Semantics of RTL and Validation of Synthesized RTL Designs using Formal Verification in Reconfigurable Computing Systems
|
|
- Melissa Ball
- 6 years ago
- Views:
Transcription
1 emntis of TL nd Vlidtion of ynthesized TL Designs using Forml Verifition in eonfigurle Computing ystems Phn C. Vinh nd Jonthn P. Bowen London outh Bnk University Centre for Applied Forml Methods, Institute for Computing eserh Fulty of BCIM, 13 Borough od, London E1 AA, UK UL: Emil: Astrt The funtionl vlidtion of stte-of-the-rt reonfigurle omputing system design is usully lorious, d ho nd open-ended tsk. It n e omplished through two si pprohes: simultion nd forml verifition. In vlidtion using forml verifition pproh, it ttempts to estlish tht the egister Trnsfer Level (TL design synthesized from the lgorithmi ehviorl speifition is mthemtilly orret. Therefore, finding the verifition methods to provide urte nd fst vlidtion esily would e very useful. In this pper, we develop semntis sed on Prtil Order Bsed Model (POM for TL nd, through this semntis, propose forml verifition method to prove the orretness of the TL synthesis result. This method n e used to hieve the following. On one hnd, it n urtely verify n TL desription with respet to ehviorl speifition of the system; on the other hnd, it n deide whether two proesses, whih re supposed to implement the sme funtion, hve the sme intertive ehviors so tht one n e repled y the other. 1 Introdution As high-level synthesis (HL in reonfigurle omputing systems eome more sophistited nd synthesized designs get more omple, it is importnt tht we develop systemti pproh to the vlidtion of synthesized egister Trnsfer Level (TL designs [13, 14]. Funtionl vlidtion of synthesized TL design n e usully omplished through two si pprohes: simultion nd forml verifition [4, 6, 7]. In this pper, we present our efforts to develop Prtil Order Bsed Model (POM sed semntis for TL nd, through these, to verify formlly the TL designs generted y n HL system tht epts lgorithmi ehviorl speifitions written in suset of VHDL nd genertes register trnsfer level design, lso epressed in VHDL. Vlidtion using forml verifition methods ttempts to estlish tht the TL design synthesized from the lgorithmi ehviorl speifition is mthemtilly orret. Theorem proving nd model heking re two populr forml verifition pprohes. Our pproh is to model the ehvior using POM notion nd use this model to develop interesting properties of the model tht should hold oth t the ehvior nd register trnsfer levels. After the TL design is synthesized, we will verify whether the sme properties ontinue to hold for the TL design. This pper is orgnized s follows. etion will drw some min shpes of relted work. etion 3 will present some si definitions of POM notion. etion 4 gives the POM semntis for the TL model. A verifition lgorithm for the TL synthesis results in reonfigurle omputing systems using POM is presented in setion 5 nd short onlusion is given in setion 6. elted Work Another vlidtion methodology of synthesized TL designs is to use simultion pproh, s presented informlly in [6, 7], for ompring simultion results of n lgorithmi VHDL speifition nd synthesized TL design, lso represented s VHDL desription, whih llows hnge in the yle-y-yle ehvior without signifint limittions. To enle this omprison, ommon set of simultion vetors is used. The informtion given in this vetor set serves s input to its POM, with whih the omprison is eeuted. eently, numer of reserhers hve een investigting tehniques known s prtil-order methods tht n signifintly redue the running time of forml vlidtion y voiding redundnt eplortion of eeution senrios. The results in [4] desrie the design of prtil-order lgorithm for the vlidtion tool nd disuss its effetiveness. It shows
2 tht reful ompile-time stti nlysis of proess ommunition ehvior yields informtion tht n e used during vlidtion to drmtilly improve its performne. 3 ome definitions Definition 3.1 (Chu spe A Chu spe is inry reltion etween two sets A nd X. It is written s triple (A, X,, where : A X {, 1} is the inry reltion s hrteristi funtion of suset of A X. A Chu spe [5] does not impose ny rdinlity restritions on A nd X. Thus ll rguments given elow will work for ll rdinlities. We n think of A s the set of events (representing the tions nd X s the set of sttes (representing the possile or permitted sttes. A stte is defined in terms of n ourrene reltion (, tht is true when the event hs ourred in the stte. Thus, eh stte is suset of A ontining the events tht hs ourred in the stte. The visul wy to write out Chu spe epliitly is s inry mtri of dimension A X, with eh entry giving the vlue of the reltion on its pir of oordintes. A is written t the top nd X on the side. Figure 1 gives emples of some Chu spes. The elements of A re denoted y,,, d,..., nd u, v, w,,... for elements of X. Mtri 1 1 Hsse Digrm Mtri Mtri Hsse Digrm Hsse Digrm Figure 1. Chu spes nd its representtions Definition 3. (POM A POM is Chu spe C given y the tuple (A, X,, where A = {, 1,..., n } is set of events, X = {, 1,..., n } is set of sttes, nd : A X {, 1} represents the ourrene reltion; i.e., (, = 1 if the event hs ourred in the 11 1 stte nd (, =, otherwise. Eh stte i A is defined in terms of s: i = { A nd (, i = 1} The POM in Figure hs three events, {,, }, nd si sttes, {, 1,, 3, 4, 5 }. It represents system where ny event of {,, } ours nd ording to the informtion omputed y tht event, one of two rest events will our. In stte, the event hs ourred; represents the stte where hs ourred fter or hs ourred fter. And so on. We n represent the POM in the form of mtri or s logil formul or s Hsse digrm [11]. In the mtri, eh entry (, ontins the vlue of the ourrene reltion. Thus, the rows of the mtri orrespond to the sttes of the POM nd the olumns orrespond to the events of the POM. Considering the mtri s truth tle, we n hve logil formul representing the POM. The pitoril representtion s Hsse digrm illustrtes the prtil order eisting etween the sttes. POM=(A,X, where A={,,} X={{},{},{,},{},{,},{,}} (A,X is represented y the following inry mtri. Mtri Logil formul Hsse Digrm f POM= 1 {} {} {,} {} {,} {,} 1 1 Figure. POM nd its representtions Definition 3.3 (Logil epresenttion We define the logil representtion f C of the POM C s: f C = <i<n f i where n = X nd f i is the logil formul orresponding to i X, defined s follows: f i = ( { A} nd (, i = 1 ( { A} nd (, i = In the logil representtion, we hve the events s vriles, the sttes s terms of the formul nd the reltion determines whether the vrile ppers omplemented or not. The logil formul f is true for eh stte tht is permitted in the POM. 11 1
3 4 POM-sed semntis for TL Ck in i Ck in j As strting point, we del with the prolem of oneto-one synhroniztion with vlue ehnge, irrespetive of the vlue tully ehnged [3, ]. ynthesis of the omple synhroniztion into TL form requires the use of severl signls to gurntee the semnti orretness of the synthesis. o eh synhroniztion opertion (event is ssoited with three signls, one for the ehnge of the dt itself nd two others to mnge the synhroniztion ( redy nd n knowledgment signl. The need for two signls for synhroniztion is due to the ft tht ommunition is rendezvous etween events. Let us ssume we hve two proesses, nd, whih respetively offer nd re le to ept vlue v through gte g t ertin time. In this se, two gtes re involved in the synhroniztion, one of whih offers vlue (epressed y the symol!, while the other epts vlue (indited y?. This sitution is epressed s in the two following sets of events: g where = {... g!v...} = {... g?v...} hemtilly, synthesis of the events g!v nd g?v n e represented s in Figure 3. The signl in i (in j represents the signl enling eeution of lok i(j nd signl out i (out j represents the termintion of lok i(j (whih oinides with the signl enling eeution of the lok i + 1. The signl g n is needed when hoie opertion is involved in the synhroniztion. The loks i nd j re synthesized into the TL lnguge s in Figure 3. The trnsmitter wits for the reeiver to e ville for synhroniztion, fter whih it knowledges the synhroniztion nd ehnges the vlue (if ny; v T represents the vrile ontining the vlue to e trnsferred, whih in TL is equivlent to register. The ehvior of the reeiver omplements tht of the trnsmitter; v is the register tht, following synhroniztion, will ontin the vlue ehnged. Aording to the synthesis sheme used, the trnsmitter is trnslted in four TL steps nd the reeiver in three steps. 4.1 et of events Let us onsider the one-to-one synhroniztion with vlue ehnge desried ove. Eh synhroniztion event is ssoited with three signls: one for the ehnge of the dt itself (g n, nd two others to mnge the synhroniztion (g rdy nd g k. In the sequel, we onsider the finite event set A = {( g rdy, g k, g n, ( g rdy, g k, g n, ( g rdy, g k, g n, ( g rdy, g k, g n, (g rdy, g k, g n, (g rdy, g k, g n i! ( out i g k g v g rdy g k g v g rdy out j... : : if( g rdy ; g rdy goto( ; +1 // Wit for the reeiver to e redy to +1 : g k 1 synhronize // Aknowledges the synhroniztion + : if( g n ; g n goto( ; +3 // Wit for the synhroniztion to e orretly +3 : g v v T onluded y the reeiver (g n=1 // Ehnges the vlue... :... : y : g rdy 1 ; if( g k ; g k goto(y ; y+1 ( Trnsltion of 3( j? ( g n // Wrns the trnsmitter to e redy for synhroniztion nd simultneously sends n k signl y+1 : g n 1 // Informs trnsmitter tht synhroniztion hs tully ourred y+ : v g v // Aepts the vlue... : (d Trnsltion of 3( Figure 3. The si intertion events nd TL lnguge g n, (g rdy, g k, g n, (g rdy, g k, g n }. At eh rising edge of the lok, n tion must e eeuted. The mening of the event set is tht the tion ( g rdy, g k, g n is eeuted when no g rdy, no g k nd no g n our; ( g rdy, g k, g n is eeuted when no g rdy nd no g k our ut only g n ours; nd so on. Definition 4.1 (Computtion A finite sequene of tions is omputtion over A nd the set of ll omputtions is denoted y A*. 4. et of ttes Let X e the set of sttes representing the possile or permitted sttes. A stte y X is defined in terms of trnsition reltion T (, when the event A n mke trnsition from the stte X to y. Thus, eh stte y is suset of A ontining the events tht n mke trnsition from tht stte, tht is y = { A nd T (, = y} Definition 4. (uessor nd Predeessor ttes A stte i X is predeessor of stte j X if T (, i = j. Thus, j is suessor of i.
4 Definition 4.3 (Initil tte A stte i X is initil stte when it hs no predeessors; i.e., there is no stte j X suh tht T (, j = i. Definition 4.4 (Finl tte A stte i X is finl stte when it hs no suessors; i.e., there is no stte j X suh tht T (, i = j. Indeed, the triple (A, X, T is POM s defined in setion 3 nd the omputtion in definition 4.1 n e lso understood s follows: omputtion Γ of POM (A, X, T is prtil order on X under trnsition reltion T ; i.e., Γ = (, 1,..., n where for ll i, i+1 Γ, i+1 is suessor of i. Definition 4.5 (POM Eeution An eeution α of POM (A, X, T is n infinite sequene of omputtions Γ i of (A, X, T. From this oservtion we will develop the notion of POMutomt in the setions elow. 4.3 POM-Automt A POM-utomton is triple A = (X,, T where X is finite set of sttes, is the initil stte, T is funtion from X A into X { }, the trnsition funtion. If T (, =, no trnsition leled y n e fired from stte. ( n e viewed s sink stte. A omputtion Γ = 1... n is epted y the utomton if there eists 1,..., n X suh tht: T (, 1 = 1 i > 1, T ( i 1, i = i This will e denoted y: n 1 n n If it is not the se, there eists 1 k n nd sequene of sttes suh tht: k k 1 uh pth through n utomton is lled the run of the utomton over the omputtion Γ. The set of ll omputtions epted y A will e denoted y L(A. The POMutomt of trnsmitters nd reeivers onsist of four sttes nd three sttes, respetively, s in Figure 4. In Figure 4(, the stte nmed stte (or orresponds to witing for the reeiver to e redy to synhronize (g rdy =, stte 1 (or 1 orresponds to the enled trnsmitter due to redy signl from the reeiver (g rdy = 1, stte (or to knowledgement of the synhroniztion (g k = 1 nd stte 3 (or 3 to the to the orret onlusion of the synhroniztion y the reeiver (g n = 1. In Figure 4(, the stte nmed stte (or orresponds to wrning the trnsmitter to e redy for synhroniztion nd simultneously sends n k signl (g rdy = 1; stte 1 (or 1 orresponds to knowledgement of the synhroniztion (g k = 1, nd stte (or to informing the trnsmitter tht synhroniztion hs tully ourred (g n = 1.! g rdy g k g n tte ( z z tte 1( 1 1 z tte ( 1 1 tte 3( where: z {,1} (,z,z (1,,z 1 (1,1, (1,1,1 (1,,z (1,1, (1,1,1 (? g rdy g k g n tte ( 1 z tte 1( tte ( where: z {,1} (1,,z (1,1, 1 (1,1,1 (1,1, (1,1,1 ( Figure 4. POM-utomton representtions We now present n utomt produt tht llows modulr desription of more omple proess. Eh suproess n e modeled n utomton nd the model of the omplete proess n e otined y omputing the produt of ll su-proess utomt. 4.4 POM-Automt Produt The proesses nd n e onneted s in Figure 5. Let = (X ; ; T nd = (X ; ; T e the utomt tht model the proesses nd respetively. We define the produt,, of nd to model the proess otined y linking to. We wnt to synhronize outputs 3
5 of with inputs of so tht when dt trnsfer etween nd is possile then this trnsfer must hppen. This leds to the following definition of the produt of nd, over the sme event set A: = (X,, T where X = X X = T is defined in the following wy: Let i = ( j, k e in X nd in A. If there eist g n! g v g k g rdy g v? g n j+1 X nd k+1 X suh tht T ( j, = j+1 nd T ( k, = k+1, we set Otherwise, we set T (( j, k, = ( j+1, k+1 T (( j, k, = Let us ssume the input (output width of is equl to the output (input width of, so tht these proesses n e onneted s in Figure 5. Eh stte in is pir onsisting of stte from nd stte from. The run of over the epted omputtion Γ = (1, 1, (1, 1, 1 is denoted s elow: ( 1, (1,1, (, 1 (1,1,1 ( 3, This mens tht in the stte ( 1, on event (1,1,, the utomton proeeds y eeuting from 1 nd in prllel, eeuting from, nd so on. 4.5 Equivlene of proesses Let A nd A e two proesses nd A nd A e their ssoited POM-utomt. A nd A re equivlent L(A = L(A In other words, the proesses A nd A re equivlent if they nnot e distinguished y their eternl ehviors. 4.6 POM emntis A POM n e interpreted s POM-utomton with set of events A nd set of possile sttes X. Being in stte, the POM-utomton eeutes some trnsitions over events to reh suessor stte of. Eh possile omputtion of the POM orresponds to eh run of the POM-utomton nd the eeution of the POM represents the set of POMutomton runs. A POM-utomton in terms of POM model is represented y the set of events A nd the events our t eh stte; i.e., the trnsition reltion T. A more prtil pproh sed on reltions etween sttes is tht eh POMutomton is modeled s set of reltions etween sttes nd for eh suh reltion we hve orresponding POM. ( g rdy g k g n = ( 1 1 z 1 = ( = ( where: z {,1} (1,,z (1,1, (1,1, 1 ( (1,1,1 (1,1,1 Figure 5. The onnetion of proesses nd nd its POM-utomton The POM of eh reltion etween sttes is onsidered s property of the POM-utomton. Thus, onjuntion of the properties will result in the POM-utomton. 5 Verifition Algorithm for the TL ynthesis esults 5.1 teps of the lgorithm Our verifition lgorithm is shown digrmmtilly in Figure 6. The steps of the lgorithm will e onsidered in the susetions elow. 5. Algorithmi ehviorl speifition The lgorithmi desription is speified using n pproprite high-level lnguge. A mjor tsk during this step is the reliztion of the different sheduling modes. In other words, progrm is reted in this step.
6 Algorithmi ehviorl speifition TL Dynmilly reonfigurle omputing epresses the notion tht the dynmi seletion of if... then... else desries the reonfigurtion in similr wy to the C MUX dynmi reonfigurtion strtion proposed y Luk et l., reported in [1] nd to lesser etent in [9], whih requires ll the lterntives to hve inputs of the sme type nd n output of the sme type. This dynmi seletion is lso similr to the sheme presented in [1], whih is little more generl thn C MUX mehnism. The sheme in [1] n desrie dynmi seletion etween ehviors with totlly different types. The design of Fleile Arry Bloks (FABs [8] nd edued Fleile Arry Bloks (FABs [15] n e epressed using dynmi seletion euse the reonfigurtion ehvior is ontrolled y four onfigurtion its, whih re inputs to the dynmi seletions nd essentilly enle dynmi seletion. 5.3 Creting the POM P EC Crete POM PEC Crete POM PEC - utomton Comprison esult Crete POM TL - utomton Figure 6. The verifition lgorithm the TL synthesis results using POM In the VHDL emple shown in the Figure 7, we hve proess P with five events relted to the sttements of P, in whih eh sttement is onsidered s n event. The min tsk is the genertion of the prtil order desription of the progrm sttements reted in setion 5.. In other words, this desription is used to indite the dt dependenies of sttements neessry for the genertion of the possile POM desription of the lgorithmi speifition, nmely (POM P EC. To fulfill this tsk, we need to eplin some terminologies in terms of the following si reltions etween events: independene, preedene, onflit nd disjuntive enle reltion [11]. Definition 5.1 (Independene reltion The independene reltion ( represents the independent eeution of two events nd. The POM for this reltion is shown in Figure 8, where ll sttes re permitted; tht is, ll susets of A re vlid sttes. No order is imposed to the ourrene of the events nd. P: proess egin events red(a,b; if (A>B then C:=A+B; else C:=s(A-*B; end if; send(c, hnnel1; end proess; Figure 7. A proess nd its event list f = p f p = = =1 + # f #= = + den f den= = + + Figure 8. ome si reltions etween events Definition 5. (Preedene reltion The preedene reltion ( represents the ourrene of the event followed y the ourrene of the event. The POM representtion for the reltion n e seen in Figure 8. This reltion is used to model the sequentil eeution of events. Definition 5.3 (Conflit reltion The onflit reltion (# represents either the ourrene of or the ourrene of. The orresponding POM nd logil formul re shown in Figure 8. A onflit reltion etween two events nd mens tht oth nd n never our in sme omputtion of the POM. Definition 5.4 (Disjuntive enle reltion The disjuntive enle reltion permits the representtion
7 of two events, whose eeutions disjuntively enle third event; i.e., den(,, mens the eeution of O enles the ourrene of. This reltion is needed, together with the onflit reltion eplined ove, to enle the events tht follow n if... then... else sttement. Figure 8 presents the POM nd its orresponding logil formul. In our prtil pproh to reting POM P EC, we use the reltions etween events. These reltions n e etrted from the system speifition given in high-level progrmming lnguge s in setion 5.. Let there e proess P with set of events A, together with reltions etween events, whih were etrted from tht proess speifition s follows. In Figure 7, we hve five events, where preedes 1, 1 preedes, 1 preedes 3, nd 3 re in onflit, nd the eeution of O 3 enles the ourrene of 4. The onjuntion them give us the POM for the proess P. Formlly, POM P EC is desried s POM P EC = { 1, 1, 1 3, # 3, den( 4,, 3 } 5.4 Creting the POM P EC -utomton From the event list nd dependeny reltions etween the events reted in setion 5.3, n utomton of the POM P EC is reted (see Figure TL ynthesis esult This is n TL synthesis result of the lgorithmi ehviorl speifition. This result hs een reted from the synthesis stge nd is trnsferred into the urrent verifition. The TL module [1, 3] is defined y the following: Components: ontins the delrtion of the omponents tht mke up the proessing unit. Control sequene: defines the internl ommnd sequene tht must e emitted y the ontrol unit. Permnent ssignment: defines n opertion tht must e repeted every lok yle. The ontrol sequene is mde up of steps; eh one is numered nd must e eeuted in single lok unit. Figure 1 shows this ontrol sequene for the proess P. (, 1,, 3, 4 (, 1,, 3, 4, 1,, 3, 4 (, 1,, 3,, 1,, 3,,,,, ( 4 ( 4 (, 1,, 3, 4 (, 1,, 3, 4 1 (, 1,, 3, ( 1 3 4, 1,, 3, (, 1,, 3, 4 5 ( 4 (, 1,, 3, 4 Figure 1. TL ontrol of proess P POM PEC = { p 1, 1 p, 1 p 3, # 3, den( 4,, 3} nd POM PEC utomton Logil formul: f = f p 1 Λ f 1 p Λ f 1 p 3 Λ f # 3 Λ f den(4,, 3 = ( 1 ( 3 = Λ ( 1 + Λ ( Λ + Λ ( Figure 9. POM P EC -utomton of proess P 5.6 Creting the POM T L -utomton A POM-utomton needs to e generted from the TL synthesis result. From the TL of proess P s shown in Figure 1, POM T L -utomton is reted s in Figure 11. POM TL utomton y 1 where:,y {,1} nd y Figure 11. POM T L -utomton of proess P
8 5.7 Comprison etween the POM P EC - utomton nd POM T L -utomton In ompring the POM speifition nd TL utomt, we need to determine the following to verify the orretness of n TL synthesis result. Definition 5.5 (Corretness of n TL ynthesis result An TL synthesis result using POM is orret iff it stisfies ll requirements of the POM speifition. Theorem 5.1 An TL synthesis result using POM is orret iff L(T L = L(PEC. In other words, the set of ll omputtions epted y T L is equl to the ones epted y PEC. Proof 5. if prt. By definition 5.5, when L(T L is equl to L(PEC then n TL synthesis result is orret. To prove the only if prt, we need to prove tht if L(T L L(PEC then the TL synthesis result is not orret. There re two ses s follows: If L(T L L(PEC then there eists requirement of omputtion Γ L(PEC\L(T L tht is not synthesized in the TL result. By definition 5.5, the TL synthesis result is not orret. If L(PEC L(T L then the speifition nd TL synthesis re not equivlent. In onsequene of this, their omputing ehviors re distint. In other words, the TL synthesis result is not orret s well. This onsidertion shows tht if the synthesis proess hs generted vlid TL result, then omprison is rried out here s n emintion tht heks whether POM T L - utomton is equivlent to POM P EC -utomton. Indeed, Figures 9 nd 11 show tht L(T L = L(PEC; thus the verifition indites the orretness of the TL synthesis results for the proess P. 6 ummry In this pper, Prtil Order Bsed Model (POM sed semntis for egister Trnsfer Level (TL desription nd verifition lgorithm hve een developed for vlidting the TL synthesis results. ome key fetures of this pproh re tht, firstly, the notion of POM (s Chu spe is onsidered s semnti si for TL nd, seondly, the notion of POM-utomt is dedited towrds forml orretness of the synthesis result t the register trnsfer level. The forml verifition method is sed on funtionl equivlene heking to determine if the POM T L -utomton is equivlent to the POM P EC - utomton. In other words, omprison is defined s n emintion tht heks whether the synthesis proess hs generted vlid TL desription. eferenes [1] B. Biley nd D. Gjski. TL semntis nd methodology. In Pro. 14th Interntionl ymposium on ystems ynthesis, pges 69 74, Montrel, Cnd, 3 eptemer 3 Otoer 1. [] V. Crhiolo, M. Mlgeri, nd G. Mngioni. An lgorithm for diret synthesis of forml speifitions. In Pro. 8th IEEE Interntionl Workshop on pid ystem Prototyping, pges 8 38, 4 6 June [3] V. Crhiolo, M. Mlgeri, nd G. Mngioni. Hrdwre/softwre synthesis of forml speifitions in odesign of emedded systems. ACM Trnstions on Design Automtion of Eletroni ystems, 5(3:399 43, July. [4] P. Godefroid, D. Peled, nd M. tskusks. Using prtilorder methods in the forml vlidtion of industril onurrent progrms. IEEE Trnstions on oftwre Engineering, (7:496 57, July [5] V. Gupt. Chu pes: A Model of Conurreny. PhD thesis, tnford University, UA, [6] C. Hnsen, A. Kunzmnn, nd W. osenstiel. Verifition y simultion omprison using interfe synthesis. In Pro. Design, Automtion nd Test in Europe (DATE, pges , 3 6 Ferury [7] C. Hnsen, F. A. M. D. Nsimento, nd W. osenstiel. Verifying high level synthesis results using prtil order sed model. In Pro. Hrdwre Lnguges, Design, Verifition nd Test (HLDVT, n Diego, CA, UA, Novemer [8]. D. Hynes nd P. Y. K. Cheung. A reonfigurle multiplier rry for video imge proessing tsks, suitle for emedding in n FPGA struture. In Pro. IEEE ymposium on FPGAs for Custom Computing Mhines, pges 6 34, April [9] W. Luk, N. hirzi, nd P. Y. K. Cheung. Compiltion tools for run-time reonfigurle designs. In Pro. 5th Annul IEEE ymposium on FPGAs for Custom Computing Mhines, pges 56 65, April [1] W. Luk, N. hirzi, nd P. Y. K. Cheung. Modelling nd optimising run-time reonfigurle systems. In Pro. IEEE ymposium on FPGAs for Custom Computing Mhines, pges , April [11] F. A. M. D. Nsimento nd W. osenstiel. Prtil order sed modeling of onurreny t the system level. In Pro. Interntionl Workshop on Conjoint ystems Engineering (CONYE, BdTölz, Germny, Mrh [1]. ingh. Interfe speifition for reonfigurle omponents. In Pro. IEEE/ACM Interntionl Conferene on Computer Aided Design (ICCAD, pges 1 19, n Jose, UA, 1 14 Novemer. [13] P. C. Vinh nd J. P. Bowen. An lgorithmi pproh y heuristis to dynmil reonfigurtion of logi resoures on reonfigurle FPGAs. In Pro. ACM/IGDA 1th Interntionl ymposium on Field Progrmmle Gte Arrys, pge 54, Monterey, UA, 4 Ferury 4. [14] P. C. Vinh nd J. P. Bowen. On the visul representtion of onfigurtion in reonfigurle omputing. Eletroni Notes in Theoretil Computer iene (ENTC, 19:3 15, 4. [15] C. Visvkul, P. Y. K. Cheung, nd W. Luk. A digit-seril struture for reonfigurle multipliers. In G. J. Brener nd. Woods, editors, Field-Progrmmle Logi nd Applitions, volume 147 of Leture Notes in Computer iene, pges pringer-verlg, 1.
NON-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 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 informationExercise 3 Logic Control
Exerise 3 Logi Control OBJECTIVE The ojetive of this exerise is giving n introdution to pplition of Logi Control System (LCS). Tody, LCS is implemented through Progrmmle Logi Controller (PLC) whih is lled
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 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 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 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 informationEngr354: 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 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 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 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 informationNondeterministic Finite Automata
Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)
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 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 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 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 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 informationBehavior Composition in the Presence of Failure
Behvior Composition in the Presene of Filure Sestin Srdin RMIT University, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Univ. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re
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 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 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 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 informationPetri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering
Petri Nets Ree Alreht Seminr: Automt Theory Chir of Softwre Engeneering Overview 1. Motivtion: Why not just using finite utomt for everything? Wht re Petri Nets nd when do we use them? 2. Introdution:
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 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 informationDiscrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α
Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw
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 informationDescriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata
Desriptionl Complexity of Non-Unry Self-Verifying Symmetri Differene Automt Lurette Mris 1,2 nd Lynette vn Zijl 1 1 Deprtment of Computer Siene, Stellenosh University, South Afri 2 Merk Institute, CSIR,
More informationBottom-Up Parsing. Canonical Collection of LR(0) items. Part II
2 ottom-up Prsing Prt II 1 Cnonil Colletion of LR(0) items CC_LR(0)_I items(g :ugmented_grmmr){ C = {CLOURE({ })} ; repet{ foreh(i C) foreh(grmmr symol X) if(goto(i,x) && GOTO(I,X) C) C = C {GOTO(I,X)};
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
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 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 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 informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
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 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 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 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 informationCS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata
CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or
More informationSemantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4
Semnti nlysis SI 16 Priniples of Progrmming Lnguges Fulty of omputer Siene Dlhousie University Winter 2012 Reding: hpter 4 Motivtion Soure progrm (hrter strem) Snner (lexil nlysis) Front end Prse tree
More informationCompiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Siene Deprtment Compiler Design Spring 7 Lexil Anlysis Smple Exerises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sienes Institute 47 Admirlty Wy, Suite
More informationMATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE
U.P.B. Si. Bull., Series C, Vol. 75, Iss. 2, ISSN 86 354 MATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE An-Mri CALFA, Gheorghe ŞTEFAN 2 Designed for emedded omputtion in system on hip design, the Connex
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 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 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 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 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 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 informationAutomatic Synthesis of New Behaviors from a Library of Available Behaviors
Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u
More informationLIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon
LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 One-wy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N
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 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 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 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 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 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 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 informationTransition systems (motivation)
Trnsition systems (motivtion) Course Modelling of Conurrent Systems ( Modellierung neenläufiger Systeme ) Winter Semester 2009/0 University of Duisurg-Essen Brr König Tehing ssistnt: Christoph Blume In
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationHybrid Systems Modeling, Analysis and Control
Hyrid Systems Modeling, Anlysis nd Control Rdu Grosu Vienn University of Tehnology Leture 5 Finite Automt s Liner Systems Oservility, Rehility nd More Miniml DFA re Not Miniml NFA (Arnold, Diky nd Nivt
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 informationFormal Languages and Automata
Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University
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 informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationThe DOACROSS statement
The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows prouer-onsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete
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 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 informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
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 informationResources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations
Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding
More informationFoundations of Computer Science Comp109
Reding Foundtions o Computer Siene Comp09 University o Liverpool Boris Konev konev@liverpool..uk http://www.s.liv..uk/~konev/comp09 Prt. Funtion Comp09 Foundtions o Computer Siene Disrete Mthemtis nd Its
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 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 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 informationCSCI565 - Compiler Design
CSCI565 - Compiler Deign Spring 6 Due Dte: Fe. 5, 6 t : PM in Cl Prolem [ point]: Regulr Expreion nd Finite Automt Develop regulr expreion (RE) tht detet the longet tring over the lphet {-} with the following
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 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 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 informationCompositional Specification of Functionality and Timing of Manufacturing Systems
Compositionl Speifition of Funtionlity nd iming of Mnufturing Systems Brm vn der Snden, João Bstos, Jeroen Voeten, Mr Geilen, Mihel eniers, wn Bsten, John Jobs, nd mon Shiffelers Eindhoven University of
More informationCompositional Specification of Functionality and Timing of Manufacturing Systems
Compositionl Speifition of Funtionlity nd iming of Mnufturing Systems Brm vn der Snden, João Bstos, Jeroen Voeten, Mr Geilen, Mihel eniers, wn Bsten, John Jobs, nd mon Shiffelers Eindhoven University of
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 informationCARLETON UNIVERSITY. 1.0 Problems and Most Solutions, Sect B, 2005
RLETON UNIVERSIT eprtment of Eletronis ELE 2607 Swithing iruits erury 28, 05; 0 pm.0 Prolems n Most Solutions, Set, 2005 Jn. 2, #8 n #0; Simplify, Prove Prolem. #8 Simplify + + + Reue to four letters (literls).
More informationBehavior Composition in the Presence of Failure
Behior Composition in the Presene of Filure Sestin Srdin RMIT Uniersity, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Uni. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re t
More informationLearning Partially Observable Markov Models from First Passage Times
Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).
More informationRegular languages refresher
Regulr lnguges refresher 1 Regulr lnguges refresher Forml lnguges Alphet = finite set of letters Word = sequene of letter Lnguge = set of words Regulr lnguges defined equivlently y Regulr expressions Finite-stte
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 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 informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationCompression of Palindromes and Regularity.
Compression of Plinromes n Regulrity. Kyoko Shikishim-Tsuji Center for Lierl Arts Eution n Reserh Tenri University 1 Introution In [1], property of likstrem t t view of tse is isusse n it is shown tht
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 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 informationSolutions - Homework 1 (Due date: September 9:30 am) Presentation and clarity are very important!
ECE-238L: Computer Logi Design Fll 23 Solutions - Homework (Due dte: Septemer 2th @ 9:3 m) Presenttion nd lrity re very importnt! PROBLEM (5 PTS) ) Simpliy the ollowing untions using ONLY Boolen Alger
More informationChapter 2 Finite Automata
Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationFormal Methods for XML: Algorithms & Complexity
Forml Methods for XML: Algorithms & Complexity S. Mrgherit di Pul August 2004 Thoms Shwentik Shwentik XML: Algorithms & Complexity Introdution 1 XML Exmple Doument Composer Nme Clude Debussy /Nme Vit Born
More informationReversible space-time simulation of cellular automata. J r me O. Durand-Lose 1
Lortoire Bordelis de Reherhe en Informtique, ur nrs 304, Universit Bordeux I, 35, ours de l Li rtion, 33 405 Tlene Cedex, rne. Rpport de Reherhe Num ro 77-97 Reversile spe-time simultion of ellulr utomt
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
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 information