1 J SHIMURA Masato jcd02773@nifty.ne.jp 2008 12 8 1 J 1 2 J 4 3 5 4 8 5 /de Morgan law 11 6 16 7 19 8 Reference 21 A 21 J 5 1 J J Atom ) APL J
1 J 2 tasu =: + (Tacit definition) (Explicit definition) 1.1 (&) (@) x u&v y Fork 1.1.1 Bond & Bond(&) (Atop(@) 0&{ u u v v v y x y 1&{ ( p) ( q) x v&u y is (v x) u (v y) 1.1.2 Atop & u v y x u v y _8 3+:@- 7
1 J 3 1.2 2 3 Train J 3 2 1.2.1 Hook 1.2.2 Hook y g h x g y y -(+/ % #) -"1(+/ % #) NB. h 1.2.3 Fork mean=:+/#% g f h y y g f h x y x y 5.5 (+/%#) >:i.10 1.2.4 Capped fork [: u v Hook Fork
2 J 4 g h y x g h y George Boole (1815-1864) England 16 19 29 34 Queens College sir George Everest 5 *1 1854 An investigation into the Laws o f Thought, on Which are f ounded the Mathematical Theoties o f Logic and Probabilities 1938 2 J J J J formula Bool AND *.. 0 1 0 0 0 1 0 1 1 is 1 1 only 0 0 0 1 d=: 0 1 d *./ d 24 *. 60 120 NB.LCM *1
3 5 OR +. +. 0 1 0 0 1 1 1 1 1 is 0 0 only 0 1 d +./ d 24 +. 60 12 NB. GCD 1 1 Not AND *: : 0 1 0 1 1 1 1 0 reverse AND 1 1 d *:/ d 1 0 Not OR +: + : 0 1 0 1 0 1 0 0 reverse OR 1 0 d +:/ d 0 0. NOT -. 0 1 1 0 -. 0 1 1 0 (i.9)-. 2 3 5 7 0 1 4 6 8 abcdefg -. aiueo bcdfg 2 3 4 5 -. ABCDEF 2 3 4 5 3 S o f tware AND & OR NOT!
3 6 *2 AND,OR,NOT 3 (XOR) J AND and logis and logis0 OR or logis or logis0 NOT not logis not logis0 R0 1 1 0 0 1 0 1 0 T T F F T F T F J formula Bool Example p q p q. 0 1 T T T AND *. 0 0 0 T F F 1 0 1 1 is both 1 0 0 0 1 d *./ d F T F F F F T and_logis R0 TTT TFF T FTF FFF p q p q +. 0 1 T T T OR +. 0 0 1 T F T 1 1 1 1 is either 1 0 1 1 1 d +./ d F T T F F F T or_logis R0 TTT TFT T FTT FFF *2 & pipe
3 7 p q pv q Not Equal (XOR exclusive or) V 0 1 0 0 1 1 1 0 d-.@=/ d 0 1 1 0 T T F T F T F T T F F F nequal_logis R0 TTF TFT FTT FFF reverse equal p p NOT -.. 0 1 1 0 -. 0 1 1 0 T F T F F T F T not_logis {. R0 TF TF FT FT 3.1 Script TF 2 0 and logis0, and logis and_logis0=: 3 : 0 NB. calc AND A0 B0 =: y A0 *. B0 ) and_logis=: 3 : 0 NB. calc AND A0 B0 =: y (trans_tf_sub y),.trans_tf_sub A0 *. B0 )
4 8 or_logis0=: 3 : 0 NB. calc OR A0 B0 =: y A0 +. B0 ) not_logis0=: 3 : 0 NB. calc Not equal A0 =:; y -. ; y ) NB. --------------------- R0=: 1 1 0 0 ; 1 0 1 0 not logis 1 picl_not 0 1 2 3 pick_pq_not R0 -------+-------+ 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 -------+-------+ p q p q 0 1 2 3 4 2 5 IMPLY cond pq cond pq0 EQUIV condw pq condw pq0
4 9 4.1 p q ( p) q 1 1 0 0 ((-.@[) +. ]) 1 0 1 0 NB. use fork 1 0 1 1 cond_pq0 R0 1 0 1 1 trans_tf_sub 1 0 1 1 TFTT p q q p p q q p cond_pq R0 cond_pq. R0 cond_pq -. L:0 R0 cond_pq.-. L:0 R0 TTT TTT TTT TTT TFF TFT TFT TFF FTT FTF FTF FTT rorate not not and rotate cond_pq0=: 3 : or_logis0 (<-.;{.y),{: y 4.2 *3 p q p q q p p q (p q) (q p) *3 if and only if
4 10 = equal(=) condw_pq R0 1 1 0 0 = 1 0 1 0 TTT 1 0 0 1 TFF FTF condw_pq0 R0 1 0 0 1 condw_pq0=: 3 : 0 TMP=: y,-.&.> y NB. close<-calc<-open and_logis0 (or_logis0 2 1{ TMP);or_logis0 3 0{TMP ) J Grammar J NOT TEXT J J 1 : 0 (+: ; -: ; *: ; %:) 2j_1 +----+------+----+-----------------+ 4j_2 1j_0.5 3j_4 1.45535j_0.343561 +----+------+----+-----------------+ logis=. 5 not_logis0&{.;and_logis0;or_logis0;cond_pq0;condw_pq0 logis R0 ------- 0 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 ------- not and or cond condw
5 /de Morgan law 11 :( L:0) 1 0 1 0 0 0 0 <;.1 : logis trans_tf0 R0 pq naocw +--+-----+ TT FTTTT TF FF FT TFTTF FF TT +--+-----+ NB. pick not ( p) q 0/1/2/3 R0,-.&.> R0 -------+-------+ 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 -------+-------+ 0 1 2 3 p q p q (2 1&pick_pq_not) R0 0 0 1 1 1 0 1 0 cond_pq0&(2 1&pick_pq_not) R0 1 1 1 0 5 /de Morgan law 5.1 (p q) = ( p) ( q) (p q) = ( p) ( q)
5 /de Morgan law 12 p, q P, Q p q P Q (p q) (P Q) August De Morgan 1806-1871 7 antilogalism James Dodson 10 14 Ox f ord 16 trinity algebra MA 22 On the study o f mathematics U.S.A. 1837 S ophia Elizabeth 3 4 London Mathematics S ociety Ox f ord, Cambridge, Royal S ociety (p q) not_logis0&or_logis0 0 0 0 1 R0 not_logis0&or_logis0 trans_tf0 R0 pq* --- TTF TFF FTF
5 /de Morgan law 13 ( p) ( q) 1 1 0 0 (*.& -.) 1 0 1 0 0 0 0 1 not_logis0 L:0 R0 0 0 1 1 0 1 0 1 and_logis0&(not_logis0(l:0)) R0 0 0 0 1 and_logis0&(not_logis0(l:0)) trans_tf0 R0 pq* ---- TTF TFF FTF
5 /de Morgan law 14 (p q) not_logis0&and_logis0 R0 0 1 1 1 not_logis0&and_logis0 trans_tf0 R0 TTF TFT FTT ( p) ( q) 1 1 0 0 (+.& -.) 1 0 1 0 0 1 1 1 (not_logis0(l:0)) R0 0 0 1 1 0 1 0 1 or_logis0&(not_logis0(l:0)) R0 0 1 1 1 or_logis0&(not_logis0(l:0)) trans_tf0 R0 TTF TFT FTT
5 /de Morgan law 15 5.2 /tautology T 30 (p q) (p q) T ( (p q)) ((( p) q) ( p) p) ((p q) (p q) ) (p q) (p q) ( (p q)) ((( p) q) ( p) p) p q p q q p p q p q p q T T T T T T F F F T F T F F T F F F T T OR 0,0 1 tt0=. and_logis0;condw_pq0 tt0 R0 1 0 0 0 1 0 0 1 cond_pq0 1 1 1 1 tt0 R0
6 16 cond_pq0 trans_tf0 awt ---- TTT FTT tt0 R0 (( p) q) (p q) (( p) q) (p q) (( p) q) = (p q) condw_pq0 (cond_pq0 (2 1&{ R0, -.&.> R0));or_logis0 R0 1 1 1 0 1 1 1 0 condw_pq0 (cond_pq0 (2 1&{ R0, -.&.> R0));or_logis0 R0 1 1 1 1 p q p, q 6 6.1 2 range suspect p q q p
6 17 p q p q q p T T T F F T F F T F F T T F T F F T T T TTT p q q p logis=. cond_pq0;(3 2&pick_pq_not) logis R0 -------+ 1 0 1 1 0 1 0 1 0 0 1 1 -------+ logis trans_tf0 R0 pqcnn ------ TTTFF TF FTTFT TT NB. all T 6.2 3 p, q, r 3 2!) p, q, r p, q, r p q q r p r 3 T
6 18 p q q r p r p q r p q q r p r T T T T T T T T F T F F T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T. 2 2 2 #: i.8 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 A0=. {@ :@. 2 2 2 #: i.8 +---------------+---------------+---------------+ 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 +---------------+---------------+---------------+ logis=. cond_pq0&(0 1&{);cond_pq0&(1 2&{);cond_pq0&(0 1&{) logis A0 +---------------+---------------+---------------+ 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 +---------------+---------------+---------------+ logis trans_tf0 A0 pqrccc ------ TTTTTT NB. T
7 19 TTFTFT TFTFTF TFF FTTTTT NB. T FTFTFT TTT NB. T FTT NB. T 7 1. (p p) p 2. q (p q) 3. (p q) (q p) 4. p (q r) q (p r) 5. (q r) ((p q) (p r)) 1. (p p) p 2. p (p q) 3. (p q) (q p) 4. (p q) ((r p) (p r)) (4) 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. ( p q) (q p) 6 (p p) p r0=. and_logis0&(0 0& pick_pq_not); 0&pick_pq_not r0 R0
7 20 1 1 0 0 1 1 0 0 cond_pq0 1 1 1 1 r0 R0 q (p p) r1=.([: > 1&pick_pq_not);or_logis0 r1 R0 1 0 1 0 1 1 1 0 cond_pq0 r1 R0 1 1 1 1 q (p q) is ( q) (p q) q p p (p q) (q p) h2=. or_logis0;or_logis0&. h2 R0 1 1 1 0 1 1 1 0 cond_pq0 1 1 1 1 h2 R0
8 Reference 21 8 Reference 1966 1962 A position 84 70 a. i. TF T a. 84 sequential machine a=. T F > ;: a T F antibase #: 2 2 #: i.4 0 0 NB. 0 0 1 NB. 1 1 0 NB. 2 1 1 NB. 3 {@ :@. 2 2 #: i.4 1 1 0 0 1 0 1 0