Timed Circuits. Asynchronous Circuit Design. Timing Relationships. A Simple Example. Timed States. Timing Sequences. ({r 6 },t6 = 1.

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1 Timed Circuis Asynchronous Circui Design Chris J. Myers Lecure 7: Timed Circuis Chaper 7 Previous mehods only use limied knowledge of delays. Very robus sysems, bu exremely conservaive. Large funcional unis do no have zero delay. Gaes and wires do no have an infinie delay. Timing analysis can idenify addiional unreachable saes. These unreachable saes are addiional don cares. Timed circuis use his informaion o opimize he design. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 1 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 2 / 1 A Simple Example Timing Relaionships Shopkeeper acively calls winery and paron. Calls he paron immediaely afer calling he winery wihou waiing for he wine o arrive. The shopkeeper does he following: Calls he winery, Calls he paron, Peers ou he window unil he sees boh he wine delivery boy and he paron, Les hem in, and Complees he sale. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 3 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 4 / 1 Timed Saes Timing Sequences There is a imer i associaed wih each arc in he graph. A imed sae is an unimed sae and value of all acive imers. ({r 6 }, = 0) A imer is allowed o advance by any amoun less han is upper bound resuling in a new imed sae. ({r 6 }, = 1.1) ({r 6 }, = 2.22) ({r 6 }, = ) When a imer reaches is lower bound, i becomes saisfied. When a imer reaches is upper bound, i becomes expired. An even enabled by a single rule mus happen someime afer is imer becomes saisfied and before i becomes expired. When an even is enabled by muliple rules, i mus happen afer all of is rules are saisfied, bu before all of is rules are expired. Exend he noion of allowed sequences o imed saes paired wih he ime of he sae ransiion. Sae ransiion can be eiher ime advancemen or a change in he unimed sae. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 5 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 6 / 1

2 Example: Timing Sequence Timed Sae Space Exploraion ( ([{r 6 }, 6 = 0], 0), ([{r 6 }, 6 = 2.22], 2.22), ([{r 1,r 2 }, 1 = 2 = 0], 2.22), ([{r 1,r 2 }, 1 = 2 = 2.1], 4.32), ([{r 1,r 3 }, 1 = 2.1, 3 = 0], 4.32),... Since ime can ake on any real value, here is an uncounably infinie number of imed saes and imed allowed sequences. Mus eiher group imed saes ino finie number of equivalence classes or resric he values of he imers. Several possible mehods for imed sae space exploraion: Region mehod Discree-ime mehod Zone mehod POSET mehod Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 7 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 8 / 1 Regions Possible Timed Saes Using Regions A region is described by he ineger componen of each imer and he relaionship beween he fracional componens. f( 1 ) = f( 2 ) = 0: region is a poin. f( 1 ) = 0 and f( 2 ) > 0: region is a verical line segmen. f( 1 ) > 0 and f( 2 ) = 0: region is a horizonal line segmen. f( 1 ) = f( 2 ) > 0: region is a diagonal line segmen. f( 1 ) > f( 2 ) > 0: region is an lower riangle. f( 2 ) > f( 1 ) > 0: region is an upper riangle. 2 0,5 1 0,5 171 disinc imed saes Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 9 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 10 / 1 Timed Sequence Using Regions Timed Sequence Using Regions ==0 f()=f()=0 ==0 f()=f()=0 ==0 f()=f()>0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 11 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 12 / 1

3 Timed Sequence Using Regions Timed Sequence Using Regions ==0 f()=f()>0 ==1 f()=f()=0 ==1 f()=f()=0 ==1 f()=f()>0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 13 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 14 / 1 Timed Sequence Using Regions Timed Sequence Using Regions ==1 f()=f()>0 ==2 f()=f()=0 ==2 f()=f()=0 ==2 f()=f()>0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 15 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 16 / 1 Timed Sequence Using Regions Timed Sequence Using Regions ==2 f()=f()>0 =0,=2 f() > f()=0 =0,=2 f() > f()=0 =0,=2 f() > f() > 0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 17 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 18 / 1

4 =0,=2 f()=f()=0 =0,=2 f()=f()>0 =1,=3 f()=f()=0 =1,=0 f()=f()=0 =0,=2 f() > f() > 0 =0,=0 f() > f()=0 =0,=2 f() > f()=0 =0,=3 f() > f()=0 ==0 f()=f()=0 ==0 f()=f()>0 ==1 f()=f()=0 ==1 f()=f()>0 ==2 f()=f()=0 ==2 f()=f()>0 =0,=2 f() > f()=0 =0,=2 f() > f() > 0 ==3 f()=f()=0 ==0 f()=f()=0 =0,=3 f() > f()=0 =0,=0 f() > f()=0 =0,=3 f()=f()=0 =0,=2 f()=f()=0 =0,=2 f()=f()>0 =0,=3 f()=f()=0 =1,=3 f()=f()=0 =1,=0 f()=f()=0 Timed Sequence Using Regions Timed Sequence Using Regions =0,=2 f() > f() > 0 =0,=3 f() > f()=0 =0,=3 f() > f()=0 =0,=0 f() > f()=0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 19 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 20 / 1 Timed Sae Space Using Regions Timed Sae Space Explosion Requires 26 imed saes o represen all he iming relaionships for only 4 unimed saes. Wors-case complexiy is: S n! ( ) n k 4 1/k ln2 ln2 where S is number of unimed saes, n is he number of rules enabled concurrenly, and k is maximum iming consrain. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 21 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 22 / 1 Discree-Time Possible Timed Saes Using Discree-Time For imed Peri-nes and TEL srucures, all iming requiremens are of he form or, since iming bounds are inclusive. In his case, fracional componens are no necessary. Only need o rack discree-ime saes. Wors-case complexiy is now: S (k + 1) n Reducion by a facor of more han n!. 2 0,5 1 0,5 36 disinc imed saes Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 23 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 24 / 1

5 Timed Sae Space Using Discree-Time Timed Sae Space Explosion Again ==0 ==1 Unforunaely, he discree-ime echnique is sill exponenial in he number of concurren imers and size of he iming bounds. =0,=2 ==2 ==3 =0,=2 Changing each iming bound of o [19,31] and o [53,inf], number of imed saes goes from 69 o more han Changing each iming bound of o [191,311] and o [531, inf], number of imed saes goes o over 300,000! =1,=3 =0,=3 =0,=3 =1,=3 =1,=0 ==0 =1,=0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 25 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 26 / 1 Zones Represening Zones using Linear Inequaliies Anoher approach is o use convex polygons, called zones, o represen equivalence classes of imed saes. One zone is represening 171 regions or 36 discree-ime saes. Convex polygons can be represened using linear inequaliies. Inroduce a dummy imer 0 which always akes he value 0. For each pair of imers, inroduce inequaliy of he form: j i m ij 2 0,5 1 0,5 Example: Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 27 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 28 / 1 Difference Bound Marices icalizaion Se of inequalies can be colleced ino a daa srucure called a difference bound marix (DBM). The difference bound marix for his example is shown below: Many DBMs represen he same zone. Need unique DBM represenaion o deermine when a zone has been seen before during he deph firs search. Each zone has a canonical DBM represenaion when all enries are minimal. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 29 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 30 / 1

6 icalizaion Example DBM as Digraph Add las wo equaions o ge: icalizaion equivalen o all pairs shores pah problem. Creae a labeled digraph where: There is a verex for each imer i, An arc from i o j for each linear inequaliy of he form j i m ij when i j. Each arc is labeled by m ij Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 31 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 32 / 1 Floyd s Algorihm Floyd s Algorihm Example recanonicalizaion(m) for k = 1 o n for i = 1 o n for j = 1 o n if (m ij > m ik + m kj ) hen m ij = m ik + m kj ; Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 33 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 34 / 1 Zone Creaion Resric Afer a rule fires: Resric o reflec minimum firing ime. Recononicalize Projec ou informaion on rule ha fired. Exend marix wih new rows and columns for new rules. Advance ime Recononicalize Example: firing of a rule r k = e k,f k,l k,u k where e k is he enabling even, f k is he enabled even, l k is he lower bound of he cooresponding imer k, and u k is he upper bound on he imer. Consrain DBM o indicae rule has reached is lower bound. 0 k l k, so se m k0 o l k. DBM may no longer be maximally igh. icalize DBM using Floyd s algorihm. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 35 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 36 / 1

7 Projec Advance Time Remove he row and column cooresponding o k. If rule firing causes an even, new rules may be enabled. For newly enabled rules, inroduce a new imer l wih a row and column in he DBM. Iniialize m l0 and m 0l o 0. Iniialize each m lj o m 0j. Iniialize each m il o m i0. Se all imers o heir upper bound (i.e., m 0j = u j ). icalize he DBM using Floyd s algorihm. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 37 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 38 / 1 Updae Zone updae_zone(m,r j,even_fired,r en,r new ) if m j0 > l j hen m j0 = l j recanonicalize(m) projec(m,r j ) if (even_fired) hen foreach r i R new m i0 = m 0i = 0 foreach r k R new m ik = m ki = 0 foreach r k (R en R new) m ik = m 0k m ki = m k0 foreach r i R en m 0i = u i recanonicalize(m) normalize(m,r en) normalize(m,r en) foreach r i R en if (m i0 < premax(r i)) hen foreach r j R en m ij = m ij (m i0 + premax(r i)) m ji = m ji +(m i0 + premax(r i)) foreach r i R en if (m 0i > premax(r i)) hen m 0i = max j(min(m 0j,premax(r j)) m ij) recanonicalize(m) Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 39 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 40 / 1 Iniial Zone Iniial Zone Iniial Iniial AdvTime/ / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 41 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 42 / 1

8 Zone Afer Winery Called Zone Afer Winery Called Resric/ Resric/ Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 43 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 44 / 1 Zone Afer Winery Called Zone Afer Winery Called Projec Exend Exend AdvTime/ / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 45 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 46 / 1 Zone Afer Zone Afer Resric Resric Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 47 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 48 / 1

9 Zone Afer Zone Afer Projec Projec Exend Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 49 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 50 / 1 Zone Afer Zone Afer Exend AdvTime AdvTime / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 51 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 52 / 1 Zone Afer and Paron is Called Zone Afer and Paron is Called Resric/ Reconon Resric/ Reconon Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 53 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 54 / 1

10 Zone Afer and Paron is Called Zone Afer and Paron is Called Projec Exend Exend AdvTime Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 55 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 56 / 1 Zone Afer and Paron is Called Zone Afer Rule Expires AdvTime Reconon/ Norm Resric Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 57 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 58 / 1 Zone Afer Rule Expires Zone Afer Rule Expires Resric Reconon Reconon Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 59 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 60 / 1

11 Zone Afer Rule Expires Zone Afer Rule Expires Projec AdvTime/ Reconon AdvTime/ Reconon Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 61 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 62 / 1 Zone Afer Zone Afer Resric/ Resric/ Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 63 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 64 / 1 Zone Afer Zone Afer Projec Exend Exend AdvTime/ / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 65 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 66 / 1

12 Zone Afer Wine is Purchased Zone Afer Wine is Purchased Resric/ Resric/ Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 67 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 68 / 1 Zone Afer Wine is Purchased Zone Afer Wine is Purchased Projec Exend Exend AdvTime/ / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 69 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 70 / 1 Zone Afer Paron is Called Zone Afer Paron is Called Iniial Iniial Resric Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 71 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 72 / 1

13 Zone Afer Paron is Called Zone Afer Paron is Called Resric Projec Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 73 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 74 / 1 Zone Afer Paron is Called Zone Afer Paron is Called Projec Exend Exend AdvTime Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 75 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 76 / 1 Zone Afer Paron is Called Zone Afer Paron is Called and AdvTime / Norm Resric/ Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 77 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 78 / 1

14 Zone Afer Paron is Called and Zone Afer Paron is Called and Resric/ Projec Projec Exend Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 79 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 80 / 1 Zone Afer Paron is Called and Zone Afer Paron is Called and Exend AdvTime AdvTime / Norm Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 81 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 82 / 1 Zone Afer Rule Expires Zone Afer Rule Expires Resric Resric Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 83 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 84 / 1

15 Zone Afer Rule Expires Zone Afer Rule Expires Projec Projec AdvTime / Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 85 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 86 / 1 Zone Afer Rule Expires Timed Sae Space using Zones AdvTime / rule expires rule expires 0 inf -1 0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 87 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 88 / 1 Adverse Example Order versus Causaliy [1,40] a [1,40] b a 40 [1,40] c [1,40] d [a,b] 1 unimed sae 2,825,761 discree-ime saes 219,977,777 zones 0 0 [b,a] 40 b Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 89 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 90 / 1

16 POSET Timing POSET Graph/Marix/Zone Using linear races inroduces fake orderings. Need o separae concurrency from casualiy. Find zones on POSETs raher han linear races. Represen POSETs using graph/marix. rese [1,40] [1,40] a b a 40 [a,b] & [b,a] b r a b r a b a b a b Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 91 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 92 / 1 POSET Timing Creaing New Zones [1,40] a [1,40] [1,40] c [1,40] d 1 unimed sae 2,825,761 discree-ime saes 219,977,777 zones 1 zone found using POSET iming b If even occurs, updae POSET marix and creae zone: Se minimums o 0 (i.e., m i0 = 0). Se maximums o he upper bound (i.e., m 0j = u j ). Copy relaven ime separaions from POSET marix o zone (i.e., m ij = p ij ). icalize. Oherwise, projec ou imer cooresponding o rule ha fired. Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 93 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 94 / 1 Iniial Zone using POSETs Zone afer he Winery is Called using POSETs Iniial POSET Iniial zone / icalize / r r rese Exend POSET Projec POSET Iniial zone / icalize / r cw r 0 3 cw -2 0 cw cw rese Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 95 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 96 / 1

17 POSET afer he Zone afer he using POSETs rese Exend POSET cw wa cw 0 3 wa -2 0 Iniial zone icalize / Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 97 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 98 / 1 POSET afer he Paron is Called Zone afer he Paron is Called using POSETs Exend POSET icalize Projec POSET cw wa cp cw wa -2 0 cp -2 0 cw wa cp cw wa cp wa cp wa 0 1 cp 1 0 rese Iniial zone icalize / Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 99 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 100 / 1 Zone afer he Rule Expires using POSETs POSET afer he Projec zone Advance ime / icalize Exend POSET icalize Projec POSET wa cp pa wa 0 1 cp 1 0 pa -5 0 wa cp pa wa 0 1 cp 1 0 pa pa pa 0 rese Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 101 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 102 / 1

18 Zone afer he using POSETs POSET afer he Wine is Purchased rese Iniial zone / icalize / Exend POSET Projec POSET pa wp pa 0 3 wp -2 0 wp wp 0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 103 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 104 / 1 Zone afer he Wine is Purchased using POSETs POSET afer he Paron is Called rese Iniial zone / icalize / Exend POSET cp cw cp 0-2 cw 3 0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 105 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 106 / 1 Zone afer he Paron is Called using POSETs POSET afer he Iniial zone icalize / Exend POSET icalize Projec POSET wa cp cw wa 0-2 cp 0-2 cw wa cp cw wa cp cw wa cp wa 0 1 cp 1 0 rese Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 107 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 108 / 1

19 Zone afer he using POSETs Timed Sae Space using POSETs Zone icalize / rule expires 0 inf 0 0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 109 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 110 / 1 Wine Shop Example: TEL Srucure Wine Shop Example: Unimed Sae Graph F010R req_wine+/1 [ack_wine] req_wine /1 [~ack_wine] csc+/1 ack_wine+/1 [0,1] [0,1] [~req_wine] [0,1] [req_wine] $3+/1 [0,1] [~ack_paron] req_wine- CSC0+ R00R1 0010R F0101 req_wine+ req_paron+ ack_wine- CSC0+ req_wine- 100R1 RR01F 00F01 req_paron+ req_wine+ CSC0- ack_paron+ 1R01F RR010 R101F ack_paron+/1 [~req_paron] ack_paron /1 [req_paron] ack_wine /1 [0,1] req_paron+/1 req_paron /1 csc /1 [0,1] [0,1] [ack_paron] ack_wine+ CSC0- ack_paron+ req_wine+ ack_paron+ 1R F R10F0 ack_paron+ CSC0- req_wine+ req_paron- 110F0 RF000 req_paron- req_wine+ ack_paron- 1F000 R0000 ack_paron- req_wine+ req_wine+ CSC0-10R00 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 111 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 112 / 1 Wine Shop Example: Reduced Sae Graph Wine Shop Example: Speed-Independen Circui R0000 ack_wine reqwine+ C CSC0 10R00 req_paron ackwine+ ackparon- F010R 1F000 CSC0+ reqparon- F F0 reqwine- ackparon+ 00F01 1R010 ackwine- reqwine+ CSC0- ack_wine CSC0 ack_paron CSC0 req_wine ack_paron req_paron CSC0 C req_paron R00R1 RR010 1R01F C ack_wine reqparon+ RR01F CSC0- reqwine+ req_wine CSC0 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 113 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 114 / 1

20 Wine Shop Example: Timed Circui Summary ack_wine C CSC0 req_paron Regions CSC0 ack_wine C req_paron Discree-ime saes Zones Zones + POSETs ack_paron Timed circuis req_wine ack_paron req_paron ack_wine Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 115 / 1 Chris J. Myers (Lecure 7:Timed Circuis) Asynchronous Circui Design 116 / 1