap = == = swap equal == equal
With swap one can express that elements are left
unit elements for binary operations. In the case of operations
such as addition and the Boolean operations this already follows from
their commutativity. On the other hand, concatenation of strings or
lists is not commutative, but the empty string and the empty list are
both right and left unit elements for concatenation. They are also
both right unit elements for merge.
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 == uncons
There 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 merge
Their 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 combinators.
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 5
In 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) + k
If 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
size (or length) of string function which maps the size of
concatenations onto sums of sizes. The De Morgan laws are
another example.
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 g
Some 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 or
In the above, charset transforms a string of characters
into a set of characters, and the or operator computes
set union in this case. Another homomorphism is the sort
operator which maps unordered lists under concatenation onto ordered
lists under a binary merge operator which preseves
ordering:
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
each other. Here is the arithmetic law:
(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 * k
The 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 left distribution
laws.
+ * == [*] 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 dup
The 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)) = x
The 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 25
The 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 sort
The law cannot be expressed in conventional notation because
split produces two results.
A category consists of a collection of objects and for any two objects a collection of morphisms, 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 monoids: 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 functors. 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 constructors.
They take integers, characters, truth values and so on into
LISTs of integers, LISTs of characters,
SETs of integers and so on. The functors must
also take integer functions such as squaring into corresponding
functions on LISTs or SETs 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 transformations. 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 reverse function: for all functions
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 unitlist
Note that in the third equation on the right the app2
combinator has to use a constructed program. The last two
laws again say the same thing.
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 cons
This 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 cons
But 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 concat
For 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 sums and
products 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 set types. The simplest is in terms of
bitstrings with potential elements 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 or applies. The
naturality of these operators is expressed by
[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 elements
For 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 bags; 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 can be defined by
mmap == [map] cons map
and 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 transpose
The zip function is natural, zipping the two
lists and then mmaping has the same effect as
mapping and then zipping:
[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 trees, also called
recursive lists. 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 treeflattened to form a one-level list. For example
[ [1 [2 3] [] 4] [5] ] treeflatten == [ 1 2 3 4 5 ]
The treeflattening function is a natural transformation
between the TREE and LIST functors, the
order of treeflattening and mapping does not matter:
[treeflatten] dip map == treemap treeflatten
The 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 stripped 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 treereverse
Once 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] map
which 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] map
which 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 flattened to produce a
single level list. The same original list of list can first be
flattened 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
mapped 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 flatten
It satisfies the following laws:
[unitlist] dip bind == i
[unitlist] bind == id
[K [H] bind] bind == [K] bind [H] bind
The 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 bind
Wadler 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.