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
falsenulco [nullary [false]] dip branchTODO: 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
true0 \>=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[] swapTuck an empty list just under the first item on the stack.
See lshift.
Function
... b a <{}
-----------------
... [] b a[] rollupTuck 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
trueFunction
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] genrecThe 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 firstThis 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 firstUnlike 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 first3 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 firstThis 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.)
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[P] [Q] b
[P] [Q] [i] dip i
[P] i [Q] i
P [Q] i
P QThis combinator 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
-----------------------
... Aunary popdRuns any other quoted function and returns its first result while consuming exactly two items from the stack.
Basis Function Combinator
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
Gentzen diagram.
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Basis Function Combinator
Use a Boolean value to select one of two quoted programs to run.
branch == roll< choice i
False [F] [T] branch
--------------------------
F
True [F] [T] branch
-------------------------
TGentzen diagram.
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Basis Function Combinator
ccons ccons
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(Function)
Given two items and a list, append the items to the list to make a new list.
B A [...] ccons
---------------------
[B A ...]cons consDoes cons twice.
Basis Function Combinator
Use a Boolean value to select one of two items. :
A B false choice
----------------------
A
A B true choice
---------------------
BCurrently Python semantics are used to evaluate the "truthiness" of the Boolean value (so empty string, zero, etc. are counted as false, etc.)
Gentzen diagram.
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Basis Function Combinator
clear == stack [pop stack] loop
... clear
---------------Gentzen diagram.
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Basis Function Combinator
fork popdd
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Basis Function Combinator
cleave popdd
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Basis Function Combinator
cmp takes two values and three quoted programs on the stack and runs 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
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Basis Function Combinator
cons dip
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Basis Function Combinator
codi reco
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Basis Function Combinator
Concatinate the two lists on the top of the stack. :
[a b c] [d e f] concat
----------------------------
[a b c d e f]Gentzen diagram.
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Basis Function 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 clause 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.
It works by rewriting into a chain of nested ifte expressions, e.g.:
[[[B0] T0] [[B1] T1] [D]] cond
-----------------------------------------
[B0] [T0] [[B1] [T1] [D] ifte] ifteGentzen diagram.
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(Basis Function)
Given an item and a list, append the item to the list to make a new list.
A [...] cons
------------------
[A ...]func(cons, [list(A), B|S], [list([B|A])|S]).Cons is a venerable old function from Lisp. It doesn’t inspect the item but it will not cons onto a non-list. It’s inverse operation is called uncons.
Basis Function Combinator
dip infrst
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Basis Function 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 xGentzen diagram.
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Basis Function Combinator
Like dip but expects two items. :
... y x [Q] dip
---------------------
... Q y xGentzen diagram.
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Basis Function Combinator
Like dip but expects three items. :
... z y x [Q] dip
-----------------------
... Q z y xGentzen diagram.
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Basis Function Combinator
The disenstacken operator expects a list on top of the stack and makes that the stack discarding the rest of the stack.
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See floordiv.
Basis Function Combinator
divmod(x, y) -> (quotient, remainder)
Return the tuple (x//y, x%y). Invariant: q * y + r == x.
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Basis Function Combinator
[0 >] [dup --] while
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Basis Function Combinator
drop == [rest] timesExpects 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]Gentzen diagram.
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Basis Function Combinator
(a1 -- a1 a1)Gentzen diagram.
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Basis Function Combinator
(a2 a1 -- a2 a2 a1)Gentzen diagram.
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Basis Function Combinator
(a3 a2 a1 -- a3 a3 a2 a1)Gentzen diagram.
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Basis Function Combinator
[F] dupdip == dup [F] dip
... a [F] dupdip
... a dup [F] dip
... a a [F] dip
... a F aGentzen diagram.
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Basis Function Combinator
dup dipd
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Basis Function Combinator
stack [clear] dip
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Basis Function Combinator
Same as a == b.
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Basis Function Combinator
([a1 ...1] -- a1)Gentzen diagram.
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Basis Function Combinator
([a1 a2 ...1] -- a1 a2)Gentzen diagram.
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Basis Function Combinator
<{} [concat] step
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Basis Function Combinator
Return the floor of x as an Integral.
This is the largest integer <= x.
Gentzen diagram.
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Basis Function Combinator
Same as a // b.
Gentzen diagram.
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Basis Function Combinator
[i] app2
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Basis Function Combinator
([a1 a2 a3 a4 ...1] -- a4)Gentzen diagram.
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Basis Function Combinator
true [tuck mod dup 0 >] loop pop
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Basis Function Combinator
Compiled GCD function.
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Basis Function Combinator
Same as a >= b.
Gentzen diagram.
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Basis Function Combinator
General Recursion Combinator. :
[if] [then] [rec1] [rec2] genrec
---------------------------------------------------------------------
[if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifteFrom "Recursion Theory and Joy" (j05cmp.html) 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] genrecIf the [I] if-part fails you must derive R1 and R2 from: :
... R1 [F] R2Just 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] iftePrimitive recursive functions are those where R2 == i. :
P == [I] [T] [R] tailrec
== [I] [T] [R [P] i] ifte
== [I] [T] [R P] ifteGentzen diagram.
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Basis Function Combinator
getitem == drop firstExpects 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] 0 getitem
-------------------------
aGentzen diagram.
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Basis Function Combinator
<{} [cons] times
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Basis Function Combinator
[stack popd] dip
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Basis Function Combinator
Same as a > b.
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Basis Function Combinator
Accepts a quoted symbol on the top of the stack and prints its docs.
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Basis Function Combinator
[sqr] ii + sqrt
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(Basis Combinator)
Append a quoted expression onto the pending expression.
[Q] i
-----------
Qcombo(i, [list(P)|S], S, Ei, Eo) :- append(P, Ei, Eo).This is probably the fundamental combinator. You wind up using it in all kinds of places (for example, the x combinator can be defined as dup i.)
Basis Function Combinator
The identity function.
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Basis Function Combinator
If-Then-Else Combinator :
... [if] [then] [else] ifte
---------------------------------------------------
... [[else] [then]] [...] [if] infra select i
... [if] [then] [else] ifte
-------------------------------------------------------
... [else] [then] [...] [if] infra first choice iHas the effect of grabbing a copy of the stack on which to run the if-part using infra.
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Basis Function Combinator
... a [Q] ii
------------------
... Q a QGentzen diagram.
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(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.)
... [a b c] [Q] infra
---------------------------
c b a Q [...] swaackswons swaack [i] dip swaackThis 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.
Basis Function Combinator
infra first
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Basis Function Combinator
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. for example:
[sqr dup mul] inscribe
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Basis Function Combinator
Same as a <= b.
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Basis Function Combinator
Basic loop combinator. :
... True [Q] loop
-----------------------
... Q [Q] loop
... False [Q] loop
------------------------
...Gentzen diagram.
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Basis Function Combinator
Same as a << b.
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Basis Function Combinator
Same as a < b.
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Basis Function Combinator
[codireco] ccons
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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.
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Basis Function Combinator
Given a list find the maximum.
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Basis Function Combinator
Given a list find the minimum.
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Basis Function Combinator
Same as a % b.
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See mod.
Basis Function Combinator
Same as a * b.
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Basis Function Combinator
Same as a != b.
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Basis Function Combinator
Same as -a.
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Basis Function Combinator
Same as not a.
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“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
true0 >=Basis Function Combinator
[nullary] cons
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(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
... AA very useful function that runs any other quoted function and returns it’s first result without disturbing the stack (under the quoted program.)
Basis Function Combinator
swap at
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Basis Function Combinator
Same as a | b.
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Basis Function Combinator
(a2 a1 -- a2 a1 a2)Gentzen diagram.
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Basis Function Combinator
[i] map
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See getitem.
Basis Function Combinator
Plus or minus :
a b pm
-------------
a+b a-bGentzen diagram.
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Basis Function Combinator
(a1 --)Gentzen diagram.
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Basis Function Combinator
(a2 a1 -- a1)Gentzen diagram.
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Basis Function Combinator
(a3 a2 a1 -- a2 a1)Gentzen diagram.
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Basis Function Combinator
(a2 a1 --)Gentzen diagram.
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Basis Function Combinator
(a3 a2 a1 -- a1)Gentzen diagram.
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Basis Function Combinator
(a4 a3 a2 a1 -- a2 a1)Gentzen diagram.
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Basis Function Combinator
pop popop
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Basis Function Combinator
Same as a ** b.
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Basis Function Combinator
Decrement TOS.
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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 RecurGentzen diagram.
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Basis Function Combinator
1 swap [*] step
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Basis Function Combinator
[unit] dip
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Basis Function Combinator
[0 <=] [1 - dup] anamorphism
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Basis Function Combinator
unit [down_to_zero] infra
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Basis Function Combinator
rest cons
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See mod.
See mod.
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.
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Basis Function Combinator
([a1 ...0] -- [...0])Gentzen diagram.
if not basis.
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Basis Function Combinator
Reverse the list on the top of the stack. :
reverse == [] swap shuntGentzen diagram.
if not basis.
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if basis
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Basis Function Combinator
(a1 a2 a3 -- a2 a3 a1)Gentzen diagram.
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Lorem ipsum.
Basis Function Combinator
(a1 a2 a3 -- a3 a1 a2)Gentzen diagram.
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See rollup.
See rolldown.
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.
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Basis Function Combinator
([a1 a2 ...1] -- [...1])Gentzen diagram.
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Basis Function Combinator
Same as a >> b.
Gentzen diagram.
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Basis Function Combinator
<{} infra
Gentzen diagram.
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Lorem ipsum.
Basis Function Combinator
([a1 a2 ...1] -- a2)Gentzen diagram.
if not basis.
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if basis
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Lorem ipsum.
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
-----------------------
BThe 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.
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Basis Function Combinator
Print redistribution information.
Gentzen diagram.
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Basis Function Combinator
uncons [swons] dip
Gentzen diagram.
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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.
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Basis Function Combinator
[pop ++] step_zero
Gentzen diagram.
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Basis Function Combinator
Given a list return it sorted.
Gentzen diagram.
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Basis Function Combinator
[[[abs] ii <=] [[<>] [pop !-] ||] &&] [[!-] [[++]] [[--]] ifte dip] [[pop !-] [--] [++] ifte] ifte
Gentzen diagram.
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Basis Function Combinator
[drop] [take] clop
Gentzen diagram.
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Basis Function Combinator
[take reverse] [drop] clop
Gentzen diagram.
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Basis Function Combinator
dup *
Gentzen diagram.
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Basis Function Combinator
Return the square root of the number a. Negative numbers return complex roots.
Gentzen diagram.
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Basis Function Combinator
(... -- ... [...])Gentzen diagram.
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Basis Function Combinator
[stack] dip
Gentzen diagram.
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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] stepThe step combinator executes the quotation on each member of the list on top of the stack.
Gentzen diagram.
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Basis Function Combinator
0 roll> step
Gentzen diagram.
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Basis Function Combinator
(... a1 -- ... a1 a1 [...])Gentzen diagram.
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Lorem ipsum.
Basis Function Combinator
(... a2 a1 -- ... a2 a1 a1 a2 [...])Gentzen diagram.
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if basis
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Lorem ipsum.
Basis Function Combinator
Same as a - b.
Gentzen diagram.
if not basis.
if not basis.
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Basis Function Combinator
Increment TOS.
Gentzen diagram.
if not basis.
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Basis Function Combinator
Given a quoted sequence of numbers return the sum. :
sum == 0 swap [+] stepGentzen diagram.
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Lorem ipsum.
Basis Function Combinator
([...1] -- [...0])Gentzen diagram.
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Lorem ipsum.
Basis Function Combinator
(a1 a2 -- a2 a1)Gentzen diagram.
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Basis Function Combinator
[swap] dip
Gentzen diagram.
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Basis Function Combinator
swap concat
Gentzen diagram.
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Basis Function Combinator
([...1] a1 -- [a1 ...1])Gentzen diagram.
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Basis Function Combinator
[i] genrec
Gentzen diagram.
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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.
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(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
-------------------------
... Abinary popdRuns any other quoted function and returns its first result while consuming exactly three items from the stack.
Basis Function Combinator
([a1 a2 a3 ...1] -- a3)Gentzen diagram.
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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] timesGentzen diagram.
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See bool.
Basis Function Combinator
(a2 a1 -- a1 a2 a1)Gentzen diagram.
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(Combinator)
Run a quoted program using exactly one stack value and leave the first item of the result on the stack.
... x [P] unary
---------------------
... Anullary popdRuns 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 [...]func(uncons, Si, So) :- func(cons, So, Si).This is the inverse of cons.
Basis Function Combinator
Given a list remove duplicate items.
Gentzen diagram.
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Basis Function Combinator
(a1 -- [a1 ])Gentzen diagram.
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Basis Function Combinator
[i] dip
Gentzen diagram.
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Basis Function Combinator
([a1 ...1] -- [...1] a1)Gentzen diagram.
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Lorem ipsum.
Basis Function Combinator
True if the form on TOS is void otherwise False.
Gentzen diagram.
if not basis.
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if basis
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Lorem ipsum.
Basis Function Combinator
Print warranty information.
Gentzen diagram.
if not basis.
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Basis Function Combinator
swap nulco dupdipd concat loop
Gentzen diagram.
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Basis Function Combinator
Print all the words in alphabetical order.
Gentzen diagram.
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(Combinator)
[F] x
-----------
[F] Fdup iThe x combinator …
Basis Function Combinator
Same as a ^ b.
Gentzen diagram.
if not basis.
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if basis
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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.
if not basis.
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Lorem ipsum.