Here are my latest thoughts on negative binary numbers. a=:1 0 0 1 1 0 1 #.a 77 b=:-a b _1 0 0 _1 _1 0 _1 #.b _77
The challenge is to get from _77 to _1 0 0 _1 _1 0 _1 which should be the result that #: provides. But that is not what happens: #:#.b 0 1 1 0 0 1 1 #.#:#.b 51 So here is a script which does that. It has lots of possibilities for simplification but it does give an idea of a possible approach. If someone can write the parts in loops or recursive style it would be interesting to compare results. I'm not a J programmer, but rather a J user who would like #: to work. Have fun, share your ideas. ]n=:15 NB.Number of places like 4 for a 4 digit number d=: 13 :'-?y#2' d ]dd=:d n NB. d creates an n digit negative binary number w=: 13 :'2^|.i.y' w w n NB. w is a weighting list for each place value ]nd=:#.dd NB. nd converts a negative binary to a negative decimaldecimal number NB.nd=:_432532 insert a specific decimal to convert it ]p=:+/nd <-w 20 ]b1=:-1=<.nd%-w n ]r1=:nd-#.b1 ]b2=:-1=<.r1%-w n ]r2=:nd-#.b1+b2 ]b3=:-1=<.r2%-w n ]r3=:nd-#.b1+b2+b3 ]b4=:-1=<.r3%-w n ]r4=:nd-#.b1+b2+b3+b4 ]b5=:-1=<.r4%-w n ]r5=:nd-#.b1+b2+b3+b4+b5 ]b6=:-1=<.r5%-w n ]r6=:nd-#.b1+b2+b3+b4+b5+b6 ]b7=:-1=<.r6%-w n ]r7=:nd-#.b1+b2+b3+b4+b5+b6+b7 ]b8=:-1=<.r7%-w n ]r8=:nd-#.b1+b2+b3+b4+b5+b6+b7+b8 ]b9=:-1=<.r8%-w n ]nb=:b1+b2+b3+b4+b5+b6+b7+b8+b9 #.nb NB. nb has created the correct negative binary number ]dd #.dd ]p NB.nb id the binary created from a negative decimal number #:nd #.#:nd NB. results with problems Linda -----Original Message----- From: programming-boun...@jsoftware.com [mailto:programming-boun...@jsoftware.com] On Behalf Of Kip Murray Sent: Saturday, December 17, 2011 2:02 AM To: Programming forum Subject: Re: [Jprogramming] How #: should have been designed Linda, if you look over my original post, shown way below, you will see I used a verb of Raul's named twoscomplement and then a verb of mine named tc. Don't overlook Raul's part at the very bottom. Raul has since presented a corrected version of twoscomplement. It appears to me what you got from my verb tc is the same as what I got. I will think over your work below, in particular the part about f and #.f . Kip On 12/16/2011 9:16 PM, Linda Alvord wrote: > > Kip, I get a different result from tc than you show in your original post. > Using J I get: > > tc =: #:~2 #~ [:>./ 1 + [:>. (2 ^. |)`(2 ^.>:)@.(0<: ])"0 > ]a=:tc i:6 > 1 0 1 0 > 1 0 1 1 > 1 1 0 0 > 1 1 0 1 > 1 1 1 0 > 1 1 1 1 > 0 0 0 0 > 0 0 0 1 > 0 0 1 0 > 0 0 1 1 > 0 1 0 0 > 0 1 0 1 > 0 1 1 0 > > This is copied from your post: > > The verb tc (two's complement) appears to work > > tc =: #:~ 2 #~ [:>./ 1 + [:>. (2 ^. |)`(2 ^.>:)@.(0<: ])"0 > > twoscomplement i: 6 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > 1 1 1 > 0 0 0 > 0 0 1 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > > Call this b: > > ]b=: }."1 a > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > 1 1 1 > 0 0 0 > 0 0 1 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > > c=:#:i:6 > c > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > 1 1 1 > 0 0 0 > 0 0 1 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > (}."1 tc i:6)-:(#:i:6) > 1 > b-:c > 1 > tcinv=:[: #. (* (_1 ^ 0= i.@#))"1 > tcinv a > _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 > > tcinv b > 2 3 _4 _3 _2 _1 0 1 2 3 _4 _3 _2 > > So I thought you really meant the four column result in a . > > However the array in b matches the result of using #: on i: > > ]c=:#:i:6 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > 1 1 1 > 0 0 0 > 0 0 1 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > > b-:c > 1 > > No matter if it was created by #: or by tc , it is an array of unique > numbers. It only represents the following list. > > #.b > 2 3 4 5 6 7 0 1 2 3 4 5 6 > #.c > 2 3 4 5 6 7 0 1 2 3 4 5 6 > > This is incorrect. > > Now consider: > > (8+i.15)-15#15 > _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 > > d=:(#:8+i.15)-((5#2)#:,15#15) > d > 0 0 _1 _1 _1 > 0 0 _1 _1 0 > 0 0 _1 0 _1 > 0 0 _1 0 0 > 0 0 0 _1 _1 > 0 0 0 _1 0 > 0 0 0 0 _1 > 0 0 0 0 0 > 1 _1 _1 _1 _1 > 1 _1 _1 _1 0 > 1 _1 _1 0 _1 > 1 _1 _1 0 0 > 1 _1 0 _1 _1 > 1 _1 0 _1 0 > 1 _1 0 0 _1 > > The array in d is a beautiful representation of negative binary numbers. > They include negative signs. If the leftmost digit is negative, the number > is negative. Look only at the top and if you recognize the pattern as 5 it > will > Be negative 5 > > The array follows traditional place value. > > |.2^i.5 > 16 8 4 2 1 > > d*($d)$|.2^i.5 > 0 0 _4 _2 _1 > 0 0 _4 _2 0 > 0 0 _4 0 _1 > 0 0 _4 0 0 > 0 0 0 _2 _1 > 0 0 0 _2 0 > 0 0 0 0 _1 > 0 0 0 0 0 > 16 _8 _4 _2 _1 > 16 _8 _4 _2 0 > 16 _8 _4 0 _1 > 16 _8 _4 0 0 > 16 _8 0 _2 _1 > 16 _8 0 _2 0 > 16 _8 0 0 _1 > > +/"1 d*($d)$|.2^i.5 > _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 > > And the sum of the value of the digits is correct. Furthermore, #. > Understands it with no change in code. > > #.d > _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 > > > My contention is that if we, the users, could get used to negative numbers, > as we have learned to like _123.132j432.23 we would embrace these six > binary representations: > > f=:0,.0,.(?6 7$3){_1 0 1 > f > 0 0 _1 1 1 0 _1 _1 1 > 0 0 1 _1 0 1 1 _1 0 > 0 0 0 _1 _1 _1 _1 1 _1 > 0 0 0 0 _1 1 0 0 1 > 0 0 1 0 _1 _1 0 0 0 > 0 0 1 _1 1 _1 1 0 1 > > The two columns of 0's are necessary for this to work correctly. > > Using place value and J we can "read" these numbers correctly > > f*($f)$|.2^i.9 > > 0 0 _64 32 16 0 _4 _2 1 > 0 0 64 _32 0 8 4 _2 0 > 0 0 0 _32 _16 _8 _4 2 _1 > 0 0 0 0 _16 8 0 0 1 > 0 0 64 0 _16 _8 0 0 0 > 0 0 64 _32 16 _8 4 0 1 > > +/"1 f*($f)$|.2^i.9 > _21 42 _59 _7 40 45 > > And with no change in code, #. Gives the same results > > #.f > _21 42 _59 _7 40 45 > > It is only #: that must be changed. I would think this would be fairly > straightforward if these representatives of negative binary numbers were > adopted. > > #:#.f > > 1 1 1 1 1 0 0 > 0 1 0 0 0 1 0 > 0 1 1 1 1 0 1 > 1 0 1 0 0 1 0 > 0 1 1 1 0 1 1 > 1 1 0 0 0 1 0 > > Since the current representation that #: provides has only positive numbers > it is where the changes are necessary. > > Finally ou would never see the displays above if you only consider positive > integers. > > ]e=:#:i.7 > 0 0 0 > 0 0 1 > 0 1 0 > 0 1 1 > 1 0 0 > 1 0 1 > 1 1 0 > #.e > 0 1 2 3 4 5 6 > > So, my take on this is that the user must learn to accept a new way of > understanding of what binary numbers look like. They will be unique. You > can identify the sign by the leading element. If you can read the number in > binary, it will be the same number but negative. > > I keep trying to explain, and as I do it is getting easier for me to > understand what I am saying. That's I guess how you learn, by trying to > teach. > > Linda > > > > > > > > > > -----Original Message----- > From: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] On Behalf Of Kip Murray > Sent: Friday, December 16, 2011 1:53 PM > To: Programming forum > Subject: Re: [Jprogramming] How #: should have been designed > > You need to use Marshall's two's complement inverse > > tcinv > [: #. (* (_1 ^ 0 = i.@#))"1 > tcinv tc i: 6 > _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 > > In emails I insert an extra space in front of J output -- this seems to > protect it from the vagaries of email. > > I will look at your post, thanks for repeating it. > > On 12/16/2011 2:40 AM, Linda Alvord wrote: >> Here's what I get. The line will split but when I paste it back together: >> >> tc =: #:~ 2 #~ [:>./ 1 + [:>. (2 ^. |)`(2 ^.>:)@.(0<: ])"0 >> ]a=:tc i:6 >> 1 0 1 0 >> 1 0 1 1 >> 1 1 0 0 >> 1 1 0 1 >> 1 1 1 0 >> 1 1 1 1 >> 0 0 0 0 >> 0 0 0 1 >> 0 0 1 0 >> 0 0 1 1 >> 0 1 0 0 >> 0 1 0 1 >> 0 1 1 0 >> #.a >> 10 11 12 13 14 15 0 1 2 3 4 5 6 >> >> Not good. But your display does not have the first column, so maybe you > need >> to post the function again. >> >> I tried to remove the first column to match your dislay: >> >> ]b=:1}."1 a >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> 1 1 1 >> 0 0 0 >> 0 0 1 >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> >> #.b >> 2 3 4 5 6 7 0 1 2 3 4 5 6 >> >> This is also not working. >> >> Did you really read my post in a reply to a message to Devon? I'll repeat > it >> here as I am interested in your comments: >> >> ]a=:8+i.15 >> 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 >> ]b=:15#15 >> 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 >> ]c=:a-b >> _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 >> f=: 13 :'(x#2)#:y' >> f >> ] #:~ 2 #~ [ >> >> The function f converts the lists to binary numbers. The largest number > 22 >> is less than 32 so only 5 digits are require. >> >> ]aa=:5 f a >> 0 1 0 0 0 >> 0 1 0 0 1 >> 0 1 0 1 0 >> 0 1 0 1 1 >> 0 1 1 0 0 >> 0 1 1 0 1 >> 0 1 1 1 0 >> 0 1 1 1 1 >> 1 0 0 0 0 >> 1 0 0 0 1 >> 1 0 0 1 0 >> 1 0 0 1 1 >> 1 0 1 0 0 >> 1 0 1 0 1 >> 1 0 1 1 0 >> >> ]bb=:5 f b >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> 0 1 1 1 1 >> >> The result below is the difference in J between the two arrays above. > It >> is not a conversion using #: >> >> ]cc=:aa-bb >> 0 0 _1 _1 _1 >> 0 0 _1 _1 0 >> 0 0 _1 0 _1 >> 0 0 _1 0 0 >> 0 0 0 _1 _1 >> 0 0 0 _1 0 >> 0 0 0 0 _1 >> 0 0 0 0 0 >> 1 _1 _1 _1 _1 >> 1 _1 _1 _1 0 >> 1 _1 _1 0 _1 >> 1 _1 _1 0 0 >> 1 _1 0 _1 _1 >> 1 _1 0 _1 0 >> 1 _1 0 0 _1 >> >> For comparison, the result cc2 is obtained as a J result using #: >> ]cc2=: 5 f c >> 1 1 0 0 1 >> 1 1 0 1 0 >> 1 1 0 1 1 >> 1 1 1 0 0 >> 1 1 1 0 1 >> 1 1 1 1 0 >> 1 1 1 1 1 >> 0 0 0 0 0 >> 0 0 0 0 1 >> 0 0 0 1 0 >> 0 0 0 1 1 >> 0 0 1 0 0 >> 0 0 1 0 1 >> 0 0 1 1 0 >> 0 0 1 1 1 >> >> Or: >> >> ]cc3=:#:c >> 0 0 1 >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> 1 1 1 >> 0 0 0 >> 0 0 1 >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> 1 1 1 >> >> Both cc2 and cc3 seem implausible because all numbers are known positive >> numbers. Now look at cc : >> >> w=: 13 :'(x,y)$|.2^i.y' >> w >> , $ [: |. 2 ^ [: i. ] >> 15 w 5 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> 16 8 4 2 1 >> >> cc*15 w 5 >> 0 0 _4 _2 _1 >> 0 0 _4 _2 0 >> 0 0 _4 0 _1 >> 0 0 _4 0 0 >> 0 0 0 _2 _1 >> 0 0 0 _2 0 >> 0 0 0 0 _1 >> 0 0 0 0 0 >> 16 _8 _4 _2 _1 >> 16 _8 _4 _2 0 >> 16 _8 _4 0 _1 >> 16 _8 _4 0 0 >> 16 _8 0 _2 _1 >> 16 _8 0 _2 0 >> 16 _8 0 0 _1 >> >> >> +/"1 cc*15 w 5 >> _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 >> >> This is c! >> >> Not only that, J knows it is c. >> >> #.cc >> _7 _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6 7 >> >> So what do you think? >> >> Linda >> >> -----Original Message----- >> From: programming-boun...@jsoftware.com >> [mailto:programming-boun...@jsoftware.com] On Behalf Of Kip Murray >> Sent: Friday, December 16, 2011 2:43 AM >> To: Programming forum >> Subject: Re: [Jprogramming] How #: should have been designed >> >> The verb tc (two's complement) appears to work >> >> tc =: #:~ 2 #~ [:>./ 1 + [:>. (2 ^. |)`(2 ^.>:)@.(0<: ])"0 >> >> twoscomplement i: 6 >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> 1 1 1 >> 0 0 0 >> 0 0 1 >> 0 1 0 >> 0 1 1 >> 1 0 0 >> 1 0 1 >> 1 1 0 >> >> tc i: 6 >> 1 0 1 0 >> 1 0 1 1 >> 1 1 0 0 >> 1 1 0 1 >> 1 1 1 0 >> 1 1 1 1 >> 0 0 0 0 >> 0 0 0 1 >> 0 0 1 0 >> 0 0 1 1 >> 0 1 0 0 >> 0 1 0 1 >> 0 1 1 0 >> >> >> On 12/15/2011 12:52 PM, Kip Murray wrote: >>> twoscomplement i: 4 NB. error >>> 1 0 0 >>> 1 0 1 >>> 1 1 0 >>> 1 1 1 >>> 0 0 0 >>> 0 0 1 >>> 0 1 0 >>> 0 1 1 >>> 1 0 0 >>> >>> rp =: [: }: [: i: 2 ^<: NB. representable in y bits >>> >>> rp 3 >>> _4 _3 _2 _1 0 1 2 3 >>> >>> twoscomplement rp 3 NB. OK >>> 1 0 0 >>> 1 0 1 >>> 1 1 0 >>> 1 1 1 >>> 0 0 0 >>> 0 0 1 >>> 0 1 0 >>> 0 1 1 >>> >>> >>> On 12/14/2011 9:13 AM, Raul Miller wrote: >>>> The subject line of this thread is arguably wrong -- there are a >>>> variety of "good ways" of decomposing integers to binary. >>>> >>>> That said, it's interesting to think about the various proposals >>>> expressed in terms similar to those which could be used to implement >>>> monadic #: >>>> >>>> antibase2=: #:~ 2 #~ 1 + 2<.@^. 1>.>./@,@:|@:<. >>>> twoscomplement=: #:~ 2 #~ 1 + 2<.@^. 1 +>./@,@:|@:<. >>>> signwithbits=: #:~ 0, 2 #~ 1 + 2<.@^. 1>.>./@,@:|@:<. >>>> >>>> (In all cases the #: here is dyadic, so these definitions are >>>> independent of the definition of monadic #:) >>>> >>>> antibase2 i: 3 >>>> 0 1 >>>> 1 0 >>>> 1 1 >>>> 0 0 >>>> 0 1 >>>> 1 0 >>>> 1 1 >>>> twoscomplement i: 3 >>>> 1 0 1 >>>> 1 1 0 >>>> 1 1 1 >>>> 0 0 0 >>>> 0 0 1 >>>> 0 1 0 >>>> 0 1 1 >>>> signwithbits i: 3 >>>> _1 0 1 >>>> _1 1 0 >>>> _1 1 1 >>>> 0 0 0 >>>> 0 0 1 >>>> 0 1 0 >>>> 0 1 1 >>>> >>>> There's also (* * #:) but that one assumes the antibase2 >>>> implementation... >>>> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
