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N-Terminal Switching Circuits (ate Read Fal 14, 881 {ee ice condone ave scl nies, ad ae speed by eg Si SUAS Seat int oka Bh en ‘eerle aniehng Bote Sich othe there tp oir nce In a ret ite af The Bell Sytem Technical Jounal, C. . Shannan sicased the sgubeas of Iworenrinal reny conta. memory. Some of bis rls willbe generate ith paper to ternal ek high tre select evteher ith any tamer of postions insta! of the fem of rey, The kil of eee ehh wile olen may be vial 8 8 bck bow with A acces terns and ih A wall extending rm Tench shalt operas a selector ate (which wi wally cou several slp lector srches gran toarther inside the bor, The rovers nd cen ‘cts inside the hoe are commer eletially to one anoter and to the teeminnst se that ach way of eting the Meals determines peter of faterconnecton ofthe ers ‘Werdo not permit the blac ox to canta ay magnets or athe devices which woul ly the cist tn ere sexptillye Rese of ths me Etction oar ete sey uly tthe singles Kd of sltehing che ‘nbich the stste of the ternal depends oly othe present sate of the 3 shafts, a at un the past istry of che halts, We may thes use dhe Acsra trina eng foton tomo a ule which aie ve each 839 ti sing the If stasis of tne sri, Wee cccrn nit the probe of synthesis given an Neti seiching fet f to fd Evitching cies for which the state of Lae sll and leis comespond inthe wey Faecal ly f Tet yess pu he the mambers of yoo which the AF she can acsume,'Taen theese poe ilicent ales of Ue sats ad the sate 1 Baan, AI, 2. pp e389 88. V armas srs eres co tity Gn his paper “log” stand for “Jogsiton to ase 2”, The rents to follow include an estimate of the minitum nutter uf cntaeln nae for lmost all -teminal switching functions ade uetwork sya ntl] hice oes a suer a ontario the sme oer of magnitude ae the ‘isimutn nueber. The maui of opts eae fr alee Y-teminal 2 og 2" H log 8 "alent are yl hee inthe see hat the fsetion of sitchin fan tiots which can be synthesied using femer contact thas the sven name> led a ewe aor nee Too mumber of cosets usd hy Che BP spore Ps the numer of post syntveinntte nln: 22" tone is he mare of pans on the ssichng fein a when 7 wn ace large. Te wards Ig ont itr a at ns hg reson ota Shune' sapere? a whe is mie svtcking aa, Gh of mm eusing ats nt ing wets te ttn ean So acon ese wea iw emia nee tye stor ces tes Te ths cme ae "iss emt ec si’ Uy vg hashing nts ngewe much tle: fete at ins song rs michel eps, ecards a te etn tee “rt fer veg ny att asa water wl be 9 Meh im pve Ace Mh facia eine il thems) nd al offi ead one Paar I: Two-Teanvat Neracnns Select Sitches A typi selector switch is shown in lg. 1 TH ais of nomber of rotors turned by a haf whieh cen bestia any ane a p potions, In each poiton of the shaft, certain of the totes Gn’ sry shershy elerng ‘thas brenches inthe networks cantaaing the tel contacts. Hoeven, ‘he ony Kc of enitees to be cnsidere re sce chose with The rope 670, aw wran sxc neon rounNay JODY 1851 ‘st if cont stoma ya rotor when te haf iin peton mabe then thie etact remains sucked fret athr tions of the shat "Networks built rom lwerpaton switches are sualyed withthe id of ‘Bosiean Asin Is publ to catalan which aypaprste for eekctor solic ctuits, A deal account of tis algebra a bon ‘sven by HE Ph “The ae o's switch wih p pentane can essed with 2 switching syrable + which eenges over the ols Ly 2,-"~ fe Tha *# = BY means Uh sume ss “the sich ein tsb pein,” ‘he stale 2 ewe-seemina eto, wsg Bf switches with fx, aN Pe pens hindrance Fanetinn fnrss" ta) of fie MT aniteng variables sy, ee ithe ranging Yom 1 to fu. Ab wean f = | remus the reat ope un f = 0 Ineans te cial vege, Then fl, 4a) Sy = yap the Sane to repressing the seticscommetin a to nts: whos unetons ate Pend g hile fn oo) ovo) Fepresess the nebworks jane In paral ‘The cz wh 2 pein es 1: coos of jus «toto whieh tches contact in ts nance function pees exe) > lots 86 ra yp I (859, 2Frncisas swireuey econ on Tore oa simple ieni'y which coresponde to the Bealean Algebra "exurson by coms. An identity nalogeus to the Hooesn Algebra expansion owe hy pdt ie 8 faye en) = feted 190 seve the rage a. it rom It 9. To prove (1) we aged emlyobseeve Sat i the prt sl he terms for which # = bave the ylue I, Te wal Ing seem, for which # = ay, hus the vas fly). Tas eviteing Interpretation of (1) easton ig 2, By repeated ase of (1) fallow hak say faction fla, => yy) ca Be wt a& a caren invelving f a os) . fr — Sf NN Fig 2 span afin, a eat trent, wdition signs, multiplication siane, he ex), anil nothing the. Suck exjeesions may be ropnded a Rooke futons il the 2) “svarlables they may be eerie an factored according tote usu ues OF Benenm Algebr, Havever, oe shoud keep nmin thet toe ee) age Subject to the constrains that a selector switch cu Le i nly ae lion atsny ven tne, The elect of ther conse ety all satel le aie) hele) ed ‘Te inverse Gof she Mole vavible ei) iv the Baan faction 11 tea ade) = 0 tea = + Li ten ta 69pm eran sxezen noms rman, guns 1981 Regard as windnre fuselion a te sitehing wap fe vam nt La wren ay = Tren by 1 the vite has 9s positon, i tes fy — contacts to bulla ciel with htlance Fneton 6 Synthase Suppose thet Af shally goveraed by avitching variables ah. 5 ey ae gen, together we two termina! hirano fant J °° The eynthets pecilem i to desi & network with hindrance function fits yal ulingoutable rotors nd eontats to Fors selector itn from the gv sat ‘One sl ian ci be found immediate!y {G1 As deveibed above, expr the rsatee funtion fr, ++» #3) sen Hekoes function (very) of Ripean waiabes Qo" Ee fwbich sce tee wth new Ibels Here fy Fy bf il Ay ache ell hn shin af eyes re ay ness ca ue wed to design aebwnes nrated By t3e Monon warbler yo Be sd wit hire Feta IMB yh (Ge) In ve mtn din CQ sepa ack ntact by ee aperop ate eg) andro lack soctact& By ehe sorrel). "The slation found in [} wil ordinate se wp a number of contis ‘shih buns kenge by many ores of wayside. Foor Skarnoe's theorems on rsby ete ne know tt he pity i igh chat as many 38 contacts wi be te ine i). Thea ecu i) wil have even maze Contacts if sme eels ey) ave sod. "The syuthene prose which Glows eepleres the egwnent R= Disb, in the eaimate ofthe purer sf catacts bythe smaer number Dt, Tig» The reader te ory Ve proces as ental the same athe fine given by Stannnm fo tw ical ring newer. The network wil eae take the form of tse connected so a sat wih reduces all Fametinns of the sritcbing vavaliss waisa gover i aisegeuar sumer cuacerts on ore expands sta, rut outa 2 one is f= TL tetas batt pated bite tty seal vst tho muitipe prada is sakem aver Ee ene ange fe vaca Doce res Tw ier Sunatars eet) os — ala ce all Screlzed with ove lm cv asso Hg, $s the expan ad ese petermed hors al the siales yoo ye, em hy (2) wd i EE ype cena ‘Aww that Ue dosnt forstion J vm be syeeaed bp connecting ene terminal to th Spat lees ee He ey cakes, “Tbe ruber splat sa wl Tae ber ha Tye uated tie ol sil ee une Te uetausk mlseh we got spose te forme fslanen is Fg consists of the tee of Hs Bhs haste cogs the at Ieee f'n tion Ce igs shh fe designed ao Pat the Firemans ftom ts inp el bs ete ave he fete rn eve her tein ocean 64m wma samme acmoncar orn, roty 151 sd Fin ven vals roy ty, 10° 20 of che switching varie ae ee By nla ch at the Ee: = yale at Cas wtp le Kel se "The hee fm tr ptt fot op ka oi gst at wack is the hindrance of the nelvark of Fig 4 This bindiauce ust fleas crane aac oo aha) tae dat the eounestions te tre die Junctive tre clo no come ay inform ccs song the othe led of ae right fond reser eich ren thal the volrth hast squid Finda ly orpre cvcice of the veambee ff switches tho eee RC will oben 2 mami sig Nteonk bs Pre 4 Pantone ingore ll fg 50 Sgn °> A) ae te ld a which pedusen every Sanction of Goa," yale Let thew vatables be [mE relied 91, °°> 3am Rage ringee yo pe Tek the neat Pisce fe teal ‘Panto TA hoor shih potuces esr fonetion of ©) em Dw w nerf of eet sate @ ves PPO ‘The proof is by inductinn on L, Sappoce that a reser ta poet were function of {945 oa) age brea Bult with yy ycontals and try > Dull one for every fanctisa a «= 3) Tre rele a fete wie, he newark muse pooduce fs 2°", dor there are p= rier way sf scsing ths optches ad to cacao {1 fo the vai the font foreach sine othe stra, OF these fans, Pie a ae Hea Tove 27 °°" funy rw as nail ona sage Be ens independent of). Anyone oft remaining (7 — 2" 7 Fane Nema smarcaise eucuas os on fem he oboe by someting tothe Funetons fn = hy BD, ea, Bd throug ey) = 9) a8 ebowe Sa Hig. 8 Tn ifr wh psc al funeions of tej rere this al utes ¢contarte where a dom fe < PQ aot ve now asuine hst formula (3) ld for Ys we obtain ws PE Ths the Ueore wil follow by induction when we prove (8) forthe ease LL Sines = {no contacisane nota fo ayathsiethe to netons Dad 1) the dreunlty () nedunes, wher = 1,29 the Esceus prove — - rn iv Ex § Sein tepntal Racin on, 7. ‘The inutins pve we have jas Gent wll a uy the erat sauner of contacts whon the lrg ences reused up det ad the sal sritches lst, IE, nthe proces, p> fon, then the nuuber of contarts Ishish would ave been saved by making éwiteh soy precede switch 3, found tobe pare iy slings in one of osrnsing inthe incution pron, che factor Pin [3 cle ro to nenty Pe the arallet of 1 ange, This efcenen i uamacenary So the theney shih falls "gr, 616mm wrt. geste uemstoar oun, jure 1981 Te Tas The numb of eas wa inthe te of Fe 3 monet tana te) Te cam be show cht che rost economia way to build the tee ist pt the stl syitehes at the row endo the ue. 1 te srs ple af ‘estons of any of the sweckes i he ter then the pumber of contae's inthe tre os Ca ® BeBe mon Foper Bows leing court the nunter of cantare whine ate used Sn he wee and fhe actwors whieh produces all anetons is Fi 4, eal remains to Fes hom many of the ven sitchen, ~~" 2 AF to Be pat it ext fo hea te pris Theor I Tes P be th Spt of he mamers ba af aes hic saesertatis than am sa ssine, Thon any eohing Suction of sess ya) toh Be ete! ering we ere a ® Th eove the theorem we cgi to nso annoy as Ps grace or vos than = 2g Gro ts (PSHE 2 og I Te hs cee me we the syuthsi proces decsiba!above, putting al the svitches int the tse and nen inher whi produces al fenton. "The numberof nevacty wae is fee Can 22" and toe fone fll esnuse pH 2a It ue ts (2H 2g 1) “ihe rane ke use the syne gcse deserbed slows, pug into the right fond selork ecllesin Sof sitebessostanon that Eg: cames Uavicectpoutble to 2 vg H without actually exon i. Taon I p= t= 2hog in, ve non Aso Oi ny se nEPet Togtt Since 0 lms thas F> Pf 1 1 adn nator mits the euiton Sil ese LL pwns mang Feed Pim ‘ing (5), the i of natin he ta ee then + ar Theme <(me> teH) Slane = 2 log, wasn tose tun the thane it poved ‘Oly sual fraction of the function wil we thi may ont, In tury partial case the umber of eats dl be aot ae (etau “rope dary ieee sis ete ates he ator Sl fn about hall the ese Past Us W-Penesas, Neaaons Syetass Tat he arene rae be sell 1, 2, <-> 7 Tove poi of \erminal fan terminal nemo there corresponds & hindrance funtion Botany ty) slic tls whether genie eck! th Bren aed J. The By cine a cerene: mien FR p Rah Bot By = imple By =F The iter of consseney ponents ‘ a Raw