Thun Function Reference

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Version -10.0.0

Each function, combinator, or definition should be documented here.

/

Binary Boolean and.

bool not /

/

Binary Boolean or.

bool not [/]

abs

Take an integer from the stack and replace it with its absolute value.

add

Take two integers from the stack and replace them with their sum.

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.) The quoted programs are run with nullary.

   [A] [B] and
----------------- A -> true
        B


   [A] [B] and
----------------- A -> false
     false

TODO: this is derived in one of the notebooks I think, look it up and link to it, or copy the content here.

or

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

Example

The range function generates a list of the integers from 0 to n - 1:

[0 <=] [-- dup] anamorphism

joy? 5

5

joy? [0 <=] [– dup]

5 [0 <=] [– dup]

joy? anamorphism

[4 3 2 1 0]

Note that the last value generated (0) is at the bottom of the list. See the Recursion Combinators notebook.

app1

“apply one”

Combinator

Given a quoted program on TOS and anything as the second stack item run the program without disturbing the rest of 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.

Discussion

Just a specialization of nullary really. Its parallelizable cousins are more useful.

app2 app3 appN unary

app2

Combinator

Like app1 with two items.

   ... y x [Q] . app2
-----------------------------------
   ... [y ...] [Q] . infra first
       [x ...] [Q]   infra first

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.

app1 app3 appN unary

app3

Combinator

Like app1 with three items.

     ... z y x [Q] . app3
-----------------------------------
   ... [z ...] [Q] . infra first
       [y ...] [Q]   infra first
       [x ...] [Q]   infra first

Discussion

See app2.

app1 app2 appN unary

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

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.

app1 app2 app3 unary

*

See mul.

at

See getitem.

average

Compute the average of a list of numbers. (Currently broken until I can figure out what to do about “numeric tower” in Thun.)

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.

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

Discussion

Runs any other quoted function and returns its first result while consuming exactly two items from the stack.

nullary ternary unary

b

Combinator

Run two quoted programs

   [P] [Q] b
---------------
      P Q

Discussion

This combinator may seem trivial but it comes in handy.

dupdip ii

bool

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.

not

branch

Combinator

Use a Boolean value to select and run one of two quoted programs.

   false [F] [T] branch
--------------------------
          F

   true [F] [T] branch
-------------------------
             T

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.

choice ifte select

ccccons

   a b c d [...] ccccons
---------------------------
       [a b c d ...]

Do cons four times.

ccons cons times

ccons

   a b [...] ccons
---------------------
      [a b ...]

Do cons two times.

cons ccons

choice

Use a Boolean value to select one of two items.

   a b false choice
----------------------
          a

   a b true choice
---------------------
          b

Discussion

It’s a matter of taste whether you implement this in terms of branch or the other way around.

branch select

clear

Clear everything from the stack.

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.

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

Discussion

Like cleave but consumes an additional item from the stack.

   1 2 3 4 [+] [-] clop
--------------------------
         1 2 7 -1

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

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

Discussion

This is one of those weirdly specific functions that turns out to be useful in a few places.

appN codireco

codireco

Combinator

This is part of the make_generator function. You would not use this combinator directly.

Discussion

See make_generator and the “Using x to Generate Values” notebook as well as Recursion Theory and Joy by Manfred von Thun.

make_generator

concat

Concatinate two lists.

   [a b c] [d e f] concat
----------------------------
       [a b c d e f]

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 expressions, e.g.:

      [[[B0] T0] [[B1] T1] [D]] cond
-----------------------------------------
   [B0] [T0] [[B1] [T1] [D] ifte] ifte

ifte

cons

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.

uncons

dinfrirst

Combinator

Specialist function (that means I forgot what it does and why.)

dipddd

Combinator

Like dip but expects four items. :

   ... z y x w [Q] . dipddd
-------------------------------
             ... . Q z y x w

Discussion

See dip.

dip dipd dipdd dupdip dupdipd infra

dipdd

Combinator

Like dip but expects three items. :

   ... z y x [Q] . dipdd
-----------------------------
             ... . Q z y x

Discussion

See dip.

dip dipd dipddd dupdip dupdipd infra

dipd

Combinator

Like dip but expects two items.

   ... y x [Q] . dipd
-------------------------
           ... . Q y x

Discussion

See dip.

dip dipdd dipddd dupdip dupdipd infra

dip

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.)

dipd dipdd dupdip dupdipd infra

disenstacken

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

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.

enstacken stack unstack

div

Divide.

divmod

    x y divmod
------------------
     q      r
   (x/y)  (x%y)

Invariant: qy + r = x.

down_to_zero

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

range

drop

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]

take

dupdd

dup the third item down on the stack.

   a b c dupdd
-----------------
     a a b c

dup dupd dupdip dupdipd

dupdipd

Combinator

Run a copy of program F under the next item down on the stack.

   a [F] dupdipd
-------------------
      F a [F]

dupdip

dupdip

Combinator

Apply a function F and dup the item under it on the stack.

   a [F] dupdip
------------------
      a F a

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.

dupdipd

dupd

dup the second item down on the stack.

   a b dupd
--------------
    a a b

dup dupdd dupdip dupdipd

dup

“Dup”licate the top item on the stack.

   a dup
-----------
    a a

dupd dupdd dupdip dupdipd

enstacken

Put the stack onto the stack replacing the contents of the stack.

   ... a b c enstacken
-------------------------
       [c b a ...]

Discussion

This is a destructive version of stack. See the note under disenstacken about the apparent but illusory reversal of the stack.

stack disenstacken unstack

eq

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 neq

=

See eq.

!=

See neq.

!-

Not negative.

    n !-
----------- n < 0
   false


   n !-
---------- n >= 0
   true

Discussion

Return a Boolean value indicating if a number is greater than or equal to zero.

first

Replace a list with its first item.

   [a ...]
--------------
      a

second third fourth rest

first_two

Replace a list with its first two items.

   [a b ...] first_two
-------------------------
           a b

first second third fourth rest

flatten

Given a list of lists, concatinate them.

Example

   [[1 2] [3 [4] 5] [6 7]] flatten
-------------------------------------
          [1 2 3 [4] 5 6 7]

Discussion

Note that only one “level” of lists is flattened. In the example above [4] is not unquoted.

concat first first_two fourth getitem remove rest reverse rrest second shift shunt size sort split_at split_list swaack third zip

fork

Combinator

Run two quoted programs in parallel and replace them with their results.

   ... [F] [G] fork
----------------------
       ... f g

Discussion

The basic parallelism combinator, the two programs are run independently.

cleave clop map

fourth

Replace a list with its fourth item.

   [a b c d ...] fourth
--------------------------
          d

first second third rest

gcd

Take two integers from the stack and replace them with their Greatest Common Denominator.

Discussion

Euclid’s Algorithm

ge

Greater-than-or-equal-to comparison of two numbers.

   a b ge
--------------
   Boolean
   (a >= b)

cmp eq gt le lt neq

genrec

Combinator

General Recursion Combinator.

                      [if] [then] [rec1] [rec2] genrec
---------------------------------------------------------------------
   [if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte

Discussion

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

getitem

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

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.)

concat first first_two flatten fourth remove rest reverse rrest second shift shunt size sort split_at split_list swaack third zip

grabN

Expect a number on the top of the stack and cons that many items from under it onto a new list.

Example

   a b c d e 3 grabN
-----------------------
      a b [c d e]

grba

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.

Discussion

This function “grabs” an item from the stack along with a copy of the stack. It’s part of the app2 definition.

app2

>=

See ge.

>>

See rshift.

>

See gt.

gt

Greater-than comparison of two numbers.

   a b gt
--------------
   Boolean
   (a > b)

cmp eq ge le lt neq

help

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.

–

See pred.

-

See sub.

id

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

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

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

app2 b

i

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.

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


... [a b c] [F] swons swaack [i] dip swaack
... [[F] a b c]       swaack [i] dip swaack

c b a [F]   [...] [i] dip swaack
c b a [F] i [...]         swaack
c b a  F    [...]         swaack
d e         [...]         swaack
... [e d]

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.

swaack

infrst

Combinator

Does infra and then extracts the first item from the resulting list.

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

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 neq

<=

See le.

<>

See neq.

<

   ... a <{}
----------------
   ... [] a

Discussion

Tuck an empty list just under the first item on the stack.

<<{}

<<

   ... b a <{}
-----------------
   ... [] b a

Discussion

Tuck an empty list just under the first two items on the stack.

<{}

<<

See lshift.

<

See lt.

loop

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.

branch fork while

lshift

Logical Left-Shift

   a n lshift
----------------
     (a×2ⁿ)

rshift

lt

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 neq

make_generator

Given an initial state value and a quoted generator function build a generator quote.

   state [generator function] make_generator
-----------------------------------------------
     [state [generator function] codireco]

Example

   230 [dup ++] make_generator
---------------------------------
     [230 [dup ++] codireco]

And then:

   [230 [dup ++] codireco] 5 [x] times pop
---------------------------------------------
             230 231 232 233 234

Discussion

See the “Using x to Generate Values” notebook.

codireco

map

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.

Example

   5 [1 2 3] [++ *] map
--------------------------
       5 [10 15 20]

Discussion

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

max

Given a list find the maximum.

Example

   [1 2 3 4] max
-------------------
         4

min size sum

min

Given a list find the minimum.

Example

   [1 2 3 4] min
-------------------
         1

max size sum

mod

Return the remainder of a divided by b.

   a b mod
-------------
    (a%b)

divmod mul

modulus

See mod.

mul

Multiply two numbers.

   a b mul
-------------
    (a×b)

div product

neg

Invert the sign of a number.

   a neg
-----------
    -a

neq

Not-Equal comparison of the two items on the top of the stack, replacing them with a Boolean value.

   a b neq
-------------
   Boolean
   (a = b)

cmp eq ge gt le lt

not

Invert the Boolean value on the top of the stack.

   true not
--------------
    false

   false not
---------------
     true

bool

nulco

Take the item on the top of the stack and cons it onto [nullary].

     [F] nulco
-------------------
   [[F] nullary]

Discussion

Helper function for or and and.

and or

null

True if the item on the top of the stack is an empty list, false if it’s a list but not empty, and an error if it’s not a list.

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

Example

... [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.)

unary binary ternary

of

Like getitem but swaps the order of arguments.

Example

   2 [a b c d] of
--------------------
         c

getitem

or

Logical bit-wise OR.

and

over

dup the second item on the stack over the first.

   a b over
--------------
    a b a

Definition

There are many many ways to define this function.

swap tuck

[pop] nullary

[dup] dip swap

unit dupdip

unit dupdipd first

And so on…

Discussion

A fine old word from Forth.

tuck

pam

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.)

Example

   5 7 [[+][-][*][/][%]] pam
-------------------------------
      5 7 [12 -2 35 0 5]

Discussion

A specialization of map that runs a list of functions in parallel (if the underlying map function is so implemented, of course.)

map

%

See mod.

pick

See getitem.

+

See add.

++

See succ.

pm

Plus or minus. Replace two numbers with their sum and difference.

      a b pm
-----------------
   (a+b) (a-b)

popdd

pop the third item on the stack.

   a b c popdd
-----------------
       b c

pop popd popop popopd popopdd popopop

popd

pop the second item down on the stack.

   a b popd
--------------
      b

pop popdd popop popopd popopdd popopop

pop

Pop the top item from the stack and discard it.

   a pop
-----------

popd popdd popop popopd popopdd popopop

popopdd

   a b c d popopdd
---------------------
        c d

pop popd popdd popop popopd popopop

popopd

pop the second and third items from the stack.

   a b c popopd
------------------
        c

pop popd popdd popop popopdd popopop

popop

pop two items from the stack.

   a b popop
---------------

pop popd popdd popopd popopdd popopop

popopop

pop three items from the stack.

   a b c popopop
-------------------

pop popd popdd popop popopd popopdd

pow

Take two numbers a and n from the stack and raise a to the nth power. (n is on the top of the stack.)

   a n pow
-------------
    (aⁿ)

Example

   2 [2 3 4 5 6 7 8 9] [pow] map
-----------------------------------
    2 [4 8 16 32 64 128 256 512]

pred

Predecessor. Decrement TOS.

succ

primrec

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

Discussion

Simple and useful specialization of the genrec combinator from the original Joy system.

genrec tailrec

product

Just as sum sums a list of numbers, this function multiplies them together.

Definition

1 swap [mul] step

Or,

[1] [mul] primrec

?

Is the item on the top of the stack “truthy”?

Discussion

You often want to test the truth value of an item on the stack without consuming the item.

bool

quoted

“Quote D” Wrap the second item on the stack in a list.

   a b quoted
----------------
     [a] b

Discussion

This comes from the original Joy stuff.

unit

range

Expect a number n on the stack and replace it with a list: [(n-1)...0].

Example

     5 range
-----------------
   [4 3 2 1 0]

   -5 range
--------------
      []

Discussion

If n is less than 1 the resulting list is empty.

range_to_zero

range_to_zero

Take a number n from the stack and replace it with a list [0...n].

Example

   5 range_to_zero
---------------------
    [0 1 2 3 4 5]

Discussion

Note that the order is reversed compared to range.

down_to_zero range

reco

Replace the first item in a list with the item under it.

   a [b ...] reco
--------------------
     [a ...]

codireco make_generator

remainder

See mod.

rem

See mod.

remove

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]

Definition

See the “Remove Function” notebook.

rest

   [a ...] rest
------------------
      [...]

first uncons

reverse

Reverse the list on the top of the stack.

Example

   [1 2 3] reverse
---------------------
       [3 2 1]

rolldown

   a b c rolldown
--------------------
       b c a

rollup

roll>

See rollup.

roll<

See rolldown.

rollup

   a b c rollup
------------------
      c a b

rolldown

rrest

   [a b ...] rrest
---------------------
        [...]

rest

rshift

Logical Right-Shift

   a n rshift
----------------
     (a∕2ⁿ)

lshift

run

Run a quoted program in a list.

Example

   [1 2 +] run
-----------------
       [3]

second

   [a b ...] second
----------------------
          b

first third fourth

select

Use a Boolean value to select one of two items from a sequence. :

   [a b] false select
------------------------
           a

   [a b] true select
-----------------------
           b

Discussion

The sequence can contain more than two items but not fewer.

choice

sharing

Print redistribution information.

Discussion

Mathematically this is a form of id, but it has the side-effect of printing out the GPL notice.

warranty

shift

Move the top item from one list to another.

Example

   [x y z] [a b c] shift
---------------------------
      [a x y z] [b c]

shunt

shunt

Like concat but reverse the top list into the second.

Example

   [a b c] [d e f] shunt
---------------------------
       [f e d a b c]

Discussion

This is more efficient than concat so prefer it if you don’t need to preserve order.

concat reverse shift

size

Replace a list with its size.

Example

   [23 [cats] 4] size
------------------------
           3

/

See div.

small

Return true if the item on the top of the stack is a list with zero or one item in it, false if it is a list with more than one item in it, and an error if it is not a list.

null

sort

Given a list return it sorted.

Example

   [4 2 5 7 1] sort
----------------------
      [1 2 4 5 7]

spiral_next

Example code.

Discussion

See the “Square Spiral Example Joy Code” notebook.

split_at

Split a list (second on the stack) at the position given by the number on the top of the stack.

Example

   [1 2 3 4 5 6 7] 4 split_at
--------------------------------
       [5 6 7] [4 3 2 1]

Discussion

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

split_list

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]

Discussion

Compare with split_at. This function does extra work to ensure that concat would reconstruct the original list.

split_at

sqr

Square the number on the top of the stack.

   n  sqr
------------
     n²

stackd

Grab the stack under the top item and put it onto the stack.

Example

   ... 1 2 3 stackd
------------------------
  ... 1 2 [2 1 ...] 3

stack

stack

Put the stack onto the stack.

      ... c b a stack
---------------------------
   ... c b a [a b c ...]

Discussion

This function forms a pair with unstack, and together they form the complement to the “destructive” pair enstacken and disenstacken.

enstacken disenstacken stackd unstack

step

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

Discussion

See the Recursion Combinators notebook.

step_zero

step_zero

Combinator

Like step but with 0 as the initial value.

   [...] [F] step_zero
-------------------------
     0 [...] [F] step

Discussion

size and sum can both be defined in terms of this specialization of step.

step

stuncons

Take the stack and uncons the top item.

Example

   1 2 3 stuncons
--------------------
   1 2 3 3 [2 1]

stununcons

Take the stack and uncons the top two items.

Example

   1 2 3 stununcons
----------------------
    1 2 3 3 2 [1]

stuncons

sub

Subtract the number on the top of the stack from the number below it.

   a b sub
-------------
    (a-b)

add

succ

Successor. Increment TOS.

pred

sum

Combinator

Given a quoted sequence of numbers return the sum.

Example

   [1 2 3 4 5] sum
---------------------
         15

size

swaack

Swap stack. Take a list from the top of the stack, replace the stack with the list, and put the old stack onto it.

Example

   1 2 3 [4 5 6] swaack
--------------------------
   6 5 4 [3 2 1]

Discussion

This function works as a kind of “context switch”. It’s used in the definition of infra.

infra

swapd

Swap the second and third items on the stack.

   a b c swapd
-----------------
      b a c

over tuck

swap

Swap the top two items on the stack.

   a b swap
--------------
     b a

swapd

swoncat

concat two lists, but swap the lists first.

concat

swons

Like cons but swap the item and list.

   [...] a swons
-------------------
      [a ...]

tailrec

Combinator

A specialization of the genrec combinator.

Discussion

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

take

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]

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] ternary
-------------------------
         ... a

Discussion

Runs any other quoted function and returns its first result while consuming exactly three items from the stack.

binary nullary unary

third

   [a b c ...] third
-----------------------
           c

first second fourth rest

times

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

Discussion

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.

tuck

dup the item on the top of the stack under the second item on the stack.

   a b tuck
--------------
    b a b

over

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

Discussion

Runs any other quoted function and returns its first result while consuming exactly one item from the stack.

binary nullary ternary

uncons

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 [...]

Discussion

This is the inverse of cons.

cons

unique

Given a list remove duplicate items.

unit

   a unit
------------
    [a]

unstack

Take a list from the top of the stack and concat it to the stack.

joy? 1 2 3 [4 5 6]

1 2 3 [4 5 6]

joy? unstack

1 2 3 6 5 4

stack disenstacken enstacken

unquoted

Combinator

Unquote (using i) the list that is second on the stack.

Example

   1 2 [3 4] 5 unquoted
--------------------------
         1 2 3 4 5

unit

unswons

   [a ...] unswons
---------------------
       [...] a

||

Combinator

Short-circuiting Boolean OR

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

and

void

True if the form on TOS is void otherwise False.

Discussion

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

warranty

Print warranty information.

while

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

loop

words

Print all the words in alphabetical order.

Discussion

Mathematically this is a form of id.

help

x

Combinator

Take a quoted function F and run it with itself as the first item on the stack.

   [F] x
-----------
   [F] F

Discussion

The simplest recursive pattern.

See the Recursion Combinators notebook. as well as Recursion Theory and Joy by Manfred von

zip

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.

Example

   [1 2 3] [4 5 6] zip
-------------------------
   [[1 4] [2 5] [3 6]]