Typesetting Karnaugh Maps with L A TEX and TikZ

Transcription

1 Typesetting Krnugh Mps with L A TEX n TikZ Luis Pulo Lus emil: lus@utfpr.eu.r Version:., Version te: 89 Astrt Krnugh mps re use to simplify logi equtions leing to the most ompt expression of two levels for given truth tle. The rwing of them use to e oring, nnoying n errorprone tsk. This set of mros inten to simplify the tsk. They n typeset Krnugh mps with up to twelve vriles, whih is more thn you might likely nee. You only hve to provie list of vrile ientiers plus the truth tle of your logi funtion. The mros lso llow to highlight the simplitions of your logi funtion iretly within Krnugh mp. This pkge is se on kvmros.tex from krnugh pkge referre herein s the originl one. The moitions rrie out intene to use Tik Z inste of ntive LATEX ommns llowing esier ustomistion, esier extension when you nee to rw other elements long with the mp n leing to higher grph qulity. Introution Krnugh mps n Veith hrts re use to simplify logi equtions. They re mp representtions of logi funtions, n in tht they re equivlent. Veith hrts re not supporte y this pkge, ut it shoul not e ig prolem to port Anres W. Wieln's veith mro, ville in krnugh pkge, if you nee it. Plese note tht this pkge, inluing its oumenttion, is se on Anres W. Wieln's previous work n the uthor wishes to register his knowlegment.. Comprison with Other Pkges If you sk yourself why nother Krnugh mp typesetting pkge? the nswer is esy: euse I ws not ompletely hppy with the pkges ville I know n those re:. krnugh: it is gret pkge tht uses ntive LATEX ommns to rw the mp. It supports Krnugh mps n Veith hrts. It employs reursive lgorithm with no size limit whih les to n interesting kin of symmetry. Rememer, Krnugh mps re ll out symmetry. It is not ustomisle, for instne, one nnot hnge the istne etween rs (the mrks showing roun the mp with vrile ientiers on them) n if the vrile ientier is long it will overlp nother r. Also, I wnt to use TikZ to rw olourful semitrnsprent gures on to top of the mp to highlight groups (prime implints) n, lthough it is possile, it is rther iult n the result is not very goo euse they lwys look it o. I hve longtime experiene with this pkge n I hve lso written jv progrm to rw the mps euse, though typesetting simple mps is esy, highlighting the prime implints is not.. krnughmp: it uses TikZ so you got lot of options for ustomistion. It is limite to eight vrile whih, to e honest, shoul e enough for nyone. The prolem is tht it only rws rs (those mrking mentione ove) up to four vriles. Also, the orer in whih the vrile list is inputte is ierent from the orer employe y krnugh. The tul limit my e ierent for you. A twelve vriles mp ontins of 9 ells in gri. They re simply too ig to hnle mnully n you shoul onsier to use softwre. It works until you low the memory out whih will hppen out ten to twelve vriles. Those rs hve een unerppreite long the history. Krnugh (9) himself lle them simplie lels n use them only to reple the Gry oe numers showing roun the mp. Their true strength is the ese wy they point out whih vrile elong to prime implint n whih oes not. An pproh muh esier thn interpreting the Gry oe numers.

2 . skmps: this pkge genertes ongurle Amerin style Krnugh mps for,, n vriles. In Ameri, inste of rs enoting the one vlue of vriles, they use Gry oe inries on the top n left sie of the mp. This ehviour n e mimi with tikzkrnugh (see Setion ), though, in my twenty yers of experiene tehing the sujet, I hve foun out tht rs re muh more intuitive. The skmps ontins four mros, one for eh numer of vriles, n it n e use to highlight the prime implints in the very sme wy tht krnugh oes.. krnughmp: uses TikZ to rw up to four mps of four vriles leing to D six vriles mp. It ontins ommns for rwing implints on top of the mp. Like skmps, this pkge uses Gry oe inste of rs. With tikzkrnugh you n typeset ig (up to twelve vriles or 9 ells) goo looking mps. Using jv softwre, you n o it utomtilly, inluing highlighting the solution.. Introutory exmple Let us strt with n introution on how to use these mros. The rst thing you hve to o is to lo Ti kz. For this type \usepkge{tikz} in the premle of your oument. Then, if the pkge is somewhere TEX n n it, lo the lirry with the ommn \usetikzlirry{krnugh}. If it is not, you n use something like \input tikzlirrykrnugh.oe. You my nee to provie the full or reltive pth to le tikzlirrykrnugh.oe.tex. Suppose now you hve logi funtion f with the following truth tle: Inex f Inex f 8 9 This logi funtion n esily e put into Krnugh mp y using the \krnughmp mro in Tik Z environment (\egin{tikzpiture}) or inline ommn (\tikz). The \krnughmp mro hs ve mntory prmeters:. the numer of vriles in the mp;. n ientier for the funtion;. list of vrile ientiers for the vriles;. the list of vlues of f for eh line in the truth tle; n. possily empty set of TikZ ommns tht will e rown efore the funtion vlues so the vlues will pper on top of them. The vrile ientiers in the thir prmeter re orere from highest to lowest signine (the sme wy s in the truth tle, with hving signine of = 8 n hving signine of = ). The list of vlues of f ws re from lowest to highest inex. The fth prmeter remins empty in this exmple, it will e isusse further on: \tikz[krnugh,enle inies] \krnughmp{}{f(,,,)}{}{ }{}; f(,,,) 8 9

3 The inies in the upper left orner of eh ell orrespon to the inies in the truth tle: The mros tht re the vrile list n the list of logi vlues (i.e., prmeters # n #) work reursively. Eh entry hs to e one hrter long n spes re llowe, otherwise like vrile ientier enlose in $s you hve to put it into urly rkets: \egin{tikzpiture}[krnugh] \krnughmp{}{$f(,,,)$}{{$$}{$$}{$$}{$$}}% f(,,, ) { }{} \en{tikzpiture} Oserve tht the lels re ll in mth moe in this exmple. Also, TikZ environment ws use so there is no semiolon (;) in the en of \krnughmp mro. Moreover, the inies were omitte y removing enle inies from the options list. Krnugh Mp Lirry Tik Z Lirry krnugh \usepgflirry{krnugh} % L A T E X n plin T E X n pure pgf \usepgflirry[krnugh] % ConT E Xt n pure pgf \usetikzlirry{krnugh} % L A T E X n plin T E X when using Tik Z \usetikzlirry[krnugh] % ConT E Xt when using Tik Z This lirry provies TEX mros to typeset Krnugh mps. This lirry enes the following key: /tikz/krnugh (no vlue) This key shoul e psse s n option to piture or sope tht ontins mp, i.e., tht lls \krnughmp mro. It will o some internl setups. \krnughmp{ num vr }{ funtion }{ vr list }{ ontents }{ eortion } This mro retes Knugh mp of num vr vriles for vrile funtion s funtion of the vriles liste in vr list for the vlues given in ontent n pplying the speie eortion. Any ut the rst prmeter n e empty. \krnughmpvert{ num vr }{ funtion }{ vr list }{ ontents }{ eortion } Similr to \krnughmp, ut mp will e trnspose (like in mtrix trnsposition). /tikz/every krnugh (style, initilly empty) The style utomtilly pplie to every Krnugh mp. Cn e gloly set using \tikzset. \kminexounter A TEX ounter for ell inex. See kmell/.style for n exmple of pplition. \kmunitlength={ length } This length sets the size of n iniviul ell in the mp. Must e set efore krnugh is use. /tikz/isle rs= oolen (efult true, initilly flse) The inies n esily e lulte from the vrile vlue in the truth tle, e.g., row : the inex equls = = =. White spes re relly usle to mke the string more rele leing to fst verition.

4 Boolen swith tht isles the typesetting of ll rs n the funtion ientier. Usle when you wnt n Amerin style mp. The initil vlue is flse mening tht the rs will e typeset unless they re expliitly isle. \egin{tikzpiture}[krnugh,isle rs] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} Note tht $f(,,)$ n {$$}{$$}{$$} re not use n oul e empty. /tikz/kmr (style, initilly ) The style use for the top n sie rs relte to the vriles n enoting the rows n olumns for whih the respetive vrile is. The initil vlue is mening they ll will e represente s line with T hppe tips. f(,, ) \egin{tikzpiture}[krnugh,kmr/.style={lue,<>}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmr lel (style, initilly empty) The style use for the vrile ientiers on the rs. f(,, ) \egin{tikzpiture}[krnugh, kmr lel/.style={lue,font=\lrge}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmr sep= with (no efult, initilly.\kmunitlength) The istne etween the r loser to the mp n the mp itself. It epens minly on the line tip use in kmlines/.style. \egin{tikzpiture}[krnugh,kmr sep=\kmunitlength] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} f(,, ) /tikz/kmr top sep= with The istne etween two rs on top of mp. kmr lel/.style. (no efult, initilly \kmunitlength) It epens minly on the font height use in

5 \egin{tikzpiture}[krnugh,kmr top sep=\kmunitlength] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} f(,, ) /tikz/kmr left sep= with (no efult, initilly \kmunitlength) The istne etween two rs t the left sie of mp. It epens minly on the vrile ientier with n the font size use in kmr lel/.style. f(,,, ) \egin{tikzpiture}[krnugh,kmr left sep=\kmunitlength] \krnughmp{}{$f(,,,)$}{{$$}{$$}{$$}{$$}}% { }{} \en{tikzpiture} /tikz/enle inies= oolen (efult true, initilly flse) Boolen swith tht enles the typesetting of ll inies. The initil vlue is flse mening tht the inies will not e typeset unless they re expliitly enle. \egin{tikzpiture}[krnugh,enle inies] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} f(,, ) \en{tikzpiture} /tikz/kminex The style use for ell inex if enle (see lso enle inies). (style, initilly re,font=\tiny) \egin{tikzpiture}[krnugh,enle inies, kminex/.style={lue,font=\sriptsize\itshpe}] f(,, ) \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kminexposx= imension The horizontl istne from the ell left sie to the inex entre. (no efult, initilly.\kmunitlength)

6 f(,, ) \egin{tikzpiture}[krnugh,enle inies, kminexposx=.8\kmunitlength] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kminexposy= imension The vertil istne from the ell ottom to the inex entre. (no efult, initilly.8\kmunitlength) f(,, ) \egin{tikzpiture}[krnugh,enle inies, kminexposy=.\kmunitlength] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kminexpos={ x oorinte }{ y oorinte } Sets \kminexposx n \kminexposy to x n y oorintes mesure in \kmunitlength from the ell ottom left orner. \egin{tikzpiture}[krnugh,enle inies, kminexpos={.8}{.}] f(,, ) \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/inry inex= oolen (efult true, initilly flse) Boolen swith tht sets the inex presenttion to inry oe. It is onvenient to lso set the inex oorintes. In the following exmple, the signine orer is, n, mening, is the most signint it n is the lest signint it. Therefore, the left most it of the inies is one only in the two left olumns elow r, the mile it is one in the ottom row s r extens n the right most it is one in the entrl olumns elow r. \egin{tikzpiture}[krnugh,enle inies, inry inex,kminexpos={.}{.8}] f(,, ) \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmell The style use for ell ontents. (style, initilly empty)

7 f(,, ) \egin{tikzpiture}[krnugh,kmell/.style={lue,font=\lrge}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} Some interesting pplitions of kmell/.style involves the ell inex given y \the\kminexounter. You n nme every ell for future use or ple lel within the ell inex just like enle inies oes. In the following exmple, kmell/.style is use to ple lel within eh ell with the eiml vlue of the ell inex n enle inies is for the inry vlue. Moreover, the ell ontent is lso numer tht orrespon to the ell inex (mnully ple) just to show the orreltion. f(,, ) \egin{tikzpiture}[krnugh,enle inies, inry inex,kminexpos={.}{.8}, kmell/.style={lel={[font=\sriptsize,lue, lel istne=.\kmunitlength] elow left:\the\kminexounter}, green!!lk}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmvr The style use for the vrile nme (funtion) of the mp. (style, initilly empty) f(,, ) \egin{tikzpiture}[krnugh,kmvr/.style={lue,font=\lrge}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmox The style use for the ox surrouning the mp. (style, initilly empty) f(,, ) \egin{tikzpiture}[krnugh,kmox/.style={lue,very thik}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture} /tikz/kmlines The style use for the lines seprting jent rows n olumns insie the mp. (style, initilly empty) f(,, ) \egin{tikzpiture}[krnugh,kmlines/.style={lue,very thik}] \krnughmp{}{$f(,,)$}{{$$}{$$}{$$}}{ }{} \en{tikzpiture}

8 You n options to the grphis y setting the every krnugh style whih is utomtily pplie. Mrking simplitions The lrey mentione fth prmeter n e use if you wnt to rw something insie the Krnugh mp. For exmple, this is useful if you wnt to show how you simplie logi funtion highlighting the prime implints: \egin{tikzpiture}[krnugh,x=\kmunitlength,y=\kmunitlength, thik, f(,,, ) grp/.style n rgs={}{#,fill=#!, minimum with=#\kmunitlength, minimum height=#\kmunitlength, roune orners=.\kmunitlength, fill opity=., retngle,rw}] \krnughmp{}{$f(,,,)$}{{$$}{$$}{$$}{$$}}% { }% { \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \rw[lue] (n.north) to [en left=] (n.north) (n.south) to [en right=] (n.south); \noe[grp={re}{.9}{.9}](n) t (.,.) {}; \noe[grp={re}{.9}{.9}](n) t (.,.) {}; \rw[re] (n.west) to [en right=] (n.west) (n.est) to [en left=] (n.est); } \en{tikzpiture} n the orresponing expression is: f(,,, ) = + where olours were use to relte the suexpression with the prime implint highlighte on the mp. Inste of LATEX's grphis mros you n use TikZ for this purpose. In this exmple, new style grp ws ene in orer to rw semitrnsprent retngles with speie olour, with n height (oth given in \kmunitlength). The Krnugh mp hs its tum t the lower left point extly. The entre point oorintes of those retngles re speie using the t ommn. The length of single ell within the Krnugh mp is equl to \kmunitlength. Thus, the x n y units re set to \kmunitlength so the oorintes n e written without the unit n the retngles will fll in the preise position even if one hnges the mp size y hnging the \kmunitlength. Ajusting the mp size Possily the most importnt feture tht you n hnge is the size of the igrms n it is one y hnging the size of the ells within the mp, simply y typing: As in the originl pkge. 8

9 f \kmunitlength=mm \egin{tikzpiture}[krnugh] \krnughmp{}{$f$}{}{ }{} \en{tikzpiture} The setting of the \kmunitlength remins tive until you hnge it gin; 8 the efult \kmunitlength is 8 mm: Complete exmple In this n in the next setions, exmples of how iniviul vriles n ell ontents n e formt re presente. The syntx relies on squre rkets ([]) to enlose Tik Z fetures tht hnge the pperne n more grphs to n iniviul vrile or ell ontent. Let us see more interesting n olourful exmple: 8 Or, of ourse, until you leve the group in whih you reene the vlue. 9

10 \kmunitlength=.em \egin{tikzpiture}[krnugh,x=\kmunitlength,y=\kmunitlength, } thik, grp/.style n rgs={}{#,fill=#!, minimum with=#\kmunitlength, minimum height=#\kmunitlength, roune orners=.\kmunitlength, fill opity=., retngle,rw}, kmr/.style={lue,<>,oule=white,semithik}, kmr left sep=.\kmunitlength, kmr sep=.\kmunitlength, kmr lel/.style={re!!lk,font=\lrge}, kminex/.style={ornge,font=\tiny}, enle inies, kmell/.style={yn!8!lk}, kmox/.style={rown,thik}, kmlines/.style={rown,thin}, kmvr/.style={green!!lk,font=\huge}, ll/.style={left,lign=right,text with=.\kmunitlength}] \krnughmp{}{$z_{}$}{% {$x_{}$}{$x_{}$}{$x_{}$}{$x_{}$}% {[yellow!!lk,nme=nv,,oule=re,very thik, lel={[font=\tiny,green!!lk]ove:vr.}, text=lue!!re]$x_{}$}% {$x_{}$}}% {{}{}{}{}{}{}{}{}{}{}% {[re,nme=n,lel={[nme=nl,ornge!9!lk, lel istne=\kmunitlength]left:speil}, irle,inner sep=pt,rw=green!!lue]}% {}{}{}{}{}{}{}{}{}}% { \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \noe[grp={lue}{.9}{.9}](n) t (.,.) {}; \rw[lue] (n.est) to [en left=] (n.west) (n.est) to [en left=] (n.west) (n.est) to [en right=] (n.west) (n.south) to [en right=] (n.north) (n.south) to [en left=] (n.north) (n.south) to [en right=] (n.north) (n.south) to [en left=] \noe[grp={re}{.8}{.8}](n) t (.,.) {}; \noe[grp={re}{.8}{.8}](n) t (.,.) {}; \rw[re] (n.north) to [en left=] (n.north); (n.north) (n.south) to [en right=] (n.south); \noe[grp={ornge}{.9}{.9}](n) t (.,.) {}; \noe[grp={ornge}{.9}{.9}](n) t (.,.) {}; \rw[ornge] (n.west) to [en right=] (n.west) (n.est) to [en left=] \noe[grp={tel}{.8}{.8}](n) t (.,.) {}; \noe[grp={tel}{.8}{.8}](n) t (.,.) {}; (n.est); \rw[tel] (n.west) to [en right=] (n.west) \rw[<] (Nv) +(,) (n.est) to [en left=] (n.est); noe[ll]{vrile on its r}; \rw[<] (N) +(,) noe[ll]{speil ell}; \rw[<] (Nl.) +(.,.) \en{tikzpiture} The orresponing Krnugh mp looks like this: noe[ll]{lel for speil ell};

11 x x z x x lel for speil ell vrile on its r x vr. x x Speil vr. x speil ell 9 8 en the logi expression 9 is z = x x + x x + x x x + x x x. You my notie tht the zeros were omitte (reple y {} in the list). Also, the ell is speil euse {[re,nme=n, lel={[nme=nl,ornge!9!lk, lel istne=\kmunitlength ]left: Speil}, irle, inner sep=pt, rw=green!!lue]}. You n put lmost nything insie ell using urly rkets n you n ustomize the ell style using squre rkets. The formt is: {[opt]string} where opt is n optionl set of styles (mong other Tik Z prmeters) whih will e psse s the lst option of TikZ ommn \noe n string will e written insie the ell y tht ommn. To use this syntx, it is impertive tht the very rst hrter fter the opening urly rkets ( {) e the opening squre rkets ([). Mthing pirs of squre rkets re llowe insie the optionl sequene provie tht they re protete insie pir of urly rkets. In this se, the proper ontent of ell is just the numer ner the en, ll the rest is the style pplie to this single, therefore oe etween squre rkets. The style uses TikZ syntx in orer to hnge olour, font size, lel, gure, eortion n nme it for future referene. In this se, two noes re nme N n Nl for future referene. Ner the TikZ environment en, those nmes re use to ple rrows pointing to the noes with esription. The \rw ommn tht rws those rrows nnot e ple insie the fth rgument of mro \krnughmp euse the fth rgument is typeset efore the ells ontents (the fourth rgument), therefore no nme woul e rete t the time the fth rgument is typeset. The vriles ientiers (the thir rgument) n lso e formtte iniviully using style, ut note tht the ustom style will e pplie to oth the r line n the noe for the vrile ientier. If r gets segmente, just like x r, the nme noe will e the top most if the r is vertil or the right most if the r is horizontl. The x r is ierent from the other rs euse [yellow!!lk,nme=nv,,oule=re, very thik, lel={[font=\tiny,green!!lk]ove:vr.}, text=lue!!re] hnges its pperne. The noe nme Nv is lso not ville t the time the fth rgument is typeset. So ny ommn tht mkes use of it will nee to e ple fter the en of mro \krnughmp. The istne etween rs on the left sie ws set to.\kmunitlength to prevent overlpping etween x (the lel) n x r n x n x r, ut the istne etween the rs on top ws left unhnge. The istne etween the mp n the rs losest to it ws set to.\kmunitlength to prevent overlpping etween the r tip ( ) n the mp itself. 9 This is not of ny importne here, ut I ouln't hol myself k. By the wy, if you re urious, there re nother two miniml solutions.

12 The inies n e ompute y x + 8 x + x + x + x + x whih is it izrre. The truth tle vlues ought to e rrnge oring to this inex orer. This izrreness is the prie we py to hve the vriles ple in positions whih re more intuitive. See Setion for jv softwre tht n help on this mtter. Amerin style If you relly wnt n Amerin style mp n you re not fri of mitting it pulily, you n still use this pkge to typeset it. The rst thing to o is to isle the rs n the funtion ientiers. Therefore, this option hs to e inlue in the Tik Z environment: isle rs. Then you will nee rows n olumns lels in Gry oe n ption for the mp n vriles ientiers. In the lst exmple, these n e hieve y ppening the following oe in the fth rgument of the krnughmp mro: \rw[kmox] (,8) noe[elow left,mgent]{$x_,x_,x_$} noe[ove right,violet]{$x_,x_,x_$} +(,) noe[ove left,green!!lk] {$z_$}; \foreh \x/\l in % } {/,/,/,/,/,/,/,/} { \noe[violet] t (\x+.,8.) {\l}; \noe[mgent] t (.,.\x) {\l}; The result shoul e: z x, x, x x, x, x Note, however, tht the inex insie ell oes not mth the Gry oe vlue of the respetive row n olumn. The inies n still e ompute y 8 Vertil moe x + 8 x + x + x + x + x. For n o numer of vriles, the Krnugh mp is retngulr n mro krnughmp will typeset it twie s wie s it is high (not tking into ount the rs). Like this single vrile mp: Do not use Gry oe tle to ompute the inex.

13 f() \tikz[krnugh,enle inies]% \krnughmp{}{$f()$}{{$$}}{}{}; This lyout is goo for presenttions euse the projetion re is usully wier thn higher. Pper sheets, on the other hn, re usully higher thn wier, so for ig mp you my nee something like : f() \tikz[krnugh,enle inies]% \krnughmpvert{}{$f()$}{{$$}}{}{}; This is lle, for lk of etter nme, vertil moe n it is one y the krnughmpvert mro. Note tht krnughmpvert mro rrnges the vriles in ierent orer. Compre the two squre (four vriles) mps elow in the norml (on the left) n vertil moe (on the right) pying ttention to the inies n vriles ientier. f(,,, ) f(,,, ) Norml (horizontl) moe Vertil moe The inies re lulte in the sme wy, ut their position insie the mp re ierent euse the vriles positions re ierent. It is like one mp is mirrore n then rotte 9 o (mirrore horizontlly n rotte lokwise or mirrore vertilly n rotte ounterlokwise.) Extly like mtrix trnsposition. One interesting pplition of vertil moe is when you wnt to keep onsisteny in vrile ientier position mong mps with o n even numer of vriles. For exmple, if you wnt the most signint vrile ppering on top of the mps you n use norml (horizontl) moe for mps of o numer of vriles n vertil moe for even mounts, like this: g(,,, ) 8 f(,, ) 9 Norml (horizontl) moe Vertil moe A more generl pproh is to use the jv softwre esrie in Setion to rete mps with ritrry vriles positioning. Suppose tht you esire the most signint vrile to pper t the left sie of three vriles mp. You n o the opposite of wht ws one in the lst exmple, ut you will en up with vertil mp of three vriles n mye it is not wht you wnt. Using the softwre esrie in Or you n use lnspe. Not to e onfuse with TEX vertil moe.

14 Setion llows to e ple t the left in norml (horizontl) moe mp. This will hnge the inies euse it reorers the truth tlet suh tht will no longer e the most signint vrile, ut without hnging the logi funtion. 9 If you use the originl version of the mros you ertinly hve notie numer of hnges. The most importnt one is tht now you ontrol the pperne of ell, inex, et. y hnging the style n not through mros. Also, you nee Ti kz piture environment or inline ommn. Finl remrks This is not even nerly ll you nee to know out the usge of these mros, ut it is goo strt. In se you n ug, or if you hve omments or suggestions, plese sen me n emil. The mximum size mp I oul proue ws Krnugh mp with vriles; with igger mps I only exeee TEX's min memory. This is ue to the mros' reursive lgorithm. Quite likely you will exee TEX's pity with even smller mps if they our in lrge ouments. If you nee help to typeset Krnugh mps with or without the prime implints highlighte, you n try JQM Jv Quine MCluskey for minimiztion of Boolen funtions ville on net/projets/jqmjvquinemluskey/. It n generte the solution n rete the orresponing mp se on given truth tle. One very useful feture of this softwre is tht you n reorer the vriles on the mp to suite your prtiulr pplition inste of rely exlusively on the mro to stter your vriles roun.