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Functor Reference

Version -10.0.0

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

&

Basis Function Combinator

Same as a & b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

&&

Basis Function Combinator

nulco [nullary [false]] dip branch

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

*

Basis Function Combinator

Same as a * b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

•

Basis Function Combinator

The identity function.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

^

Basis Function Combinator

Same as a ^ b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

=

Basis Function Combinator

Same as a == b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

!=

Basis Function Combinator

Same as a != b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

!-^^^^

Basis Function Combinator

0 >=

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

>

Basis Function Combinator

Same as a > b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

>=

Basis Function Combinator

Same as a >= b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

>>

Basis Function Combinator

Same as a >> b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

Functor Reference

Version -10.0.0

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

-^^^

Basis Function Combinator

Same as a - b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

--^^^^

Basis Function Combinator

Decrement TOS.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<

Basis Function Combinator

Same as a < b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<=

Basis Function Combinator

Same as a <= b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<>

Basis Function Combinator

Same as a != b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<

Basis Function Combinator

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<<

Basis Function Combinator

Same as a << b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

<<

Basis Function Combinator

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

%

Basis Function Combinator

Same as a % b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

+

Basis Function Combinator

Same as a + b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

++

Basis Function Combinator

Increment TOS.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

?

Basis Function Combinator

dup bool

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

/

Basis Function Combinator

Same as a // b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

//

Basis Function Combinator

Same as a // b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

/floor

Basis Function Combinator

Same as a // b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

||

Basis Function Combinator

nulco [nullary] dip [true] branch

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

abs

Basis Function Combinator

Return the absolute value of the argument.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

add

Basis Function Combinator

Same as a + b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

anamorphism

Basis Function Combinator

pop \[

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

and

Basis Function Combinator

Same as a & b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

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

Definition

nullary popd

Discussion

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

app2

Basis Function Combinator

Like app1 with two items.

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

app3

Basis Function Combinator

Like app1 with three items.

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

appN

Basis Function Combinator

grabN

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

at

Basis Function Combinator

getitem == drop first

Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0. :

[a b c d] 0 getitem
-------------------------
 a

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

average

Basis Function Combinator

sum

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

b

(Combinator)

Run two quoted programs

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

Definition

[i] dip i

Derivation

[P] [Q] b
[P] [Q] [i] dip i
[P] i [Q] i
 P    [Q] i
 P     Q

Discussion

This combinator comes in handy.

Crosslinks

dupdip ii

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

nullary ternary unary

bool

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.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

branch

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
-------------------------
      T

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ccccons

Basis Function Combinator

ccons ccons

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ccons

(Function)

Given two items and a list, append the items to the list to make a new list.

   B A [...] ccons
---------------------
      [B A ...]

Definition

cons cons

Discussion

Does cons twice.

Crosslinks

cons

choice

Basis Function Combinator

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

A B false choice
----------------------
A


A B true choice
---------------------
  B

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

Crosslinks

clear

Basis Function Combinator

Clear everything from the stack.

clear == stack [pop stack] loop

   ... clear
---------------

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

cleave

Basis Function Combinator

fork popdd

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

clop

Basis Function Combinator

cleave popdd

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

cmp

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
     L

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

codi

Basis Function Combinator

cons dip

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

codireco

Basis Function Combinator

codi reco

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

concat

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.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

cond

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]{.title-ref} expressions, e.g.:

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

cons

(Basis Function)

Given an item and a list, append the item to the list to make a new list.

   A [...] cons
------------------
     [A ...]

Source

func(cons, [list(A), B|S], [list([B|A])|S]).

Discussion

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.

Crosslinks

ccons uncons

dinfrirst

Basis Function Combinator

dip infrst

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dip

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 x

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dipd

Basis Function Combinator

Like dip but expects two items. :

... y x [Q] dip
---------------------
  ... Q y x

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dipdd

Basis Function Combinator

Like dip but expects three items. :

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

disenstacken

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.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

div

Basis Function Combinator

Same as a // b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

divmod

Basis Function Combinator

divmod(x, y) -> (quotient, remainder)

Return the tuple (x//y, x%y). Invariant: q * y + r == x.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

down_to_zero

Basis Function Combinator

0 \>

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

drop

Basis Function Combinator

drop == [rest] times

Expects an integer and a quote on the stack and returns the quote with n items removed off the top. :

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dup

Basis Function Combinator

(a1 -- a1 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dupd

Basis Function Combinator

(a2 a1 -- a2 a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dupdd

Basis Function Combinator

(a3 a2 a1 -- a3 a3 a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dupdip

Basis Function Combinator

[F] dupdip == dup [F] dip

... a [F] dupdip
... a dup [F] dip
... a a   [F] dip
... a F a

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

dupdipd

Basis Function Combinator

dup dipd

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

enstacken

Basis Function Combinator

stack [clear] dip

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

eq

Basis Function Combinator

Same as a == b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

first

Basis Function Combinator

([a1 ...1] -- a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

first_two

Basis Function Combinator

([a1 a2 ...1] -- a1 a2)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

flatten

Basis Function Combinator

<{} [concat] step

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

floor

Basis Function Combinator

Return the floor of x as an Integral.

This is the largest integer <= x.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

floordiv

Basis Function Combinator

Same as a // b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

fork

Basis Function Combinator

i

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

fourth

Basis Function Combinator

([a1 a2 a3 a4 ...1] -- a4)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

gcd

Basis Function Combinator

true [tuck mod dup 0 >] loop pop

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

gcd2

Basis Function Combinator

Compiled GCD function.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ge

Basis Function Combinator

Same as a >= b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

genrec

Basis Function Combinator

General Recursion Combinator. :

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

From "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] 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

Primitive recursive functions are those where R2 == i. :

P == [I] [T] [R] tailrec
  == [I] [T] [R [P] i] ifte
  == [I] [T] [R P] ifte

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

getitem

Basis Function Combinator

getitem == drop first

Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0. :

[a b c d] 0 getitem
-------------------------
 a

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

grabN

Basis Function Combinator

<{} [cons] times

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

grba

Basis Function Combinator

stack popd

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

gt

Basis Function Combinator

Same as a > b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

help

Basis Function Combinator

Accepts a quoted symbol on the top of the stack and prints its docs.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

hypot

Basis Function Combinator

sqr

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

i

(Basis Combinator)

Append a quoted expression onto the pending expression.

   [Q] i
-----------
    Q

Source

combo(i, [list(P)|S], S, Ei, Eo) :- append(P, Ei, Eo).

Discussion

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.)

id

Basis Function Combinator

The identity function.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ifte

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 i

Has the effect of grabbing a copy of the stack on which to run the if-part using infra.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ii

Basis Function Combinator

... a [Q] ii
------------------
 ... Q a Q

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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.)

   ... [a b c] [Q] infra
---------------------------
    c b a Q [...] 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

swaack

infrst

Basis Function Combinator

infra first

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

inscribe

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

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

le

Basis Function Combinator

Same as a <= b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

loop

Basis Function Combinator

Basic loop combinator. :

... True [Q] loop
-----------------------
   ... Q [Q] loop

... False [Q] loop
------------------------
   ...

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

lshift

Basis Function Combinator

Same as a << b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

lt

Basis Function Combinator

Same as a < b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

make_generator

Basis Function Combinator

codireco

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

max

Basis Function Combinator

Given a list find the maximum.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

min

Basis Function Combinator

Given a list find the minimum.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

mod

Basis Function Combinator

Same as a % b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

modulus

Basis Function Combinator

Same as a % b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

mul

Basis Function Combinator

Same as a * b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

ne

Basis Function Combinator

Same as a != b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

neg

Basis Function Combinator

Same as -a.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

not

Basis Function Combinator

Same as not a.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

!-

"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

nullary

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

unary binary ternary

of

Basis Function Combinator

swap at

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

or

Basis Function Combinator

Same as a | b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

over

Basis Function Combinator

(a2 a1 -- a2 a1 a2)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

pam

Basis Function Combinator

i

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

pick

Basis Function Combinator

getitem == drop first

Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0. :

[a b c d] 0 getitem
-------------------------
 a

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

pop

Basis Function Combinator

(a1 --)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popd

Basis Function Combinator

(a2 a1 -- a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popdd

Basis Function Combinator

(a3 a2 a1 -- a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popop

Basis Function Combinator

(a2 a1 --)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popopd

Basis Function Combinator

(a3 a2 a1 -- a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popopdd

Basis Function Combinator

(a4 a3 a2 a1 -- a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

popopop

Basis Function Combinator

pop popop

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

pow

Basis Function Combinator

Same as a ** b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

pred

Basis Function Combinator

Decrement TOS.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

product

Basis Function Combinator

1 swap [*] step

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

quoted

Basis Function Combinator

unit

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

range

Basis Function Combinator

0 \<=

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

reco

Basis Function Combinator

rest cons

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

rem

Basis Function Combinator

Same as a % b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

remainder

Basis Function Combinator

Same as a % b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

rest

Basis Function Combinator

([a1 ...0] -- [...0])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

rolldown

Basis Function Combinator

(a1 a2 a3 -- a2 a3 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

rollup

Basis Function Combinator

(a1 a2 a3 -- a3 a1 a2)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

roll>

Basis Function Combinator

(a1 a2 a3 -- a3 a1 a2)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

roll<

Basis Function Combinator

(a1 a2 a3 -- a2 a3 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

rrest

Basis Function Combinator

([a1 a2 ...1] -- [...1])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

rshift

Basis Function Combinator

Same as a >> b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

run

Basis Function Combinator

<{} infra

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

second

Basis Function Combinator

([a1 a2 ...1] -- a2)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

sharing

Basis Function Combinator

Print redistribution information.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

shift

Basis Function Combinator

uncons [swons] dip

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

size

Basis Function Combinator

pop ++

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

sort

Basis Function Combinator

Given a list return it sorted.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

spiral_next

Basis Function Combinator

\[\[abs
\[++

ifte

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

split_at

Basis Function Combinator

drop

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

split_list

Basis Function Combinator

take reverse

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

sqr

Basis Function Combinator

dup *

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

stack

Basis Function Combinator

(... -- ... [...])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

stackd

Basis Function Combinator

stack

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

step_zero

Basis Function Combinator

0 roll> step

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

stuncons

Basis Function Combinator

(... a1 -- ... a1 a1 [...])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

stununcons

Basis Function Combinator

(... a2 a1 -- ... a2 a1 a1 a2 [...])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

sub

Basis Function Combinator

Same as a - b.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

succ

Basis Function Combinator

Increment TOS.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

swaack

Basis Function Combinator

([...1] -- [...0])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

swap

Basis Function Combinator

(a1 a2 -- a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

swapd

Basis Function Combinator

swap

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

swoncat

Basis Function Combinator

swap concat

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

swons

Basis Function Combinator

([...1] a1 -- [a1 ...1])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

tailrec

Basis Function Combinator

i

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

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

binary nullary unary

third

Basis Function Combinator

([a1 a2 a3 ...1] -- a3)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

truthy

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.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

tuck

Basis Function Combinator

(a2 a1 -- a1 a2 a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

binary nullary ternary

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

cons

unique

Basis Function Combinator

Given a list remove duplicate items.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

unit

Basis Function Combinator

(a1 -- [a1 ])

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

unquoted

Basis Function Combinator

i

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

unswons

Basis Function Combinator

([a1 ...1] -- [...1] a1)

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

warranty

Basis Function Combinator

Print warranty information.

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

while

Basis Function Combinator

swap nulco dupdipd concat loop

Gentzen diagram.

Definition

if not basis.

Derivation

if not basis.

Source

if basis

Discussion

Crosslinks

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

Crosslinks

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

Crosslinks

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

Crosslinks