Synthesis of asynchronous control circuits with automatically generated relative timing assumptions

Transcription

1 Syntsis o asynronous ontrol iruits wit automatially gnrat rlativ timing assumptions 1 Jori Cortalla, 2 Mial Kisinvsky, 2 Stvn M. Burns an 2 Kn Stvns 1 Univ. Politènia Catalunya, Barlona, Spain an 2 Stratgi CAD La, Intl Corporation, USA Astrat Tis papr sris a mto o syntsis o asynronous iruits wit rlativ timing. Asynronous ommuniation twn gats an mouls typially utilizs ansaks to nsur untionality. Rlativ timing assumptions in t orm vnt a ours orvnt an us to rmov runant ansaks an assoiat logi. Tis papr prsnts a mto or automati gnration o rlativ timing assumptions rom t untim spiiation. Ts assumptions an us or ara an lay optimization o t iruit. A st o rlativ timing onstraints suiint or t orrt opration o t iruit is ak-annotat to t signr. Eprimntal rsults or ontrol iruits o a prototyp ia32 instrution lngt oing an string unit all RAPPID ( Rvolving Asynronous Pntium R Prossor Instrution Dor ) sows signiiant improvmnts in ara an lay ovr sp-inpnnt iruits. 1 Introution Asynronous ommuniation utilizs ansaking to nsur untionality tat rquir som ara an lay pnalty wit rspt to synronous sign. Timing inormation an us to omat t ull ansak ovra in ara an lay y rmoving runant ansaks an assoiat logi. Sin asolut timing inormation is mostly unknown until layout is omplt, rlativ timing inormation in t orm vnt a ours or vnt is a natural rprsntation o timing tat an us in t sign low. Rlativ timing (RT) was us or sign o a prototyp ia32 instrution lngt oing an string unit all RAPPID ( Rvolving Asynronous Pntium R Prossor Instrution Dor ) tat was ariat an tst sussully [15, 16]. Silion rsults sow signiiant avantags - in partiular, prorman o instrutions pr ns - wit managal risks using tis sign tnology. RAPPID aivs tr tims astr prorman an al t latny issipating only al t powr an rquiring a minor ara pnalty as a omparal 4MHz lok iruit. Anotr primnt wit a iruit as on timing assumptions is sri in [2]. T sign low or syntsizing rlativ timing iruits is as ollows. Rlativ timing assumptions ar provi y t usr or trat y t algoritm prsnt in tis papr. T iruits ar tn sign using t assumptions or ara an lay optimization. RT iruits an optimiz wit rspt to t untim iruits or two rasons: RT assumptions ru t st o raal stats an n inras t numr o on t ar stats or logi optimization o all signals. It is possil to tn t st o stats in wi a signal is nal witout anging t st o raal stats i otr nal signals ar known to or an ma astr tan t arly nal (a.k.a. lazy) signal. Tis aitional liility as loal on t ars tat an ir rom on signal to anotr. Tis work was suport y a grant rom Intl Corporation an was on uring a visit to SCL in summr X /99/$ IEEE. A (possily rla) sust o timing assumptions us or optimization is ak-annotat y t tool an om timing onstraints. Dirnt vali ntlists rquir irnt timing onstraints. T iruits ar tn sign to mt t rlativ orrings, or vrii tat t rstritions ar alray part o t lays in t systm. Mtos as on sparation analysis [6], gomtri timing [1], an rlativ timing an ploy or vriiation [12]. In [3] it is sown tat rlativ timing syntsis an automat using lazy transition systms in wi naling an iring rgions or signal transitions ar istinguis. Tis papr nans t mto o [3] in tr major ways. A mto or automati gnration o timing assumptions starting rom a sp-inpnnt (untim) spiiation is prsnt. Most o t timing assumptions us in RAPPID iruits an automatially trat. Only arittural or nvironmntal assumptions on t inputs n to spii y t usr. A mto or automati akannotation o RT onstraints suiint or t orrt opration o a iruit is vlop. A mto or timing awar stat noing is ploy. It rus t numr o stat signal an gnrats timing assumptions or stat signals i nssary. It as a signiiant positiv t on ot ara an prorman. Stion 2 prsnts asi tory an mols. Stion 3 sri a mto or automati gnration o RT assumptions. Stion 4 prsnts tniqu or trating timing onstraints or a riv RT ntlist an rily sri timing-awar stat noing. Stion 5 prsnts primntal rsults. 2 Basi notions For rvity, w assum t rar to amiliar wit Ptri nts, a ormalism us to spiy onurrnt systms. W rr to [9] or a gnral tutorial on Ptri nts. Lazy transition systms an lazy stat graps wr introu in mor tail in [3]. 2.1 Transition Systms an Stat Graps A transition systm (TS) is a quarupl [13] TS = (S;E;T;s in ),wrsis a non-mpty st o stats, E is a st o vnts, T S E S is a transition rlation, an s in is an initial stat. T lmnts o T ar all t transitions o TS an will otn not y s! s insta o (s;;s ). Stat Graps ar inary intrprt transition systms: vry stat is assign a inary vtor o signal valus in t spii iruit; vry vnt is intrprt as a rising (a+) or alling (a,) transition o a signal a. Notation a is us i on is not spii aout t irtion o t signal transition. T st o signals o an SG is all X = I [ O, wr I an O not t st o input an output signals rsptivly. A laling untion v : S!;1g n assigns a vtor o signal valus to a stat (n = jxj). W will all v a (s) t valu o signal a in stat s. An SG is onsistnt i rising

2 z+ - + z- g (a) y- (a) a s a s1 s2 s11 s3 s12 s4 s5 s13 s6 s14 s15 s7 s16 s8 s9 s17 s18 g s1 s19 g s2 s21 g s22 s23 Fig. 1: (a) Ptri nt, () Transition Systm. y- z+ - z () + 1 z y y () () z z () y z+ z Fig. 2: (a) STG or t ampl, (,) SGs wit timing omains, (,,) Ciruits. an alling transitions altrnat or vry signal on any pat in t SG. An ampl o a TS an a SG ar givn in Figur 1.() an Figur 2.(), orrsponingly. 2.2 Signal Transition Grap A Signal Transition Grap (STG) is a Ptri nt (PN) in wi transitions ar lal wit rising an alling signal transitions lik in a SG. An ampl o a PN is sown in Figur 1.(a). Tis PN orrspons to a TS in Figur 1.(). An STG as an assoiat SG in wi a raal marking orrspons to a stat an a transition twn a pair o markings to an ar lal wit t sam vnt o t transition. Figur 2.(a) pits an STG wit tr signals, ;y,anzorrsponing to t SG in Figur 2.(). For simpliity, plas wit only on input an output transitions ar otn omitt in STGs. 2.3 Eitation an quisnt rgions T itation rgion o an vnt a, not y ER(a ),is t st o stats su tat s 2 ER(a ), s!. a Tquisnt rgion o a+, not y QR(a+), is t st o stats su tat s 2 QR(a+), v a (s) =1 ^ s 62 ER(a,). Similarly, s 2 QR(a,), v a (s) = ^ s62 ER(a+). In Figur 2.(), ER(,) =11;111g an QR(,) =1;11;1g. 2.4 Lazy transition systms T main istintiv atur o a lazy systm is tat it an assum a non-zro lay twn naling o transition an its iring. Du to tis, t st o stats in wi a transition is nal migt largr tan t st o stats in wi t transition irs. Dinition 2.1 (Enaling an iring rgions) T naling rgion, EnR(a), oa signaltransitiona is a t st o stats in wi transition a is nal. T iring rgion, FR(a), o a signal transition a is t st o stats z- () 1 y () z rom wi a an ir, i.. s 2 FR(a),9s :s a!s. A potntially naling rgion, PEnR, givs an uppr oun or a st o stats wi an slt as an atual naling rgion in t RT-implmntation. T rom in oosing t naling rgion witin t PEnR givs aitional possiilitis or logi optimization. It is asy to s t ollowing orrsponn twn t introu rgions: FR(a) EnR(a) PEnR(a). W will r isussion o ampls until Stions Dinition 2.2 (Lazy stat grap) A transition a is all lazy i EnR(a) 6= FR(a). A stat grap is all lazy (lazy SG) i at last on transition is lazy 1. T orrtnss proprtis o SGs an asily transrr onto lazy SGs. A lazy SG is onsistnt, trministi an ommutativ i t unrlying SG as ts proprtis. Prsistny proprty must gnraliz or naling an irings as isuss in Stion Timing assumptions Timing assumptions oul onsrvativly in in t orm tlling tat on vnt is appning or or atr anotr. Dirn assumptions. A irn assumption < a (ras or a), involving two potntially onurrnt vnts a an, assums tat, u to rtain timing aratristis, wnvr an a ar ot nal, always irs arlir tan a. InanSG tis assumption an rprsnt y t onurrny rution o a wit rspt to. RT irn assumptions allows on to liminat stats unraal in timing omain similar to stat limination as on asolut timing inormation in [1, 11]. Ty ar not suiint owvr or prssing lazy avior o signals. Early naling assumption. Suppos tat transition a triggrs t iring o transition,i..aan ar orr in t spiiation. Assum tat a an ma astr tan in t iruit. Tn t naling o an start arlir,.g., rom t vnts triggring a, an t propr orring o a or will still nsur y t timing proprtis o t implmntation. In lazy SG tis rsults in t akwar pansion o PEnR() into FR(a). Simultanity assumption. T simultanity assumption is a rlativ notion, wi is in on a st o onurrnt transitions T wit rspt to a rrn transition a. It tlls tat rom t point o viw o a t skw o irings tims o transitions rom T is ngligil. Tis assumption an viw as a loal unamntal mo o T wit rspt to a an n as a gnralization o urst-mo mains [14, 17]. An ampl o t appliation o simultanity assumption is isuss in Stion 4.2. Assumptions rlating only input vnts annot automatially gnrat rom t iruit avior an an provi y t signr or gnrat rom t implmntation o t nvironmnt. 2.6 Nt-stat untions T implmntation o an SG as a logi iruit is on troug t inition o t nt-stat untion or a output signal an inary vtor. For SGs itisinas ollows: ( 1 i9s2 ER(a+) [ QR(a+) s.t. v(s) =Z a (Z) = i9s2 ER(a,) [ QR(a,) s.t. v(s) =Z, otrwis 1 As w ar targt at optimization o output signals o a iruit lazy aviors o input signals is not onsir.

3 T nt-stat untion a is orrtly in wn t SG as t CSC proprty, i.. tr is no pair o raal stats (s;s ) su tat v(s) =v(s ) an (s 2 ER(a+) [ QR(a+) or s 2 ER(a,) [ QR(a,)). Not tat a is an inompltly spii untion wit a on t ar (DC) st orrsponing to tos inary vtors witout any assoiat stat in t SG. T logi ntlist is sp-inpnnt i SG is trministi, ommutativ an output-prsistnt[4]. In t SG o Figur 2.(), t DC st is mpty sin all inary vtors av a orrsponing stat in t SG. As an ampl, (11) =; y (11) = z (11) =1sin signals an y ar nal, an z is stal in tat stat. T Karnaug maps or t nt-stat untions ar pit in Figur 3.(a). For a lazy SG t nt-stat untions ar in irntly: ( 1 i9s2 FR(a+) [ QR(a+) s.t. v(s) =Z a (Z) = i9s2 FR(a,) [ QR(a,) s.t. v(s) =Z, otrwis Not tat tis inition gnrally givs mor on t ar vtors tat t inition or a SG u to two rasons: Mor stats ar unraal, sin timing assumption an ru onurrny Stats in (PEnR, FR) o not long to itr FR, or QR, an n ar inlu into t DC-st. As an ampl, in t lazy SG o Figur 2.(), (11) =,; y (11) = z (11) =1asplaininStion4.2. T onitions or sp-inpnnt implmntaility an trivially tn to lazy SGs. 2.7 Logi syntsis From t nt-stat untions o a SG, a sp-inpnnt iruit an riv y implmnting t oolan quation o a output signal as an atomi ompl gat [8] or as a gnraliz C-lmnts [1, 7]. For ampl, a spinpnnt ompl gat implmntation or t STG in Figur 2.(a) is a ntlist: = z + y; y = + z; z = + zy: Similarly, rom t nt-stat untion spiiation orrsponing to a lazy SG, an RT-iruit an riv in t orm o ompl gats or gnraliz C-lmnts as illustrat y an ampl in Stions Monotoni ovrs Not vry logi untion riv rom t inition o t nt-stat untion satisis azar-rom onitions, an n vali. T ollowing inition is rlat to azars in t avior o asynronous iruits. Givn two sts o stats S 1 an S 2 o an SG su tat S 2 S 1, w will say tat S 1 is a monotoni ovr o S 2 i or a transition s! a s : (s 2 S 1, S 2 ) s 2 S 1 ) ^ (s 2 S 2 ) s 62 S 1, S 2 ) Only monotoni ovrs o FRs an slt as EnRs or azar-r solutions or logi ntlist [3]. I S 1 = EnR(a) an S 2 = FR(a), tn (1) no isaling o a is possil an (2) tr ar no transitions rom FR(a) to EnR(a), FR(a), i.., no isaling o irings or a is possil itr. Hn, prsistny o a in t RT implmntation is guarant. For ampl, in t SG o Figur 2.(), t st 11;11;111g is a monotoni ovr o ER(,). Howvr, tst1;11;111g is not, sin t transition 1,! 11 violats t onitions or monotoniity. 3 Automati gnration o rlativ timing assumptions 3.1 Orring rlations Lt TS =(S;T;E;s ) a transition systm. Assum tat vry vnt in E orrspons to a singl onnt itation rgion. Dinition 3.1 (Conlit) An vnt 1 2 E isals anotr vnt 2 2 Ei9s 1! 1 s2 su tat s 1 2 ER( 2 ) an s 2 62 ER( 2 ). Two vnts 1 ; 2 2 Earinirt onlit i 1 isals 2 or 2 isals 1. Dinition 3.2 (Conurrny) Two vnts 1 ; 2 2 Ear onurrnt (not y 1 k 2 ) i ty orm a stat iamon, i.. 1. ER( 1 ) \ ER( 2 ) 6= /, 2. 8s 2 ER( 1 ) \ ER( 2 ) : (s 1! s 1 ) 2 T ^ (s 2! s 2 ) 2 T )9s 3 2S: (s 2!s3 1 )2T ^ (s 1!s3 2 )2T. Dinition 3.3 (Triggr) An vnt 1 2 E triggrs anotr vnt 2 2 E (not y 1,! 2 )i9s 1! 1 s2 su tat s 1 62 ER( 2 ) an s 2 2 ER( 2 ). Dinition 3.4 (Enal or) Lt 1 ; 2 2 E two onurrnt vnts. 1 an nal or 2 (not y 1 2 )i9s 1! s 2 su tat s 1 2 ER( 1 ),ER( 2 ) an s 2 2 ER( 1 ) \ ER( 2 ). Dinition 3.5 (Enal simultanously) Lt 1 ; 2 2 E two onurrnt vnts. 1 an 2 an nal simultanously (not y )i9s 1! s 2 su tat s 1 62 ER( 1 ) [ ER( 2 ) an s 2 2 ER( 1 ) \ ER( 2 ). Dinition 3.4 an tn to sts o vnts as ollows. Dinition 3.6 (Enal or a st o vnts) Lt 2 E an vnt pairwis onurrnt wit all t vnts in t st X = 1 ;:::; n ge. an nal or X(not y X) i 9s 1! s 2 su tat s 1 2 ER(),ER(X ), s 2 2 ER()\ER(X ) an 62 X, wr ER(X )=ER( 1 )[ :::[ER( n ). Figur 1. pits t transition systm riv rom t Ptri nt o Figur 1.a. T ollowing ats an riv using t initions aov: :(a k ), k,, 3, t. Evnt annot nal or ; g, ut an nal or ; ;gg sin tr is a transition s 9! s19 su tat s 9 2 ER(),ER(; ;gg), s 19 2 ER() \ ER(; ;gg) an 62 ; ;gg. 3.2 Dlay mol A lay mol or vnts prsnt in tis stion givs an inormal intuitiv motivation or t automati gnration o timing assumptions. Tis mol rrs to t lay o t vnts in t TS. T lay o an vnt is in as t irn twn its naling tim an its iring tim. Tr typs o vnts ar onsir: Non-input vnts: its lay is in t intrval [1, ε; 1 + ε] Fast input vnts: its lay is in t intrval (1 + ε; ) Slow input vnts: its lay is in t intrval [ ; ) T syntsis approa also assums tat (1) t lay o a gat implmnting a non-input vnt an lngtn y lay paing or transistor sizing, (2) t lay o two gats an always ma longr tan t lay o on gat.

4 Hn, on an assum tat ε < 1=3, (3) t iruit will nvr tak longr tan tim units (minimum lay o a slow input vnt) in oming stal rom any stat o t systm an a quisnt nvironmnt. T prvious assumptions on t timing avior o t iruit an translat into assumptions on t iring orr o t vnts. 3.3 Ruls or riving timing assumptions W prsnt ruls or riving timing assumptions in t ollowing ormat: (1) orring rlations tat must satisi in a (Lazy) SG or a rul to appli, (2) automati timing assumption tat an gnrat, an (3) inormal justiiation o a rul as on t aov lay mol Assumptions twn non-input vnts T ollowing ruls an appli or riving timing assumptions twn non-input vnts, 1 ; 2 ; 3 2 E: I. Evnt nal or anotr vnt. Orring rlations: ( 1 k 2 ) ^ ( 1 2 ) ^ ( ) ^ ( ). Dirn timing assumption: 1 irs or 2 Dlay assumptions: on gat sortr tan two gats. II. Evnts simultanously nal. Orring rlations: ( 1 k 2 ) ^ ( ) ^ ( ). Dirn timing assumption: 1 irs or 2 Dlay assumptions: lay o 2 longr tan lay o 1. III. Evnt triggr y vnts simultanously nal. Orring rlations: ( 1 k 2 ) ^ ( ) ^ ( ) ^ [( 1 ) 3 ) _ ( 2 ) 3 )]. Simultanity timing assumption: 1 an 2 simultanous wrt 3. Dlay assumptions: on gat sortr tan two gats. IV. Early (spulativ) naling or orr vnts. Orring rlations: ( 1,! 2 ). Early naling timing assumption: 1 irs or 2 (ut 2 an nal onurrntly wit 1 ). Dlay assumptions: lay o 1 sortr tan lay o 2. Lt us illustrat t prvious ass wit t ampl o Figur 1 assuming tat all vnts ar non-input. Timing assumptions o typ I an riv or t pairs o vnts (; ), (;g) an (;), wr t irst lmnt o t pair is assum to ir or t son. Timing assumptions o typ II an appli to t pairs (;) an (;). Timing assumptions o typ III an appli,.g., to t vnts triggr y t pair (;) tat triggrs t vnts, an g. Timing assumptions o typ IV an appli,.g., to t vnt triggr y t vnt. I tis assumption applis, tn potntial naling rgion or inlus stats s2;s5;s8;s12;s15;s18;s21g as on t ar stats or t valus o t nt stat untion or signal in aition to t originally prsnt stats s3; s6; s9; s13; s16; s19; s22g Assumptions twn non-input an input vnts Assum tat 1 ; 2 2 E ar a non-input an an input vnt rsptivly an ty ar onurrnt. V. Input not nal or non-input vnt. Orring rlations: ( 1 k 2 ) ^ Dirn timing assumption: 1 irs or 2. Dlay assumptions: lay o nvironmnt longr tan lay o on gat. Tis assumption ovrs t ons o typ I an II or t as in wi 2 is an input vnt. T lay assumption us in tis as stats tat t rspons tim o t nvironmnt will always longr tan t lay o on gat Assumptions twn non-input vnts an slow input vnts Assum tat 2 E is a slow input vnt, X = 1 ;:::; n g E is a st o non-input vnts an is pairwis onurrnt wit all t vnts in X. VI. Slow nvironmnt not nal or non-input vnts. Orring rlations: (8 i 2 X : k i ) ^ 6X. Dirn timing assumptions: X irs or. Dlay assumptions: lay o slow input vnt longr tan (lay o stailizing t iruit unr a quisnt nvironmnt). To illustrat t maning o tis timing assumption w will onsir tat is an input vnt an is a slow input vnt in t ampl o Figur 1. T rst o vnts ar non-input. Atr iring t vnts a, an a stat in wi, an ar nal is ra (stat s 3 ). At tis point it an assum tat an will ir or (two gat lays vs. slow nvironmnt). Howvr, no assumptions an maaoutt iring orrtwn ang sin g is pr y an input vnt () or wi no uppr oun on its lay an assum. In as woul a non-input vnt, woul assum to ir or an g also. 4 Bakannotation o timing onstraints Atr logi syntsis, t valiity o t timing assumptions must vrii or valiat to nsur t orrt untion o t iruit. Howvr, t iruit may orrt or a st o stats largr tan t on in y t tim omain, wi an otain y a st o lss stringnt timing assumptions. In otr wors, som o t timing assumptions ar runant or a partiular logi syntsis solution, wil som otr an rla. Tis stion attmpts to answr t ollowing qustion: Can w riv a minimal st o timing assumptions suiint or a iruit to orrt? Tis st o timing assumptions akannotat or a givn logi syntsis solution is all timing onstraints. Timing assumptions (ot manual an automati) ar part o t spiiation an provi aitional rom or logi syntsis, wil timing onstraints is a part o t implmntation, sin ty onstitut rquirmnts to mt suiint or a partiular ntlist solution to vali. 4.1 Eampl 1 Lt us anal t ampl in Figur 2. T saow stats in SG o Figur 2.() orrspon to t tim omain trmin y t timing assumptions z+ < y + an <, Unr ts assumptions, logi syntsis an prorm y onsiring t stats 11 an 1 unraal, i.. in t on t ar st o t logi untions or all signals ;y;z. T iruits o Figurs 2.() an 2.() av a orrt avior unr t prvious assumptions. Looking at t iruit o Figur 2.() w osrv tat: T gats = z + y an y = + z ar orrt implmntations or t wol untim omain. T gat z = is a orrt implmntation or all t stats pt or 1. In tis stat = an z, soul av n nal aoring to t nt stat untion o t implmntation, ut it is not nal in tis stat aoring to t original stat grap spiiation. Tus, vn t iruit may av n otain using t two prvious assumptions, only on rlativ timing onstraint <, must nsur or t iruit to orrt. In gnral, a gat o t iruit is orrt or a sust o t untim omain wi is also a suprst o t tim omain. T iruit is orrt or tos stats in wi all gats ar orrt.

5 y z * * * * * * * * * * 1 * * * Spiiation Implmntation: * LEGEND: 11 - gloal DC 11 * - loal DC - rquir 11 -onurrny timing onstraints 11 rution (a) () () () () () -unraal stats Fig. 3: Nt stat untions or ampl: (a) Original untim spiiation; () Spiiation or RT assumptions z+ < an <, ; (,) Implmntations rom Figurs 2.(,); () Spiiation or RT assumption ;z+ simultanous wit rspt to, ; () Implmntationrom Figur 2.(). 4.2 Eampl 2 Lt us onsir t sam ampl unr a simultanity assumption + an ar simultanous wit rspt to,. Unr tis assumption stat 1 is unraal an oms a on t ar or all signals. In aition stats 11 an 11 oms on t ars or signal, sin ot long to t potntial EnR(,) aoring to t smantis o t simultanity assumption. Only on timing onstraints, z+ <,, is suiint or t iruit in Figur 2.() to orrt. Gat = y is not nal in 11, n onurrny is ru in tis stat wit rspt to t original untim SG an stat 1 oms unraal unr any gat lays. Stat 11 on t ontrary orrspons to t onurrny pansion or naling o,. Tis naling is lazy sin 11 2 EnR(,) ^ FR(,). Figur 3 sows Karnaug maps or t nt stat unstions o signals ;y, anzor spiiations an implmntations orrsponing to t ampls aov. A lgn sows tat timing assumptions provi two typs o on t ar vtors in RT spiiations: gloal on t ars orrsponing to stats unraal u to timing assumptions, an loal on t ars tat ir or irnt signals. In t RT implmntations som stats om unraal u to untim onurrny rution an tror isrpanis in t orrsponing valus o t nt stat untions ompar wit t original untim spiiation an ignor; som isrpanis orrspons to onurrny rution (isaling o signal transitions witout prsistny violation), an inally, otr isrpanis orrspon to lazy naling an rquir timing onstraints or orrt iruit avior. 4.3 Corrtnss o RT iruit Lt S an original untim SG wit a init st o raal stats U 2 an initial stat s. Lt G a iruit ntlist implmnting S unr timing onstraints C. Apair < G;C > is all a rlativ timing iruit (RT iruit). It ins a lazy SG, L<G;C>, wit a st o raal stats U L. T RT-iruit implmntation an ontain mor signals tan t original spiiation S i som stat signals ar insrt or rsolving stat onlits. Lt us assum tat S as n signals an L as k;k n; signals. Tn or omparing stats on ns to us a omomorism : B k 7! B n, tat givn an implmntation stat is (k, n) nw intrnal signals an otains a spiiation stat. Homomorism,, is naturally tn to sts o stats. A RT-iruit is sai to orrt i t ollowing onitions ar satisi: 1. (U L ) U, i.. no stats outsi original untim omain ar raal y t RT-iruit. 2. All signals prsistnt in S ar also prsistnt in lazy SG L<G;C>. All stat signals insrt in L<G;C> ar prsistnt. Commutativity an trminism ar prsrv. 3. T initial stat is prsrv wit rspt to t I/O intra, i.., i s 2 S an s 2 L <G;C> ar t initial stats o t original SG an t lazy SG orrsponing to t implmntation, tn tr is a pat τ s ) (s ) or (s ) ) τ s in S su tat squn τ ontains only vnts o intrnal signals, not osrval y t nvironmnt. 4. No vnts isappar: I ER S () 6= /, tnfr L () 6= / ^ (FR L ()) ER S () 5. No nw alok stats appar in L<G;C>. 4.4 Tory or akannotation For t as o position lt us assum tat no stat signals ar insrt in t RT iruit, an tror t numr o signals stays t sam or S an L. W will rily isuss ow stat signal insrtion is on in Stion 4.8. Lt U t st o stats raal in t untim omain o a stat grap an T U t st o stats raal unr a st o timing assumptions, manual - provi y t usr an automati - riv or syntsis aoring to t ruls o Stion 3. Lt us assum tat w av a iruit wit m output signals, a 1 ;:::;a m. Lt G = g a1 (X);:::;g am (X)g (wr X is t st o signals) a st o gats implmnting t RT iruit, wr g ai (X ) nots t oolan untion implmnt y t gat o signal a i. Raal stats in t untim omain Lt us all R (G) t st o stats raal in t untim omain or t iruit G. Not tat, in gnral, U, R (G) 6= / u to t rution o onurrny impos y t iruit, an R (G), U 6= / u to pansion o onurrny or naling or lazy transitions. T lattr stats ar not gnrat y our prour sin ty must unraal in RT omain anyway. T ormr stats ar o intrst, sin ty o not rquir any timing onstraints (s ampls 4.1 an 4.2). Lt us not U G = R (G) \ U: U G an alulat as ollows: 1. For a output signal a i, alulat isal(a i )= s2u js62 EnR G (a i ) ^ s 2 ER S (a i )g, i.. stats in 2 Our implmntation is urrntly limit y t oun untim STGs an SGs. It an asily tn to unoun untim STGs y making unoun (ininitly growing) markings o STGs unraal in RT omain.

6 1 1 1 Untim omain C Unraal Tim omain Corrt Inorrt 11 avior avior C Fig. 4: Formulation o t akannotation prolm. C 1 ;C 2 ;C 3 g is t st o timing onstraints suiint or orrtnss o RT solution. wi a i was nal in t untim omain in SG, S, ut ma stal y t iruit. C1 2. For a output signal a i, rmov all ars s a i! rom t SG or all stats s 2 isal(a i ). 3. Calulat t nw st U G = R (G) \ U o raal stats. Stats wit inorrt avior Lt us all inorrt(g) U G t st o stats insi U G tat ar rquir to unraal or t orrtnss o t iruit. Ts stats an alulat as ollows: 1. For a output signal a i, alulat inorrt(a i )= s2(u,t )js2enr G (a i ) ^ s 2 QR S (a i )g, i.. stats in wi a i was stal in t untim omain, ut nal in t iruit. 2. inorrt(g) =U, S G \ a i inorrt(a i ) Bakannotation: prolm ormulation W n a st o onstraints tat mak t stats in inorrt(g) unraal. A trivial solution to tis prolm is to tak t omplt st o timing assumptions us or logi syntsis, i.. tos or wi T is t st o raal stats. Our goal, owvr, is to in t lss stringnt st o onstraints suiint to mak t iruit orrt. Givn a st o timing onstraints C = C 1 ;:::;C p g, w will all R (C) U t st o stats raal atr applying C in t untim omain. In gnral, t prolm an ormulat as ollows (s Figur 4): Fin a st o onstraints C wit t largst R (C) su tat 1. T R (C) U, inorrt(g) 2. 8s 2 T : (s 2 EnR G (a i ) ^ s 62 ER S (a i )) )9a j : s a j!s ^ s 2T ^ (a j <a i )2C T irst onition guarants tat no inorrt stats insi U ar raal (onstraints C 1 ;C 2 in Figur 4), wras t son maks sur tat no stats outsi U an ra in t RT iruit (onstraint C 3 in Figur 4). 4.5 Fining a st o timing onstraints Rlativ timing onstraints ar in in trms o iring orr o vnts. Constraining a iring orr twn a pair o vnts maks only sns wn ty an nal simultanously an ir in any orr, i.. wn ty ar onurrnt. Tus, a timing onstraint C i an not y an orr pair o onurrnt vnts,.g. C i =( j < k ). Givn a onstraint C i =( j < k ), w in t st o ars isal(c i ) as isal(c i )=s k!s j9s!s 1!:::!s n : s 1 ;:::;s n,1 2ER( k ) ^ s n 2 ER( k ) \ ER( j )g s3 s2 s6 inorrt avior a s s1 s5 s9 s1 tim omain s4 s7 s8 inorrt avior a orr < < < < < < < < unraal {s4,s7} {s7} {s4,s5,s7,s8} {s7,s8} {s2,s3} {s3} {s2,s3,s5,s6} {s3,s6} Fig. 5: Eampl or akannotation wit tal o unraal stats or a pair o orr vnts. In partiular, t pat s 1! :::! s n an mpty i s 2 ER( j ) \ ER( k ). isal(c i ) is t st o ars wit lal k tat must not ir in orr or j to ir or k, i.. tos ars wit sour stats in wi ot vnts ar onurrnt or pring ER( j ) \ ER( k ) insi ER( k ). Givn a st o onstraints C = C 1 ;:::;C p g, R (C) is t st o raal stats atr rmoving t ars in [ C i 2C isal(c i ) 4.6 Eampl 3 Figur 5 sows an ampl or riving a st o timing onstraints or akannotation. Initially w av U = s ;:::;s 1 g an T = s ;s 1 ;s 2 ;s 5 ;s 8 ;s 9 ;s 1 g. Lt us assum tat s 6 an s 7 ar t stats in wi t avior o t iruit is inorrt. T tal in Figur 5 ontains t st o stats tat om unraal y ruing t onurrny twn a pair o onurrnt vnts 3.For ampl, y imposing t orr <, t stats s 2 an s 3 om unraal. T prolm to solv is t ollowing: in a st o orring onstraints twn pairs o vnts su tat t nw st o raal stats ovrs T an os not intrst t st o inorrt stats s 6 ;s 7 g. Morovr, w want to maimiz t st o raal stats, i.. to in t lss stringnt st o timing onstraints. T prolm an pos as a ovring prolm. T lls o t tal in ol orrspon to tos onstraints tat o not rmov any stat rom T. T ovring prolm an ormulat as ollows: ( < ) ^ ( < _ < ) wit t minimum-ost solution C = < ; < g an R (C) =s ;s 1 ;s 2 ;s 4 ;s 5 ;s 8 ;s 9 ;s 1 g 4.7 Solving t ovring prolm T ovring prolm or akannotation os not orrspon to a unat ovring prolm, sin t ost o t inal solution (numr o isal ars) is not t sum o t ost o a onstraint. Currntly, ptriy uss a gry approa to solv t ovring prolm tat an asily implmnt y symoli BDD-as tniqus. It mrly onsists in oosing t onstraint tat rmovs t maimum numr o ars wos stination is in inorrt(g) an tat av not n rmov y prvious onstraints. Tis pross is itrativly rpat until all inorrt stats om unraal. 3 For simpliity, unraal stats ar rport in t tal or tis ampl. In gnral, t analysis must prorm y alulating t rmov isal ars. In tis partiular as, t rsulting analysis is t sam.

7 li- li lo li+ lo+ FIFO () (a) ri- ro ri ri+ lo- ro+ ro- li- li- li+ + ri- li+ lo+ () + ro- - () OR Fig. 6: (a) FIFO ontrollr, () Spiiation, () Spiiation wit stat noing signal, () RT implmntation wit gc lmnts, () Timing onstraints suiint or orrtnss. In som ass, not all t inorrt stats an ma unraal sin t tim stat spa as n prou y arly naling som vnts. In tos ass, a similar itrativ pross is ut to ovr tos inorrt stats tat an lgaliz y arly naling. As an ampl, onsir t stat s 6 in Figur 5. Assum tat s 6 is inorrtsintnt-statuntioniniatstat is nal in tat stat. T stat oul ma orrt y tning EnR( ) towars s 6 an imposing t typ-iv onstraint <. 4.8 Timing awar stat noing T prolm o stat noing is in insrting stat signals or rsolving CSC onlits. Stat noing in our implmntation is automatially solv using an tnsion o t mto prsnt in [4]: Only tos noing onlits raal in t RT omain ar onsir in t ost untion su tat no ort is invst in solving onlits unraal in RT omain, T. Automati timing assumptions an gnrat or insrt stat signals using ruls rom Stion 3 implying tat t stat signals an implmnt as RT logi. 5 Eprimntal rsults 5.1 Aami ampls T rsults or t wll-known nmarks us at aamia ar prsnt in Tal 1. Tals 1.(a) an 1.() prsnt t rsults or spiiations wit an wit stat oing onlits rsptivly. SI a,si t an TI rprsnt ara an lay optimization or sp-inpnnt sign, an rlativ timing rsults, orrsponingly. For a primnt, ara is stimat as t numr o litrals o t st an rst ntworks o gnraliz C lmnts. Dlay (rspons tim) is stimat as t avrag numr o non-input vnts in t ritial pat twn t iring o two input vnts. Comparing t olumns SI t an TI, w osrv a rution o aout 4% in ara. T rution in rspons tim is lss tan 5% i w onsir all vnts to av a lay o on tim unit. Howvr, t prorman improvmnt is mu mor pronoun i it wr valuat wit atual lays, givn tat t logi o t tim implmntation is mu simplr. W rport tis analysis in Stion Eampl: a FIFO ontrollr In tis stion w tra t vlopmnt o a FIFO ll (spii in Figurs 6.(a),()), a simplii astration o a part o t RAPPID sign. T mouls at t lt an rigt sis o t ontrollr av a similar sp as t ontrollr itsl. In at, ts vnts ar gnrat y twin mouls onnt at a si. For tis rason, it is not wis to assum tat t input vnts ar slow. W simulat our FIFOs using irnt implmntations o t FIFO ll an masur a yl tim o t ri- lo- lo+ ro ro- ri+ lo- ro+ ri+ li lo () ri ro Dsign FIFO yl tim Cll orwar latny RT SI RT rsul SI rsul Tal 2: Prorman omparison o FIFOs normaliz to a an-out our invrtr lay FIFO an a orwar latny (an avrag vnt propagation tim rom li to ro) o a ll. T rsults normaliz to t lay o an invrtr wit an-out our in a givn tnology ar sown in Tal 2. For t irst rlativ timing FIFO (rport in t irst row) w us a RT iruit riv y ptriy using only automati timing assumptions prsnt in Figurs 6.(). A propr transistor sizing is rquir or orrt opration o t iruit. No usr-in assumptions on t nvironmnt ar us. T timing analysis plain in Stion 3 as n appli to t spiiation, an stat noing as n automatially solv as sri in Stion 4.8. Wit tis stratgy, only on aitional stat signal,, was rquir as sown in Figur 6.() 4. Tr ar som intrsting aspts o tis implmntation: T stat signal is is switing onurrntly wit otr ativity in t iruit.tis is a rsult o t stat noing stratgy o ptriy tat attmpts to inras t onurrny o nw stat signals until ty isappar rom t ritial pats aoring to t lay mol plain in Stion 3. T rspons tim o t iruit wit rgar to t nvironmnt is only on vnt (two invrtrs), i.. as soon as an output vnt is nal it irs witout rquiring t iring o any otr intrnal vnt. Finally, t implmntation o Figur 6.() rquirs som timing onstraints to orrt. Atr applying t mto propos in Stion 4, iv timing onstraints twn pairs o onurrnt vnts av n riv tat ar suiint or t iruit to orrt. Ty ar grapially rprsnt in Figur 6.(). T onstraints l o + <, an r o + <, ar not inpnnt. Sin t implmntation o is = l o + r o,it is always guarant tat on o tm will ol, wras t otr must nsur. Sin l o + an r o + ar nal simultanously, ts onstraints will always ol i t lay o two gats is longr tan t lay o on gat. From t rst o onstraints, t most stringnt is, < r i +. In t worst as, ot r i + an, will nal simultanously y r o +. In tis as, it is rquir t lay o, to sortr tan t lay o r i + (rom t nviromnt). In as o a vry ast nvironmnt, it an or y irnt tniqus,.g. transistor sizing or lay paing or gat. For t son FIFO (t son row o t tal) w riv a sp-inpnnt iruit using ptriy in t mo o automati onurrny rution [5] witout onstraining I/O onurrny o t ll. Baus o onurrny rution only on stat signal was rquir [4] lik in t as o t automati RT solution. Howvr, t stat signal was on a ritial yl an t implmntation o lo an ro ontain aitional p-transistors, wi ma t sp-inpnnt iruit 2-3% slowr tan t RT on. 5.3 RAPPID ontrol iruits In tis stion w ompar manually optimiz RT ontrol iruits us or RAPPID [16, 15] wit tos riv automatially wit ptriy. For a ampl, Tal 3, rports: manual (otain y applying rlativ timing manually), automati (otain automatially y ptriy 4 Tis nw spiiation is not stritly a Ptri nt, sin t ars rom l o + an r o + to t OR pla iniat an or-ausality rlation:, is triggr y t irst vnt to ir, wras t tokn prou y t latst vnt is impliitly onsum. An quivalnt Ptri Nt is a it mor umrsom an is omitt or simpliity.

8 Ara Rspons tim Stat signals iruit SI a SI t TI SI a SI t TI SI a SI t TI aast allo-outoun mastr-ra mmu mmu mr mr nak-pa nowik ram-ra-su su-ram-writ su-ra-tl sq sq-mi vmus Total Ara iruit SI TI u u onvrta rgn al 8 7 azar 8 8 mslat 24 2 trimos-sn 3 21 var v v5 1 1 v6a v wrata Total (a) () Tal 1: Eprimntal rsults: spiiations witout CSC (a) an wit CSC (). Dsign Ara (# tr.) Worst as Avrag as rspons tim rspons tim m a s m a s m a s FIFO-A FIFO-B Byt-ntr Tag-unit Summary Tal 3: Comparison or two gnri rprsntativ ampls (io) an two ontrol iruits rom RAPPID (ytontrol, tag-unit). Rspons tim is masur in gat lays, ara in transistors. m: manual,a: automati,s: spinpnnt. an applying rlativ timing) an sp-inpnnt (otain automatially y ptriy witout onurrny rution). From t tal it an u tat automati solutions ar quit omparal wit manually optimiz RT signs. T improvmnt in rspons tim y applying rlativ timing is aout a ator o 2, sustantially ttr tan or t ampls o Tal 1. Tis is aus t signrs o ts iruits a a strongr intration wit t tool an provi aggrssiv timing assumptions on t nvironmnt tat oul not riv automatially. 6 Conlusions T mto or automati gnration o timing assumptions prsnt in tis papr allows t signr to onntrat on ining tos timing assumptions tat an only u rom a tail knowlg o t nvironmnt. T tniqu or automati ak-annotation o timing onstraints rlativ to a partiular RT iruit provis nssary timing inormation or t own-stram tools. Timing-awar stat noing allows ara/lay optimization o RT iruits. Rlativ timing prsnts a mil-groun twn lok an asynronous iruits, an is a rtil ara or CAD vlopmnt. Bot urst-mo[14, 17] an spinpnnt spiiations ar at opposit trms o a mor gnral lass o rlativ timing spiiations. Akowlgmnts W woul lik to tank Sai Rotm, Luiano Lavagno, Al Konratyv an Alanr Yakovlv or tir ontriutions in motivating tis work an vloping t tory or syntsis wit rlativ timing. Rrns [1] S. Burns. Gnral onitions or t omposition o stat oling lmnts. In Intrnational Symposium on Avan Rsar in Asynronous Ciruits an Systms, Aizu, Japan, Mar [2] W. S. Coats, J. K. Lau, I. W. Jons, S. M. Fairanks, an I. E. Sutrlan. A io ata swit sign primnt. In Pro. Intrnational Symposium on Avan Rsar in Asynronous Ciruits an Systms, pags 4 17, [3] J. Cortalla, M. Kisinvsky, A. Konratyv, L. Lavagno, A. Tauin, an A. Yakovlv. Lazy transition systms: appliation to timing optimization o asynronous iruits. In Proings o t Intrnational Conrn on Computr-Ai Dsign, pags , Novmr [4] J. Cortalla, M. Kisinvsky, A. Konratyv, L. Lavagno, an A. Yakovlv. A rgion-as tory or stat assignmnt in spinpnnt iruits. IEEE Transations on Computr-Ai Dsign, 16(8): , August [5] J. Cortalla, M. Kisinvsky, A. Konratyv, L. Lavagno, an A. Yakovlv. Automati syntsis an optimization o partially spii asynronous systms. In DAC, pags 1 115, Jun [6] Hnrik Hulgaar an Stvn M. Burns. Boun lay timing analysis o a lass o CSP programs wit oi. In Pro. Intrnational Symposium on Avan Rsar in Asynronous Ciruits an Systms, pags 2 11, Novmr [7] Alain J. Martin. Syntsis o asynronous VLSI iruits. In J. Straunstrup, itor, Formal Mtos or VLSI Dsign, aptr 6, pags Nort-Hollan, 199. [8] D. E. Mullr an W. C. Bartky. A tory o asynronous iruits. In Annals o Computing Laoratory o Harvar Univrsity, pags , [9] T. Murata. Ptri Nts: Proprtis, analysis an appliations. Proings o t IEEE, pags , April [1] Cris J. Myrs. Computr-Ai Syntsis an Vriiation o Gat- Lvl Tim Ciruits. PD tsis, Dpt. o El. Eng., Stanor Univrsity, Otor [11] Cris J. Myrs an Trsa H.-Y. Mng. Syntsis o tim asynronous iruits. IEEE Transations on VLSI Systms, 1(2):16 119, Jun [12] Rau Ngulsu an A Ptrs. Vriiation o sp-pnns in singl-rail ansak iruits. In Pro. Intrnational Symposium on Avan Rsar in Asynronous Ciruits an Systms, pags , [13] M. Nilsn, G. Roznrg, an P.S. Tiagarajan. Elmntary transition systms. Tortial Computr Sin, 96:3 33, [14] S.M. Nowik. Automati Syntsis o Burst-Mo Asynronous Controllrs. PD tsis, Stanor Univrsity, Dpt. o Computr Sin, [15] S. Rotm, K. S. Stvns, R. Ginosar, P. A. Brl, C. J. Myrs, K. Yun, R. Kol, C. Dik, M. Ronkn, an B. Agapiv. RAPPID: An asynronous instrution lngt or. In Pro. ASYNC, April [16] K. S. Stvns, S. Rotm, an R. Ginosar. Rlativ timing. In Pro. ASYNC, April [17] Knnt Yi Yun. Syntsis o Asynronous Controllrs or Htrognous Systms. PD tsis, Stanor Univrsity, August 1994.