64 KiB
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
Definition
nulco [nullary [false]] dip branch
Derivation
TODO: this is derived in one of the notebooks I think, look it up and link to it, or copy the content here.
Discussion
This is seldom useful, I suspect, but this way you have it.
Crosslinks
*
See mul.
•
See id.
^
See xor.
=
See eq.
!=
See ne.
!-
Function
Not negative.
n !-
----------- n < 0
false
n !-
---------- n >= 0
true
Definition
0 \>=
Discussion
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
Definition
[] swap
Discussion
Tuck an empty list just under the first item on the stack.
Crosslinks
<<
See lshift.
<<{}
Function
... b a <{}
-----------------
... [] b a
Definition
[] rollup
Discussion
Tuck an empty list just under the first two items on the stack.
Crosslinks
%
See mod.
+
See add.
++
See succ.
?
Function
Is the item on the top of the stack "truthy"?
Definition
Discussion
You often want to test the truth value of an item on the stack without consuming the item.
Crosslinks
/
See floordiv.
//
See floordiv.
/floor
See floordiv.
||
Combinator
Short-circuiting Boolean OR
Definition
Discussion
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
Crosslinks
abs
Function
Return the absolute value of the argument.
Definition
add
Basis Function
Add two numbers together: a + b.
anamorphism
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
Definition
[pop](#pop) \[
Example
The range function generates a list of the integers from 0 to n - 1:
0 <=
Discussion
See the Recursion Combinators notebook.
and
Basis Function
Logical bit-wise AND.
Crosslinks
app1
"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.
Definition
Discussion
Just a specialization of nullary really. Its parallelizable cousins
are more useful.
Crosslinks
app2
Combinator
Like app1 with two items.
... y x [Q] . app2
-----------------------------------
... [y ...] [Q] . infra first
[x ...] [Q] infra first
Definition
[grba] [swap] [grba] [swap]
Discussion
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.
Crosslinks
app3
Combinator
Like [app1] with three items.
... z y x [Q] . app3
-----------------------------------
... [z ...] [Q] . infra first
[y ...] [Q] infra first
[x ...] [Q] infra first
Definition
3 [appN]
Discussion
See [app2].
Crosslinks
appN
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
Definition
[grabN]
Discussion
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].
Crosslinks
at
See getitem.
average
Function
Compute the average of a list of numbers. (Currently broken until I can figure out what to do about "numeric tower" in Thun.)
Definition
[sum]
Discussion
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.
b
Combinator
Run two quoted programs
[P] [Q] b
---------------
P Q
Definition
[i]
Discussion
This combinator may seem trivial but it comes in handy.
Crosslinks
binary
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
Definition
[unary] [popd]
Discussion
Runs any other quoted function and returns its first result while consuming exactly two items from the stack.
Crosslinks
bool
Basis Function
Convert the item on the top of the stack to a Boolean value.
Discussion
For integers 0 is false and any other number is true; for lists the
empty list is false and all other lists are true.
Crosslinks
[not]
branch
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
Definition
[rolldown] [choice] [i]
Discussion
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].
Crosslinks
[choice] [ifte] [select]
ccccons
Function
a b c d [...] ccccons
---------------------------
[a b c d ...]
Do [cons] four times.
Definition
[ccons] [ccons]
Crosslinks
[ccons] [cons] [times]
ccons
Function
a b [...] ccons
---------------------
[a b ...]
Do [cons] two times.
Definition
[cons] [cons]
Crosslinks
[cons] [ccons]
choice
Basis Function
Use a Boolean value to select one of two items.
a b false choice
----------------------
a
a b true choice
---------------------
b
Definition
[pop]
Discussion
It's a matter of taste whether you implement this in terms of [branch] or the other way around.
Crosslinks
[branch] [select]
clear
Basis Function
Clear everything from the stack.
Definition
[stack] [bool] pop] [stack] [bool [loop]
Crosslinks
[stack] [swaack]
cleave
Combinator
Run two programs in parallel, consuming one additional item, and put their results on the stack.
... x [A] [B] cleave
------------------------
... a b
Derivation
[fork] [popdd]
Example
1 2 3 [+] [-] cleave
--------------------------
1 2 5 -1
Discussion
One of a handful of useful parallel combinators.
Crosslinks
[clop] [fork] [map]
clop
Combinator
Run two programs in parallel, consuming two additional items, and put their results on the stack.
... x y [A] [B] clop
--------------------------
... a b
Definition
[cleave] [popdd]
Discussion
Like [cleave] but consumes an additional item from the stack.
1 2 3 4 [+] [-] clop
--------------------------
1 2 7 -1
Crosslinks
[cleave] [fork] [map]
cmp
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
Discussion
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
Crosslinks
TODO: link to tree notebooks where this was used.
codi
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
Definition
[cons] [dip]
Discussion
This is one of those weirdly specific functions that turns out to be useful in a few places.
Crosslinks
[appN] [codireco]
codireco
Combinator
This is part of the [make_generator] function. You would not use this combinator directly.
Definition
[codi] [reco]
Discussion
See [make_generator] and the
"Using x to Generate Values" notebook
as well as
Recursion Theory and Joy by Manfred von Thun.
Crosslinks
[make_generator]
concat
Function
Concatinate two lists.
[a b c] [d e f] concat
----------------------------
[a b c d e f]
Crosslinks
[first] [first_two] [flatten] [fourth] [getitem] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [split_at] [split_list] [swaack] [third] [zip]
cond
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
Discussion
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
Crosslinks
[ifte]
cons
Basis Function
Given an item and a list, append the item to the list to make a new list.
a [...] cons
------------------
[a ...]
Discussion
Cons is a venerable old function from Lisp. Its inverse operation is [uncons].
Crosslinks
[uncons]
dinfrirst
Combinator
Specialist function (that means I forgot what it does and why.)
Definition
[dip] [infrst]
dip
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
Discussion
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.)
Crosslinks
[dipd] [dipdd] [dupdip] [dupdipd] [infra]
dipd
Combinator
Like [dip] but expects two items.
... y x [Q] . dipd
-------------------------
... . Q y x
Discussion
See [dip].
Crosslinks
[dip] [dipdd] [dupdip] [dupdipd] [infra]
dipdd
Combinator
Like [dip] but expects three items. :
... z y x [Q] . dip
-----------------------------
... . Q z y x
Discussion
See [dip].
Crosslinks
[dip] [dipd] [dupdip] [dupdipd] [infra]
disenstacken
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
Definition
[clear]
Discussion
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.
Crosslinks
[enstacken] [stack] unstack
div
See floordiv.
divmod
Function
x y divmod
------------------
q r
(x/y) (x%y)
Invariant: qy + r = x.
Definition
[floordiv]
down_to_zero
Function
Given a number greater than zero put all the Natural numbers (including zero) less than that onto the stack.
Example
3 down_to_zero
--------------------
3 2 1 0
Definition
0 \>
Crosslinks
[range]
drop
Function
Expects an integer and a quote on the stack and returns the quote with n items removed off the top.
Example
[a b c d] 2 drop
----------------------
[c d]
Definition
[rest]
Crosslinks
[take]
dup
Basis Function
"Dup"licate the top item on the stack.
a dup
-----------
a a
Crosslinks
[dupd] [dupdd] [dupdip] [dupdipd]
dupd
Function
[dup] the second item down on the stack.
a b dupd
--------------
a a b
Definition
[dup]
Crosslinks
[dup] [dupdd] [dupdip] [dupdipd]
dupdd
Function
[dup] the third item down on the stack.
a b c dupdd
-----------------
a a b c
Definition
[dup]
Crosslinks
[dup] [dupd] [dupdip] [dupdipd]
dupdip
Combinator
Apply a function F and [dup] the item under it on the stack.
a [F] dupdip
------------------
a F a
Definition
[dupd] [dip]
Derivation
a [F] dupdip
a [F] dupd dip
a [F] [dup] dip dip
a dup [F] dip
a a [F] dip
a F a
Discussion
A very common and useful combinator.
Crosslinks
[dupdipd]
dupdipd
Combinator
Run a copy of program F under the next item down on the stack.
a [F] dupdipd
-------------------
F a [F]
Definition
[dup] [dipd]
Crosslinks
[dupdip]
enstacken
Function
Put the stack onto the stack replacing the contents of the stack.
... a b c enstacken
-------------------------
[c b a ...]
Definition
[stack] clear [dip]
Discussion
This is a destructive version of [stack]. See the note under [disenstacken] about the apparent but illusory reversal of the stack.
Crosslinks
[stack] [unstack] [disenstacken]
eq
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)
Crosslinks
[cmp] [ge] [gt] [le] [lt] [ne]
first
Function
Replace a list with its first item.
[a ...]
--------------
a
Definition
[uncons] [pop]
Crosslinks
[second] [third] [fourth] [rest]
first_two
Function
Replace a list with its first two items.
[a b ...] first_two
-------------------------
a b
Definition
[uncons] [first]
Crosslinks
[first] [second] [third] [fourth] [rest]
flatten
Function
Given a list of lists, concatinate them.
Example
[[1 2] [3 [4] 5] [6 7]] flatten
-------------------------------------
[1 2 3 [4] 5 6 7]
Definition
[<{}] concat [step]
Discussion
Note that only one "level" of lists is flattened. In the example above
[4] is not unquoted.
Crosslinks
[concat] [first] [first_two] [fourth] [getitem] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [split_at] [split_list] [swaack] [third] [zip]
floor
Basis Function
Return the largest integer <= x.
Discussion
This function doesn't make sense (yet) to have because there are (as yet) only integers in the system.
floordiv
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)
Discussion
All the division commands need to be revisited when the "numeric tower" for Thun gets nailed down.
Crosslinks
[divmod]
fork
Combinator
Run two quoted programs in parallel and replace them with their results.
... [F] [G] fork
----------------------
... f g
Definition
[i]
Discussion
The basic parallelism combinator, the two programs are run independently.
Crosslinks
[cleave] [clop] [map]
fourth
Function
Replace a list with its fourth item.
[a b c d ...] fourth
--------------------------
d
Definition
[rest] [third]
Crosslinks
[first] [second] [third] [rest]
gcd
Function
Take two integers from the stack and replace them with their Greatest Common Denominator.
Definition
true tuck] [mod] [dup] 0 [> [loop] [pop]
Discussion
Euclid's Algorithm
gcd2
Function
Compiled GCD function.
Discussion
See [gcd].
Crosslinks
[gcd]
ge
Basis Function
Greater-than-or-equal-to comparison of two numbers.
a b ge
--------------
Boolean
(a >= b)
Crosslinks
[cmp] [eq] [gt] [le] [lt] [ne]
genrec
Combinator
General Recursion Combinator.
[if] [then] [rec1] [rec2] genrec
---------------------------------------------------------------------
[if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte
Definition
\[[genrec]
(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.)
Discussion
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
Crosslinks
[anamorphism] [tailrec] [x]
getitem
Function
Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0.
Example
[a b c d] 2 getitem
-------------------------
c
Definition
[drop] [first]
Discussion
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.)
Crosslinks
[concat] [first] [first_two] [flatten] [fourth] [remove] [rest] [reverse] [rrest] [second] [shift] [shunt] [size] [sort] [split_at] [split_list] [swaack] [third] [zip]
grabN
Function
Expect a number on the top of the satck and put that many items from uner it onto a new list.
Example
a b c d e 3 grabN
-----------------------
a b [c d e]
Definition
[<{}] cons [times]
grba
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.
Definition
[stack] [popd]
Discussion
This function "grabs" an item from the stack along with a copy of the stack. It's part of the [app2] definition.
Crosslinks
[app2]
gt
Basis Function
Greater-than comparison of two numbers.
a b gt
--------------
Boolean
(a > b)
Crosslinks
[cmp] [eq] [ge] [le] [lt] [ne]
help
Function
Accepts a quoted symbol on the top of the stack and prints its documentation.
[foo] help
----------------
Discussion
Technically this is equivalent to pop, but it will only work if the
item on the top of the stack is a quoted symbol.
hypot
Function
x y hypot
---------------------------
sqrt(sqr(x) + sqr(y))
Definition
[sqr]
Discussion
This is another function that has to wait on the numeric tower.
Crosslinks
[sqrt]
i
Basis Combinator
Append a quoted expression onto the pending expression.
[Q] . i
-------------
. Q
Discussion
This is a fundamental combinator. It is used in all kinds of places. For
example, the [x] combinator can be defined as dup i.
id
Basis Function
The identity function.
Discussion
Does nothing. It's kind of a mathematical thing, but it occasionally comes in handy.
ifte
Combinator
If-Then-Else combinator, a common and convenient specialization of [branch].
[if] [then] [else] ifte
---------------------------------------
[if] nullary [else] [then] branch
Definition
[nullary]
Crosslinks
[branch] [loop] [while]
ii
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
Definition
[dip]
Example
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 •
Discussion
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
Crosslinks
[app2] [b]
infra
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
Definition
[swons] [swaack] i [dip] [swaack]
Discussion
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.
Crosslinks
infrst
Combinator
Does [infra] and then extracts the [first] item from the resulting list.
Definition
[infra] [first]
inscribe
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.
Example
[sqr dup mul] inscribe
Discussion
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.
le
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)
Crosslinks
[cmp] [eq] [ge] [gt] [lt] [ne]
loop
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
Discussion
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,
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].
Crosslinks
[branch] [fork] [while]
lshift
Basis Function
a n lshift
----------------
(a×2ⁿ)
Crosslinks
[rshift]
lt
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)
Crosslinks
[cmp] [eq] [ge] [gt] [le] [ne]
make_generator
Basis Function Combinator
codirecoGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
map
Basis Function Combinator
Run the quoted program on TOS on the items in the list under it, push a new list with the results in place of the program and original list.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
max
Basis Function Combinator
Given a list find the maximum.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
min
Basis Function Combinator
Given a list find the minimum.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
mod
Basis Function Combinator
Same as a % b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
modulus
See mod.
mul
Basis Function Combinator
Same as a * b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
ne
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)
Crosslinks
[cmp] [eq] [ge] [gt] [le] [lt]
neg
Basis Function Combinator
Same as -a.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
not
Basis Function Combinator
Same as not a.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
!-
"not negative"
(Function, Boolean Predicate)
Integer on top of stack is replaced by Boolean value indicating whether it is non-negative.
N !-
----------- N < 0
false
N !-
---------- N >= 0
true
Definition
0 >=
nulco
Basis Function Combinator
nullaryGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
nullary
(Combinator)
Run a quoted program without using any stack values and leave the first item of the result on the stack.
... [P] nullary
---------------------
... A
Definition
[stack] dip infra first
Derivation
... [P] nullary
... [P] [stack] dip infra first
... stack [P] infra first
... [...] [P] infra first
... [A ...] first
... A
Discussion
A very useful function that runs any other quoted function and returns it's first result without disturbing the stack (under the quoted program.)
Crosslinks
of
Basis Function Combinator
swap at
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
or
Basis Function Combinator
Same as a | b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
over
Basis Function Combinator
(a2 a1 -- a2 a1 a2)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
pam
Basis Function Combinator
iGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
pick
See getitem.
pm
Basis Function Combinator
Plus or minus :
a b pm
-------------
a+b a-b
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
pop
Basis Function Combinator
(a1 --)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popd
Basis Function Combinator
(a2 a1 -- a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popdd
Basis Function Combinator
(a3 a2 a1 -- a2 a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popop
Basis Function Combinator
(a2 a1 --)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popopd
Basis Function Combinator
(a3 a2 a1 -- a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popopdd
Basis Function Combinator
(a4 a3 a2 a1 -- a2 a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
popopop
Basis Function Combinator
pop popop
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
pow
Basis Function Combinator
Same as a ** b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
pred
Basis Function Combinator
Decrement TOS.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
primrec
Basis Function 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.:
n [Base] [Recur] primrec
0 [Base] [Recur] primrec
------------------------------
Base
n [Base] [Recur] primrec
------------------------------------------ n > 0
n (n-1) [Base] [Recur] primrec Recur
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
product
Basis Function Combinator
1 swap [*] step
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
quoted
Basis Function Combinator
unitGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
range
Basis Function Combinator
0 \<=Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
range_to_zero
Basis Function Combinator
unit [down_to_zero] infra
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
reco
Basis Function Combinator
rest cons
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rem
See mod.
remainder
See mod.
remove
Basis Function Combinator
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]
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rest
Basis Function Combinator
([a1 ...0] -- [...0])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
reverse
Basis Function Combinator
Reverse the list on the top of the stack. :
reverse == [] swap shunt
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rolldown
Basis Function Combinator
(a1 a2 a3 -- a2 a3 a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rollup
Basis Function Combinator
(a1 a2 a3 -- a3 a1 a2)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
roll>
See rollup.
roll<
See rolldown.
round
Basis Function Combinator
Round a number to a given precision in decimal digits.
The return value is an integer if ndigits is omitted or None. Otherwise the return value has the same type as the number. ndigits may be negative.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rrest
Basis Function Combinator
([a1 a2 ...1] -- [...1])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
rshift
Basis Function
a n rshift
----------------
(a∕2ⁿ)
Crosslinks
[lshift]
run
Basis Function Combinator
<{} infra
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
second
Basis Function Combinator
([a1 a2 ...1] -- a2)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
select
Basis Function Combinator
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. Currently Python semantics are used to evaluate the "truthiness" of the Boolean value (so empty string, zero, etc. are counted as false, etc.)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sharing
Basis Function Combinator
Print redistribution information.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
shift
Basis Function Combinator
uncons [swons] dip
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
shunt
Basis Function Combinator
Like concat but reverses the top list into the second. :
shunt == [swons] step == reverse swap concat
[a b c] [d e f] shunt
---------------------------
[f e d a b c]
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
size
Basis Function Combinator
pop ++Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sort
Basis Function Combinator
Given a list return it sorted.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
spiral_next
Basis Function Combinator
\[\[abs\[++ifte
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
split_at
Basis Function Combinator
dropGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
split_list
Basis Function Combinator
take reverseGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sqr
Basis Function Combinator
dup *
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sqrt
Basis Function Combinator
Return the square root of the number a. Negative numbers return complex roots.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
stack
Basis Function Combinator
(... -- ... [...])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
stackd
Basis Function Combinator
stackGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
step
Basis Function 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
The step combinator executes the quotation on each member of the list on top of the stack.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
step_zero
Basis Function Combinator
0 roll> step
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
stuncons
Basis Function Combinator
(... a1 -- ... a1 a1 [...])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
stununcons
Basis Function Combinator
(... a2 a1 -- ... a2 a1 a1 a2 [...])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sub
Basis Function Combinator
Same as a - b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
succ
Basis Function Combinator
Increment TOS.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
sum
Basis Function Combinator
Given a quoted sequence of numbers return the sum. :
sum == 0 swap [+] step
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
swaack
Basis Function Combinator
([...1] -- [...0])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
swap
Basis Function Combinator
(a1 a2 -- a2 a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
swapd
Basis Function Combinator
swapGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
swoncat
Basis Function Combinator
swap concat
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
swons
Basis Function Combinator
([...1] a1 -- [a1 ...1])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
tailrec
Basis Function Combinator
iGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
take
Basis Function Combinator
Expects an integer and a quote on the stack and returns the quote with just the top n items in reverse order (because that's easier and you can use reverse if needed.) :
[a b c d] 2 take
----------------------
[b a]
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
ternary
(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] unary
-------------------------
... A
Definition
binary popd
Discussion
Runs any other quoted function and returns its first result while consuming exactly three items from the stack.
Crosslinks
third
Basis Function Combinator
([a1 a2 a3 ...1] -- a3)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
times
Basis Function Combinator
times == [-- dip] cons [swap] infra [0 >] swap while pop :
... n [Q] . times
--------------------- w/ n <= 0
... .
... 1 [Q] . times
-----------------------
... . Q
... n [Q] . times
------------------------------------- w/ n > 1
... . Q (n - 1) [Q] times
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
truthy
See bool.
tuck
Basis Function Combinator
(a2 a1 -- a1 a2 a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
unary
(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
Definition
nullary popd
Discussion
Runs any other quoted function and returns its first result while consuming exactly one item from the stack.
Crosslinks
uncons
(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 [...]
Source
func(uncons, Si, So) :- func(cons, So, Si).
Discussion
This is the inverse of cons.
Crosslinks
unique
Basis Function Combinator
Given a list remove duplicate items.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
unit
Basis Function Combinator
(a1 -- [a1 ])
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
unquoted
Basis Function Combinator
iGentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
unswons
Basis Function Combinator
([a1 ...1] -- [...1] a1)
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
void
Basis Function Combinator
True if the form on TOS is void otherwise False.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
warranty
Basis Function Combinator
Print warranty information.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
while
Basis Function Combinator
swap nulco dupdipd concat loop
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
words
Basis Function Combinator
Print all the words in alphabetical order.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
x
(Combinator)
[F] x
-----------
[F] F
Definition
dup i
Discussion
The x combinator ...
xor
Basis Function Combinator
Same as a ^ b.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.
zip
Basis Function Combinator
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.
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
Lorem ipsum.
Crosslinks
Lorem ipsum.