Fujitsu Laboratories of America. 77 Rio Robles, San Jose CA happens when one attempts to compare the functionality. inputs is neglected.
|
|
- Derick Taylor
- 6 years ago
- Views:
Transcription
1 VERIFUL : VERItion using FUntionl Lerning Rjrshi Mukherjee y Dept. of Eletril nd Computer Engineering University of Texs t Austin Austin TX 7872 Astrt It is well known tht lerning (i.e., indiret implitions) sed tehniques perform very well in mny instnes of omintionl iruit verition when the two iruits eing veried hve mny orresponding internl equivlent points. We present some results on omintionl iruit design verition using powerful, nd highly generl lerning tehnique lled funtionl lerning. Funtionl lerning is sed on OBDDs nd hene n eiently lern novel implitions sed on funtionl mnipultion. Introdution Anlysis of logi design, onstituting of prolems suh s, representtion, verition, ATP, et., poses one of the most fundmentl hllenges in the eld of omputer-ided design. For exmple, logi verition of two dierent reliztions of the sme Boolen funtion during the proess of iruit synthesis is of utmost importne to gurntee orretness of the iruit eing implemented. As disussed in [6], digitl iruit synthesis typilly onsists of sequene of tomi opertions through whih the iruit is ltered lolly to suit spei needs, keeping the funtionlity sme. ene, it n e rgued tht fter eh suh tomi hnge the two versions of the iruit efore nd fter the hnge remin very similr. This ft immeditely lures one to try nd extrt these internl equivlenes nd use them eetively in order to simplify the prolem of logi verition. Bermn et. l. [5] proposed the rst method of using these internl equivlent points to estlish the equivlene of two iruits. In [5] deomposition is found using the min/ut lgorithm tht filittes deomposing the prolem of verition of the whole iruit into muh smller nd simpler prolems. Cerny nd Murs presented in [8] further oservtions to estlish ross-reltions etween two pproprite uts in the two iruits. Lerning tehniques [8, 2] n often e eiently used to extrt the internl equivlent points whih re used susequently to speed up the proess of design veri- tion. owever, s shown in [5], diret use of these equivlent points to prove the outputs of two iruits equivlent n use the prolem of flse negtives. This The rst uthor ws supported in prt during the erly prt of this work y the ONR under grnt N4-92-J-366 nd the Ntionl Siene Foundtion under grnt MIP y Currently lso with Fujitsu Lortories of Ameri, 77 Rio Roles, Sn Jose CA Jwhr Jin Mshiro Fujit Fujitsu Lortories of Ameri 77 Rio Roles, Sn Jose CA hppens when one ttempts to ompre the funtionlity of the two iruits using these equivlent points s pseudo primry inputs. It is esy to see tht this kind of omprison n erroneously prove the two iruits to e funtionlly inequivlent even though they my tully e equivlent. This is euse the interdependene of the pseudo primry inputs in terms of the true primry inputs is negleted. A tehnique for extrting nd utilizing internl equivlent points for logi verition without hving to fe the prolem of flse negtives ws presented in [4]. Another interesting tehnique to rry out design verition using internl equivlenes nd oservility don't res hs een presented in [7]. owever, tehniques suh s [7] my not disover even ll neessry ssignments. Tehniques of [4] n sle poorly with inresing iruit sizes s ws found in some industril iruits [3]. Most suh tehniques nd it very diult to detet reltionships etween points not in struturl proximity. Importntly, the types of reltionships tht these tehniques n disover re limited. For exmple, these methods n lern Constnt-Vlue Reltionships: if onstnt Boolen vlue v 2f;gt given gte implies nother onstnt Boolen vlue t nother gte. owever, they nnot detet more involved reltionships etween set of funtions with nother set of funtions. For exmple, gte f = my simply imply tht disjuntion tken over some given set of funtions must e. Or, under f =, set of gtes must ssume identil vlue. Clerly, these tehniques nnot esily lern onditions implied y more omplex Boolen reltions mong two or more gtes in the iruit. We suggest one possile solution to the omintionl verition prolem through lerning tehnique sed on OBDDs. We will show tht this tehnique n work s eiently or even etter thn typil lerning tehniques. The pility of OBDDs to model vrious funtions nd lso symolilly mnipulte them is well epted. ene it is not surprising tht OBDDs n e lso esily used to derive more involved internl reltionships implitions s well. Our results show tht the sizes of the OBDDs tht re required to e uilt re extremely smll. ene, there is no memory explosion. Although not used in the present work, use of dynmi reordering [] n mkeiteven further eient for most omintionl iruits enountered in rel life. In this pper we will hiey fous on demonstrting tht funtionl lerning is n eient tool for deteting internl orrespondenes. Sine onstnt-vlue reltionships re the only kind whih existing lerning tehniques n detet, our hief tsk in this pper is to show tht funtionl lerning n e s eient s exist-
2 ing lerning tehniques in the determintion of onstntvlue reltionships. 2 On Funtionl Lerning The lerning tehniques involve the temporry injetion of logi vlues t ritrry signls in digitl iruit nd the susequent exmintion of its logil onsequenes. Sortes [8] rried out stti lerning nd the method of lerning ws further improved in [2]. Funtionl lerning, introdued in [] is esily seen s the superset of ll the previous lerning methods [8, 2]. Funtionl lerning is omplete method, i.e. given suient time, it n identify ll neessry ssignments from given sitution of vlue ssignments in iruit. The onept of funtionl lerning is explined elow with the help of n exmple. We will illustrte the dded pility of funtionl lerning using n exmple in Figure Theorem 2.. If =)=then V =. Conversely, if V =, then =)=. By the Lw of Contrpositum =)=.. If =)=then V =. The onverse is lso true. Tht is if V =, then =) =. By the Lw of Contrpositum =)=. As shown in [], the ove two opertions sue in heking ny simple onstnt-vlue reltionship etween two points nd. As shown in the Figures 4, if Boolen is injeted t the gte in the iruit shown in the Figure 3, then the gte eomes unjustied. The OBDD for this wire in terms of (,,) is shown in the Figures 4(), nd the OBDDs for oth the wires, nd I, in terms of (,,,d) re shown in the Figure 4(). The result of the AND opertion etween the OBDD nd the OBDD is OBDD R nd is shown in Figure 4(). It n e seen tht there is no unique lerning ut we lern tht under the ondition =,oth Y nd Y2 must rry identil Boolen vlues. Note tht heking for the equivlene of two OBDDs is onstnt time opertion. 3 CUT () 4 () 6 Ciruit A () () (unjust.) d Ciruit B I m Y = = BDD opertion used : = Figure : Funtionl lerning Ciruit C n Y2 Consider the wire 23 in the iruit shown in Figure to e unjustied to. nd re the OBDDs for the wires 23 nd 6 respetively, uilt in terms of the pseudo inputs, nd. The two OBDDs re shown in Figure 2 []. The result of the AND opertion etween nd is. This implies tht when is Boolen, is Boolen. ene, it is lerned tht Boolen on the wire 23 implies Boolen on the wire 6. By forwrd implition, it is lerned tht the wire 24 must rry Boolen. A Boolen on the wire 6 nd Boolen on the wire 24 re the neessry onditions for Boolen on the wire 23. () Figure 2: OBDDs for wire 23 nd wire 6 The theorem tht governs the lerning opertions in funtionl lerning is presented elow []. () Figure 3: Funtionl lerning d R d () () () Figure 4: BDDs in Ciruit of Figure 3. () BDD ; () BDD ; () BDD ^ 2. Preise mrking of potentil lerning re iven n unjustied wire, funtionl lerning rst hooses n pproprite ut. The present tehnique for hoosing ut is sed on simple redth-rst trversl of the trnsitive fn-in one of the unjustied wire, strting t the unjustied wire. Next, the OBDD for the unjustied wire is uilt in terms of the ut vriles. One the OBDD is uilt, pproprite AND opertions,
3 s explined in the previous setion, must e performed in order to lern indiret implitions. But, the wires where lerning will e possile under the given sitution of vlue ssignments in the iruit re not known efore hnd. In order to preisely demrte the potentil lerning res in the iruit preproessing of the OBDD is rried out. Denition 2. A justition vetor in n OBDD is pth from the root vrile in the OBDD to tht terminl node whose vlue is equl to the vlue of the unjustied wire. During the preproessing of the OBDD for the unjustied wire, onstnt k numer of justition vetors re extrted. These vetors re pplied to the iruit nd omplete implition is rried out. After the pplition of ll the k justition vetors, the wires tht rry ommon Boolen vlues for ll the onsistent justition vetors re mrked s the potentil lerning res. Only these wires re sujeted to the lerning opertions using the proedure explined erlier. The numer of justition vetors tht need to e pplied in order to demrte the potentil lerning res with high mount of preision is mtter of heuristi. The proess of justition vetor evlution is stopped if either no new wires re eliminted from the list of potentil lerning res fter two suessive evlutions or if user-dened upper limit on the numer of justition vetors to e extrted is rehed or if evlution of ll the justition vetors in the OBDD is ompleted. In the se of lrger OBDD for whih omplete enumertion of ll the justition vetors is not fesile, the proedure outlined ove gives the wires whih omprise the potentil lerning re. One the potentil lerning re hs een mrked, lerning opertions sed on the theorem stted ove re rried out t these wires. 3 Algorithm for Verition As is ustomry in suh pprohes, the prolem of design verition hs een onverted to the prolem of heking for stisility. The two iruits to e veried re joined t their primry inputs. Their orresponding primry outputs re fed in pirs to 2-input XOR gtes. This new iruit will e heneforth referred to s the omposite iruit. Thus, if the two iruits re to e proven inequivlent, ll tht needs to e done is to prove the stisility of the output of ny of the ove XOR gtes. In this work it is ssumed tht there re no externl don't res in the iruits. owever, this work n e extended to inorporte externl don't res s well. In this ontext, results presented in [9] n e esily inorported. The omplete ow digrm for the logi verition lgorithm is shown in Figure 5. In the lerning phse, indiret implitions from one iruit to the other iruit re lerned y injeting Boolen vlues t dierent wires in the iruit. At eh gte Boolen vlue is injeted suh tht the gte eomes unjustied. For exmple Boolen is injeted t the output of n AND gte nd Boolen t the output of n OR gte. Similrly, oth Boolen nd Boolen re injeted t the output of n XOR/XR gte. This phse is rried out using YES START Lerning phse : injeting Boolen vlues nd storing indiret implitions Cheking phse : hek if ll pirs of POs of the two iruits re equivlent Is the verifition of ll the POs over? Is L > L C Cmx? STOP L = L + C C L = L + L L Is L > L L Lmx? ATP YES Figure 5: Flow Digrm for Verition ertin initil level of lerning L L. In our implementtion, we strt with n initil lerning level of. All the indiret implitions lerned during this phse re stored long with the dt struture of the wire from whih itws lerned. In this phse it is importnt to hve nished proessing ll the wires in the trnsitive fn-in of the present wire. This is needed to ensure tht during the proessing of the present wire we n mke use of the pre-stored implitions of the wires in its trnsitive fn-in to speed up the proess of lerning indiret implitions. In the heking phse, we hek for the equivlene of the orresponding primry outputs of the two iruits. In this phse Boolen is injeted t the output of the pproprite XOR gte nd prestored implitions re used to try nd prove onit of this sitution of vlue ssignment. An OBDD for the output of the XOR gte is uilt in terms of n pproprite ut t level L C in the two iruits. Next, the indiret implitions mong the ut vriles re utilized in order to try nd prove tht ll the pths in the OBDD strting from the topmost vrile nd terminting t the node re inonsistent. If onit is proved, then the two orresponding primry outputs re equivlent. This phse is initited with n initil level L C =. If the two outputs nnot e proved to e equivlent, then L C is inremented nd the proess is repeted till L C exeeds preset L Cmx. If fter this phse, we still hve primry outputs remining tht hve not een shown to e equivlent, the lerning level for the lerning phse is inremented nd lerning for indiret implitions is rried out with higher preision. If the lerning level L L for lerning phse exeeds preset L Lmx, then for the remining primry outputs we use n ATP tool tht tries to generte test for the fult s-- t the
4 output of the pproprite XOR gte t the output of the omposite iruit. The ATP tool mkes use of ll the pre-stored indiret implitions, thus eetively reduing the serh spe for test. The two orresponding primry outputs re equivlent if the fult is proved to e redundnt. Otherwise, test vetor is generted whih is the distinguishing vetor for the two iruits. If the ATP tool fils to prove the fult to e redundnt nd lso fils to nd test vetor for the fult, then VERIFUL orts. It should e noted tht the lerning phse is y itself omplete lgorithm for logi verition. iven enough time, the lerning phse itself n prove the iruits to e equivlent or inequivlent s the se my e. Now, we riey disuss some other interesting hrteristis of funtionl lerning. owever, detiled proofs of the theorems hve een omitted due to spe onstrint. Assume tht S f (L) nd S r (L) re the set of onditions lerned respetively y the funtionl lerning (FL) tehnique when operting t distne L from n initil vlue ssignment (or \operting t level L" s we will use in the following), nd reursive lerning (RL) tehnique t reursion level L. It n e proved tht Theorem 3. S f (L) S f (L ) [ :::[S f (L 2) [ S f () An importnt dierene etween FL nd RL tehniques n e notied in following. Assume t f (L), t r (L) re the time required to lern operting t level L in FL, nd reursion level L in RL. ene T f (L) = P i=l i= t f (i) is the time required if the method egins y lerning t eh level eginning from level. owever, in FL one n hoose to egin t level L nd ypss the eorts required t erlier levels. Theorem 3.2 In Reursive lerning, to lern onditions in S r (L) -treursion level L, one needs to spend t lest T r (L) = P i=l i= t r(i) units of time. owever, in funtionl lerning, only t f (L) units of time n sue in providing the lower ound on disovering S f (L). Ignoring the rguments out BDD vrile ordering, n RL method pprently hs higher omputtionl omplexity. As shown in [], for some funtions, FL tkes time polynomil in numer of gtes nlyzed ut RL requires exponentil time resoures. The reverse is not true. It does not pper possile one n require polynomil time for n RL method ut exponentil time for the FL method. Interestingly, our experiments show tht FL performs eqully well s RL sed verition even on 6288, multiplier, hving n exponentil representtion using OBDDs. Note, typilly BDD hs exponentilly mny pth in it. ene it is wiser to write the onjuntion of ll lerned onditions themselves s BDD. ene lerning =)= n e written s ( ^ ). Let eh of the lerning ondition i e expressed s BDD, nd LC represent the set of suh BDDs. Let e some utset suh tht it seprtes the output F of iruit C nd output F2 of iruit C2 from the primry inputs. Consider F() nd F2() to e the orresponding output OBDDs when expressed in terms of gtes t the ut. In other words, the gtes on ut re now the (pseudo) inputs through whih outputs F nd F2 re expressed. Let LC() e the set of lerning ondition BDDs, gin expressed in terms of ut-points (gtes) on. Theorem 3.3 F nd F2, orresponding output funtions of the iruits C nd C2 re equivlent if (F() F2()) ^ i, i 2 LC() is not stisle. For pth enumertion mehnism, following simple oservtion is quite useful. Oservtion: A sheme working y pth extrtion n esily exhustively nlyze funtion without extrting ll pths sine we wish to test for stisility only y determining whether onit exists on ll stisle pths. Let! e some utset seprting the root nd the terminls in BDD F() F2(). Theorem 3.4 If we (redth-rst) enumerte ll pths from root leding up to utset! within BDD F() F2() then F nd F 2 re not equivlent if there is onit on eh suh pth. 4 Results In this setion the results on ISCAS 85 enhmrk iruits re presented. These results re extremely enourging nd prove tht funtionl lerning sed verition n e very powerful tool tht designers n use to verify their designs during the synthesis proedure. We present here the results where the redundnt omintionl iruits re veried ginst their nonredundnt versions. To otin orret perspetive of our results, note tht the orresponding spe s well s times required for veritions using OBDDs is out three orders of mgnitude higher for iruits suh s 6288! The experiments hve een rried out on Sun Spr Sttion. The times reported re in seonds. The results show tht funtionl lerning is n extremely fst nd eient method to extrt internl equivlenes nd indiret implitions in digitl logi. Note, Theorems 3.3 nd 3.4 were not employed for the results otined in this pper; only simple pth enumertion ws used in the heking phse. An dded dvntge of funtionl lerning tht we intend to utilize in our future work is its ility to work on iruits where prts of the logi re not represented s logi gtes ut simply s Boolen funtions. This is euse of its pility to mnipulte informtion on Boolen funtions y using BDDs. Note tht the other existing lerning tehniques [8, 2] do not hve this pility. We lso intend to use dditionl reltionships (indiret implitions whih re not onstnt-vlue reltions) tht re esily desried in our pproh. 5 Conlusion In this pper we hve presented the preliminry results on VERIFUL, design verition tool sed on funtionl lerning, n extremely powerful lerning tehnique for digitl logi, whih ws rst introdued in []. We hve onduted experiments on the ISCAS 85 enhmrk iruits. These experiments indite tht Note, the numer of prestored implitions re indiret implitions exept smll numer of dispensle, diret implitions tht our preliminry version of progrm stores.
5 Tle : Verition of some enhmrk iruits Ciruits eing veried L L L C # prestored implitions CPU time (min:se) 432 vs. 432nr : vs. 499nr 56 : vs. 355nr 499 vs :3.5 : vs. 98nr 267 vs. 267nr :5.76 : vs. 354nr 535 vs. 535nr :5 6: vs. 6288nr 7552 vs. 7552nr :24.5 3:5 funtionl lerning sed verition n eome very powerful nd verstile tool for use during iruit synthesis. Further, in this pper we hve lso riey pointed out theoretilly the superiority nd greter power of funtionl lerning over the other existing lerning tehniques [8, 2]. Mny other interesting properties of funtionl lerning n lso e shown s well s the ft tht these tehniques n esily e integrted with onventionl OBDD sed verition methods to yield very powerful verition methodologies [2]. After further improved integrtion of urrent progrms with sophistited ATP tool, our susequent reserh will e direted towrds the pplition of funtionl lerning to the elds of OBDD sed sequentil iruit testing [3] nd verition, nd optimiztion of oth omintionl nd sequentil logi iruits. We lso intend to use vrious new kinds of lernings suh s indiret implitions etween sets of funtions nd generlized reltionships in iruit tht n e esily extrted y our method. Referenes [] Mukherjee R., Jin J., Prdhn D. K., \Funtionl Lerning: A new pproh to lerning in digitl iruits", Pro. IEEE VLSI Test Symp., pp , April 994 [2] Jin J., Mukherjee R., Fujit M., \Advned Verition Tehniques Bsed on Lerning" Sumitted for Pulition. [3] Entren L., Cheng K-T., \Sequentil Logi Optimiztion By Redundny Addition And Removl", ICCAD, 993, pp [4] oel P., \An Impliit Enumertion Algorithm to enerte Tests for Comintionl Logi Ciruits", IEEE Trnstions on Computers, vol. C-3, pp , Mr. 98. [5] Mlik S. et l., \Logi Verition using Binry Deision Digrms in Logi Synthesis Environment", ICCAD, 988, pp [6] Fujit M., Fujisw., Kwto N., \Evlution nd Improvements of Boolen Comprison Method Bsed on Binry Deision Digrms", ICCAD, 988, pp [7] Fujiwr., Shimono T., \On the Aelertion of Test enertion Algorithms", Pro. 3th Int. Symp. Fult Tolernt Computing, pp. 98-5, 983. [8] Shulz M., Trishler E., Sfert T., \SOCRATES: A highly eient utomti test pttern genertion system", Int. Test Conf., 987. [9] Brynt R. E., \rph-bsed Algorithms for Boolen Funtion Mnipultion", IEEE Trnstions on Computers, vol. C-35, no. 8, Aug [] Bre K. S., Rudell R. L., Brynt R. E., \Eient Implementtion of BDD Pkge", Pro. 27th Design Automtion Conf., pp [] Rudell R. L., \Dynmi Vrile Ordering for Ordered Binry Deision Digrms", Pro. ICCAD, 993, pp [2] Kunz W., Prdhn D. K., \Reursive Lerning: An Attrtive Alterntive to the Deision Tree for Test enertion in Digitl Ciruits", Pro. Int. Test Conf., pp , 992. [3] Moondnos, J. nd Arhm, J. A., \Sequentil Redundny Identition Using Verition Tehniques", Pro. Int. Test Conf., pp , 992. [4] Kunz W., \ANNIBAL: An Eient Tool for Logi Verition Bsed on Reursive Lerning", Pro. Int. Conf. Computer Aided Design, 993. [5] Bermn C. L., Trevyllin L.., \Funtionl Comprison of Logi Designs for VLSI Ciruits", ICCAD, 989, pp [6] As E. J., Klingsheim K., Steen T., \Quntifying Design Qulity: A Model nd Design Experiments", Pro. of EURO ASIC 992, pp [7] Brnd D., \Verition of Lrge Synthesized Designs", ICCAD, 993, pp [8] Cerny E., Murs C., \Tutology Cheking Using Cross- Controllility nd Cross- Oservility Reltions", IC- CAD, 99, pp [9] Shiple T. R., ojti R., Bryton R. K., \euristi minimiztion of BDDs using don't res", DAC, 994, pp
Engr354: Digital Logic Circuits
Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost
More informationy1 y2 DEMUX a b x1 x2 x3 x4 NETWORK s1 s2 z1 z2
BOOLEAN METHODS Giovnni De Miheli Stnford University Boolen methods Exploit Boolen properties. { Don't re onditions. Minimiztion of the lol funtions. Slower lgorithms, etter qulity results. Externl don't
More informationLogic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 22 Reding: Logi Synthesis in Nutshell Setion 2 most
More informationDorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationUnit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
More informationImplication Graphs and Logic Testing
Implition Grphs n Logi Testing Vishwni D. Agrwl Jmes J. Dnher Professor Dept. of ECE, Auurn University Auurn, AL 36849 vgrwl@eng.uurn.eu www.eng.uurn.eu/~vgrwl Joint reserh with: K. K. Dve, ATI Reserh,
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationAlgorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:
Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationNecessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )
Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us
More informationLogic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most
More informationChapter 4 State-Space Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationHardware Verification 2IMF20
Hrdwre Verifition 2IMF20 Julien Shmltz Leture 02: Boolen Funtions, ST, CEC Course ontent - Forml tools Temporl Logis (LTL, CTL) Domin Properties System Verilog ssertions demi & Industrils Proessors Networks
More information1. Logic verification
. Logi verifition Bsi priniples of OBDD s Vrile ordering Network of gtes => OBDD s FDD s nd OKFDD s Resoning out iruits Struturl methods Stisfiility heker Logi verifition The si prolem: prove tht two iruits
More informationSystem Validation (IN4387) November 2, 2012, 14:00-17:00
System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
More informationCS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
More informationAlgorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationNondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationGlobal alignment. Genome Rearrangements Finding preserved genes. Lecture 18
Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re
More informationPropositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches.
Propositionl models Historil models of omputtion Steven Lindell Hverford College USA 1/22/2010 ISLA 2010 1 Strt with fixed numer of oolen vriles lled the voulry: e.g.,,. Eh oolen vrile represents proposition,
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More informationMetodologie di progetto HW Technology Mapping. Last update: 19/03/09
Metodologie di progetto HW Tehnology Mpping Lst updte: 19/03/09 Tehnology Mpping 2 Tehnology Mpping Exmple: t 1 = + b; t 2 = d + e; t 3 = b + d; t 4 = t 1 t 2 + fg; t 5 = t 4 h + t 2 t 3 ; F = t 5 ; t
More informationLecture 11 Binary Decision Diagrams (BDDs)
C 474A/57A Computer-Aie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, K-mp, iruit, et
More informationComputational Biology Lecture 18: Genome rearrangements, finding maximal matches Saad Mneimneh
Computtionl Biology Leture 8: Genome rerrngements, finding miml mthes Sd Mneimneh We hve seen how to rerrnge genome to otin nother one sed on reversls nd the knowledge of the preserved loks or genes. Now
More informationAlgorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More informationTest Generation from Timed Input Output Automata
Chpter 8 Test Genertion from Timed Input Output Automt The purpose of this hpter is to introdue tehniques for the genertion of test dt from models of softwre sed on vrints of timed utomt. The tests generted
More informationAppendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities
Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the
More informationGeneralization of 2-Corner Frequency Source Models Used in SMSIM
Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More informationFinite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions
More informationCS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014
S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More information, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1.
Exerise Genertor polynomils of onvolutionl ode, given in binry form, re g, g j g. ) Sketh the enoding iruit. b) Sketh the stte digrm. ) Find the trnsfer funtion T. d) Wht is the minimum free distne of
More informationm2 m3 m1 (a) (b) (c) n2 n3
Outline LOGIC SYNTHESIS AND TWO-LEVEL LOGIC OPTIMIZATION Giovnni De Miheli Stnford University Overview of logi synthesis. Comintionl-logi design: { Bkground. { Two-level forms. Ext minimiztion. Covering
More informationA Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version
A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment
More informationInstructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.
ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationCounting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
More informationReview Topic 14: Relationships between two numerical variables
Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationIntermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationCSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4
Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones
More informationChapter 8 Roots and Radicals
Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi
More information= state, a = reading and q j
4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationBisimulation, Games & Hennessy Milner logic
Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory
More information22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)
More informations the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p
Skeptil Rtionl Extensions Artur Mikitiuk nd Miros lw Truszzynski University of Kentuky, Deprtment of Computer Siene, Lexington, KY 40506{0046, frtur mirekg@s.engr.uky.edu Astrt. In this pper we propose
More informationLecture 6. CMOS Static & Dynamic Logic Gates. Static CMOS Circuit. PMOS Transistors in Series/Parallel Connection
NMOS Trnsistors in Series/Prllel onnetion Leture 6 MOS Stti & ynmi Logi Gtes Trnsistors n e thought s swith ontrolled y its gte signl NMOS swith loses when swith ontrol input is high Peter heung eprtment
More informationArrow s Impossibility Theorem
Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationAlpha Algorithm: Limitations
Proess Mining: Dt Siene in Ation Alph Algorithm: Limittions prof.dr.ir. Wil vn der Alst www.proessmining.org Let L e n event log over T. α(l) is defined s follows. 1. T L = { t T σ L t σ}, 2. T I = { t
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationTIME AND STATE IN DISTRIBUTED SYSTEMS
Distriuted Systems Fö 5-1 Distriuted Systems Fö 5-2 TIME ND STTE IN DISTRIUTED SYSTEMS 1. Time in Distriuted Systems Time in Distriuted Systems euse eh mhine in distriuted system hs its own lok there is
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationFault Modeling. EE5375 ADD II Prof. MacDonald
Fult Modeling EE5375 ADD II Prof. McDonld Stuck At Fult Models l Modeling of physicl defects (fults) simplify to logicl fult l stuck high or low represents mny physicl defects esy to simulte technology
More informationNEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE
NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE V.S. Gordeev, G.A. Myskov Russin Federl Nuler Center All-Russi Sientifi Reserh Institute of Experimentl Physis (RFNC-VNIIEF)
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}
More informationThermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R
/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationEFFICIENT SYMBOLIC COMPUTATION FOR WORD-LEVEL ABSTRACTION FROM COMBINATIONAL CIRCUITS FOR VERIFICATION OVER FINITE FIELDS
EXTENDED VERSION OF THE PAPER ACCEPTED TO APPEAR IN IEEE TRANS ON CAD, PAPER ACCEPTANCE OCTOBER 2015 1 EFFICIENT SYMBOLIC COMPUTATION FOR WORD-LEVEL ABSTRACTION FROM COMBINATIONAL CIRCUITS FOR VERIFICATION
More informationFor a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
More informationI 3 2 = I I 4 = 2A
ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationArrow s Impossibility Theorem
Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties
More informationCS 573 Automata Theory and Formal Languages
Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple
More information2.4 Theoretical Foundations
2 Progrmming Lnguge Syntx 2.4 Theoretil Fountions As note in the min text, snners n prsers re se on the finite utomt n pushown utomt tht form the ottom two levels of the Chomsky lnguge hierrhy. At eh level
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More informationElectromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors
LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly
More informationANALYSIS AND MODELLING OF RAINFALL EVENTS
Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.
More informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
More informationexpression simply by forming an OR of the ANDs of all input variables for which the output is
2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output
More informationAbstraction of Nondeterministic Automata Rong Su
Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1 Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering,
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationDynamic Template Matching with Mixed-polarity Toffoli Gates
Dynmi Templte Mthing with Mixed-polrity Toffoli Gtes Md Mzder Rhmn 1, Mthis Soeken 2,3, nd Gerhrd W. Duek 1 1 Fulty of Computer Siene, University of New Brunswik, Cnd 2 Deprtment of Mthemtis nd Computer
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationSynthesis of Hazard-Free Multilevel Logic Under Multiple-Input Changes from Binary Decision Diagrams
Synthesis o Hzrd-Free Multilevel Logi Under Multiple-Input Chnges rom Binry Deision Digrms Bill Lin, Memer, IEEE, Srinivs Devds, Memer, IEEE Astrt We desrie new method or diretly synthesizing hzrdree multilevel
More informationLing 3701H / Psych 3371H: Lecture Notes 9 Hierarchic Sequential Prediction
Ling 3701H / Psyh 3371H: Leture Notes 9 Hierrhi Sequentil Predition Contents 9.1 Complex events.................................... 1 9.2 Reognition of omplex events using event frgments................
More information