;#+

An extended version of ;#

;#+ is basically ;# but Turing Complete.

Memory layout

;#+ has two accumulators: primary and secondary. They are also referred to as A and B, respectively. There is a delta value, or Δ. Furthermore, ;#+ has a memory tape of size 1000 by default. This can be written to and read from. To read from this tape, there is a pointer, P.

Commands

Without further ado:

+    add accumulators together into the current accumulator
;    increase accumulator by the delta value
:    increase pointer by delta value
-    negate the delta value
#    mod 127 the accumulator, output ascii char, and reset accumulator
<    read cell from memory into accumulator
>    save accumulator into memory
~    swap accumulator with secondary accumulator
.    go to the (accumulator)th position in memory
*    read character of input into accumulator
(    if the accumulator is zero, skip to the next matching `)`
)    if the accumulator is not zero, rewind to the previous matching `(`
p    print the numeric value of the accumulator
`    show debug information
!    set accumulator to 1 if zero; otherwise, 0

Proof of Turing completeness

One can show that there is a one-to-one correspondence between ;#+ and BF:

BF Command	;#+ Equivalent
>	:
<	-:-
+	;
-	-;-
[	(
]	)
,	*
.	#

Technically, . and # have different output behaviours, but the functionality remains the same.