Essays/Bitwise Functions on Characters
m b. for m e. 16+i.16 computes bitwise functions on (machine-word) integers, whence m b.&.(a.&i.) computes bitwise functions on characters.
Problem: suppose y is computed in one of two ways:
y=: (c{a.) m b.&.(a.&i.) a.
y=: a. m b.&.(a.&i.) c{a.
where c is an integer between 0 and 255. Given y , compute integers c1 and m1 such that
y -: (c1{a.) m1 b.&.(a.&i.) a.
The problem can be solved as follows:
f1=: 3 : 0 " 1
'i j'=. a.i.0 255{y
q=. _2 ]\ 0,j, i,i, i,255, i,(255-i), i,0, 255,j
p=. _2 ]\ j,17, i,19, i,23, i,22, i,18, j,27
p {~ q i. i,j
)
f2=: 3 : 0 " 1
assert. ((,256)-:$y) *. 2=3!:0 y
'i j'=. a.i.0 255{y
if. i=0 do. cm=. j,17
elseif. j=i do. cm=. i,19
elseif. j=255 do. cm=. i,23
elseif. j=255-i do. cm=. i,22
elseif. j=0 do. cm=. i,18
elseif. i=255 do. cm=. j,27
elseif. 1 do. assert. 0 [ 'y is not generated by m b.' end.
assert. y -: (c{a.) m b.&.(a.&i.) a. [ 'c m'=. cm
)
f1 and f2 are equivalent; f2 is less concise than f1 but incorporates checks and is more readily translated into a lower-level language. If y is not a result of m b. , then an error is signalled.
Examples:
] 'c m'=: 0 16+?256 16
110 19
y=: (c{a.) m b.&.(a.&i.) a.
f2 y
110 19
y -: (110{a.) 19 b.&.(a.&i.) a.
1
] 'c m'=: 0 16+?256 16
54 25
y=: a. m b.&.(a.&i.) c{a.
f2 y
201 22
y -: (201{a.) 22 b.&.(a.&i.) a.
1
f1 a.{~ ?~ 256
|index error: f1
| p {~q i.i,j
f2 a.{~ ?~ 256
|assertion failure: f2
| 0['y is not generated by m b.'
Test all possibilities:
ya=: (16+i.16) 4 : 'y x b.&.(a.&i.) a.'"0/ a. ay=: (16+i.16) 4 : 'a. x b.&.(a.&i.) y '"0/ a. $ ya 16 256 256 $ ay 16 256 256 1 [ f2 ya 1 1 [ f2 ay 1 (f1 -: f2) ya 1 (f1 -: f2) ay 1
The choice of 0 255 as the y entries to use in the decision process is not unique. In fact, as the following expressions demonstrate, any pair i,255-i would do.
t=: ya ,&(,/) ay NB. all possibilities
$ t
8192 256
# ~. t NB. # of all unique possibilities
1528
# ~. 0 255{"1 t NB. 0 255 capture them all
1528
j=: (,.|.) i.256 NB. all pairs i,255-i
*./ 1528 = j #@~.@({"1"_)"1 _ t
1
Contributed by Roger Hui. An earlier version of the text appeared in the J Forum on 2007-10-22.