Toi

Toi (the name means nothing) is an esoteric language designed by Oklopol.

Overview
Toi stores all data in sets (of sets of sets of ...). There is a context S (which is a set) that all instructions modify. Execution is Brainfuck-style, left to right, one character instructions, loops. Brackets have to match, otherwise everything's a legal program. There is no set called the error set. The language is suspected to be Turing complete.

Instructions
Toi has 16 instructions (counting loops as single instructions). In the following "ordinal" refers to the Von Neumann ordinals. Zero is the empty set, and the number n is represented by the set containing the sets corresponding to all numbers below n.

The instructions are the following:
 * '<...>' adds the set '<...>' to S. The syntax is usual LISP style, '<<> <<> <>>>' ~ ( ( )), except they're sets. Also to clarify, '<<>>' adds the set –, not the set {}. Base 10 numbers are interpreted as synonyms for the corresponding ordinal.
 * '-<...>' removes the set '<...>' from S. Numbers work as expected here too.
 * '(A{B}' is a type of for-each. For each s in S the Toi program A is run with s as the context. If after running A the context of A is a nonempty set, B is run with s as context, and s is replaced in S by the resulting context after running B. Changes to S are done after the for-each.
 * '-(A{B}' works like '(A{B}' but the condition is running A results in an empty set.
 * '(A[B]' is a while loop. While running A with context S results in a nonempty set, run B with context S, changing S to B's result after running B.
 * '-(A[B]' works like '(A[B]', but the condition on A's result is that it's empty.
 * '.' prints '.'.
 * ':' prints ':'.
 * 'e' works like '<>'.
 * 'E' does not add the error set to S.
 * 'n' prints a newline.
 * 'd' prints the set S in the format in which sets are usually input in code. A subset that is an ordinal is printed as a base 10 number corresponding to that ordinal.
 * 'r' changes S to {t | t ∈ s, s ∈ S} (this is decrement if S is an ordinal).
 * 'a' changes S to S ∪ {t | t ∈ s, s ∈ S}.
 * 'u' changes S to {S}.

Printing the naturals
<> ([(<>{.} uan ] Should be obvious how this works. 'ua' works as the successor function on the ordinal S. '(<>{B}' maps B over all of S.

Useful algorithms
(Essentially from a discussion on the #esoteric irc channel)

To clear S regardless of its initial value: ([r] To turn any S except the empty set into – : ({([r]} To remove every element of S except the empty set (can be used to test if S has the empty set as an element): ({([r]u}-<<>>

An ordered tuple (x,y) of two sets x and y can be implemented as the set { {{},}, } To turn S into the tuple (S,S): uuueua-e To apply A to just the first coordinate of a tuple in S: ( ({([r]u}-<<>> { rrr A uuue } To apply B to just the second coordinate of a tuple in S: -( ({([r]u}-<<>> { rr B uu } To extract the first coordinate of a tuple in S: -( ({([r]u}-<<>> { ([r] } rrrr To extract the second coordinate of a tuple in S: ( ({([r]u}-<<>> { ([r] } rrr

Addition: Turn S = (x,y) into S = x+y, for ordinals x and y. ( ( ({([r]u}-<<>> { ([r] } rrr [ ( ({([r]u}-<<>> { rrr ua uuue } -( ({([r]u}-<<>> { rr r uu } ] -( ({([r]u}-<<>> { ([r] } rrrr

External resources

 * Toi interpreter (the interpreter does not currently implement all of Toi, and is definitely not a reference implementation, it is also incredibly space consuming, hopefully the author will fix it tomorrow)