1 J 2 tasu =: + (Tacit definition) (Explicit definition) 1.1 (&) x u&v y Fork Bond & Bond(&) 0&{ u u v v v y x y 1&{ ( p) ( q) x v&
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1 1 J SHIMURA Masato [email protected] J 1 2 J /de Morgan law Reference 21 A 21 J 5 1 J J Atom ) APL J
2 1 J 2 tasu =: + (Tacit definition) (Explicit definition) 1.1 (&) (@) x u&v y Fork 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) Atop & u v y x u v y _8 3+:@- 7
3 1 J Train J Hook Hook y g h x g y y -(+/ % #) -"1(+/ % #) NB. h Fork mean=:+/#% g f h y y g f h x y x y 5.5 (+/%#) >:i Capped fork [: u v Hook Fork
4 2 J 4 g h y x g h y George Boole ( ) England Queens College sir George Everest 5 * An investigation into the Laws o f Thought, on Which are f ounded the Mathematical Theoties o f Logic and Probabilities J J J J formula Bool AND * is 1 1 only d=: 0 1 d *./ d 24 * NB.LCM *1
5 3 5 OR is 0 0 only 0 1 d +./ d NB. GCD 1 1 Not AND *: : reverse AND 1 1 d *:/ d 1 0 Not OR +: + : reverse OR 1 0 d +:/ d 0 0. NOT (i.9) abcdefg -. aiueo bcdfg ABCDEF S o f tware AND & OR NOT!
6 3 6 *2 AND,OR,NOT 3 (XOR) J AND and logis and logis0 OR or logis or logis0 NOT not logis not logis0 R T T F F T F T F J formula Bool Example p q p q. 0 1 T T T AND * T F F is both d *./ d F T F F F F T and_logis R0 TTT TFF T FTF FFF p q p q T T T OR T F T is either d +./ d F T T F F F T or_logis R0 TTT TFT T FTT FFF *2 & pipe
7 3 7 p q pv q Not Equal (XOR exclusive or) V d-.@=/ d T T F T F T F T T F F F nequal_logis R0 TTF TFT FTT FFF reverse equal p p NOT 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 )
8 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=: ; not logis 1 picl_not pick_pq_not R p q p q IMPLY cond pq cond pq0 EQUIV condw pq condw pq0
9 p q ( p) q ((-.@[) +. ]) NB. use fork cond_pq0 R trans_tf_sub 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
10 4 10 = equal(=) condw_pq R = TTT TFF FTF condw_pq0 R 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_ j_2 1j_0.5 3j_ j_ logis=. 5 not_logis0&{.;and_logis0;or_logis0;cond_pq0;condw_pq0 logis R not and or cond condw
11 5 /de Morgan law 11 :( L: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,-.&.> R p q p q (2 1&pick_pq_not) R cond_pq0&(2 1&pick_pq_not) R /de Morgan law 5.1 (p q) = ( p) ( q) (p q) = ( p) ( q)
12 5 /de Morgan law 12 p, q P, Q p q P Q (p q) (P Q) August De Morgan antilogalism James Dodson Ox f ord 16 trinity algebra MA 22 On the study o f mathematics U.S.A S ophia Elizabeth 3 4 London Mathematics S ociety Ox f ord, Cambridge, Royal S ociety (p q) not_logis0&or_logis R0 not_logis0&or_logis0 trans_tf0 R0 pq* --- TTF TFF FTF
13 5 /de Morgan law 13 ( p) ( q) (*.& -.) not_logis0 L:0 R and_logis0&(not_logis0(l:0)) R and_logis0&(not_logis0(l:0)) trans_tf0 R0 pq* ---- TTF TFF FTF
14 5 /de Morgan law 14 (p q) not_logis0&and_logis0 R not_logis0&and_logis0 trans_tf0 R0 TTF TFT FTT ( p) ( q) (+.& -.) (not_logis0(l:0)) R or_logis0&(not_logis0(l:0)) R or_logis0&(not_logis0(l:0)) trans_tf0 R0 TTF TFT FTT
15 5 /de Morgan law /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 R cond_pq tt0 R0
16 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 R condw_pq0 (cond_pq0 (2 1&{ R0, -.&.> R0));or_logis0 R p q p, q range suspect p q q p
17 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 R logis trans_tf0 R0 pqcnn TTTFF TF FTTFT TT NB. all T p, q, r 3 2!) p, q, r p, q, r p q q r p r 3 T
18 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 #: i A0=. {@ :@ #: i logis=. cond_pq0&(0 1&{);cond_pq0&(1 2&{);cond_pq0&(0 1&{) logis A logis trans_tf0 A0 pqrccc TTTTTT NB. T
19 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
20 cond_pq r0 R0 q (p p) r1=.([: > 1&pick_pq_not);or_logis0 r1 R cond_pq0 r1 R q (p q) is ( q) (p q) q p p (p q) (q p) h2=. or_logis0;or_logis0&. h2 R cond_pq h2 R0
21 8 Reference 21 8 Reference A position a. i. TF T a. 84 sequential machine a=. T F > ;: a T F antibase #: 2 2 #: i NB NB NB NB. 3 {@ :@. 2 2 #: i
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Masato Shimura [email protected] 2008 7 23 1 2 1.1....................................... 2 1.2..................................... 2 2 3 2.1 Example...................................... 3 2.2 Script...........................................
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論理回路 ( 基礎 ) 法政大学 情報科学部 大森健児 参考書 論理演算 () AND,OR,NOT,XOR AND OR NOT XOR 論理演算 (2) NAND,NOR NAND NOR 前提 結論 If A then B は A が真のとき B が真であるならば この文は真であり A が偽のときは B が真であろうとなかろうとこの文は真である A が真のとき B が偽であればこの文は偽である
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0 2 SHIMURA Masato [email protected] 2009 12 8 1 1 1.1................................... 2 1.2.......................................... 3 2 2 3 2.1............................... 3 2.2.......................................
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