Dialectical Logic K-Model: A Mathematical Model for Machine

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Piers Mills
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1 Joual o ahmac Rach; ol. 9 o. 6; Dcmb 7 ISS EISS Publhd by Caada C o Scc ad Educao Dalccal Logc odl: ahmacal odl o ach Shazhuag Hghch dc Hb Cha Yaozh Jag Coodc: Yaozh Jag Shazhuag Hghch dc Hb Cha. Emal: agyaozh@6.com Rcd: Smb 7 ccd: Ocob 8 7 Ol Publhd: Ocob 7 7 do:.5539/m.968 URL: h://do.og/.5539/m.968 bac axom ym o dalccal logc modl whch bad o chho gymhod ablhd by auho h a. h auho dcb ha ubclaw h mo mag lcd om obclaw ad d ha h h whch amd by ao abaco ad hg acal llgc. am m dcb ha axom ym o dalccal logc modl whch coa uch a logcaabl gy coao law oz cl mmax cl) ad obdd law c. I h axom ym alo coa uch a a couou ualuuco ym alud o al ad h gah o logcaabl. d dcb h gg alu mhod by max bad o gah ad chho law o h logc aabl. h auho dcb mly h la ad ola logc aabl ym. d dcb mly h logc aabl old hdmo Euclda ac ad oology wo ac aaly. Dalccal logc modl would uly a comuao algohm da o mach o ha h mach abl o h by dalccal logc mhod hu a moa omaoad mhod mayb h dalccal logc. ywod: cal llgc Dalccal logc modl gah chholaw Coadcouco max SSC): 3B99. Ioduco Obc wold co o aco whch a ER EERY IORIO SPCE ad IE. mog h aco h a may o ou law o cauao ca b calld a obclaw alo ca b calld a obclogc. Coodg o obclaw h a mo mag wh mach o acal llgc whch ca b calld a ubclaw alo ca b calld a ubclogc. h mach o acal llgc oly hold obclaw hough ubclaw o wh aoh wod oly hold obclogc hough ubclogc. Subclogc bul o m cculao by h h whch amd Sao baco ad hg. Sao uably o mach om obc omao wh o. baco coco clacaoably o omao ad o am. hg ably o mach achg om cococlacao omao o g h ubclogc. h dalccal logc wa ablhd by Hgl a ga hlooh Bcga E. ; oo. 966). By h oo o dalccal logc h obc law a dyamcal ad ad d by h caualy law. I ma oo : coadco a h cau o obc ay. h ablhm o mahmacal modl o dalccal logc h cay codo od o ma ha h dalccal logc comuabl mach. Dalccal logc modl ablhd h a would uly a comuaoda o mach o ha h mach abl o h by dalccal logc mhod o a moa omaoad mhod would b h dalccal logc. h dalccal logc modl combd wh h ma cogo cholog mag cogo chology oc cogo chology x cogo chology) daa ba chology ad machlogammg chology wll a a ga og o acal llgc.. xom Sym ad hom. Coco ad h Cocodmo... h Coco h am Whch b amd h Obchg a Poy ad glcd coday o. Coco h ba o acal llgc.... h Cocodmo h cocodmo h dmoumb o coco. o xaml HORSE a odmo 8

2 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 coco; WHIE HORSE a wodmo coco; RUI WHIE HORSE a hdmo coco; OE RUI WHIE HORSE a oudmo coco; c...3. oca Daaba h h ma cogo cholog wll u h daa ba alhough h a oh daa ba o xaml hycal daa ba ad chmcal daa ba c. ) h h daa ba a aocad by h cocodmo.. Sal Law... Do Do obcdoma by B ad ubcdoma by ; d ach oao ach oao a ahmc o g ubclogc om obclogc) ad ach oao ach oao a ahmc whch ma ubclogc o obclogc od o chc u o al) ad hu B B B B B B coulo h B B abo h h cycl ach oao obouly a logcaomao bw doma ad doma B. Somm ca do by. Symbol dod ha h obc logc ad h ubc logc a mo mag ach oh. Do ubclogc aabl by ) { a ) a ) a a abo m aabl gh uc do h logc aabl coa coul o coadcg ub aabl a ) calld h aumb o ) gh ubc do h h cauao law. dod h ualu uco o logc aabl ) ad. Do h mo mag obcdoma o ) by B ) B.... Obc Rachbl Law o y obclogc aabl B ) B h mu alway x logcaomao ma..3. Rach Eo lagcogc Law o y coc logcaomao alway ma ) B }.. o y lm C ouco ) B ) lm C B B ) E ) B B ) 83

3 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 h omulao a abo C=+ =+; o C= = = * I ac coc logcaomao ca bg h achg dgc. h a h cogc o h oao h bgh h oao...4. Logc aabl Egy Coao Law o logc aabl ) oducggy E qual o h coumd wo J by coadcouco h.4 coadcouco a blow).. E J..5.oz Pcl ax Pcl)Joho Ia ) Logc aabl ) chagg mu b ad o mu b ad aymocally by ha couco C mmum ad gauco maxmum. Do h u gauco by L.. o..6. moya Law L max lm L max m C ) m C..6.. Lam moy Law o m quc do h mmoy cwghuco o coadco ub aabl a a ) om m by R R a ad R a R a I ad oly h mmoy h a ) o ha o by h la m Imoac moy Law o moac quc a aq a So h mmoy h a ) ad oly a bgg o Bgg Pobably moy Law o oy h coodg obably a ad a P P P ad m o h mmoy..3 ach Slogammabl ad Slcocabl Law I h achg oc h mach mu ha ably o logammabl ad lcocabl whou h oao by huma bg..4 obdd Law h a wo d o logc: o o hm dalccallogc coodg o llgc quo ad aoh o hm 84

4 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 maglogc coodg o mooal quo. h maglogc would b obdd o u o mach bcau h maglogc wll ma mach o oduc mooal quo o ha h auoomoumd wll blog o h mach. h acal llgc wh h auoomoumd wll o l o b a wllg ool o huma bg o a w cao wll b cad o cau h hamul o huma bg..5 Logc aabl Iaablchaacc Law h logc aabl aabl...6 Logcal Iduc ad Dduc hod hom o cycl ach oao ahmc ma ) lm{ a ) a ) a ) B )}.6.. duc) a abo u o.6.. hyoh) uo a abo u o.6.3. dduc) o ha a abo u o all. Poo: hough o o h alway x ouco bad o h..3. ach o alagcogc law h.6.. hyoh) al h oducg h al o h.6.3. dduc).hu h uh o y o would b caully chcd. Poo o..7 Do o Logc lgohmboola oao) o ) ) ) do h u alu uco by coodg o.7.. Logc Boola OR) Do ; } logc ad by a blow commua law aoca law ) ).7.. Logc Boola D) do m{ ; } logc ad by a blow commua law aoca law ) ).7.3. Logc Hybd hmc Boola OR & Boola D) Logc hybd ahmc ad a blow 85

5 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 86 dbu law ) ) ).7.4. Logc Boola O) o o o h ; ga o o h. Do ) ) logc ad by a blow dmo law ) ) ; ) ) m.8 D oga hom ) ) Poo: } ; } ; m{ ) h oo o omulao) o. } ; m{ } ; m{ } ; ) h oo o omulao) o. Rma: how abo wh m dgad o a coa ad doy h coadco h omulao h h omulao ) ad ) wll b dgad om h couou u alu uco Boola algba alud o al + o h mahmacal modl o omal logc.. bay Boola algba..9 Logc u alud uco Comoo hom o logc u alud uco a blow h a comoo hom: I quao x a oo } { m ad

6 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 87 m o ha } ; } ; } ; } ; ; m a mla a abo alo } ; { m } ; { m } ; { m } ; m{ m I quao whch h obl augm oo ad comlx oo wll b mod ou. I o h oo ha } ; o } ; m{ whch lc a maxmum o mmum o u uco. Poo:obouly bad o do o logc ahmc. Poo o.. La Logcal Sym d ola Logcal Sym ccodg o h... a how abo obouly w ha h omulao a blow... Do Logc aabl ad a amd la logcal ym ad oly hy a ad by... ad...3 a blow; ohw ca b amd ola logcal ym: Rma : h og aco luc by huma bg o h obc would oduc ola logc; h wa aco luc o zo aco luc by huma bg o h obc would oduc la logc. Bcau o h o aco luc o huma bg would oduc h ola o. xaml o la logc cc ad a xaml o ola logc gg o chology. mla a abo aoh xaml o la logc ogal cology ym ad aoh xaml o ola logc a lad ym agculu.. Exo by gah o logc aabl Bg C. 973)... chho Powuco Law d chho lowuco Law... D a cocg dcd ml gah wh +)od ad E dg } ;E { )

7 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 I gah y od wll b g a owuco ) o od ad od dg E E h o h dg E x a owuco ) ) ) 3) I omulao 3) low ach od l ad u ach od gh h oo dco would g a ga h o o ). I h gah y cycl W ad by...3. chho lowuco Law E W I h gah y od ad by ) E a abo u low ) o ad ouu low ) ga Coadco uco D R ) ) ) h coadco uco o dg E gah Wo d Egy Law I h gah y dg E ma W ) ) d W calld ha wo do by coadcouco R ) o dg E.... Sucu ah o Logcaabl )... ah ha Coul o od ad wo od oal +);... ah a oloo omull dg dcd ml gah;...3. I h gah + o od couc a c ub gah aoh + ga od couc aoh c ub gah; 88

8 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; od coc oly o od o od q coc oly o ga od q whch gh ubc qual h owuco o od a coa +h owuco o a coa h owuco o dg E a coa + h lowuco o dg E a coa I;...6. I h gah oh od ad dg all b dd owuco ) owuco ) lowuco ) ad coadcouco R ) o hm h a all ad by chho law a abo g. h ucu o gah..3. h gg alu o logc aabl by gah Coodg o y logc aabl h mu x a gah od o g h u alu o h logc aabl. h u alu g by h omulao a blow 4) h h o aco ad ga aco whch omd h wo d o h coadco a balac; h h o aco h coadco uo aco; h h ga aco h coadco uo aco. 89

9 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 9. Exo by ow uco max E low uco max ad coadco uco max C o logc aabl h xo by max ca ma logc aabl comuabl mach.... Pow uco max E h lm o ow uco max E a: a a h ow h ow uco alu o h od ; b. a h colum h aod co o h ow ; c. h oh max lm a: dg x; ohw. hu h ow uco max E... low uco max h max lm o h low uco max a a. a h ow h od co ; b. a h colum aod co o h ow; c. h oh lm o low uco max a: h dg x;ohw. I whch h low uco u o ouu h od. S h...3. h h low uco max

10 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 9 I I..3. coadco uco max C h max lm o h coadco uco a a. a h ow od co ; b. a h colum aod co o h ow; c. h oh lm o coadco uco max C a: I h dg x h h max lm ; ohw. hu h coadco uco max I I C..4. Po o h max E C h ablhd xo o ax E ad C ma h gah comuabl mach. ow om o abou h mac xlad a blow: a h ma dagoal lm o h h mac xc ow ad colum all a zo; b bcau o h gah a dcd gah h h h mac xc h ow ad colum a ga ymmcal max bad o h ma dagoal l a h ax o ymmy; c h lm o h mac ga.. dco o dg wog uod uoo;

11 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7.3 h Logc aabl xyz; cd o Euclda Sac O d o gam old h hdmo Euclda ac uch a ooball baball ad c. wa by huma bg. h d o gam acg o h Euclda ac u alu uco o oly ddd o m bu alo ddd o h oo Euclda ac. Do h hdmo o Euclda ac by x y z h h logc aabl dod by xyz; h ualuuco dod by xyz;..4. h Logc aabl w; cg o oologcal wo Sac oh d o gam old oologcal wo ac w uch a alway hghway chal c. ollow.3 do h logc aabl o h d gam by w; w;. ad h u alu uco o h logc aabl by d h hd d o gam old oologcal wo ac oly by la m umb o by abolu m uch a gam o chboad ym. ollow.3. do h logc aabl by w; u alu uco by w;. ad do h.5 Dc aabl ym o h logc aabl logc aabl acg o Euclda ac xyz; ad logc aabl acg o oologcal wo ac w; o h aal omao o h amlddaa o h logc aabl logc aabl acd o Euclda ac xyz; ad logc aabl acd o oologcal wo ac w; h wll b oducd h d o dc ym do hm aaly by Δ x y z; w ; hm h Δ do h aml m al. Rma3: h dal abou h h d o gam ad h dc m aabl ym wll b xlad ucco a by auho..6 Hdy ad aao hom.6.. o h od da o coadcouco ) h Q ) calld aao o o calld hdy..6.. Ey coadcouco ).. ad dq ) d Q x ad R alway ca b a algbaum o a coa C ad uco R ) R ) Q ) C 5) 9

12 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 dq ) d h omulao 5) Q ) aao ad C hdy Hdy ad aao hom Hdy ad aao a bac law obclogc o ha a bac law o cau a a mo mag alo ubclogc. Poo: combg h.6.. ad.6.. how a abo whch ca b od obouly..7 h Ccalo hom o h coadcouco ) ma R mod da x ad h o d R ) d o m h h o ad om callcd o a calld ccalo..7.. Poyuco Ccalo hom h xc o ccalo wll ma wha om w oyuco bo o om old oyuco dad. Poo: obouly. Poo o..8 Iomohcqualy I h aumb o logc aabl ) ) omohcqualy do omohcqualy a qual.. h ) ad ) calld ) ) Iomohcqualy ad by a blow lx law ) ) ) ) ymmcal law ) a law ) ) h ) ) ) ad ) ) h ) ) m m.8.. Smlay bw logc aabl ) ad ) hg aalogy mhod.8... Do: h wo logc aabl ) ) a. ) ; ad ) calld mlay b. h gah ad gah h coadcouco o coodg dg oooal.. R ) CR ) R ) R ) omulao a abo C a coa I ) ad ) calld mlay wha dod a ) ). h lao 93

13 h://m.cc.og Joual o ahmac Rach ol. 9 o. 6; 7 ad by lx law ymmcal law ad a law a am a omohcqualy how abo hg alogyabl hom wo o logc aabl ) ad ) a aalogyabl hg ad oly ) ) ) C Poo: bad o h do o mlay ) h ) ) I omulao a abo C a coa hu hy a aalogyabl hg. Poo o. 3. Cocluo how abo auho ha ablhd a axom ym o dalccal logc modl dd o al law om do gah ad oz cl ad ha od om hom o dalccal logc modl. uho alo ha ablhd h ggalu mac o logc aabl. I ac auho ha ablhd a comuabl mahmacal modl o coadco by ma o Hgl coadco oo o mach. h modl would uly a uul ma o HII ad RESOI acal llgc. am m auho alo ha dcbd mly h logc aabl old hdmo Euclda ac oology wo ac ad dc logc aabl ym. h adacd o ad dal o h a how a o dalccal logc modl wll b xlad ucco a by h auho. Rc Bcga E. ). Hgl Dalccal Logc Oxod Uy P. oo. 966). h omalzao o Hgl Dalccal logc Iaoal Phloohcal Qualy IPQ64): Jauay. h://do.og/.584/q Joho Ia ). h oz: coml alaohogog:h Ch Uy P Bg C. 973). ah ad hy gah_mdam:ohhollad Publhg Comay Coygh Coygh o h acl ad by h auho) wh ublcao gh gad o h oual. h a oacc acl dbud ud h m ad codo o h Ca Commo buo lc h://cacommo.og/lc/by/4./). 94

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