Crazy J
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| Paradigm(s) | functional |
|---|---|
| Designed by | User:Blashyrkh |
| Appeared in | 2025 |
| Computational class | Turing complete |
| Reference implementation | Github |
| Influenced by | Lazy K Unlambda |
| File extension(s) | .crazy |
Crazy J, designed and implemented by User:Blashyrkh in late 2025, is a crippled little brother of Lazy K. It resembles the syntax of its older brother (two of them, actually) and adds a new one, uses the same IO model (with tiny modification), but based on different set of combinators. Which makes it slightly... different.
To understand the semantics (and, probably, even the syntax) of the language it's recommended to get familiar with Lambda calculus and Combinator calculus first.
In some sense Crazy J may be considered a joke language, but it was created to get an answer to the serious question: is it possible to create Turing-complete combinator-based functional language using a non-complete set of combinators? And the answer is "yes"!
Instructions
Crazy J is based on non-complete IJ basis:
I = λ x . x J = λ x y z w . x y (x w z)
Nevertheless, Crazy J is proven to be Turing-complete (by implementing Cyclic tag system in it).
Syntax
Crazy J supports three different syntaxes.
- Unlambda-like parenthesis-free syntax (inspired by Unlambda and Lazy K). A typical program looks like this (but usually much longer):
````j``j```jii`j`j``jii``jiiij```````j`j````jjii`j`j``jiii`j```jjiijj```j``jii``ji``jii`j`j``jii````j```jii```j``ji```ji``j``jiiii```jjii``j`jijii``jii
- Combinator calculus syntax (inspired by Lazy K). The same program:
J(J(JII(J(J(JII))))(JII))IJ(J(J(JJII(J(J(JII)))))I(J(JJII))JJ(J(JII)(JI(JII))(J(J(JII))))(J(JII(J(JI(JI(J(JII)I)I))(JJII)(J(JI)J)))II(JII)))
-.<-.->.<->+-[--++[-.<-]<-->+[--++<-]--[-.[,.][-.<-]][-.+<,>+<,->[--++][-,-]<.->++.]]
Whitespace is ignored. The rest of a string starting from # until the end of line is a comment and is ignored as well.
Contrary to Lazy K, Crazy J doesn't allow to mix styles: the actual syntax used is determined by the first non-whitespace and non-comment character. An empty program is treated as an identity function (cat).
Unlambda-like syntax
Syntax Semantics
Expr ::= "i" (λ x . x)
| "j" (λ xyzw . xy(xwz))
| "`" Expr1 Expr2 (Expr1 Expr2)
Combinator calculus syntax
Syntax Semantics
Expr ::= "I" (λ x . x)
| "J" (λ xyzw . xy(xwz))
| Expr1 Expr2 ... ExprN ((Expr1 Expr2) ... ExprN)
| "(" Expr ")" Expr
Brainfuck mimic syntax
Written using this syntax, Crazy J program looks like valid (but not necessarily ever halting) Brainfuck program.
| Character | Meaning | CC syntax equivalent |
|---|---|---|
| > | Set parsing order* to left-to-right (the default) | None |
| < | Set parsing order* to right-to-left | None |
| [ | Start of sub-expression | (
|
| ] | End of sub-expression | )
|
| - | λ x y z w . x y (x w z) | J
|
| + | λ x . x | I
|
| , | Shortcut for [>-+] |
(JI)
|
| . | Shortcut for [>-++] |
(JII)
|
*Parsing order is a unique concept of this syntax mode. If left-to-right parsing order is active, source fragment ABC means (AB)C. In right-to-left parsing order the same string ABC is interpreted as C(BA). Parsing order doesn't change when [ is encountered ("inherited" by sub-expression), but if it's changed between [ and ], it's restored after ] (similar to environment variables of Unix processes: inherited by children, never passed to parent).
IO model
Almost exactly as of Lazy K. To be precise:
- Input is Church list of Church numerals where each element represents single byte of input stream. Contrary to Lazy K, ASCII character 0 is represented as Church numeral 1, ASCII character 1 - as Church numeral 2, and so on. The input stream is terminated with Church value 257 infinitely repeated.
- The program is called as a function with the input list passed as a parameter.
- Result of program evaluation must be a Church list of Church numerals. First value greater than or equal to 257 marks the end of output. This value minus 257 is the program's exit code. Output list is never examined beyond EOF, and it's not just implementation detail as of Lazy K. It's essential part of Crazy J specification.
- Input is as lazy as possible: next byte is read from standard input only when it's needed to apply it to an argument.
Church numeral 0 is impossible in IJ basis, and therefore all input codes are incremented on input and decremented back before output.
Lack of K combinator
Whereas Lazy K's motto is "there's more than one way to do it", Crazy J's motto is "there's no way to do it, but we do it anyway."
To be precise:
- No constant function (known as "K combinator")
- No Church booleans
- No Church zero
- No Scott numerals
- No familiar list access primitives like
car,cdrandnth
Abstraction elimination rules
As usual for languages of this family, abstraction elimination process is started from the most deeply nested lambda until no lambdas left. All free variables of the lambda in question are considered constants.
There are three approaches to abstraction elimination. They all share the same common property/drawback (which is a consequence of the chosen combinator basis): λ x . CONST can never be expressed in IJ basis. In practice it means that if your program needs it then you should look for alternative lambda expression for your program (see Examples section for the general idea how to do it).
Rule 0: eta reduction
It doesn't matter what approach is used, this is the only rule that actually eliminates an abstraction (i.e. gets rid of λ in an expression):
λ x . F x ≡ F (if x is not used in F)
Important (and obvious) special case:
λ x . x ≡ I
Straighforward approach: BCS
This approach uses known IJ expressions for B, C and S combinators:
B = λ x y z . x (y z) = J(JII)(JI(JII))(J(J(JII))) C = λ x y z . x z y = J(JII)(J(JII))(J(JII)) S = λ x y z . x z (y z) = J(J(JI(J(J(J(JII)))I)))I(J(JJII))JJ
Rule 1: B rule
λ x . F G(x) ≡ B F (λ x . G(x)), where G(x) is a function of x, F is not
Alternatively, it can be re-written using Q combinator (IJ expression for Q is a bit shorter):
λ x . F G(x) ≡ Q (λ x . G(x)) F, where Q = λ x y z . y (x z) = J(J(J(JII)))(JIII)(J(JI))
Even shorter IJ expression can be produced if this inlined version of Q combinator is used:
Q x y = B y x = λ z . y (x z) = J (J I I) x (J I y)
Rule 2: C rule
λ x . F(x) G ≡ C (λ x . F(x)) G, where F(x) is a function of x, G is not
Alternatively, it can be re-written using R combinator (IJ expression for R is much shorter):
λ x . F(x) G ≡ R G (λ x . F(x)), where R = λ x y z . y z x = J(JII)
Rule 3: S rule
λ x . F(x) G(x) ≡ S (λ x . F(x)) (λ x . G(x)), where F(x) and G(x) are both functions of x
BCS+
This approach uses known IJ expressions for many common combinators, while using B, C and S rules as a fallback.
| Combinator name | LC expression | Bruteforced Crazy J expression |
|---|---|---|
| T | λ x y . y x | ``jii
|
| Q1 | λ x y z . x (z y) | `ji
|
| R | λ x y z . y z x | `j``jii
|
| Q4 | λ x y z . z (y x) | ``ji``jii
|
| Q3 | λ x y z . z (x y) | ``j``ji``jiii
|
| Q2 | λ x y z . y (z x) | ```j``jii`ji`j``jii
|
| F | λ x y z . z y x | ```j``jiii`j`j``jii
|
| C | λ x y z . x z y | ```j``jii`j``jii`j``jii
|
| Q | λ x y z . y (x z) | ```j`j`j``jii```jiii`j`ji
|
| B | λ x y z . x (y z) | ```j``jii``ji``jii`j`j``jii
|
| V | λ x y z . z x y | ```j``j```jii`j`j``jii``jiiij
|
| F* | λ x y z w . x w z y | ```j``jii`j`j``jii`j``jii
|
| V* | λ x y z w . x z y w | ```j``jii`j``jii````jiji`ji
|
| C* | λ x y z w . x y w z | ```j`j``jii`j``jii``ji`j``jii
|
| G | λ x y z w . x w (y z) | ```j``jii``ji`j``jii`j`j``jii
|
| B1 | λ x y z w . x (y z w) | ````j```jj``j```jiiii`j``jiiiij
|
| D (B') | λ x y z w . x y (z w) | ```j`j``jii``ji``jii``ji`j`j``jii
|
| R* | λ x y z w . x z w y | ```j`j`j``jiii```j``j```jiiij`ji`jj
|
| C' | λ x y z w . x (y w) z | ```j```j`j`j`j``jiii`j`j`j`j``jiii`jj
|
| B3 | λ x y z w . x (y (z w)) | ```j`j``jii``jji```j``ji`jii`j`j``jii
|
| Combinator name or expression | LC expression | Bruteforced Crazy J expression | Notes |
|---|---|---|---|
| BJT | λ x y z w . y x (w x z) | ```j``jii``jii``jij |
|
| C(BJT)T | λ x y z . z x y x | ````j```jj``jiiii``jiji |
|
| BC(C(BJT)T) | λ x y z . y x z x | ````j`j``jii`ji```j``jjii``jiii |
|
| W | λ x y . x y y | ````j``ji``j`j``jiii`j``jii`jij |
|
| W' | λ x y . y x x | ```j``ji``j``jiii`j``jii``jij |
|
| M (sometimes called U) | λ x . x x | ````j`j``ji```jiiii``j```jjiiij |
|
| M2 | λ x y . x y (x y) | ````````jjji`j`j```j``j`jiii``jiiiiji |
|
| L | λ x y . x (y y) | `````j`j`j`j``jiii``jiji``j`jii |
|
| O | λ x y . y (x y) | ```j`j``jiii```j```j``j`jiiiiij |
|
| U | λ x y . y (x x y) | ```j`j```j``j`j```jiiiii`jii``jj`jj |
|
| λ x . x x x | `````j`j```j```jj`j``jiiii`jiiiji |
No official name, we call it XXX or X3 | |
| S | λ x y z . x z (y z) | `````j`j````jjii`j`j``jiii`j```jjiijj |
|
| H | λ x y z . x y z y | ```j`j``jiii```j```j``j`jiiii`j``jiij |
|
| Y = M2L | λ x . x (Y x) | `````j``j`j`j```j``j`jiii``jiii``jiiji`````j`j`j`j``jiii``jiji``j`jii |
IJ approach
This approach was developed specifically for IJ basis. The abstraction elimination is performed in the following order:
- Find the innermost lambda.
- Count a number of occurrences of bound variable.
- If no occurrences of bound variable are found in lambda expression then this lambda can't be expressed in IJ basis. ABORT
- If there's exactly one occurrence of bound variable, goto step 6.
- If there are more than one occurrences of bound variable, goto step 9.
- If bound variable is inside the ()-subexpression, it must be moved out using one or several applications of Q1 rule.
- If bound variable is not at the rightmost place of the expression then it must be moved there using T, R and JR rules.
- Apply eta-reduction, repeat from step 1.
- Find any two occurrences of the bound variable (the closer to each other the better). If both occurrences share identical context (say,
J I I xin both places), then the whole repeating subexpression can be deduplicated. - Use permutating rules to move two occurrences of the variable to positions suitable for one of deduplicating rules (usually J-T deduplicating rule is the universal fallback).
- Apply deduplicating rule.
- Repeat from step 2.
Permutating rules
The aim of these rules is to move bound variable either to the rightmost position of the expression (to apply eta-reduction rule) or to a position suitable for one of deduplicating rules.
Q1 rule
This rule can be used to extract bound variable from ()-subexpression:
a (x b) c d ... = Q1 a b x c d ..., where Q1 = JI
\ ^
\____________________|
Here's an example of multiple applications of the rule:
a (x b c) d e ... = Q1 a c (x b) d e ... = Q1 (Q1 a c) b x d e ...
T rule
This rule can be used to move bound variable from the leftmost position to the right by 1 position:
x a b ... = T a x b ..., where T = JII
\ ^
\______________|
R rule
This rule can be used to move bound variable to the right by 1 position:
a x b c d ... = R b a x c d ..., where R = JT = J(JII)
\ ^
\__________________|
JR rule ("castling move")
This rule can be used to move bound variable to the right by 2 positions. It's more compact that two application of R rule (5 combinators instead of 2x4):
a x b c d e ... = J R c a b x d e ...
\ ^
\________________________|
Deduplicating rules
All deduplicating rules are used to reduce number of occurrences of bound variable by 1.
J-T deduplicating rule
This is the most commonly used deduplicating rule:
a x (b x c) d e ... = J (T x) a c b d e ..., where T = JII
The important special case:
a x (b x) c d ... = a x (T x b) c d ... = J (T x) a b T c d ...
J-R deduplicating rule
a (b c x) x d e ... = J (R x) a c b d e ...
The important special cases:
a x x = a (I I x) x = J (R x) a I I x x = I (I I x) x = J (R x) I I I x x x = J (R x) x I I = J (J T x) x I I = J (R x) J T J I I
J-Q1 deduplicating rule
Almost never happens to be applicable because no permutating rule can place bound variable to the leftmost place of subexpression. Nevertheless, the pattern may occur initially, so the rule exists:
x (x (a b) c) d e ... = J (Q1 x) c a b d e ...
J deduplicating rule
Another almost-never-happening rule:
x a (x b c) d e ... = J x a c b d e ...
Church numerals
Numeric constants and arithmetics are essential for Crazy J since Church numerals are used for both input and output, and are powerful constructs per se. Here we summarize IJ expressions for arithmetics as well as provide bruteforce results for some numeric constants.
| Operation | LC expression | Crazy J (Unlambda syntax) | Crazy J (CC syntax) | Size | Notes |
|---|---|---|---|---|---|
| +1 | λ n f x . f (n f x) | ````j`j`j``jiii```j``j```jiiij`ji`jj````j`j``jiii``jij``jii |
J(J(J(JII)))I(J(J(JIII)J)(JI)(JJ))(J(J(JII))I(JIJ)(JII)) |
30 | |
| + | λ m n f . m +1 n f | ```jii````j`j`j``jiii```j``j```jiiij`ji`jj````j`j``jiii``jij``jii |
JII(J(J(J(JII)))I(J(J(JIII)J)(JI)(JJ))(J(J(JII))I(JIJ)(JII))) |
33 | |
| * | λ m n f . n (m f) | ```j`j`j``jii```jiii`j`ji |
J(J(J(JII)))(JIII)(J(JI)) |
13 | Q combinator
|
| ^ | λ m n . n m | ``jii |
JII |
3 | T combinator
|
| λ n . nn | λ n . n n | ````j`j``ji```jiiii``j```jjiiij |
J(J(JI(JIII)))I(J(JJII)I)J |
16 | M combinator
|
| n+m | λ f x . m f (n f x) | ````j`j``jiij``ji``ji```ji``j``jii m n ``jii |
J(J(JII))J(JI(JI(JI(J(JII) m) n)))(JII) |
19+sizeof(n)+sizeof(m) | Inlined version of + function
|
| n*m | λ f . n (m f) | ```j``jii m ``ji n |
J(JII) m (JI n) |
6+sizeof(n)+sizeof(m) | Inlined version of * function
|
| nm | m n | ` m n |
m (n) |
0+sizeof(n)+sizeof(m) | Inlined version of ^ function
|
| nn | n n | ````j``j``jii n iii |
J(J(JII) n)III |
8+sizeof(n) | Inlined version of M combinator
|
| n2 | 2 n | ```j``jiii``j```jii n i |
J(JII)I(J(JII n)I) |
10+sizeof(n) | Inlined version of 2 constant
|
| Constant | Crazy J (Unlambda syntax) | Crazy J (CC syntax) | Size | Notes |
|---|---|---|---|---|
| 0 | Doesn't exist | Doesn't exist | n/a | Impossible in IJ basis |
| 1 | i |
I |
1 | |
| 2 | ```j``ji``j``jiii``jii``jij |
J(JI(J(JII)I))(JII)(JIJ) |
14 | Brute-forced (there's no shorter version) |
| 3 | ```j```j`j``ji``j``jiiiiii``j`jj``jij |
J(J(J(JI(J(JII)I)))II)I(J(JJ)(JIJ)) |
19 | Brute-forced (there's no shorter version) |
| 4 | ```j``ji```ji``j``jiiii```jjii``j`jij |
J(JI(JI(J(JII)I)I))(JJII)(J(JI)J) |
19 | Brute-forced (there's no shorter version) |
| 8 | ````j```j`j``ji``j``jiiiiii``j`jj``jij```j``ji``j``jiii``jii``jij |
J(J(J(JI(J(JII)I)))II)I(J(JJ)(JIJ))(J(JI(J(JII)I))(JII)(JIJ)) |
33 | 8=23 |
| 9 | ```j``jiii``j```jii```j```j`j``ji``j``jiiiiii``j`jj``jiji |
J(JII)I(J(JII(J(J(J(JI(J(JII)I)))II)I(J(JJ)(JIJ))))I) |
29 | 9=32 |
| 16 | ``````j``j``jii```j``ji``j``jiii``jii``jijj``jiijii |
J(J(JII)(J(JI(J(JII)I))(JII)(JIJ)))J(JII)JII |
26 | 16=(λx.xxx)2 |
| 256 | ````j``j``jii```j``ji```ji``j``jiiii```jjii``j`jijiii |
J(J(JII)(J(JI(JI(J(JII)I)I))(JJII)(J(JI)J)))III |
27 | 256=44 |
| 257 | ````j`j``jiij``ji``ji```ji``j``jiii````j``j``jii```j``ji```ji``j``jiiii```jjii``j`jijiii``jii |
J(J(JII))J(JI(JI(JI(J(JII)I)(J(J(JII)(J(JI(JI(J(JII)I)I))(JJII)(J(JI)J)))III))))(JII) |
47 | EOF constant |
Examples
Empty output
Detailed description: Crazy J/Empty output
````j``j```jii`j`j``jii``jiiij````j`j``jiij``ji``ji```ji``j``jiii````j``j``jii```j``ji```ji``j``jiiii```jjii``j`jijiii``jii
This program does nothing at all!
Truth-machine
Detailed description: Crazy J/Truth-machine
```jii```j``jii`````j``jiii``j```jii``````j```jii```j``ji``j``jiii``jii``jijii``jii````j`j``jii````j``ji``j`j``jiii`j``jii`jij``ji``jii```j`j``jii``ji``jii``ji`j`j``jii`````j``ji``j`j``jiii`j``jii`jij````j`j``jii``ji``jii``ji`j`j``jii```j``ji``j``jiii``jii``jiji``jii``ji``````j``j`j`j```j``j`jiii``jiii``jiiji`````j`j`j`j``jiii``jiji``j`jii````j``jii``ji``jii`j`j``jii````j``j```jii`j`j``jii``jiiij`````j``ji``j`j``jiii`j``jii`jij````j`j``jii``ji``jii``ji`j`j``jii```j``jiii``j```jii``````j```jii```j``ji``j``jiii``jii``jijii``jii````j`j``jii````j``ji``j`j``jiii`j``jii`jij``ji``jii```j`j``jii``ji``jii``ji`j`j``jii`````j``ji``j`j``jiii`j``jii`jij````j`j``jii``ji``jii``ji`j`j``jii```j``ji``j``jiii``jii``jiji`````j``jiii`j`j``jii``ji`````j`j`j``jii```jiii`j`ji````j``j```jii`j`j``jii``jiiij`````j``ji``j`j``jiii`j``jii`jij````j`j``jii``ji``jii``ji`j`j``jii````j```jii````j```jii```j``ji``j``jiii``jii``jijii``jiiii``jii````j``j```jii`j`j``jii``jiiij```j``jiii``j```jii``````j```jii```j``ji``j``jiii``jii``jijii``jii````j`j``jii````j``ji``j`j``jiii`j``jii`jij``ji``jii```j`j``jii``ji``jii``ji`j`j``jii`````j``ji``j`j``jiii`j``jii`jij````j`j``jii``ji``jii``ji`j`j``jii```j``ji``j``jiii``jii``jiji``jii
Hello, world!
Detailed description: Crazy J/Hello, world!
[-.[-.+<,->[-.][,-][-.+<,->.[,-][-.+<,-->++<->+[-[--][,-]]]<.->[-.[-.+<,->.[,-]][-[-.],[-.[<.-->+[,-].]<,,][-.]<.-]][,.].][-[-.[-[-[-.]][.+][-,][-.+<,>+<,->[--++][-,-]][<.-->,[-.[<.-->+[,-].]<,,][-.][-[.+<,-]+[-[--++]+]-[-[-[-.+<,-]++]+[<--->[,-]]]]]]][<.-->.[,[,[<.-->,[-.[<.-->+[,-].]<,,>][-.][-.+<,->.[,-]<.->[-.<-->[.+][-,]][-[--[-.]+]+,<-->++-+].]<.-,>[-[-.[-[-.+<,-->++]+[<--->[,-]]]][<.-->,[<.-->[<.--->[.+][-,]]-[<.-->,[-.[<.-->+[,-].]<,,][-.]<.-]<,][-.]][-.+<,->.[,-]]+]<,>[-[-[-.+<,-]++]+[-[--][,-]]]][<.-->,[-.[<.-->+[,-].]<,,][-.]]][-.[-.+<,->.[,-]]<->[-.+<,-->++<->+[-[--][,-]]<.-][<.-->,[-.[<.-->+[,-].]<,,][-.]][-.][-[-.<-][.+][-,]][<.-->[-.[-.[-.<-.->.<->+-][,,]]][-.,[-.<-,]][<.-->,[<.-,,,][-.]]]++[-.[<.-->+[,-].][-.[-.+<,>+<,->[--++][-,-]<.->++.]<,]][-.+<,->.[,-]<.->[<.-->,[-.[<.-->+[,-].]<,,][-.]<.][-[-.+<,-->++]+[<--->[,-]]]+][-.+<,->.[,-]<.->[<.--->[.+][-,]][--[-.]+<->+,<-->++-+][<.-->,[-.[<.-->+[,-].]<,,][-.]<,]][-[.[<.-->,[-.[<.-->+[,-].]<,,][-.]]],[-.+<,->.[,-][-.+<,-->++<->+[-[--][,-]]]]+[-.+<,->.[,-]]][-.+<,->.[,-][-[-.+<,-->++]+[-[--][,-]]]<.->[-.[-.+<,->.[,-]][<.-->,[-.[<.-->+[,-].]<,,][-.]<.-]][,.].]]][-.+<,-->++<->+[-[--][,-]]<.->[-.[-.+<,->.[,-]][-.[<.-->,[-.[<.-->+[,-].]<,,][-.]]<->+<,]][-.].]][,-][-.[-.]]][<.-->,[-.[<.-->+[,-].]<,,][-.]<.->[-.+<,->.[,-]][-.+<,->.[,-]<.->,[-[-.+<,-->++]+[<--->[,-]]]+].][<.-->,[-.[<.-->+[,-].]<,,][-.][-.[-.+<,->.[,-]]<->[-.+<,-<->++<->+[-[--][,-]]<.-][<.--->[.+][-,]],.]]<,][,,]]+[+++++[>++++++[->++>+++>+++>+<<<<]>>-<<<-]>>.[--------<+++++>]>-.>..+++.<<<-.>>>>----.<++++++++.--------.+++.------.<-.<<<+++++[>--<<++>-]>-.<<.>]
Reverse input
Detailed description: Crazy J/Reverse input
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Sequential tag system
Detailed description: Crazy J/Sequential tag system
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Cyclic tag system
Detailed description: Crazy J/Cyclic tag system
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QUINE
Detailed description: Crazy J/QUINE
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