Version -10.0.0
Each function, combinator, or definition should be documented here.
See and.
Combinator
Short-circuiting Boolean AND
Accept two quoted programs, run the first and expect a Boolean
value, if it's true pop it and run the second program
(which should also return a Boolean value) otherwise pop the second
program (leaving false on the stack.)
[A] [B] &&
---------------- true
B
[A] [B] &&
---------------- false
false
nulco [nullary [false]] dip branch
TODO: this is derived in one of the notebooks I think, look it up and link to it, or copy the content here.
This is seldom useful, I suspect, but this way you have it.
See mul.
See id.
See xor.
See eq.
See ne.
Function
Not negative.
n !-
----------- n < 0
false
n !-
---------- n >= 0
true
0 \>=
Return a Boolean value indicating if a number is greater than or equal to zero.
See gt.
See ge.
See rshift.
See sub.
See pred.
See lt.
See le.
See ne.
Function
... a <{}
----------------
... [] a
[] swap
Tuck an empty list just under the first item on the stack.
See lshift.
Function
... b a <{}
-----------------
... [] b a
[] rollup
Tuck an empty list just under the first two items on the stack.
See mod.
See add.
See succ.
Function
Is the item on the top of the stack "truthy"?
You often want to test the truth value of an item on the stack without consuming the item.
See floordiv.
See floordiv.
See floordiv.
Combinator
Short-circuiting Boolean OR
Accept two quoted programs, run the first and expect a Boolean
value, if it’s false pop it and run the second program
(which should also return a Boolean value) otherwise pop the second
program (leaving true on the stack.)
[A] [B] ||
---------------- A -> false
B
[A] [B] ||
---------------- A -> true
true
Function
Return the absolute value of the argument.
Basis Function
Add two numbers together: a + b.
Combinator
Build a list of values from a generator program G
and a stopping predicate P.
[P] [G] anamorphism
-----------------------------------------
[P] [pop []] [G] [dip swons] genrec
The range function generates a list of the integers
from 0 to n - 1:
[0 <=] [-- dup] anamorphism
See the Recursion Combinators notebook.
Basis Function
Logical bit-wise AND.
"apply one"
Combinator
Given a quoted program on TOS and anything as the second stack item run the program without disturbing the stack and replace the two args with the first result of the program.
... x [Q] app1
---------------------------------
... [x ...] [Q] infra first
This is the same effect as the unary combinator.
Just a specialization of nullary really. Its
parallelizable cousins are more useful.
Combinator
Like app1 with two items.
... y x [Q] . app2
-----------------------------------
... [y ...] [Q] . infra first
[x ...] [Q] infra first
[[grba] [swap] [grba] [swap]] [dip] [[infrst]] [cons] [ii]
Unlike app1, which is essentially an alias for unary, this function is not the same as binary. Instead of running one program using exactly two items from the stack and pushing one result (as binary does) this function takes two items from the stack and runs the program twice, separately for each of the items, then puts both results onto the stack.
This is not currently implemented as parallel processes but it can (and should) be done.
Combinator
Like [app1] with three items.
... z y x [Q] . app3
-----------------------------------
... [z ...] [Q] . infra first
[y ...] [Q] infra first
[x ...] [Q] infra first
3 [appN]
See [app2].
Combinator
Like [app1] with any number of items.
... xN ... x2 x1 x0 [Q] n . appN
--------------------------------------
... [xN ...] [Q] . infra first
...
[x2 ...] [Q] infra first
[x1 ...] [Q] infra first
[x0 ...] [Q] infra first
[[grabN]] [codi] [map] [disenstacken]
This function takes a quoted function Q and an
integer and runs the function that many times on that many stack
items. See also [app2].
See getitem.
Function
Compute the average of a list of numbers. (Currently broken until I can figure out what to do about "numeric tower" in Thun.)
[[sum]] [[size]] [cleave] [/]
Theoretically this function would compute the sum and the size in two separate threads, then divide. This works but a compiled version would probably do better to sum and count the list once, in one thread, eh?
As an exercise in Functional Programming in Joy it would be fun to convert this into a catamorphism. See the Recursion Combinators notebook.
Combinator
Run two quoted programs
[P] [Q] b
---------------
P Q
[[i]] [dip] [i]
This combinator may seem trivial but it comes in handy.
Combinator
Run a quoted program using exactly two stack values and leave the first item of the result on the stack.
... y x [P] binary
-----------------------
... a
[unary] [popd]
Runs any other quoted function and returns its first result while consuming exactly two items from the stack.
Basis Function
Convert the item on the top of the stack to a Boolean value.
For integers 0 is false and any other number is
true; for lists the empty list is false
and all other lists are true.
[not]
Basis Combinator
Use a Boolean value to select and run one of two quoted programs.
false [F] [T] branch
--------------------------
F
true [F] [T] branch
-------------------------
T
[rolldown] [choice] [i]
This is one of the fundamental operations (although it can be defined in terms of [choice] as above). The more common "if..then..else" construct [ifte] adds a predicate function that is evaluated [nullary].
[choice] [ifte] [select]
Function
a b c d [...] ccccons
---------------------------
[a b c d ...]
Do [cons] four times.
[ccons] [ccons]
[ccons] [cons] [times]
Function
a b [...] ccons
---------------------
[a b ...]
Do [cons] two times.
[cons] [cons]
[cons] [ccons]
Basis Function
Use a Boolean value to select one of two items.
a b false choice
----------------------
a
a b true choice
---------------------
b
[[pop]] [[popd]] [branch]
It's a matter of taste whether you implement this in terms of [branch] or the other way around.
[branch] [select]
Basis Function
Clear everything from the stack.
[stack] [bool] [[pop] [stack] [bool]] [loop]
[stack] [swaack]
Combinator
Run two programs in parallel, consuming one additional item, and put their results on the stack.
... x [A] [B] cleave
------------------------
... a b
[fork] [popdd]
1 2 3 [+] [-] cleave
--------------------------
1 2 5 -1
One of a handful of useful parallel combinators.
[clop] [fork] [map]
Combinator
Run two programs in parallel, consuming two additional items, and put their results on the stack.
... x y [A] [B] clop
--------------------------
... a b
[cleave] [popdd]
Like [cleave] but consumes an additional item from the stack.
1 2 3 4 [+] [-] clop
--------------------------
1 2 7 -1
[cleave] [fork] [map]
Combinator
Take two values and three quoted programs on the stack and run one of the three depending on the results of comparing the two values.
a b [G] [E] [L] cmp
------------------------- a > b
G
a b [G] [E] [L] cmp
------------------------- a = b
E
a b [G] [E] [L] cmp
------------------------- a < b
L
This is useful sometimes, and you can [dup] or [dupd] with two quoted programs to handle the cases when you just want to deal with [<=] or [>=] and not all three possibilities, e.g.:
[G] [EL] dup cmp
[GE] [L] dupd cmp
Or even:
[GL] [E] over cmp
TODO: link to tree notebooks where this was used.
Combinator
Take a quoted program from the stack, [cons] the next item onto it, then [dip] the whole thing under what was the third item on the stack.
a b [F] . codi
--------------------
b . F a
[cons] [dip]
This is one of those weirdly specific functions that turns out to be useful in a few places.
[appN] [codireco]
Combinator
This is part of the [make_generator] function. You would not use this combinator directly.
[codi] [reco]
See [make_generator] and the
"Using x to Generate Values" notebook as well as
Recursion
Theory and Joy by Manfred von Thun.
[make_generator]
Function
Concatinate two lists.
[a b c] [d e f] concat
----------------------------
[a b c d e f]
[first] [firsttwo] [flatten] [fourth] [getitem] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [splitat] [split_list] [swaack] [third] [zip]
Combinator
This combinator works like a case statement. It expects a single
quote on the stack that must contain zero or more condition quotes
and a default quote. Each condition quote should contain a quoted
predicate followed by the function expression to run if that
predicate returns true. If no predicates return
true the default function runs.
[
[ [Predicate0] Function0 ]
[ [Predicate1] Function1 ]
...
[ [PredicateN] FunctionN ]
[Default]
]
cond
It works by rewriting into a chain of nested [ifte]{.title-ref} expressions, e.g.:
[[[B0] T0] [[B1] T1] [D]] cond
-----------------------------------------
[B0] [T0] [[B1] [T1] [D] ifte] ifte
[ifte]
Basis Function
Given an item and a list, append the item to the list to make a new list.
a [...] cons
------------------
[a ...]
Cons is a venerable old function from Lisp. Its inverse operation is [uncons].
[uncons]
Combinator
Specialist function (that means I forgot what it does and why.)
[dip] [infrst]
Basis Combinator
The dip combinator expects a quoted program on the
stack and below it some item, it hoists the item into the
expression and runs the program on the rest of the stack.
... x [Q] . dip
---------------------
... . Q x
This along with [infra] are enough to update any datastructure. See the "Traversing Datastructures with Zippers" notebook.
Note that the item that was on the top of the stack
(x in the example above) will not be treated specially
by the interpreter when it is reached again. This is something of a
footgun. My advice is to avoid putting bare unquoted symbols onto
the stack, but then you can't use symbols as "atoms" and also use
dip and infra to operate on compound
datastructures with atoms in them. This is a kind of side-effect of
the Continuation-Passing Style. The dip combinator
could "set aside" the item and replace it after running
Q but that means that there is an "extra space" where
the item resides while Q runs. One of the nice things
about CPS is that the whole state is recorded in the stack and
pending expression (not counting modifications to the
dictionary.)
[dipd] [dipdd] [dupdip] [dupdipd] [infra]
Combinator
Like [dip] but expects two items.
... y x [Q] . dipd
-------------------------
... . Q y x
See [dip].
[dip] [dipdd] [dupdip] [dupdipd] [infra]
Combinator
Like [dip] but expects three items. :
... z y x [Q] . dip
-----------------------------
... . Q z y x
See [dip].
[dip] [dipd] [dupdip] [dupdipd] [infra]
Function
The disenstacken function expects a list on top of
the stack and makes that the stack discarding the rest of the
stack.
1 2 3 [4 5 6] disenstacken
--------------------------------
6 5 4
[[clear]] [dip] [reverse] unstack
Note that the order of the list is not changed, it just looks that way because the stack is printed with the top on the right while lists are printed with the top or head on the left.
[enstacken] [stack] unstack
See floordiv.
Function
x y divmod
------------------
q r
(x/y) (x%y)
Invariant: qy + r = x.
[[floordiv]] [[mod]] [clop]
Function
Given a number greater than zero put all the Natural numbers (including zero) less than that onto the stack.
3 down_to_zero
--------------------
3 2 1 0
[0 >] [[dup] [--]] [while]
[range]
Function
Expects an integer and a quote on the stack and returns the quote with n items removed off the top.
[a b c d] 2 drop
----------------------
[c d]
[[rest]] [times]
[take]
Basis Function
"Dup"licate the top item on the stack.
a dup
-----------
a a
[dupd] [dupdd] [dupdip] [dupdipd]
Function
[dup] the second item down on the stack.
a b dupd
--------------
a a b
[[dup]] [dip]
[dup] [dupdd] [dupdip] [dupdipd]
Function
[dup] the third item down on the stack.
a b c dupdd
-----------------
a a b c
[[dup]] [dipd]
[dup] [dupd] [dupdip] [dupdipd]
Combinator
Apply a function F and [dup] the item under it on
the stack.
a [F] dupdip
------------------
a F a
[dupd] [dip]
a [F] dupdip
a [F] dupd dip
a [F] [dup] dip dip
a dup [F] dip
a a [F] dip
a F a
A very common and useful combinator.
[dupdipd]
Combinator
Run a copy of program F under the next item down on
the stack.
a [F] dupdipd
-------------------
F a [F]
[dup] [dipd]
[dupdip]
Function
Put the stack onto the stack replacing the contents of the stack.
... a b c enstacken
-------------------------
[c b a ...]
[stack] [[clear]] [dip]
This is a destructive version of [stack]. See the note under [disenstacken] about the apparent but illusory reversal of the stack.
[stack] [unstack] [disenstacken]
Basis Function
Compare the two items on the top of the stack for equality and replace them with a Boolean value.
a b eq
-------------
Boolean
(a = b)
[cmp] [ge] [gt] [le] [lt] [ne]
Function
Replace a list with its first item.
[a ...]
--------------
a
[uncons] [pop]
[second] [third] [fourth] [rest]
Function
Replace a list with its first two items.
[a b ...] first_two
-------------------------
a b
[uncons] [first]
[first] [second] [third] [fourth] [rest]
Function
Given a list of lists, concatinate them.
[[1 2] [3 [4] 5] [6 7]] flatten
-------------------------------------
[1 2 3 [4] 5 6 7]
[\<{}] [[concat]] [step]
Note that only one "level" of lists is flattened. In the example
above [4] is not unquoted.
[concat] [first] [firsttwo] [fourth] [getitem] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [splitat] [split_list] [swaack] [third] [zip]
Basis Function
Return the largest integer \<= x.
This function doesn't make sense (yet) to have because there are (as yet) only integers in the system.
Basis Function
I don't know why this is called "floor" div, I think it rounds its result down (not towards zero or up.)
a b floordiv
------------------
(a/b)
All the division commands need to be revisited when the "numeric tower" for Thun gets nailed down.
[divmod]
Combinator
Run two quoted programs in parallel and replace them with their results.
... [F] [G] fork
----------------------
... f g
[[i]] [app2]
The basic parallelism combinator, the two programs are run independently.
[cleave] [clop] [map]
Function
Replace a list with its fourth item.
[a b c d ...] fourth
--------------------------
d
[rest] [third]
[first] [second] [third] [rest]
Function
Take two integers from the stack and replace them with their Greatest Common Denominator.
true [[tuck] [mod] [dup] 0 [>]] [loop] [pop]
Euclid's Algorithm
Function
Compiled GCD function.
See [gcd].
[gcd]
Basis Function
Greater-than-or-equal-to comparison of two numbers.
a b ge
--------------
Boolean
(a >= b)
[cmp] [eq] [gt] [le] [lt] [ne]
Combinator
General Recursion Combinator.
[if] [then] [rec1] [rec2] genrec
---------------------------------------------------------------------
[if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte
[[[genrec]] [ccccons]] [nullary] [swons] [concat] [ifte]
(Note that this definition includes the genrec
symbol itself, it is self-referential. This is possible because the
definition machinery does not check that symbols in defs are in the
dictionary. genrec is the only self-referential
definition.)
See the Recursion Combinators notebook.
From "Recursion Theory and Joy" by Manfred von Thun:
"The genrec combinator takes four program parameters in addition to whatever data parameters it needs. Fourth from the top is an if-part, followed by a then-part. If the if-part yields true, then the then-part is executed and the combinator terminates. The other two parameters are the rec1-part and the rec2-part. If the if-part yields false, the rec1-part is executed. Following that the four program parameters and the combinator are again pushed onto the stack bundled up in a quoted form. Then the rec2-part is executed, where it will find the bundled form. Typically it will then execute the bundled form, either with i or with app2, or some other combinator."
The way to design one of these is to fix your base case
[then] and the test [if], and then treat
rec1 and rec2 as an else-part
"sandwiching" a quotation of the whole function.
For example, given a (general recursive) function
F:
F == [I] [T] [R1] [R2] genrec
If the [I] if-part fails you must derive
R1 and R2 from: :
... R1 [F] R2
Just set the stack arguments in front, and figure out what
R1 and R2 have to do to apply the quoted
[F] in the proper way. In effect, the
genrec combinator turns into an [ifte] combinator with
a quoted copy of the original definition in the else-part:
F == [I] [T] [R1] [R2] genrec
== [I] [T] [R1 [F] R2] ifte
Tail recursive functions are those where R2 is the
i combinator:
P == [I] [T] [R] tailrec
== [I] [T] [R [P] i] ifte
== [I] [T] [R P] ifte
[anamorphism] [tailrec] [x]
Function
Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0.
[a b c d] 2 getitem
-------------------------
c
[drop] [first]
If the number isn't a valid index into the quote
getitem will cause some sort of problem (the exact
nature of which is implementation-dependant.)
[concat] [first] [firsttwo] [flatten] [fourth] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [splitat] [split_list] [swaack] [third] [zip]
Function
Expect a number on the top of the satck and put that many items from uner it onto a new list.
a b c d e 3 grabN
-----------------------
a b [c d e]
[\<{}] [[cons]] [times]
Function
A weird function used in [app2] that does this:
... 1 2 3 4 5 grba
-------------------------------
... 1 2 3 [4 3 2 1 ...] 5
It grabs the stack under the top item, and substitutes it for the second item down on the stack.
[[stack] [popd]] [dip]
This function "grabs" an item from the stack along with a copy of the stack. It's part of the [app2] definition.
[app2]
Basis Function
Greater-than comparison of two numbers.
a b gt
--------------
Boolean
(a > b)
[cmp] [eq] [ge] [le] [lt] [ne]
Function
Accepts a quoted symbol on the top of the stack and prints its documentation.
[foo] help
----------------
Technically this is equivalent to pop, but it will
only work if the item on the top of the stack is a quoted
symbol.
Function
x y hypot
---------------------------
sqrt(sqr(x) + sqr(y))
[[sqr]] [ii] [+] [sqrt]
This is another function that has to wait on the numeric tower.
[sqrt]
Basis Combinator
Append a quoted expression onto the pending expression.
[Q] . i
-------------
. Q
This is a fundamental combinator. It is used in all kinds of
places. For example, the [x] combinator can be defined as dup
i.
Basis Function
The identity function.
Does nothing. It's kind of a mathematical thing, but it occasionally comes in handy.
Combinator
If-Then-Else combinator, a common and convenient specialization of [branch].
[if] [then] [else] ifte
---------------------------------------
[if] nullary [else] [then] branch
[[nullary]] [dipd] [swap] [branch]
[branch] [loop] [while]
Combinator
Take a quoted program from the stack and run it twice, first under the top item, then again with the top item.
... a [Q] ii
------------------
... Q a Q
[[dip]] [dupdip] [i]
It's a little tricky to understand how this works so here's an example trace:
1 2 3 4 [++] • [dip] dupdip i
1 2 3 4 [++] [dip] • dupdip i
1 2 3 4 [++] • dip [++] i
1 2 3 • ++ 4 [++] i
1 2 4 • 4 [++] i
1 2 4 4 • [++] i
1 2 4 4 [++] • i
1 2 4 4 • ++
1 2 4 5 •
In some cases (like the example above) this is the same effect as using [app2] but most of the time it's not:
1 2 3 4 [+] ii
--------------------
1 9
1 2 3 4 [+] app2
----------------------
1 2 5 6
[app2] [b]
Combinator
Accept a quoted program and a list on the stack and run the program with the list as its stack. Does not affect the stack (below the list.)
... x y z [a b c] [Q] infra
---------------------------------
c b a Q [z y x ...] swaack
[swons] [swaack] [[i]] [dip] [swaack]
This is one of the more useful combinators. It allows a quoted
expression to serve as a stack for a program, effectively running
it in a kind of "pocket universe". If the list represents a
datastructure then infra lets you work on its internal
structure.
Combinator
Does [infra] and then extracts the [first] item from the resulting list.
[infra] [first]
Create a new Joy function definition in the Joy dictionary. A definition is given as a quote with a name followed by a Joy expression.
[sqr dup mul] inscribe
This is the only function that modifies the dictionary. It's
provided as a convenience, for tinkering with new definitions
before entering them into the defs.txt file. It can be
abused, which you should avoid unless you know what you're
doing.
Basis Function
Less-Than-or-Equal-to comparison of the two items on the top of the stack, replacing them with a Boolean value.
a b le
-------------
Boolean
(a <= b)
[cmp] [eq] [ge] [gt] [lt] [ne]
Basis Combinator
Expect a quoted program Q and a Boolean value on
the stack. If the value is false discard the quoted program,
otherwise run a copy of Q and loop
again.
false [Q] loop
--------------------
true [Q] . loop
--------------------------
. Q [Q] loop
This, along with [branch] and [fork], is one of the four main
combinators of all programming. The fourth, sequence, is implied by
juxtaposition. That is to say, in Joy F G is like
G(F(...)) in a language bassed on function
application. Or again, to quote the Joy
Wikipedia entry#Mathematical_purity),
In Joy, the meaning function is a homomorphism from the syntactic monoid onto the semantic monoid. That is, the syntactic relation of concatenation of symbols maps directly onto the semantic relation of composition of functions.
Anyway, [branch], [fork], amd [loop] are the fundamental combinators in Joy. Just as [branch] has it's more common and convenient form [ifte], [loop] has [while].
[branch] [fork] [while]
Basis Function
a n lshift
----------------
(a×2ⁿ)
[rshift]
Basis Function
Less-Than comparison of the two items on the top of the stack, replacing them with a Boolean value.
a b lt
-------------
Boolean
(a < b)
[cmp] [eq] [ge] [gt] [le] [ne]
Function
Given an initial state value and a quoted generator function build a generator quote.
state [generator function] make_generator
-----------------------------------------------
[state [generator function] codireco]
230 [dup ++] make_generator
---------------------------------
[230 [dup ++] codireco]
And then:
[230 [dup ++] codireco] 5 [x] times pop
---------------------------------------------
230 231 232 233 234
[[codireco]] [ccons]
See the
"Using x to Generate Values" notebook.
[codireco]
Combinator
Given a list of items and a quoted program run the program for each item in the list (with the rest of the stack) and replace the old list and the program with a list of the results.
5 [1 2 3] [++ *] map
--------------------------
5 [10 15 20]
This is a common operation in many languages. In Joy it can be a parallelism combinator due to the "pure" nature of the language.
[app1] [app2] [app3] appN [fork]
Basis Function
Given a list find the maximum.
[1 2 3 4] max
-------------------
4
[min] [size] [sum]
Basis Function
Given a list find the minimum.
[1 2 3 4] min
-------------------
1
[max] [size] [sum]
Basis Function
Return the remainder of a divided by
b.
a b mod
-------------
(a%b)
[divmod] [mul]
See mod.
Basis Function
Multiply two numbers.
a b mul
-------------
(a×b)
[div] [product]
Basis Function
Not-Equal comparison of the two items on the top of the stack, replacing them with a Boolean value.
a b ne
-------------
Boolean
(a = b)
[cmp] [eq] [ge] [gt] [le] [lt]
Function
Invert the sign of a number.
a neg
-----------
-a
0 [swap] [-]
Function
Like [bool] but convert the item on the top of the stack to the inverse Boolean value.
true not
--------------
false
false not
---------------
true
[bool] [true] [false] [branch]
[bool]
Function
Take the item on the top of the stack and [cons] it onto
[nullary].
[F] nulco
-------------------
[[F] nullary]
[[nullary]] [cons]
Helper function for [\|\|] and [&&].
[&&] [\|\|]
Combinator
Run a quoted program without using any stack values and leave the first item of the result on the stack.
... [P] nullary
---------------------
... a
[[stack]] [dip] [infra] [first]
... [P] nullary
... [P] [stack] dip infra first
... stack [P] infra first
... [...] [P] infra first
... [a ...] first
... a
A very useful function that runs any other quoted function and returns it's first result without disturbing the stack (under the quoted program.)
Function
Like [getitem] but [swap]s the order of arguments.
2 [a b c d] of
--------------------
c
[swap] [getitem]
[getitem]
Basis Function
Logical bit-wise OR.
[and] [xor]
Function
[dup] the second item on the stack over the
first.
a b over
--------------
a b a
There are many many ways to define this function.
[swap] [tuck]
[[pop]] [nullary]
[[dup]] [dip] [swap]
[unit] [dupdip]
[unit] [dupdipd] [first]
And so on...
A fine old word from Forth.
[tuck]
Combinator
Take a list of quoted functions from the stack and replace it with a list of the [first] results from running those functions (on copies of the rest of the stack.)
5 7 [[+][-][*][/][%]] pam
-------------------------------
5 7 [12 -2 35 0 5]
[[i]] [map]
A specialization of [map] that runs a list of functions in parallel (if the underlying [map] function is so implemented, of course.)
[map]
See getitem.
Function
Plus or minus. Replace two numbers with their sum and difference.
a b pm
-----------------
(a+b) (a-b)
[+] [-] [clop]
Basis Function
Pop the top item from the stack and discard it.
a pop
-----------
[popd] [popdd] [popop] [popopd] [popopdd] [popopop]
Function
[pop] the second item down on the stack.
a b popd
--------------
b
[swap] [pop]
[pop] [popdd] [popop] [popopd] [popopdd] [popopop]
Function
[pop] the third item on the stack.
a b c popdd
-----------------
b c
[rolldown] [pop]
[pop] [popd] [popop] [popopd] [popopdd] [popopop]
Function
[pop] two items from the stack.
a b popop
---------------
[pop] [pop]
[pop] [popd] [popdd] [popopd] [popopdd] [popopop]
Function
[pop] the second and third items from the stack.
a b c popopd
------------------
c
[rollup] [popop]
[pop] [popd] [popdd] [popop] [popopdd] [popopop]
Function
a b c d popopdd
---------------------
c d
[[popop]] [dipd]
[pop] [popd] [popdd] [popop] [popopd] [popopop]
Function
[pop] three items from the stack.
a b c popopop
-------------------
[pop] [popop]
[pop] [popd] [popdd] [popop] [popopd] [popopdd]
Basis Function
Take two numbers a and b from the
stack and raise a to the nth power.
(b is on the top of the stack.)
a n pow
-------------
(aⁿ)
2 [2 3 4 5 6 7 8 9] [pow] map
-----------------------------------
2 [4 8 16 32 64 128 256 512]
Function
Predecessor. Decrement TOS.
1 -
[succ]
Combinator
From the "Overview of the language JOY"
The primrec combinator expects two quoted programs in addition to a data parameter. For an integer data parameter it works like this: If the data parameter is zero, then the first quotation has to produce the value to be returned. If the data parameter is positive then the second has to combine the data parameter with the result of applying the function to its predecessor.
5 [1] [*] primrec
Then primrec tests whether the top element on the stack (initially the 5) is equal to zero. If it is, it pops it off and executes one of the quotations, the [1] which leaves 1 on the stack as the result. Otherwise it pushes a decremented copy of the top element and recurses. On the way back from the recursion it uses the other quotation, [*], to multiply what is now a factorial on top of the stack by the second element on the stack.
0 [Base] [Recur] primrec
------------------------------
Base
n [Base] [Recur] primrec
------------------------------------------ n > 0
n (n-1) [Base] [Recur] primrec Recur
Simple and useful specialization of the [genrec] combinator from the original Joy system.
[genrec] [tailrec]
Function
Just as [sum] sums a list of numbers, this function multiplies them together.
1 [swap] [[mul]] [step]
Or,
[1] [[mul]] [primrec]
Function
"Quote D" Wrap the second item on the stack in a list.
a b quoted
----------------
[a] b
[[unit]] [dip]
This comes from the original Joy stuff.
[unit]
Function
Expect a number n on the stack and replace it with
a list: [(n-1)...0].
5 range
-----------------
[4 3 2 1 0]
-5 range
--------------
[]
[0 \<=] [1 - [dup]] [anamorphism]
If n is less than 1 the resulting list is
empty.
[rangetozero]
Function
Take a number n from the stack and replace it with
a list [0...n].
5 range_to_zero
---------------------
[0 1 2 3 4 5]
[unit] [[downtozero]] [infra]
Note that the order is reversed compared to [range].
[downtozero] [range]
Function
Replace the first item in a list with the item under it.
a [b ...] reco
--------------------
[a ...]
[rest] [cons]
[codireco] [make_generator]
See mod.
See mod.
Function
Expects an item on the stack and a quote under it and removes that item from the the quote. The item is only removed once. If the list is empty or the item isn't in the list then the list is unchanged.
[1 2 3 1] 1 remove
------------------------
[2 3 1]
See the "Remove Function" notebook.
Basis Function
[a ...] rest
------------------
[...]
[first] [uncons]
Function
Reverse the list on the top of the stack.
[1 2 3] reverse
---------------------
[3 2 1]
[\<{}] [shunt]
Function
a b c rolldown
--------------------
b c a
[swapd] [swap]
[rollup]
Function
a b c rollup
------------------
c a b
[swap] [swapd]
[rolldown]
See rollup.
See rolldown.
Function
Round a number to a given precision in decimal digits.
Another one that won't make sense until the "numeric tower" is nailed down.
Function
[a b ...] rrest
---------------------
[...]
[rest] [rest]
[rest]
Basis Function
a n rshift
----------------
(a∕2ⁿ)
[lshift]
Function
Run a quoted program in a list.
[1 2 +] run
-----------------
[3]
[\<{}] [infra]
Function
[a b ...] second
----------------------
b
[rest] [first]
[first] [third] [fourth]
Basis Function
Use a Boolean value to select one of two items from a sequence. :
[a b] false select
------------------------
a
[a b] true select
-----------------------
b
The sequence can contain more than two items but not fewer.
[choice]
Function
Print redistribution information.
Mathematically this is a form of [id], but it has the side-effect of printing out the GPL notice.
[warranty]
Function
Move the top item from one list to another.
[x y z] [a b c] shift
---------------------------
[a x y z] [b c]
[uncons] [[swons]] [dip]
[shunt]
Function
Like [concat] but [reverse] the top list into the second.
[a b c] [d e f] shunt
---------------------------
[f e d a b c]
[[swons]] [step]
This is more efficient than [concat] so prefer it if you don't need to preserve order.
[concat] [reverse] [shift]
Function
Replace a list with its size.
[23 [cats] 4] size
------------------------
3
[[pop] [++]] [step_zero]
Function
Given a list return it sorted.
[4 2 5 7 1] sort
----------------------
[1 2 4 5 7]
Function
Example code.
See the "Square Spiral Example Joy Code" notebook.
Function
Split a list (second on the stack) at the position given by the number on the top of the stack.
[1 2 3 4 5 6 7] 4 split_at
--------------------------------
[5 6 7] [4 3 2 1]
[[drop]] [[take]] [clop]
Take a list and a number n from the stack, take
n items from the top of the list and [shunt] them onto
a new list that replaces the number n on the top of
the stack.
[split_list]
Function
Split a list (second on the stack) at the position given by the number on the top of the stack such that [concat] would reconstruct the original list.
[1 2 3 4 5 6 7] 4 split_list
----------------------------------
[1 2 3 4] [5 6 7]
[[take] [reverse]] [[drop]] [clop]
Compare with [split_at]. This function does extra work to ensure that [concat] would reconstruct the original list.
[split_at]
Function
Square the number on the top of the stack.
n sqr
------------
n²
[dup] [mul]
Basis Function Combinator
Return the square root of the number a. Negative numbers return complex roots.
Another "numeric tower" hatch...
Function
Put the stack onto the stack.
... c b a stack
---------------------------
... c b a [a b c ...]
[] [swaack] [dup] [swaack] [first]
This function forms a pair with [unstack], and together they form the complement to the "destructive" pair [enstacken] and [disenstacken].
[unstack] [enstacken] [disenstacken]
Function
Grab the stack under the top item and put it onto the stack.
... 1 2 3 stackd
------------------------
... 1 2 [2 1 ...] 3
[[stack]] [dip]
Combinator
Run a quoted program on each item in a sequence.
... [] [Q] step
---------------------
...
... [a] [Q] step
----------------------
... a Q
... [a b c] [Q] . step
----------------------------------------
... a . Q [b c] [Q] step
See the Recursion Combinators notebook.
[step_zero]
Combinator
Like [step] but with 0 as the initial value.
[...] [F] step_zero
-------------------------
0 [...] [F] step
0 [roll>] [step]
[size] and [sum] can both be defined in terms of this specialization of [step].
[step]
Function
Take the [stack] and [uncons] the top item.
1 2 3 stuncons
--------------------
1 2 3 3 [2 1]
[stack] [uncons]
Function
Take the [stack] and [uncons] the top two items.
1 2 3 stununcons
----------------------
1 2 3 3 2 [1]
[stack] [uncons] [uncons]
[stuncons]
Basis Function
Subtract the number on the top of the stack from the number below it.
a b sub
-------------
(a-b)
[add]
Function
Successor. Increment TOS.
1 +
[pred]
Combinator
Given a quoted sequence of numbers return the sum.
[1 2 3 4 5] sum
---------------------
15
[+] [step_zero]
[size]
Basis Function
Swap stack. Take a list from the top of the stack, replace the stack with the list, and put the old stack onto it.
1 2 3 [4 5 6] swaack
--------------------------
6 5 4 [3 2 1]
This function works as a kind of "context switch". It's used in the definition of [infra].
[infra]
Basis Function
Swap the top two items on the stack.
a b swap
--------------
b a
[swapd]
Function
Swap the second and third items on the stack.
a b c swapd
-----------------
b a c
[[swap]] [dip]
[over] [tuck]
Function
[concat] two lists, but [swap] the lists first.
[swap] [concat]
[concat]
Function
Like [cons] but [swap] the item and list.
[...] a swons
-------------------
[a ...]
[swap] [cons]
Combinator
A specialization of the [genrec] combinator.
[[i]] [genrec]
Some recursive functions do not need to store additional data or pending actions per-call. These are called "tail recursive" functions. In Joy, they appear as [genrec] definitions that have [i] for the second half of their recursive branch.
See the Recursion Combinators notebook.
[genrec]
Function
Expects an integer n and a list on the stack and
replace them with a list with just the top n items in
reverse order.
[a b c d] 2 take
----------------------
[b a]
[\<\<{}] [[shift]] [times] [pop]
Combinator
Run a quoted program using exactly three stack values and leave the first item of the result on the stack.
... z y x [P] ternary
-------------------------
... a
[binary] [popd]
Runs any other quoted function and returns its first result while consuming exactly three items from the stack.
Function
[a b c ...] third
-----------------------
c
[rest] [second]
[first] [second] [fourth] [rest]
Combinator
Expect a quoted program and an integer n on the
stack and do the program n times.
... n [Q] . times
----------------------- w/ n <= 0
... .
... 1 [Q] . times
-----------------------
... . Q
... n [Q] . times
------------------------------------- w/ n > 1
... . Q (n-1) [Q] times
[-- dip] cons [swap] infra [0 >] swap while pop :
This works by building a little [while] program and running it:
1 3 [++] • [-- dip] cons [swap] infra [0 >] swap while pop
1 3 [++] [-- dip] • cons [swap] infra [0 >] swap while pop
1 3 [[++] -- dip] • [swap] infra [0 >] swap while pop
1 3 [[++] -- dip] [swap] • infra [0 >] swap while pop
dip -- [++] • swap [3 1] swaack [0 >] swap while pop
dip [++] -- • [3 1] swaack [0 >] swap while pop
dip [++] -- [3 1] • swaack [0 >] swap while pop
1 3 [-- [++] dip] • [0 >] swap while pop
1 3 [-- [++] dip] [0 >] • swap while pop
1 3 [0 >] [-- [++] dip] • while pop
This is a common pattern in Joy. You accept some parameters from the stack which typically include qouted programs and use them to build another program which does the actual work. This is kind of like macros in Lisp, or preprocessor directives in C.
See bool.
Function
[dup] the item on the top of the stack under the second item on the stack.
a b tuck
--------------
b a b
[dup] [[swap]] [dip]
[over]
(Combinator)
Run a quoted program using exactly one stack value and leave the first item of the result on the stack.
... x [P] unary
---------------------
... a
[nullary] [popd]
Runs any other quoted function and returns its first result while consuming exactly one item from the stack.
Basis Function
Removes an item from a list and leaves it on the stack under the
rest of the list. You cannot uncons an item from an
empty list.
[a ...] uncons
--------------------
a [...]
This is the inverse of [cons].
[cons]
Function
Given a list remove duplicate items.
Function
a unit
------------
[a]
[] [cons]
Combinator
Unquote (using [i]) the list that is second on the stack.
1 2 [3 4] 5 unquoted
--------------------------
1 2 3 4 5
[[i]] [dip]
[unit]
Function
[a ...] unswons
---------------------
[...] a
[uncons] [swap]
Basis Function
True if the form on TOS is void otherwise False.
A form is any Joy expression composed solely of lists. This represents a binary Boolean logical formula in the arithmetic of the "Laws of Form", see The Markable Mark
Basis Function
Print warranty information.
Combinator
A specialization of [loop] that accepts a quoted predicate
program P and runs it [nullary].
[P] [F] while
------------------- P -> false
[P] [F] while
--------------------- P -> true
F [P] [F] while
[swap] [nulco] [dupdipd] [concat] [loop]
[loop]
Basis Function
Print all the words in alphabetical order.
Mathematically this is a form of [id].
[help]
Combinator
Take a quoted function F and run it with itself as
the first item on the stack.
[F] x
-----------
[F] F
dup i
The simplest recursive pattern.
See the Recursion Combinators notebook. as well as Recursion Theory and Joy by Manfred von
Basis Function
Logical bit-wise eXclusive OR.
[and] [or]
Function
Replace the two lists on the top of the stack with a list of the pairs from each list. The smallest list sets the length of the result list.
[1 2 3] [4 5 6] zip
-------------------------
[[4 1] [5 2] [6 3]]