+-)
+-) is a minimalistic esolang invented by User:None1 and inspired by brainfuck.
Memory
There is a tape with three cells, each cell contains an unbounded signed integer. There is also a pointer that wraps around.
Commands
In +-), there are three commands, both + and - matches ).
+: Increment current cell unless the previous instruction executed is its matching ), then jump to the matching ) if the current cell is zero or the next character is ).-: Decrement current cell unless the previous instruction executed is its matching ), then move the pointer to the right, then jump to the matching ) if the current cell is zero or the next character is ).): Jump to the matching + (or -) if current cell is nonzero and the previous character is ).
Computational class
It is Turing complete because it can be translated from 3 cell Brainfuck (without I/O) easily. Proof is below (BF stands for 3 cell Brainfuck in the proof for convenience):
+
+ in BF can simply be translated to +) in BF.
> and <
We notice that -) in +-) is the same as -> in BF, so > is equivalent to +->, which is +)-) in +-).
Because the pointer wraps around, now that we get the translation of >, we can simply translate < by repeating > twice, that is:
+)-)+)-)
-
- is equivalent to -><. Because -) in +-) is the same as -> in BF, - can be translated to -)+)-)+)-).
[ and ]
When not before a ), + in +-) is the same as +[ in BF, so [, the same as -+[><, can be translated to -)+)-)+)-)++)-)+)-)+)-).
] can be translated to ).
Note: >< is a placeholder that does nothing to force the loop to be nonempty. If the >< doesn't exist, empty loops ([]) won't work, because it will be translated to -)+)-)+)-)+), which will do nothing because the last ) is right before a +.
| BF | +-) |
|---|---|
+ |
+) |
- |
-)+)-)+)-) |
> |
+)-) |
< |
+)-)+)-) |
[ |
-)+)-)+)-)++)-)+)-)+)-) |
] |
) |