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1 Computng Observablty Don't Cares Ecently through Polarzaton Harm rts, Mchel Berkelaar and Koen van Ejk bstract new method s presented to compute the exact observablty don't cares (ODCs) for multple-level combnatonal crcuts. new mathematcal concept, called polarzaton, s ntroduced. Polarzaton captures the essence of ODC calculaton on the otherwse dcult ponts of reconvergence. It makes t possble to derve the ODC of a node from the ODCs of ts fanouts wth a very smple formula. Expermental results for the 39 largest MCNC benchmark examples show that the method s able to compute the ODC set (expressed as a Boolean network) for all but one crcut n at most a few seconds. Keywords Logc Synthess, Don't Cares I. Introducton Observablty don't cares (ODCs) play a central role n the synthess of Boolean networks. Together wth the external don't cares (EDCs) and the satsablty don't cares (SDCs) they represent the freedom one has to optmze the network. Especally the computaton of the ODCs has been topc of research because of ts complexty. Several papers have been publshed on the subject of ODC calculaton. In [], Bartlett et al. propose to calculate the ODCs by attenng the network. Ths s, however, mpractcal for most crcuts, because of the sze needed for the representaton. In [7], Muroga et al. propose exhaustve smulatons, whch s very tme consumng. To reduce computatonal complexty t was proposed to calculate the ODC of a node from the ODCs of ts drect fanouts n [4] by Brayton et al. However, computng the ODC n ths way s not straght-forward n the presence of reconvergent fanouts. To solve ths problem, [4] proposes usng the chan rule, orgnally ntroduced by Chang et al. n [5]. s t turns out, the use of ths rule results n very complex calculatons very quckly, so [4] proposes usng approxmatons for large crcuts. In [6], Daman et al. present a method whch s computatonally less complex, but stll approxmatons are needed for the larger crcuts. In [8], Savoj et al. use an observablty relaton to calculate the ODCs. lthough the method does not need to calculate the ODC for each prmary output separately, the operatons per node are much more complex. The paper tself does not present any results, but the authors themselves comment [9]: \We mplemented the algorthm of the ICCD paper but the algorthm was not practcal for large crcuts. We concluded that ODCs could be usually [only] computed for crcuts that were collapsble n two levels". Harm rts (emal: harm@ambt.com) s wth mbt Desgn Systems Inc, 25 ugustne Drve, Ste. 2, Santa Clara, C Mchel Berkelaar (emal: m.r.c.m.berkelaar@ele.tue.nl) and Koen van Ejk (emal: c.a.j.v.ejk@ele.tue.nl) are wth the Endhoven Unversty of Technology, Dept. of EE, P.O. Box 53, 56 MB Endhoven, the Netherlands. In ths paper we present a method whch also derves the ODC of a node from the ODCs of ts drect fanouts and also does not need to calculate the ODC for each prmary output separately, but the operatons per node are very smple: only an and over the ODCs of the fanouts, and a cofactor operaton are needed. Ths s obtaned by ntroducng the concept of polarzaton. For each node the Polarzed Observablty Don't Care (PODC) s calculated. The polarzaton exactly models the reconvergence n a network such that cofactorng the PODC wll \expand" and/or \shrnk" the PODC such that the resultng ODC wll be correct. We feel the man contrbuton of ths paper s the smple mathematcal formulaton of the constructon of the ODC network wth the use of the PODCs wthout the explct use of xor or xnor operatons. nother contrbuton s the large results table. ll prevously publshed papers mentoned above are ether completely theoretcal or show very few results, whch leaves no room for comparson of derent methods. Our results secton shows that the complete ODC network can be derved wth our method even for large crcuts, and allows future papers to compare ther results to ours. nother advantage of the method presented n ths paper s that t makes the use of EDCs very smple. The PODCs calculated at the nput of a network can be handed over as EDCs to a feedng network drectly, representng the complete EDC. These PODCs also drectly mply the Boolean relatons for the equvalence classes [3][6], as s shown n secton IV. In ths paper we express the ODCs as a Boolean network. Ths network can be used drectly by the synthess system [2]. lternatvely, the ODCs could be expressed n other representatons sutable for Boolean reasonng, such as BDDs, but ths approach s not tested n ths paper. The method s mplemented and tested on the entre set of MCNC combnatonal multple-level benchmarks. II. Defntons and Notaton ODCs are commonly calculated usng a Boolean network [] to model a combnatonal crcut. In a Boolean network, each node s assocated wth a Boolean expresson (eg. a Sum Of Cubes (SOC) expresson) n terms of ts fann nodes (or fann edges). In ths paper, we wll use a network of factored forms. In such a network each node s assocated wth a smple and or or expresson, and nverters are modeled on the edges. Ths s no lmtaton snce any Boolean expresson tself s also a factored form. The advantage of usng a network of factored forms s that there s no mplct reconvergence, whch s clearly of great mportance when calculatng ODCs.

2 multple-output combnatonal crcut s modeled by a network of factored forms. The network can be speced by an acyclc graph G (N; C) (see gure ). Each node n 2 N represents ether a prmary nput or a basc Boolean operaton,.e. an and or an or operaton. There s a drected edge c j 2 C for each connecton from node n to node n j. Each connecton can have an nverter property. The prmary nput (output) nodes n N are dented by the set of ndces I (O). prmary nput (output) does not have any ncomng (outgong) edges. Denton 5: The transtve fanout s dened recursvely: T F O(p) F O(p) [ q2f O(p) T F O(q) We say that functon f p depends on every varable whch s n ts transtve fann. Denton 6: Cofactorng a functon to a varable or ts complement: c j n j fj vp f(v p ) fj vp f(v p ) n complement III. Observablty don't cares The ODC of varable v p at node n k s a Boolean functon whch gves the condtons for whch the actual value of varable v p can not be observed at node n k. Denton 7: The ODC of varable v p at node n k : Fg.. Nodes and connectons We dene two functons: Denton : OP : N n I! f; g returns the operaton represented by node n. INV : C! fflse; TRUEg returns TRUE f c has an nverter property. Varable v denotes the varable at the output of node n. Snce we want to be able to dstngush between the value at the output of a node and the value whch s at the nput of a connected node (see gure ), we also ntroduce varables for all connectons: v j denotes the varable at nput c j of node n j. The letters p and q wll be used to denote ether an ndex or an ndex par, so varable v p can denote ether a node or a connecton varable. Denton 2: The fann of a node: F I(j) fjjc j 2 Cg The fann of a connecton: F I(j) Denton 3: The fanout of a node: F O() fjjc j 2 Cg The fanout of a connecton: F O(j) j The Boolean functon f of a node or a connecton can be derved usng the followng rules: nd: f j P Q j2f I(j) v j f OP (j) j2f I(j) v j f OP (j) f j v f INV (j) FLSE v f INV (j) TRUE: Denton 4: The transtve fann s dened recursvely: T F I(p) F I(p) [ q2f I(p) T F I(q) ODC k p f k j vp f k j vp where s the xnor operator. The ODC of a varable at all prmary outputs s a Boolean functon whch gves the condtons for whch the actual value of the varable can not be observed at any prmary output. Denton 8: The ODC of varable v p at all prmary outputs: ODC p k2o ODC k p Creatng the ODC as a network of factored forms, usng these dentons, s relatvely smple, but the resultng network turns out to be very complex for many crcuts. s a result the calculaton of the ODC n ths way, by expressng t n SOC or BDDs, be t n terms of prmary nputs or local varables, s known to be computatonally very expensve []. Dervng the ODC of a node from the ODCs of ts drect fanouts, to reduce the computatonal complexty, has been topc of research before. The ODC of a varable v j can be derved easly from the ODC of varable v j and the local ODC at node n : ODC k j ODC k j ODC j j () However, dervng the ODC of a varable v from the ODCs of ts fanout varables v j s much more dcult f the number of fanouts s more than one. Use of the chan rule [5] has been proposed by [], but t becomes already very expensve for nodes wth only two fanouts. Suppose: F O() fj ; j g

3 then: ~f p (v ; : : : ; v n ) f p (~v ; : : : ; ~v n ) ODC k ODCj k ODCj k ODCj k j vj ODCj k j vj (2) In [6] a new method was presented whch needs substantally fewer xor operatons and no hgher order dervatves. Suppose: then: ODC k F O() fj ; j ; : : : ; j n g nk m ODC k j m j vj ;:::;v jm? ;v jm ;:::;v jn (3) Ths formula stll results n such a complex ODC that n practce (less complex) approxmatons of the ODC must be used. [8] ntroduced a method whch does not need the and operaton over all outputs (see denton 8). Let: ODC X j2onfg v j g j for all 2 O where g j represents the global functon of output n j n terms of the prmary nputs. Suppose: and: then: O F O() fj ; j ; : : : ; j n g O m (v jm O m? )ODC jm j vjm ;:::;v jn O m? ODC jm j vjm ;:::;v jn ODC O n j v O n j v (4) lthough [8] does not need the and operator over all prmary outputs, the operatons needed per fanout are more complex. The paper does not report expermental results. The method presented n ths paper makes t possble to calculate the ODC wthout the use of any (explct) x(n)or operatons and also wthout the and operaton over all outputs. The resultng network of factored forms s substantally less complex. IV. Polarzed observablty don't cares To calculate the ODC n a new and more ecent way we rst have to ntroduce a new operator: polarzaton. Denton 9: The polarzaton operator appled to varable v ntroduces a new varable ~v such that ~v v. Denton : The polarzed Boolean functon ~ f p s assocated wth lteral ~v p and s dened as: Polarzaton of a varable can be seen as the \twnnng" of a varable. Note that the twn of a twn of a varable s the varable tself. Consstently, a polarzed functon (network) can be seen as a twn copy of the orgnal functon (network), usng the twn copes of ts varables. The only derence we wll assume between the orgnal and ts twn s ther behavor under the cofactor operator. Cofactorng a polarzed varable wll evaluate to the opposte constant value as ts non-polarzed twn. So, we extend the denton of cofactorng to polarzed Boolean functons. Denton : fj vp f(v p ; ~v p ) fj vp f(v p ; ~v p ) Instead of explctly copyng a network to obtan ts twn, we can also model polarzaton as an edge property (see gure 2). So, f we want to calculate fj vp n a multple-level network, any varable v p whch s on a path from f to v wth an even number of polarzatons has to be substtuted wth constant. ny varable v p on such a path wth an odd number of polarzatons has to be substtuted wth constant. The followng property follows from these dentons: fj v ( fj g~ v ), whle fj v ( fj v ) Furthermore we need a way to remove all polarzatons from a network. Denton 2: Remove polarty: ]f f (~v p v p ) for all p 2 T F I(f ) Polarzaton s used to mark factors n a network, such that they wll cofactor to the opposte value as would be the case normally. For example, f we have f g h ~ (see gure 2) wth g and h not polarzed, then fj a gj a hj ~ a g(a ) h(a ~ ) and ](fj a ) ] g(a ) h(a ~ ) g(a ) h(a ) gj a hj a. Usng these dentons we can rewrte the denton of the ODC (dentons 7 and 8) as follows: ODC p ] k2o (f k ~ f k )j vp (5) Now we wll dene the Polarzed Observablty Don't Care (PODC). It s dened recursvely, so t can be constructed for all nodes by traversng the network from the outputs to the nputs n topologcal order. It wll be proved that f the PODC s cofactored and the polarzaton s removed, then t wll be equal to the ODC. Frst we dene the PODC of a prmary output (n the case that there are no external don't cares speced). Denton 3: P ODC for all 2 O

4 h g f h ~ g ~ Denton 5: P ODC j2f O() P ODC j If we cofactor the PODC and remove polarzaton we get the ODC. Theorem : ODC p ]? P ODC p j vp a b explct a ~ b ~ v f PODC f v 4 v 5 v 2 PODC 4 PODC 5 PODC 2 h g v 6 v 7 PODC 6 PODC 7 PODC 2,3 PODC PODC 2 PODC 3 v 2 a mplct b v v 3 Fg. 2. complement polarze Explct and mplct twnnng v 8 v 9 v PODC 8 complement polarze PODC 9 PODC The PODC of a connecton c j can be derved from the PODC of node n j usng the followng denton. Denton 4: P ODC j 8 >< >: P ODC j f j f OP (j) and INV (j) FLSE P ODC j f ~ j f OP (j) P ODC g j fj ~ and INV (j) FLSE f OP (j) g P ODC j f j and INV (j) TRUE f OP (j) and INV (j) TRUE The PODC of a node n can be derved from the PODC of all connectons c j usng the followng denton: Fg. 3. PODC network constructon So usng dentons 4 and 5 and theorem we can create the ODC of any node or connecton n the network. Fgure 3 shows how the PODC network s constructed for a sample logc network. Note that n the method descrbed n [6], see equaton (3), xnor operatons are needed at multple-fanout nodes, here we only need smple and operatons. Before provng theorem, we wll rst look what happens f we apply ths theorem to denton 4. For example, n the case? of an and gate, t s easy to prove that: ODC j ] P ODC j j vj ] P ODC j j vj f ~ j j vj ]? P ODC j j vj Pk6 v kj ODC j P k6 v kj (see equaton ).

5 However provng that theorem also holds for defnton 5 s not as easy: ODC ] (P ODC j v ) Q j2f P ODC O() j ::: (see equatons 2, 3 and 4). s a matter of fact,? t s mpossble to prove ths by just assumng that ] P ODC j j vj ODCj, wthout takng nto account the polarzed nformaton of P ODC j. In order to prove theorem we dene a property whch holds for every cut set through the network. Ths cut set can contan node varables as well as connecton varables. Denton 6: cut set C s dened as a set of ndces and ndex pars such that on every path from any prmary output to any prmary nput there s exactly one node or connecton whch appears n C. To reason about cutsets, we wll dene P EQV C, whch can be understood best as a \polarzed characterstc functon". Denton 7: h f q f ~ q P ODC ~fq q f q P EQV C q2c P ODCq g ny cut set dvdes the network nto two parts. We wll use the P EQV C to prove theorem. Frst we wll prove that P EQV C s nvarant for any cutset C. From ths property we wll prove theorem. Lemma : P EQV C 2O hf ~ f ~f f Proof By denton 3, f C O, the lemma holds. So P EQV O s just the xnor of the prmary outputs of the orgnal network and ts twn. Now we wll use nducton to prove that the P EQV C does not change f the cut set s moved towards the prmary nputs. C C v j Fg. 4. Movng cut set over a node Step Move cut set over a node (see gure 4). Suppose lemma holds for a gven cut set C. Now consder another cut set C F I(j) [ C n fjg. The cut set C does not yet cross any possble nverters on connecton c j. Step a ssume OP (j) ccordng to denton 4: P ODC j P ODC j f j. So: P EQV C hf j ~ f j P ODC j j2f I(j) ~fj f j P ODCj g apply denton 4 q2c nf I(j) hf j ~ f j P ODC j j2f I(j) ~fj f j P ODCj fj ~ as f j X j j2f I(j) j2f I(j) q2c nf I(j) ~f j P ODC j f j P ODCj g q2c nf I(j) f j and therefore f j j f j f ~ j P ODC ~fj j f j q2c nf I(j) P EQV C g f j P ODCj Step b ssume OP (j) ccordng to denton 4: P ODC j P ODC j ~ f j. So: P EQV C hf j ~ f j P ODC j j2f I(j) ~fj fj P ODCj g apply denton 4 q2c nf I(j) hf j ~ f j P ODC j j2f I(j) ~fj f j P j2f I(j) j2f I(j) as f j j q2c nf I(j) f j f ~ j P ODC j ~f j f j P ODCj g f j q2c nf I(j)

6 C C f j f ~ j P ODC ~fj j f j q2c nf I(j) P EQV C g P ~ f f j2f O() P ODC g j brng rst 2 terms nto product hf ~ f P ODC j j2f O() ~f f P ODCj g q2c nfg q2c nfg as we do not cross nverters, f f j hf j ~ f j P ODC j j2f O() ~fj fj P ODCj g q2c nfg Fg. 5. Movng cut set over an nverter Step 2 Move cut set over an nverter (see gure 5). Let j be n C, and j n C. ccordng to denton 4: P ODC j P ODCj g. Snce: f j f ~ j P ODC ~fj j f j g P ODCj g ~fj fj P ODCj f j f ~ j P ODC j P EQV C Usng these steps we can obtan any cut set C through the network. Proof of theorem : ccordng to lemma we know that the P EQV C remans constant for any cut set. For the ntal cut set (through all prmary outputs) we have: ]? P EQV C j vp ] 2O " # f f ~ ~f f 2O? f j vp f j vp v p agan we see that P EQV C P EQV C. ODC p For any cut set through varable v p we know that all other f q n the cut set do not depend on varable v p C C Fg. 6. v Movng cut set over a fanout connecton Step 3 Move cut set over a fanout connecton (see gure 6). Suppose lemma holds for a gven cut set C. Now consder another cut set: C fg[c nf O(). ccordng to denton 5: P ODC Q j2f O() P ODC j. The followng s now true: P EQV f f ~ j2f O() P ODC j? ] P EQV C j vp 2 ] 4 3 f q f ~ q P ODC ~fq q fq P ODCq g 5 q2c ] ] ( ) v p q2cnfpg h f p f ~ p P ODC ~fp p f p P ODCp g rst term does not depend on v p, brng n ] q2cnfpg ] ]??? fq fq P ODC q fq fq P ODC q h f p f ~ p P ODC ~fp p f p P ODCp g v h p f p f ~ p P ODC ~fp p f p P ODCp g take cofactor of second term ]? P ODC p j vp ( : : :) ]? P ODC p j vp So, ODC p ]? P ODC p j vp v p v p v p

7 From ths proof t can be derved that t s also possble to perform the cofactorng operatons (to varable v p ) already n dentons 4 and 5, and change theorem nto: ODC p ]P ODC p. Snce the PODCs on a cutset contan all the nformaton needed to derve the ODC of any node n the nput part of the cutset, t s obvous that the PODCs of all prmary nputs of a gven crcut can be handed over as (polarzed) EDC to a feedng network. Ths then represents the complete ODC of the external crcut, and from t the Boolean relaton for the equvalence classes [6] can be derved drectly: EQV v r p ;:::;v r q ]P EQV C(~v p v r p ; : : : ; ~v q v r q). V. Examples In the followng examples we wll show how the derent methods (Tradtonal, Daman and Polarzed) compute the ODC. Wth \tradtonal" we refer to the method based on the denton of the ODC (dentons 7 and 8). In all methods only constant propagaton s used to obtan the nal results. Example 3 also shows Savoj's method. B. Example 2 See gure 8. Tradtonal: v v v 2 v 3 v 6 v4 v 5 v 7 v 94 v 95 v 8 v 9 Fg. 8. Network for example 2. Example See gure 7. ODC 9 f (v 9 ) f (v 9 ) (v 3 v 6 v 8 ) (v 3 v 7 v 6 ) v v v 2 v 3 v 6 v4 v 5 v 7 v 8 v 94 v 95 v 9 Fg. 7. Network for example Daman: ODC 9 ODC 94 j v95 ODC 95 j v94 Polarzed: (v 2 v 3 v 7 )j v95 (v v 6 v 8 )j v94 (v 3 v 6 v 7 ) (v 3 v 6 v 8 ) ODC 9 ] (P ODC 94 P ODC 95 )j v9 ] ((v v ~v 4 )(v v 2 ~v 5 ))j v9 C. Example 3 (v 3 v 6 v 8 v 7 )(v 3 v 6 v 7 v 8 ) Example taken from [6], see gure 9. Tradtonal: Daman: ODC 9 f (v 9 ) f (v 9 ) (v 3 v 6 ) (v 3 v 7 v 8 v 6 ) ODC 9 ODC 94 j v95 ODC 95 j v94 Polarzed: (v 2 v 3 v 7 )j v95 (v v 6 v 8 )j v94 (v 6 v 8 v 3 v 7 ) (v 3 v 6 v 8 ) ODC 9 ] (P ODC 94 P ODC 95 )j v9 ] ((v v ~v 4 )(v v 2 ~v 5 ))j v9 (v 3 v 6 v 7 )(v 3 v 6 v 8 ) Tradtonal: ODC 6 v v 2 v 3 v 4 v 5 v 7 v 6 Fg. 9. Network for example 3 2 f j v6 f j v6 ((v 5 v 7 ) )((v 5 v 7 ) ) (v 5 v 7 )(v 5 v 7 )

8 Daman: Savoj: Polarzed: ODC 6 gven: 2 ODC 63 v64 ODC 64 v 63? (v 4 v 5 )j v64 (v 3 v 7 )j v63? (v4 v 5 )j v64 (v 3 v 7 )j v63 (v 5 (v 5 v 7 ))( (v 5 v 7 )) (v 5 (v 5 v 7 ))(v 5 v 7 ) ODC 3 v 4 g 2 v 4 g ODC 4 v 3 g 2 v 3 g then: O 6 v 3 ODC 3 j v4 v 3 g 2 O 2 6 (v 4 O 6)ODC 4 O 6 ODC 4 ODC 6 O 2 6 v6 O 2 6 v 6 ODC 6 ] (P ODC 63 P ODC 64 )j v6 ] ((~v v 2 v 3 )(~v v 2 v 4 ))j v6 v 5 v 7 VI. Results and Conclusons The descrbed method has been mplemented to generate the ODCs of all multple-fanout nodes n a network. The resultng ODCs are created as a network of factored forms, wth no optmzatons except for constant propagaton durng cofactorng. The algorthm was tested on the complete MCNC benchmark set for multple-level combnatonal networks. The results n table I are from the crcuts whch contan ntally more than 2 edges n the network of factored forms and are obtaned on a HP 9/755/99 (appr. 2 MIPS) wth 256 MB of memory. Table I also gves the result for creaton of the ODCs usng dentons 7 and 8. Note that we have only created the network for ODCs for multple-fanout ponts n the network, as all others can be obtaned trvally. The number of nodes n tables I and II refer to crcuts composed of ands and ors only. Inverters are not counted, as they are annotated as edge propertes. From table I we can see that the PODC method results n a ODC crcut wth fewer edges ( lterals) n 29 out of 39 examples. The tradtonal method wns 9 tmes, and both methods fal (run out of memory) for the multpler crcut of C6288. Run tmes are wthn seconds for all examples. The falure of C6288 s probably the result of the very hgh degree of reconvergence of the multpler structure. The reason that the PODC method n some examples results n a larger ODC crcut les n the fact that these examples contan nodes wth very large fanout and wth reconvergent paths whch contan almost all local nodes. We feel condent that the results of the PODC method could be further mproved wth the addton of some Boolean smplcaton durng the buldng phase of the network. Some ntal experments wth optmzaton after the buldng phase show a gan of at least a factor of 2. The tradtonal method cannot be mproved easly n ths way, as t expresses the ODC bascally n copes of the orgnal network, cofactored once, wth an xor at the prmary output. The orgnal network should be consdered optmzed already. We do not present comparsons wth other methods, because most papers do not present results on ODC sze at all, except for [6] whch presents an average number of lterals needed to represent the ODC sets, but t s not clear whch ODCs were computed (all nodes, only multplefanout nodes or nputs nodes). It should, however, be clear that the presented method s computatonally easer than [6] snce the algorthm traverses the network n the same way, but operatons at each step are smpler. Table II shows the sze of the PODC network tself of some of the largest results n table I. It can be shown that the sze of ths network s lnear n the sze of the orgnal network. Ths s a useful property snce the PODC network can be used to provde the don't care nformaton for a feedng network as ndvdual ODCs or as a sngle Boolean relaton. References [] K.. Bartlett, R.K. Brayton, G.D. Hachtel, R.M. Jacoby, R. Rudell,. Sangovann-Vncentell and. Wang, Mult-level logc mnmzaton usng mplct don't cares IEEE Transactons on CD/ICS, vol. CD-7, No. 6, June 988, pp [2] R.. Bergamasch, D. Brand, L. Stok, M. Berkelaar and S. Prakash, Ecent Use of Large Don't Cares n Hgh-Level and Logc Synthess, Proceedngs of the IEEE Internatonal Conference on Computer ded Desgn, 995, pp [3] Brayton, R.K., F. Somenz, n Exact Mnmzer for Boolean Relatons, Proceedngs of the IEEE Internatonal Conference on Computer ded Desgn, 989, pp [4] Brayton, R.K., G.D. Hachtel,.L. Sangovann-Vncentell, Multlevel Logc Synthess, Proceedngs of the IEEE, vol. 78, No. 2, February 99, pp [5] Chang,.C.L., I.S. Reed and.v. Banes, Path Senstzaton, Partal Derence, and utomated Fault Dagnoss, IEEE Transactons on Computers, February 972, pp [6] Daman M. and G. De Mchel, Observablty Don't Care Sets and Boolean Relatons, Proceedngs of the IEEE Internatonal Conference on Computer ded Desgn, 99, pp [7] Muroga, Saburo, ahko Kambayash, Hung Ch La and Jay Nel Cullney, The Transducton Method - Desgn of Logc Networks Based on Permssble Functons, IEEE Transactons on Computers, Vol. 38, No., October 989, pp [8] Savoj, Hamd and Robert K. Brayton, Observablty Relatons and Observablty Don't Cares, Proceedngs of the IEEE Internatonal Conference on Computer ded Desgn, 99, pp [9] Savoj, Hamd, Prvate communcaton, prl 996.

9 TBLE I CPU tme, number of nodes and number of edges for the ODC representaton of factored forms tradtonal PODC crcut #nodes #edges tme (s) #nodes #edges tme (s) #nodes #edges 9symml C C C C C C C C Out of memory Out of Memory C C alu alu apex apex b c cht comp count des example f5m frg frg k lal my adder par rot sct term too large ttt unreg vda x x x TBLE II Number of nodes and edges for the orgnal and the PODC network orgnal PODC crcut #nodes #edges #nodes #edges des C k

10 Harm M..M. rts was born on September 27, 963 n Venraj, the Netherlands. He receved hs M.S. degree n Electrcal Engneerng n 989 from the Endhoven Unversty of Technology. In 989 he joned the Desgn utomaton Secton of the department of Electrcal Engneerng of the Endhoven Unversty of Technology as a researcher. In 997 he joned mbt Desgn Systems, Santa Clara, C. Mchel R.C.M. Berkelaar was born on September 24, 959 n Noordwjkerhout, the Netherlands. He receved hs M.S. degree n Electrcal Engneerng `cum laude' n 987 from the Endhoven Unversty of Technology. In 987 he joned the Desgn utomaton Secton of the department of Electrcal Engneerng of the Endhoven Unversty of Technology as a researcher. In 992 he obtaned hs Ph.D. based on hs work on logc synthess. lso n 992 he joned the permanent sta of the Desgn utomaton Secton. In 994 and 995 he spent a year as a vstng scentst at the IBM T.J. Watson Research Center. Koen van Ejk was born on September 9, 97 n Hlvarenbeek, the Netherlands. He studed Informaton Engneerng at the Endhoven Unversty of Technology, from whch he graduated wth honors on ugust 27, 992. In 992 he joned the Desgn utomaton Secton of the department of Electrcal Engneerng of the Endhoven Unversty of Technology as a researcher. In 997 he obtaned hs Ph.D. based on hs work on formal vercaton. lso n 997 he joned the permanent sta of the Desgn utomaton Secton. Hs research nterests nclude formal vercaton and synthess of dgtal crcuts.