Discrete-Time Scheduling under Real-Time Constraints

Transcription

1 Dcr- chdulng undr Ral- onran Eduard rny Yuk Wang Moapha Aboulhad aboraor AO Dép d IRO Unvré d Monréal Dp EE oncorda Unvry Monréal Québc anada Abrac W nroduc a n hod for chdulng undr ral- conran ha uabl for ynchronou y plnaon h npu pcfcaon n h for of ng dagra n hch h occurrnc of gnal ranon or acon ar rlad by lnar conran xprng h aupon on h npu acon h nvronn and h con on h oupu acon Provdd ha h pcfcaon caual gv an algorh for drvng AAP and AAP rlav chdul for h oupu acon W hn prn a n algorh for drnng hhr a gvn clock prod corrc Bad on a chdul and a vald clock prod ranfor h pcfcaon no a dcr rlav chdul uch a chdul rv a h ba for plnng a ynchronou a-achn conrollr Kyord: ng dagra rlav chdulng ral- conran ynchronou a achn Inroducon Hgh-lvl ynh of dgal hardar for ral- applcaon 0 a ll a h ynh of nrfac randucr and conrollr 4 rqur o prfor chdulng of acon and opraon undr xplc ral- conran In opraon h unknon duraon r odld ung unboundd dlay fro h ar of uch an opraon o h ar of any uccor opraon ha dpndd on nar ng conran r ud o rrc h occurrnc of opraon rlav o ach ohr nc lnar conran ay po rrcon on h paraon of h anchor opraon hu akng h pcfcaon unralzabl for all pobl duraon of h anchor opraon h auhor of dfnd h o-calld ll-podn condon of h ng conran of h pcfcaon In nrfac randucr ynh h uaon o o xn lar n ha vn ar conrand by ng onran lnar la and arl 4 7 Hr vn uually rprn gnal ranon or nablng uaon for daa o appar on bu h ynh ha hovr uch n coon h h probl of chdulng undr ral- conran ud n hgh-lvl ynh In h papr dcrb a rlav chdulng hod for pcfcaon nprd by ng dagra a foralzd n In uary h an dnguhng characrc of our approach ar a follo: - Inad of dalng h opraon ha hav duraon and gnal ranon vn condr only nananou acon h hn can b ud n par o dl h ar and h nd of an opraon or o odl ndvdually o pcfc vn or gnal ranon n h y h approach ha bn qu ffcv n odlng y ung ral- proc algbra - W xplcly dnguh oupu or nrnal acon ho occurrnc undr h conrol of h ynhzd dvc and npu acon ho occurrnc conrolld by h nvronn In boh ca h occurrnc of an acon ay b rrcd by h occurrnc of o prcdng acon npu or oupu W allo lnar la and arl yp of ng conran hovr n h papr condr only lnar conran nc h ar h ourc of noncaualy n D pcfcaon and rqur pcal chnqu for drvng chdul - o provd an for dcrbng ng aupon ha can b ad on h occurrnc of npu acon and h l on h racon of h oupu acon dnguh o knd of ng conran au and co rpcvly h allo o a ha aupon ar ad on h nvronn and hn ak h nforaon no accoun durng ynh aupon-bad raonng; h or ralc han aung ha any duraon of ay an npu opraon pobl - Allong fn aupon conran rqur a gnralzaon of h ll-podn condon of h y of conran W hav rcognzd h n h conx of nrfac pcfcaon and hr copably vrfcaon 5 a a probl of caualy aualy condon allo u o u a or coplx rucur of a pcfcaon han n uch ha h npu and oupu acon can b nrxd a n ng dagra h conrbuon of h prn papr ar: - A hod prnd for drvng a rlav chdul for oupu and nrnal acon rlav o h o-calld rggr acon 5 W can guaran h xnc of h chdul only f h pcfcaon caual

2 - h ng conran can b gvn n dn h ral hovr hardar plnaon ofn carrd ou ung ynchronou dgn chnqu hr all opraon ar ynchronzd o a coon clock of prod W prn a hod for vrfyng ha a prod vald and hn produc a dcr- rlav chdul for h oupu acon n hch h un h clock ck uch a chdul can b ud n xng ynh hod g 0 h hodology cnrd around h o calld Hrarchcal Annoad Acon Dagra HAAD 7 8 af dagra corrpond o h ng dagra ud a h ba for ynh n h papr h hrarchcal dagra for or coplx bhavor ung copoon opraor ovr laf and ohr hrarchcal dagra h opraor ar nprd by proc algbra paralll h caual rndz-vou councaon concanaon loop dlayd choc and xcpon handlng All dagra can alo b annoad by VHD procdur varabl and prdca For a uful ub of h pcfcaon languag hav dfnd h foral anc and provdd axoazaon 9 ncludng convron o a noral for ha can b ud o ranfor h pcfcaon o a nork of d Auoaa for vrfcaon ung xng ool Furhror h vrfcaon of caualy and copably on laf-lvl dagra can b ffcnly carrd ou ung onran ogc Prograng bad on Rlaonal Inrval Arhc 0 h papr organzd a follo: con nroduc bac concp abou ng dagra caualy and chdulng con 3 dcrb h drnaon of a vald clock prod and h drvaon of a dcr- rlav chdul; alo conan a copl xapl con 4 conclud h prnaon Background Inforaon ng dagra and h caualy propry In h con nroduc o background concp abou ng dagra vn graph of ng dagra and h caualy propry 5 7 Dfnon : hr a global varabl ha ncra onooncally h currn h valu of h varabl Inally h global varabl r o o ral valu τ E { n }b a of vn A -ap varabl aocad h ach vn ; x an ha occur hn h valu of h varabl bco x urrn and ap ak on fn pobly unboundd ral valu Dfnon : E { n } b a of vn c l u rprn h conran l - u on h paraon bn h occurrnc of E W call h ourc and h nk of h conran A conran c l u a prcdnc conran f u l > 0 and a concurrncy conran f u 0 l E n E h of all npu vn and E ou E h of all oupu vn E n E ou E E n E ou Dfnon 3: A ng dagra D E drnd by of vn E and h of conran ovr E A conran c l u uch ha and ar of dffrn drcon u b a prcdnc conran h conran n hav on of o pobl nn I an au conran f h nk an npu vn; ohr a co conran Au conran dl h xpcd or aud bhavor of h nvronn hl h co conran dfn h l on h occurrnc of h oupu vn In ohr ord h dvc plnaon u afy h co conran provdd ha h nvronn af h aupon Dfnon 4: ondr a ng dagra D E h c l u h vn graph EG V Eg of D a drcd ghd graph hr h vrx V E and for ach conran c l u n D hr ar o dg n E g labld by u and labld by -l ; h labl calld h gh of h dg An dg n EG labld by gh rprn h conran ha a o-dd conran n D rprnd by o on-dd conran n EG Exapl : Fgur a ho a D hr E ou { 3 } E n { 4 } h conran l u and 3 l 3 u 3 ar au conran h ohr o ar co conran I corrpondng EG gvn by Fgur b Au o n l u l 4 u 4 -l -l 4 u ou 4 ou u 4 u 34 4 l 3 u 3 -l 3 u 3 -l 34 3 n a Fgur : a apl ng Dagra D 3 b b corrpondng Evn graph

3 A pah p n EG fro o a qunc of dg k- k k A cycl a pah p h gh of a pah h u of h gh of all dg along h pah h hor pah fro o h pah ho gh h all fro aong all h pah fro o h conran y conn f hr no cycl h ngav gh n EG ohr nconn 3 An vn graph can b chckd alon for conncy hch a nal for of ralzably I aur ha h conran y ha a oluon and hu an occurrnc can b agnd o vry vn A pond ou n h noon of conncy of h vn graph of a D nuffcn for conrucng corrc plnaon In fac nconncy a pcal ca of non-caualy W no nroduc caual D 5 7: Dfnon 5: In an EG h axu paraon fro vn o vn dfnd a ax - hr and afy h D conran 9 If ax - < 0 hn vn rcly prcd vn I ll knon ha h axu paraon fro o h hor danc fro o n EG 3 W dcrb nx h xcuon anc of h odl undrlyng a D pcfcaon hy ar bad on h noon of a block of vn Dfnon 6: ondr h vn graph EG of a ng dagra D E {EB } b a paron ovr h vn E EB EB EB EB E n or EB E ou Each EB calld an vn block E { k EB l EB uch ha hr an dg fro k o l or fro l o k } E conan all vn fro EB rlad by a conran o o vn n EB h block EB h prdcor of h block EB dnod by EB prdeb f vn E rcly prcd all vn EB ax - < 0 In h ca h vn n E ar calld h rggr of EB n EB h local conran of EB ar ho conran of ha rla par of vn n EB or rla vn n EB o rggr h paron { EB }u alo afy h follong propry: Propry : For all par of block EB EB { EB } f E hn hr EB prdeb or EB prdeb In a conn h rlaon prd nduc a paral ordr < ovr block An vn block EB nabld hn all rggr vn hav occurrd EB bco nabld a f h la rggr occurrd a An nabld vn block EB fxd hn h occurrnc of all vn ar agnd a valu uch ha h local conran of h block ar afd gvn h occurrnc of h rggr If no uch agnn x hn h block canno b fxd Dfnon 7: A paron { EB }of EG caual ff af Propry and vry vn block can b nabld and fxd A D caual f EG ha a caual paron hor 5 7: A D caual ff for ach par of rggr of ach block h axu paraon bn h rggr a copud ung h local conran of h block rcly grar han h axu paraon of ha par copud ovr h nr EG In h r of h papr au ha all D ar caual h a paron { EB } chdulng of vn undr D conran In h prcdng con nroducd o knd of conran n D: au and co W can chdul only h oupu vn conrolld by h co conran nc h npu vn rlad by h au conran ar conrolld by h nvronn In h con prn chdulng algorh for oupu vn of a caual D o of h concp nroducd hr ar lar a n 0 Dfnon 8: A chdul of a D a funcon ha agn an occurrnc o ach oupu vn uch ha all co conran n h ng dagra ar afd gvn any occurrnc of h npu vn afyng h au conran and h occurrnc of prcdng oupu vn uch an agnn of occurrnc calld a vald agnn W fr dcrb a hod o fx an oupu block aung ha all rggr hav occurrd h copl chdul for a caual D can hn b oband block by block follong any oal ordr drvd fro h paral ordr bn block con ondr a block EB { } h rggr r {r } and l h occurrnc of rggr r b h occurrnc of vn EB a funcon of gvn by h local conran of EB: k k k and k b h hor danc fro r o k and h hor danc fro o r k k a : For any vn EB and a rggr r of EB h follong rlaon hold: - and - hr and ar h gh bn and r For any o vn and n EB and an dg fro o h gh h rlaon and hold 3

4 Proof: h follo drcly fro h dfnon of h hor pah nc a pah gh and n a conn conran graph u hav 0 no ngav cycl larly for uppo ha < hrfor h hor danc fro r o no bcau h pah undrlyng and h dg fro o for a horr pah - conradcon QED a : In a caual D for all vn EB rggr and h local conran of EB h follong hold: ax { } n{ } hr h occurrnc of rggr r r r r r Proof: r and r b any o rggr of EB and EB h danc of h hor pah fro r o r ung local conran of EB and pang hrough nc h y caual by hor hav - < hch nduc h condon - < h hold for any par r r ; hrfor ax { } n{ } QED r r r r Propoon : For all vn EB and all rggr r r of EB h follong hold: ax { } n{ } r r r r Proof: For any rggr r and vn - r r hnc n{ } W can r r prov h ohr half of h nqualy n a lar fahon Q E D orollary : For all EB n{ } a vald occurrnc agnn o ax{ } r r hr n or ax ud for all bu no xd hn on block n gnral Proof: W nd o prov ha for any rggr r and any par of vn and h follong condon ar afd - and - and - W fr prov nc n{ } hr x rggr r a and R b uch ha r r r r a n{ } a and b b n{ } Bad on h dfnon of and hav r r r r a a b b b b a a and hn - a a - a a a - a and - b b - b b b - b boh by a Hnc provn Nx prov r b an arbrary rggr of EB nc a a n{ } hav r By a ax { } rr r r Hnc - I follo ha - - and - Q E D Dfnon 9: Dno h hor danc fro k o r a - k r k and b h hor danc fro r o a k k r k h nrval ax{ r } n{ r } calld h fabl nrval of dnod by Morovr and ar calld h a-oon-a-pobl AAP and h a-la-a-pobl AAP yp of rlav chdul of h oupu vn rpcvly 3 Dcr- chdul W dcu hr h convron of h dn- D pcfcaon no a of dcr- rlav chdul ha can b plnd ung apld npu ynchronou fn a achn ynchronzd by a clock of prod uch a achn can b ud a h nrfac conrollr bn h nvronn and a ynchronou dvc ha run fro h a clock a h conrollr or a a conrollr n hgh-lvl ynh undr ral- conran f h vn of a D rprn h acvaon and dacvaon of o hgh-lvl opraon Fgur llura a apld npu ynchronou FM W ll gv an algorh o drn hhr a clock prod of h FM vald for plnng h conrollr hn hall prn an algorh for ranlang h ng dagra pcfcaon no chduld vron n dcr hr h un a clock ck 4

5 ynchronzr Prary npu P I prn a P clock obnaonal crcu clock nx a N Prary oupu P O clock Fgur : A apld npu Moor FM nc n gnral h npu ar no ynchronou h h FM clock hy u b fr ynchronzd W au ha h pl ynchronzr ud conng of a ynchronou aplng rgr ha nroduc a on-cycl dlay h propod oluon can b adapd o h ca hr a or coplx ynchronzr ha nroduc a dlay of k > cycl ud In ordr no o any npu gnal ranon u ak ur ha h clock ha a uffcnly hor prod o apl h npu gnal bn any o concuv chang a dfnd by h au conran of h D For conrollng h oupu gnal n a pcfd by h D alo plac a rgr a h oupu o a o hav br conrol ovr h cobnaonal oupu dlay Du o h rgr any oupu chang u b chduld a la on clock cycl afr dcng h la rggr vn h npu o h FM ar h apld npu valu I an ha npu vn u b drnd by xanng h dffrnc bn concuv apld npu valu Evn hough can hu dc h occurrnc of npu vn canno drn hr xac occurrnc ; hn o nrval drnd by h D and h clock prod In h nx con ho ho o fnd h nrval and ho o drn hhr a gvn clock prod vald 3 lock Prod Drnaon In h hod nroducd n h clock prod pcfd by h dgnr and h ool o vrfy ha h prod conn h h conran Hovr hr no algorh gvn o do ha A lar probl x n 3 hr h ynh of d VHD proc dcud In 8 o drn f vald h a achn of h conrollr u b conrucd fr h coplx ynh ak hu u b carrd ou o fnd ou ha h oluon nfabl In addon h hod canno handl lnar au and co ng conran In our approach h valdy of drnd by analyzng h ng conran only Rcall ha an vn block EB { n } ha a rggr r {r r r } An occurrnc agnn o vn of an vn block a funcon ha drn h valu of ach uch ha all h local conran of h ng dagra ar afd gvn h occurrnc of h rggr of h block o drn ha a nubr a vald clock prod hav o chck hhr hr an occurrnc agnn ha af all h conran h rpc o W hu condr h follong o quon: Drn f a vald clock prod and fnd a dcr- chdul n hch h un of Bad on h oluon o can u a bnary arch o fnd h larg vald Evry ru rggr occurrnc ha an aocad aplng a hch dcd b h aplng of r ho ral occurrnc h ru rggr and h aocad aplng u afy h follong of conran hr r and r ar arbrary rggr > 0 an ngr 0 < 3 Rlaon an ha h aplng of rggr can only happn a ulpl of h clock prod Rlaon a ha h dffrnc bn h aplng and h ral of a rggr n h nrval 0 and Rlaon 3 conran h dffrnc bn o dffrn rggr occurrnc o b n h nrval a gvn n h vn graph Dfnon 0: h of pobl ru rggr aocad h ach aplng { 0 ; } < ax b h la valu uch ha for any 5

6 ax n b h gra valu afyng n Exapl : ondr h vn graph hon n Fgur 3 and au ha ach vn on a dffrn por r and r b npu vn and O an oupu vn Whou lo of gnraly can au ha h aplng of h npu r 0 3 h aplng of r can b 3 6 or 9 rlav o 0 r - -5 r Fgur 3: Evn Graph of Exapl For 0 and 3 h ru rggr and afy 5 7 fro 3 and 3 < 0 and 0 < 3 fro h aocad ru rggr 03 { < 0 0 < 3} h ax 03 3 n 03 ax03 - n 03-3 larly for 0 and 6 h ru rggr 06 { < 0 3 < 6} hrfor ax 06 6 and n 06 3 ax03 0 n03-3 Fnally for 0 and 9 h ru rggr 09 { < 06 < 9 } yldng ax 09 7 and n 09 6 a hon by h rangl GFE ax03 0 n03 - nc chdulng h occurrnc of any oupu vn EB { n } can b don only n ulpl of rlav o h aplng rggr u o rprn h ynchronzd occurrnc of h oupu h follong u hold for any rggr r { r r r} : k 4 h a ha all h pobl rggr n har h a chdul for h oupu vn; orovr h dffrnc bn h oupu vn and h aplng of h rggr a ulpl of h clock prod I follo fro 4 ha for any o oupu vn and dvbl by h conran can hu b odfd a h gh can b changd o O Fnally h local conran of h block u b afd For any oupu vn and and any rggr r h follong rlaon u b afd: 5 6 Bad on h rul of oupu chdulng Propoon for ach rggr uch ha h pobl occurrnc agnn o an oupu vn u hu b n h nrval ax { r } n{ r } hrfor f o b a vald clock prod hn for ach oupu vn hav ax{ r } n{ r } hr h nrcon of nrval dfnd a ab cd axac nbd If axac > nbd hn ab cd I follo ha hr ax{ r } n{ r } df n n 6

7 7 } } { ax ax{ }} {ax{ ax r r n } {ax ax r for any ax and } } { n n{ }} {n{ n r r n } {n n r for any n Propoon : Gvn a of aplng }} { { r r r r f for all oupu vn h rlaon 7 afd hn h } { EB a vald AAP occurrnc agnn for h vn n h oupu vn block If 7 do no hold hn hr no occurrnc ha af all h conran Proof: If > hn no valu n h nrval dvbl by hrfor n h ca hr no vald occurrnc agnn for a a ulpl of rlav o h apld occurrnc of h rggr No ha h AAP chdul xprd n clock cycl u b grar or qual o on o ak no accoun h dlay nroducd by h oupu rgr uppo no ha 7 hold W nd o prov ha h } } { { n EB a vald occurrnc agnn for h vn n EB for any rggr r and any par of vn and h follong condon u b afd: ; and Hovr r } {ax ax and hu hr x rggr r uch ha } {ax r and r uch ha ax r hr and ar o non-ngav nubr l han uch ha and can b dvdd by I follo ha : ax ax r r ax ax r r hrfor for an arbrary rggr r n ax ax r r r n Bad on and h follong dducon ay o follo ax ax r r r r On h ohr hand and ar dvbl by < hu W no prov Bad on h follong nqual hold r r ax n r r QED orollary : For o b a vald clock prod condon 7 u b afd for all pobl aplng of h rggr

8 Exapl 3: ondr agan h vn graph n Fgur 3 W h o drn hhr 3 vald h pobl rggr aplng ar 0; 3 6 and 9 For ach pobl vrfy 7 ung Propoon : 03 o ax{ ax03 ax03 5} ax{- 35}9 03 o o n{n03 4 n03 8 n o ax{ ax06 ax06 5} ax{0 65} 06 o o n{n06 4 n06 8 n nc 3 * 06 o3 > 06 o3 hr no fabl agnn for o: hn h aplng rggr ar 0 6 h ru rggr could b 0 6; hrfor o afy h o conran h oupu ha o b grar han and dvbl by 3 hch Furhror h ru rggr could alo b -5 3 If h oupu h dffrnc bn r and o 45 >4 volang h axu bound of 4 on h paraon bn o and r If chang h conran by rplacng o r - by o r -; r o 8 by r o 0 r o 4 by r o 5 can hn vrfy ha 3 bco a vald clock prod 03 o ax{ ax03 ax03 5} ax{- 35}0 03 o o n{n03 5 n03 0 n o ax{ ax06 ax06 5} ax{0 65} 06 o3 4 o n{n 5 n 0 n o ax{ax ax 5 ax{0 7 5} 09 o o n{n 5 n 0 ax{ 56 0} r - -5 r Fgur 4: Modfd vn graph hrfor h agnn o af all h pobl npu aplng O o drn h occurrnc of h vn n an oupu block gvn a vald only nd o kno h axu paraon r and r bn h oupu vn and h rggr hr h axu paraon ar copud ovr h co conran h gh adud o ulpl of h clock prod a Onc h axu and nu paraon ar copud hr no nd o kp h conran bn h oupu vn bcau h occurrnc agnn bad on h axu and nu paraon af all h orgnal ng conran bn h oupu vn provdd ha h npu afy all h aupon and h D caual I follo ha can odfy h vn graph o ha h conran bn h rggr and h oupu vn ar of h for r and r h rulng D hu ha no oupu o oupu conran n h a vn block bu all h vn u b chduld a AAP or a AAP h abov proc uarzd n h follong algorh ha copu h occurrnc chdul for oupu vn n a caual D gvn a vald clock prod Algorh Gvn a vald odfy h conran bn oupu vn and fro o For ach oupu block copu h axu paraon r and r of ach vn h rpc o h rggr a dfnd by h local conran of h block Rplac h conran bn vn and hr rggr by h axu paraon rlav o h rggr Rov all conran bn oupu vn n h a block 8

9 3 For ach oluon of quaon and 3 vrfy ha condon 7 hold If 7 do hold for all ca can chdul AAP h occurrnc a { EB} Exapl 4: Nx ho a or coplx xapl h D and block rucur hon n fgur 5 W apply h abov algorh o drn f 0 a vald clock prod and f y hn copu h dcr- rlav AAP chdul for h oupu vn o o o3 o o o o Fgur 5: Block rucur of xapl 4 Fgur 6: Modfd graph oupu parad by ulpl of clock prod Accordng o p odfy h conran bn oupu vn o h ulpl of 0 and oban h vn graph of Fgur 6 In h nx p calcula h hor danc bn h oupu vn and h rggr and hn odfy h conran o h oupu vn Fgur 7 gv h n odfd D h block rucur no hon n h D hovr h a a n Fgur o 5 60 o o 0 60 o o4 Fgur 7: Modfd D afr p of h algorh Whn 0 hav o pobl aplng for h npu vn and hch ar 0 0 and 0 hrfor hav ax00 0 n00-0 ax00 0 n00-0 ax00 0 n00-5 ax00 5 and n00 0 h chdul of h oupu vn o o and o 3 ar drnd a follo No ha h conran for o and o ar xacly h a hrfor only nd o calcula on of chdul o ax{ax o ax o } ax{0 50 5} o n{n00 o n00 o } n{ } 0 W can fnd h ohr valu n a lar ay: 00 o o o 0 00 o 5 00 o o 3 55 hrfor h AAP chdul : or 3 W nx condr h 0 block h only on vn 3 Gvn 0 or h aplng of or 6 Ung AAP chdulng hav 4 oncluon In h papr a n ay for chdulng vn undr ral- conran and for h ynh of nrfac conrollr bad on ng dagra pcfcaon a dcrbd h hod allo o drn a vald clock prod and uabl for ynchronou y plnaon An algorh for drvng AAP and AAP rlav chdul 9

10 for h oupu acon a prnd for caual pcfcaon Expcd applcaon of h hod rang fro hgh-lvl ynh o h ynh of apld-npu ynchronou nrfac conrollr Rfrnc M McFarland A Parkr R apoano "h hgh-lvl ynh of dgal y" Proc of h IEEE No Fbruary 990 G Borrllo R H Kaz "ynhzng randucr fro nrfac pcfcaon" VI 87 Norh Holland J Brzozok Gahlngr and F Mavadda "onncy and afably of Wavfor ng pcfcaon" Nork Vol pp G Borrllo "Foralzd ng Dagra" Proc Euro-DA 9 pp nk "Exndd ng Dagra a a pcfcaon languag" Proc Euro-DA 94 pp R chlor "A provr for VHD-bad hardar dgn" Proc IFIP HD K McMllan and D Dll "Algorh for nrfac ng vrfcaon" Proc IEEE ID 99 8 E Walkup G Borrllo "Inrfac ng Vrfcaon h Applcaon o ynh" Proc DA Yn A Ih A aavan W Wolf "Effcn Algorh for nrfac ng vrfcaon" Euro-DA G D Mchl ynh and Opzaon of Dgal rcu McGra-Hll Inc N York 994 W Gra Grob nk and W dann "ng dagra a a pcfcaon languag for nrfac crcu and hr ranforaon no ynchronou FM" BENEFI-DMM 95 pp80-35 p 995 W dann "An approach o ul-paradg conrollr ynh fro ng dagra pcfcaon" Euro- DA P Gubrl W Ronl "Inrfac pcfcaon and ynh for VHD Proc" Euro-DA K Khordoc E rny "Modlng cll procng hardar h acon dagra" Proc IA K Khordoc E rny anc and Vrfcaon of ng Dagra h nar ng onran accpd o AM ranacon on Dgn Auoaon of Elcronc y ODAE May p 6 P Mochlr H Aann F Pllandn "Hgh-vl Modlng ung Exndd ng Dagra" Proc Euro- VHD '93 Haburg FRG p 993 pp K Khordoc Acon Dagra: A Mhodology for h pcfcaon and Vrfcaon of Ral- y PhD h Dp of Elcrcal and opur Engnrng McGll Unvry March W-D dann "Inroducng lock ycl" Rpor OPRODEUPA995 Unvry of Paau Nov995 9 B Brkan Gandrabur E rny Algbra of ouncang ng har for Dcrbng and Vrfyng Hardar Inrfac Proc IFIP onf on opur Hardar Dcr anguag HD P Groda E rny WJ Oldr olvng nar Mn and Max onran y Ung P bad on Rlaonal Inrval Arhc J on hor op cnc 73 Fb97 P Groda E rny Inrfac ng Vrfcaon h Dlay orrlaon Ung onran ogc Prograng ED& 97 D Ku G D Mchl Rlav chdulng undr ng onran: Algorh for Hgh-vl ynh of Dgal rcu IEEE ran AD I 6 Jun 99 pp Acknoldgn: h ork a parally uppord by an Mcron Gran No 4M 0