Exact Template Matching Using Boolean Satisfiability

Transcription

1 2013 IEEE 43r Intrntionl Symposium on Multipl-Vlu Logi Ext Tmplt Mthing Using Booln Stisfiility Nil Assi Mthis Sokn Rort Will Rolf Drhslr Institut of Computr Sin, Univrsity of Brmn, Brmn, Grmny Cyr-Physil Systms, DFKI GmH, Brmn, Grmny Astrt Rvrsil logi is n mrging rsrh r tht hs shown promising rsults in pplitions suh s quntum omputing, low powr sign, n optil omputing. Sin th synthsis of miniml iruits is umrsom tsk, mny synthsis lgorithms pply huristis n n thrfor not provi miniml solution. As onsqun, post synthsis mthos suh s winow optimiztion n tmplt mthing r ing ppli. Tmplt mthing lgorithms xplor th iruits for gt ss tht n rpl y smllr ons using spil lss of intity iruits, so ll tmplts. Th trmintion of ss pplil for sustitution is th ottlnk of th tmplt mthing lgorithm n prolm-solving mthos hv n propos in th rnt pst. Sin ths lgorithms r s on huristis, it nnot nsur tht mthing s n lwys foun. In this ppr, w propos nw pproh tht trmins mthing ss s on Booln stisfiility n thrfor nsurs tht ths ss r lwys foun if thy xist. Exprimntl rsults monstrt tht tmplt mthing yils smllr iruits whn pplying th nw mtho for s trmintion. I. INTRODUCTION Rvrsil logi is n mrging thnology in th omin of quntum omputing [1], [2], [3], low powr sign [4], [5], optil omuting [6], s wll s nnothnologis [7]. In rvrsil iruits, ll omputtions r prform in n invrt mnnr whih provis nw promising nhnmnts for omputtion thnologis. Motivt y th nfits rought y ths iruits, rsrhrs strt to vlop nw sign mthos. Svrl synthsis pprohs hv n propos to trmin iruit rliztion for givn funtion. Signifint improvmnts n hivmnts [8], [9], [10], [11] hv n m sin th truth tl s pproh hs tking pl. Howvr, th mjority of th synthsis pprohs o not gurnt optiml rliztions, in ft, th lgorithms tht o gurnt n optiml solution (.g. [12]) r only pplil to smll iruits of out 4 to 6 lins. As rsult, svrl post synthsis optimiztion pprohs [9], [13] hv n propos to minimiz givn iruit ftr it hs n synthsiz. Th tmplt mthing lgorithm xplin in [9] is on of ths optimiztion thniqus. Givn st of tmplts whih is spil lss of intity iruits, th lgorithm tris to trmin su-iruits tht mth prt of tmplt. In this s, th trmin su-iruit n rpl with th invrt rmining prt of th tmplt u to rvrsiility. If th rmining prt is smllr, th ovrll iruit siz n ru. Th ffiiny of th tmplt mthing lgorithm highly pns on th strtgy us for mthing th tmplt in iruit. Sin th onsir srh sp tht shoul inspt in orr to fin mth for tmplt is usully vry lrg, mny huristi pprohs hv n invstigt tht work ffiintly ut nnot gurnt tht mthing suiruit n foun if it xists. To ovrom this limittion, w propos nw pproh tht xploits Booln stisfiility thniqus llowing n xhustiv ut yt ffiint trmintion of ss oring to givn st of tmplts. For this purpos, th srh for s is formult s ision prolm n no s Booln formul tht is ftrwrs solv using n SMT solvr. If th instn is stisfil, th mthing s n rpl y th son hlf of th tmplt. Othrwis, it n onlu tht th tmplt nnot us to furthr ru of th iruit ost. W hv implmnt th lgorithm n ompr it to th srh mtho prsnt in [9]. Exprimntl rsults show tht ost rutions of up to out 60% ompr with rspt to th initil iruit n up to 28% with rspt to th mtho in [9] n hiv. Th rminr of th ppr is orgniz s follows. Stion II introus th sis of rvrsil logi, th tmplt mthing lgorithm, n SMT. In Stion III th gnrl strutur of our pproh is xplin, whil tils on th SMT noing r provi in Stion IV. Exprimntl rsults r givn in Stion V n th ppr is onlu in Stion VI. II. BACKGROUND A. Rvrsil Ciruits A Booln funtion is rvrsil if it mps h input ssignmnt to istint output ssignmnt. As rsult, suh funtion must hv th sm numr of input n output vrils X := {x 1,...,x n }. A iruit rlizing rvrsil funtion is s of rvrsil gts. In th litrtur, svrl typs of rvrsil gts hv n introu. Bsis th Frkin gt [14] n th Prs gt [15], multipl ontroll X/13 $ IEEE DOI /ISMVL

2 () Rvrsil gt Fig. 1. Rvrsil iruitry () Rvrsil iruit Toffoli gts [16] r wily us. In this ppr w onsir only Toffoli gt iruits. A Toffoli gt hs th form T(C, t) with st of ontrol lins C = {x i1,,x ik } X n trgt lin t X \ C. C my mpty. Th trgt lin t is invrt if n only if ll ontrol lins r ssign 1, i.. gt mps n input ssignmnt (x 1,...,t = x j,...,x n ) to (x 1,...,x i1 x i2 x ik x j,...,x n ). Quntum ost r oftn us to msur th ost of rvrsil gt. Evry rvrsil gt n trnsform into squn of lmntry quntum gts [17] whr h lmntry gt hs quntum ost of on. Exmpl 1: Fig. 1() shows Toffoli gt with two ontrol lins. Th ontrol lins r not y, whil th trgt lin is not y. Th nnott vlus monstrt th omputtion of th gt for givn input ssignmnt. Fig.1() shows iffrnt Toffoli gts in s forming rvrsil iruit. B. Tmplt Mthing Prour Tmplts hv initilly n propos in [9] n thir finition hs n just svrl tims in th pst. Th urrnt wily pt finition stts tht tmplt is n intity iruit with m gts suh tht h su-iruit of siz lss thn m 2 nnot ru y nothr tmplt. Furthrmor, tmplt onsists of two iffrnt lin typs whih w ll C-lins n T-lins n whih shoul not mistkn for ontrol lins n trgt lins of gt (f. Fig. 2). A C-lin of tmplt is lin in whih ll gts only hv ontrol lins ut no trgt lins, ll othr lins r T-lins. This sprtion is of us sin ny C-lin n uplit, rmov, or rpl without hnging th funtionlity of th iruit. This is illustrt y mns of Figs. 3(), 3(), n 3(), rsptivly. Furthr, th orr of th gts n rott ing wrpp t th iruit ounris, gin without hnging th funtionlity. This is illustrt in Fig. 3(). () Duplition () Rmovl Fig. 3. () Rplmnt () Rottion Tmplt isposition Sin tmplt is n intity iruit it n split nywhr in th mil n th lft prt quls th invrs of th right prt n vi vrs. Th tmplt mthing lgorithm is ppli in orr to ru th iruit siz or its osts. It tks iruit n tris to fin su-iruits in it tht mth prt of tmplt. If tht prt is longr or mor xpnsiv thn th rmining prt th su-iruit is rpl y th invrs of it. Th mthing prour tris to fin th first gt n thn susquntly looks for th othr gts whih o not n to nssrily jnt. Using th moving rul, gts n mov in th iruit without hnging th iruit s funtionlity [9]. Exmpl 2: Fig. 4 illustrts th si tmplt mthing lgorithm. It shows tmplt in th lowr lft ornr with lft n right prt. A su-iruit mthing th lft prt of th tmplt is foun in th originl iruit, hn, this suiruit n rpl with th invrs of th right prt of th tmplt yiling smllr simplifi iruit of lss ost. Th gr of optimiztion in tmplt mthing pns on th onsir tmplts tht r us in th pross n th srh mtho for fining mthing su-iruits. Th lttr tsk is th most omplx prt in th tmplt mthing lgorithm n not fsil whn ing ppli in n xhustiv mnnr. In orr to trmin mthing tmplts, th iruit is rrrng y pplying th tim onsuming moving rul. As onsqun, huristis r us to fin su-squns Originl Ciruit Simplifition Simplifi iruit F Mpping C-lins T-lins F F 1 Rpling F with F 1 Fig. 2. Originl tmplt Fig. 4. Tmplt mthing

3 Ciruit k 5 Tmplt Fig. 5. Enoing k k 1 Itrtiv hk Unstisfil Solvr Stisfil Rplmnt Simplifi iruit Enoing th srh mtho s n SMT instn tht o not gurnt tht su-iruit is foun if it xists. For xmpl, som of thm tk iruit n try to mth its gts to th tmplt from th input to th output si of tht iruit, othr ons pply th srh itionlly in th rvrs irtion in orr to inrs th hns of mthing th tmplt in th iruit. C. Stisfiility Moulo Thoris Booln stisfiility (SAT) is ision prolm tht trmins if th vrils of givn propositionl formul n ssign in suh wy tht th formul vluts to tru. Eqully importnt is to trmin whthr no suh ssignmnts xist, whih woul imply tht th formul is th ontrition. In th lttr s, th funtion is si to unstisfil, othrwis it is stisfil. Most morn SAT solvrs rquir th formul to in onjuntiv norml form, i.. onjuntion of luss whih in turn r isjuntions of litrls. Stisfiility Moulo Thoris (SMT) is lso ision prolm ut with mor omplx thoris rthr thn only propositionl logi. A til introution is givn in [18]. SMT is is out hking th stisfiility of first-orr formuls ontining oprtions from vrious thoris suh s th Boolns, it-vtors, rithmti, rrys, n rursiv t-typs. SMT solvrs r vill tht hnl omplx formuls suh s Z3 [19], MthSAT [20], Booltor [21], n SWORD [22]. III. GENERAL IDEA In this work, w r proposing nw srh mtho tht trmins mthing su-iruits in iruit givn tmplt whil kping th gnrl onpts for tmplt mthing s thy hv n xplin in [9]. Inst of pplying huristis for th srh, w suggst n xt n ffiint tmplt mthing lgorithm s on SMT whih llows to xhustivly xplor th full srh sp n thrfor gurnts tht mthing su-iruit is foun if it xists. Fig. 5 outlins th propos pproh. Givn tmplt with m gts n th originl iruit, th propos lgorithm rts Booln formul noing th ision prolm () Ciruit () Tmplt Fig. 6. Rvrsil iruit n tmplt whthr thr xists su-iruit in th originl iruit tht mths th first k gts of th givn tmplt. Whil initilly stting k m, k is rmnt y 1 s long s th Booln formul is unstisfil. If th formul is stisfil th mthing su-iruit n xtrt from th stisfying ssignmnt tht is rturn from th SMT solvr. Th tmplt lngth k is only rs s long s th lft prt of th tmplt is lrgr in ost thn th right on. IV. ENCODING USING SMT This stion sris th propos pproh in til. Th noing s ision prolm is xplin y listing ll onstrints tht r nssry for srhing orrt mth of givn tmplt in iruit. A. Dision Prolm Th ovrll ision prolm n stt s follows. Lt G = g 1 g iruit of n lins n T = g 1 g tmplt of n lins with g j =(C j,t j ) n g i =(C i,t i ) for j = 1,..., n i = 1,...,, rsptivly. Th onsir lins r fin ovr th vrils {x 1,...,x n } n {x 1,...,x n }, rsptivly. Cn th first k gts of th tmplt T mth in G, i.. n positions m 1,...,m k in G foun suh tht th gts n mov togthr n rsml th first k gts of T. Noti tht lso th orr of lins in th tmplt os not nssrily n to mth th originl orr of lins in th iruit. As rsult, sis th mthing positions m 1,...,m k lso lin rorring l 1,...,l n is prt of th solution, if th tmplt n mth. Exmpl 3: Figs. 6() n 6() show iruit n tmplt, rsptivly. Th first k =4gts of th tmplt n mth to th gts in th iruit t positions m 1 =3, m 2 =4, m 3 =5, n m 4 =6whn lin mpping l 1 = 1, l 2 = 3, l 3 = 2, n l 4 = 4, i.. x 1 x 1, x 2 x 3, x 3 x 2, n x 4 x 4. B. Gt Positions n Lin Mpping Th vrils m 1,...,m k n l 1,...,l n rprsnt gt n lin inxs in th intrvls [1,] n [1,n ], rsptivly. A on-hot noing is us for ll ths vrils, i.. n m i IB for i =1,...,k lj IB n for j =1,...,n

4 1 = t 1 = = t 2 = = t 3 = = t 4 = Fig. 7. Ciruit noing 0 = = = = t 1 = = t 2 = = t 3 = = t 4 = Fig. 8. Tmplt noing suh tht ν m 1 = = ν m k = ν l j = = ν l n =1, whr th siwys sum ν nots th numr of ll 1 its in itvtor. Th singl it tht is st in th vtors nots th rsptiv inx. Two onstrints r suffiint to nfor oth th orring of th positions m i n gurnt th on-hot noing, i.. n 0 < m i < m i+1 for i =1,...,k 1 ( m i m j =0) νm = k i j with M = k i=1 m i ing th it-msk ontining ll position inxs. Th lin mpping vrils o not hv to follow n orr, howvr ll it-vtors n to on-hot no n thy must ll iffr, i.. ( l i l j =0) lj 0 lj =1...1 }{{}. i j j=1 j=1 n tims C. Ciruits n Tmplts In orr to rprsnt iruits, w r mking us of two iffrnt noings whih r oth us ltr for onstrining th mpping. Thy oth shr th proprty tht ontrol lins n trgts r sprtly rprsnt s it-msk, howvr on in vrtil n on in horizontl orinttion. For th iruit s lins, w r following th horizontl shm, i.. for h ontrol lin n for h trgt lin w it-vtors n n n i IB with i [j] x i j t i IB with t i [j] x i = t j for i =1,...,n n j =1,...,. Not tht th it-vtor inis strt from 1. Exmpl 4: Th no it-vtors for th iruit lins of th initil iruit from Fig. 6() r illustrt in Fig. 7. For th tmplts w r using slightly iffrnt noing tht pturs th iffrnt moifition possiilitis tht xists for tmplts n hv n illustrt in Fig. 3. Furthr, for th noing w ssum tht th tmplt n xtn to t most n lins suh tht it fits th iruit. Also, lt us ssum tht th tmplt hs τ T-lins. Thn th first n τ iruit lins for th tmplt r fin s n i IB k with i = t i IB k n τ j=0 j (i n τ) with t i =0...0 (i n τ) with { j IB k if j =0, with j [l] x j l othrwis. Tht is, thr r no trgts on ths lins n ontrol lin n ithr on of ll possil C-lins (no s >0 )or lso (no s 0 ) initing tht th lin is not us y th tmplt. Th lins for i>n τ r no in th xt sm mnnr s for th originl iruit lins. Exmpl 5: Th no it-vtors for th iruit lins of th tmplt from Fig. 6() r illustrt in Fig. 8. Noti tht th orr of th C-lins s it is givn in Fig. 6() os not mttr nymor n is npsult in th j vrils. From th it-vtors i n t i w r riving nw itmsks č j n ť j tht sri th iruit in vrtil orinttion, i.. h it-vtor orrspons to gt inst of lin. Ths r it-vtors n č j IB n with č j [i] = i [j] ť j IB n with ť j [i] = t i [j]. D. Mpping Givn ll it-vtor noings from ov, w n fin th onstrints tht mp tmplt gts to iruit gts with rspt to lin mpping. For this purpos, w mk us of th n IB n IB ) 0 whr, IB n. W r using th funtion only in ss whr is on-hot no, hn, th funtion vluts to tru if n only if th on it tht is st in is lso st in. Givn tht, w n formliz th most importnt mpping for th noing whih mps tmplt gts to iruit gts, xprss s č l i = m j n ť li = m j with i =1,...,n n j =1,...,k. Tking th ontrol lins, tht is th formul on th lft hn si, it mns th following. Assuming thr is ontrol lin in th j th gt of th tmplt t th lin whr lin i mps to. Thn, thr must lso ontrol lin in th j th gt hosn y th mpping m j in th originl iruit t lin i n vi vrs. Th sm pplis for trgt lins. Noti tht th ft tht w hv vrtil n horizontl noings for tmplt n iruit gts plys ky rol in this noing

5 Fig. 9. E. Moving Rul 5 Moving rul grph for iruit in Fig. 6() Th noings givn so fr r not suffiint yt, sin t th urrnt stt ritrry gts in th originl iruits n mth lthough it might not possil to mov thm togthr. As onsqun, th moving rul ns to no into th SMT instn s wll. In orr to hv forml rprsnttion of th moving rul, w r mking us of th moving rul grph tht hs n introu in [23]. In moving rul grph h vrtx rprsnts gt of th iruit n two gts r onnt y n g if n only if th gts nnot intrhng y moving. Exmpl 6: Th moving rul grph for th iruit in Fig. 6() is pit in Fig. 9. W writ g i < g j if g j nnot mov for g i n gt g i nnot mov pst g j. Sin th moving rul is trnsitiv, g i <g j if n only if thr xists pth twn th vrtx rprsnting g i n th vrtx rprsnting g j in th moving rul grph. This ls to th finl onstrint whih nsurs th moving pilitis. For ll 0 <i<j<k suh tht g i < < g j <g k n thr is no j suh tht g i <g j <g k, th onstrint M[i] M[k] M[j]. Ths onstrints n sily gnrt with th hlp of th moving rul grph. Exmpl 7: Givn th moving rul grph in Fig. 9, th xtrt onstrints r s follows: M[1] M[7] M[2] M[1] M[7] M[5] M[1] M[7] M[6] M[1] M[6] M[4] M[1] M[4] M[3] M[3] M[7] M[6] M[3] M[6] M[4] M[4] M[7] M[6] V. EXPERIMENTAL RESULTS Th propos pproh hs n implmnt in C++. In orr to r th rvrsil iruits s wll s th tmplts, w us th opn sour toolkit for rvrsil iruit sign RvKit [24]. Th SMT prolm, i.. th tmplt mthing prolm is no using th mtsmt [25] frmwork whih provis th us of SAT n SMT solvrs irtly ovr its API through th lngug. Diffrnt solvrs n us within th mtsmt frmwork, for our xprimnts th Booltor [21] SMT solvr turns out to th most ffiint on. Th pproh hs n vlut on Dul-Cor AMD Prossor with 4 GB of min mmory. W us iruits provi in RvLi [26] s nhmrks. W hv ompr our pproh to th on propos in [9] n us th sm st of tmplts tht is prsnt in th work. Tl I summrizs th otin rsults for th onut xprimnts. Th first thr olumns giv th nm of th initil not optimiz iruits s wll s its numr of gts n quntum ost (QC). In th following olumns, th otin rsults for th huristi tmplt mthing pproh n th propos pproh r prsnt. For h th rsulting numr of gts, th quntum ost, th run-tim, n th improvmnt ompr to th initil iruit is givn (Impr/IC). For th propos pproh itionlly th improvmnt ompr to th huristi pproh is list (Impr/HTM). Consiring quntum ost, for most of th iruits signifint ost rution n sn whn pplying th nw pproh. Applying th huristi pproh rus th quntum osts y roun 29.14% whn onsiring ll iruits. It is lrly osrv tht ths rsults n improv whn pplying th pproh s on Booln stisfiility. Th propos pproh ls to n itionl ost rutions of 11.42% on vrg n 27.46% in th st s. Th rsults lrly onfirm th impt of th xhustiv srh to th iruit osts. Th fft is in prtiulr osrvl for lrg iruits. Huristi tmplt mthing is lry tim onsuming pross. Howvr, s it is shown, th nw pproh ns n normous omputtion tim ompr to th huristi mtho. This is xpt u to th ft tht th mth is trmin xhustivly y th SAT solvr, whih ns highr run-tim to provi th orrsponing nswr. VI. CONCLUSION In this ppr, nw srh mtho for pplying tmplts in th tmplt mthing lgorithm hs n propos. Inst of using huristis for mthing gts, th prolm is no into n instn of Booln stisfiility n thrfor nsurs n xhustiv xmintion of th srh sp. Th propos pproh ls to improvmnts in trms of iruit osts ompr to th huristi tmplt mthing pproh. Exprimntl rsults lrly onfirm ths improvmnts. REFERENCES [1] M. Nilsn n I. Chung, Quntum Computtion n Quntum Informtion. Cmrig Univ. Prss, [2] P. W. Shor, Algorithms for quntum omputtion: isrt logrithms n ftoring, Fountions of Computr Sin, pp , [3] L. M. K. Vnrsypn, M. Stffn, G. Bryt, C. S. Ynnoni, M. H. Shrwoo, n I. L. Chung, Exprimntl rliztion of Shor s quntum ftoring lgorithm using nulr mgnti rsonn, Ntur, vol. 414, p. 883, [4] R. Lnur, Irrvrsiility n ht gnrtion in th omputing pross, IBM J. Rs. Dv., vol. 5, p. 183, [5] C. H. Bnntt, Logil rvrsiility of omputtion, IBM J. Rs. Dv, vol. 17, no. 6, pp , [6] R. Cuyknll n D. R. Anrsn, Rvrsil optil omputing iruits, Optis Lttrs, vol. 12, no. 7, pp , [7] R. C. Mrkl, Rvrsil ltroni logi using swiths, Nnothnology, vol. 4, pp ,

6 TABLE I EXPERIMENTAL RESULTS Huristi Tmplt Mthing (HTM, [9]) SMT Bs Tmplt Mthing Bnh Gts QC Gts QC Tim Impr/IC Gts QC Tim Impr/IC Impr/HTM gt mo5mils gt j hw x hw mini-lu wr hw C sym hw sym hw C C hw m mx m m m hw m m m [8] K. Fzl, M. A. Thornton, n J. E. Ri, ESOP-s Toffoli gt s gnrtion, in IEEE Pifi Rim Confrn on Communitions, Computrs n Signl Prossing, 2007, pp [9] D. M. Millr, D. Mslov, n G. W. Duk, A trnsformtion s lgorithm for rvrsil logi synthsis, in Dsign Automtion Conf., 2003, pp [10] R. Will n R. Drhslr, BDD-s synthsis of rvrsil logi for lrg funtions, in Dsign Automtion Conf., 2009, pp [11] M. Sokn, R. Will, C. Ottrstt, n R. Drhslr, A synthsis flow for squntil rvrsil iruits, in Int l Symp. on Multi-Vlu Logi, 2012, pp [12] R. Will n D. Groß, Fst xt Toffoli ntwork synthsis of rvrsil logi, in Int l Conf. on CAD, 2007, pp [13] M. Sokn, R. Will, G. W. Duk, n R. Drhslr, Winow optimiztion of rvrsil n quntum iruits, in IEEE Symp. on Dsign n Dignostis of Eltroni Ciruits n Systms, 2010, pp [14] E. F. Frkin n T. Toffoli, Consrvtiv logi, Intrntionl Journl of Thortil Physis, vol. 21, no. 3/4, pp , [15] A. Prs, Rvrsil logi n quntum omputrs, Phys. Rv. A, no. 32, pp , [16] T. Toffoli, Rvrsil omputing, in Automt, Lngugs n Progrmming, W. Bkkr n J. vn Luwn, Es. Springr, 1980, p. 632, thnil Mmo MIT/LCS/TM-151, MIT L. for Comput. Si. [17] A. Brno, C. H. Bnntt, R. Clv, D. DiVinhnzo, N. Mrgolus, P. Shor, T. Sltor, J. Smolin, n H. Winfurtr, Elmntry gts for quntum omputtion, Th Amrin Physil Soity, vol. 52, pp , [18] C. W. Brrtt, R. Sstini, S. A. Sshi, n C. Tinlli, Stisfiility moulo thoris, in Hnook of Stisfiility, 2009, pp [19] L. M. Mour n N. Bjørnr, Z3: An ffiint SMT solvr, in Tools n Algorithms for th Constrution n Anlysis of Systms, 2008, pp [20] R. Bruttomsso, A. Cimtti, A. Frnzén, A. Griggio, n R. Sstini, Th MthSAT 4 SMT solvr, in Computr Ai Vrifition, 2008, pp [21] R. Brummyr n A. Bir, Booltor: An ffiint SMT solvr for it-vtors n rrys, in Tools n Algorithms for th Constrution n Anlysis of Systms, 2009, pp [22] R. Will, G. Fy, D. Groß, S. Eggrsglüß, n R. Drhslr, SWORD: A SAT lik provr using wor lvl informtion, in VLSI of Systmon-Chip, 2007, pp [23] N. Sott, G. Duk, n D. Mslov, Improving tmplt mthing for minimizing rvrsil toffoli ss, in Int l R-Mullr Workshop, [24] M. Sokn, S. Frhs, R. Will, n R. Drhslr, RvKit: An Opn Sour Toolkit for th Dsign of Rvrsil Ciruits, in Rvrsil Computtion 2011, sr. Ltur Nots in Computr Sin, vol. 7165, 2012, pp , RvKit is vill t [25] F. Hik, S. Frhs, G. Fy, D. Groß, n R. Drhslr, mtsmt: Fous on Your Applition not on Solvr Intgrtion, in Int l Workshop on Dsign n Implmnttion of Forml Tools n Systms, [26] R. Will, D. Groß, L. Tur, G. W. Duk, n R. Drhslr, RvLi: n onlin rsour for rvrsil funtions n rvrsil iruits, in Int l Symp. on Multi-Vlu Logi, 2008, pp , RvLi is vill t