Estmatng Mean Tme to Falure n Dgtal Systems Usng Manufacturng Defectve Part Level Jennfer Dworak, Davd Dorsey, Amy Wang, and M. Ray Mercer Texas A&M Unversty IBM Techncal Contact: Matthew W. Mehalc, PowerPC Engneerng Manager Hgh Performance Processor Development, IBM Mcroelectroncs Abstract Companes spend a great deal of resources testng ther crcuts and systems to ensure that very few defectve parts reach ther customers. Unfortunately, some defectve parts always escape the testng process, and the number of these escapes s dentfed as the defectve part level. However, whle defectve part level s an nvaluable metrc for both ndustry and ts customers, another mportant metrc s MTTF (mean tme to falure) for those defectve parts. Specfcally, the shorter the tme to falure, the more vtal t s that that part be screened out as defectve durng the testng process. In ths paper we present prelmnary work for predctng MTTF from data obtaned durng manufacture testng.. Introducton No customer wants to fnd that the hardware he has purchased s defectve. Accordngly, manufacturers of ntegrated crcuts and dgtal systems expend a sgnfcant amount of effort n testng ther products before shppng them to ther customers. Unfortunately, the test pattern sets appled durng manufacture testng are not perfect, and as a result, some defectve parts wll pass all of the tests and be consdered non-defectve. Ths fracton of parts that are consdered to be nondefectve even though they actually contan defects s the defectve part level, and most IC manufacturers today try to obtan defectve part levels of less than 00 or 200 defectve parts per mllon. The defectve part level that results after all manufacture testng has been completed s an mportant metrc for both the IC manufacturer and the consumer because t s a measure of the lkelhood that a purchased IC contans a manufacturng defect. Obvously, a lower defectve part level s preferable to a hgher one. However, obtanng a defectve part level of zero s mpractcal because of the amount of tme and effort that would be requred. Thus, an mportant queston to ask s at what pont (greater than zero) s the defectve part level low enough? One novel way of answerng ths queston s to fnd the mean tme to falure (MTTF) for a defectve IC gven that t was tested wth a test pattern set that produced a gven defectve part level. If the mean tme to falure s very large on the order of the lfetme of the part or longer, then addtonal defectve part level reducton s probably unnecessary. Software or wear and tear wll probably lead to falures much sooner than any falure that would result from the manufacturng defect. On the other hand, f the mean tme to falure s relatvely short, then more testng s needed to screen out those defectve parts and further reduce the defectve part level. Thus, MTTF becomes a new metrc for evaluatng the qualty of chps and the test pattern sets that were used to test them. Ths paper wll descrbe our current work n estmatng the mean tme to falure from defectve part level and fault dctonary data. 2. Background At one tme, the amount of logc contaned on an ntegrated crcut was small, and testng t was relatvely easy. In fact, f the number of nputs was low enough, an exhaustve test pattern set contanng all possble test patterns could be used, gvng great confdence that all parts that passed the tests were actually non-defectve. However, exhaustve test pattern sets quckly become mpractcal as the number of nputs ncreases because the correspondng number of possble test patterns grows exponentally. Thus, the test patterns appled durng testng are usually a small
subset of all possble patterns. However, whch subset s chosen can have a great mpact on how many defectve parts are actually detected. In 959, R. D. Eldred noted that because defects are physcal enttes that occur n the crcut s structure, test patterns can be generated that attempt to detect the defects occurrng at dfferent crcut locatons. Ths s done by targetng faults, whch predct the effect of modeled defects on the logc operaton of the crcut. For ths purpose he proposed the sngle stuck-at fault model [Eldr 59]. In ths model, a ste n the crcut may be stuck at ether a logc one or a logc zero regardless of the value that should occur as determned by the rest of the crcut s logc. Only one fault s assumed to be present n any nstantaton of the crcut at any gven tme. Unlke the number of possble nput combnatons, whch grows exponentally, the number of possble stuck-at faults grows lnearly wth crcut sze. In addton, a sngle test pattern that s targeted to a specfc stuck-at fault (generated to specfcally detect that fault) wll often fortutously detect many other sngle stuck-at faults as well. Thus, the number of test patterns requred to detect all faults n the crcut s generally sgnfcantly smaller than the number of possble stuck-at faults. Tradtonally, the sngle stuck-at fault coverage of a test set has been used as a metrc of test set qualty. Thus, a test set that acheved 99% stuck-at fault coverage would be consdered superor to one that acheved only 95% stuck-at fault coverage. However, the actual goal of testng s to dentfy all of the defectve parts and reduce the defectve part level. Thus, an accurate defectve part level predctor would be a better metrc for comparng the effectveness of test pattern sets. One of the most famous and wdely used defectve part level models s the Wllams Brown model, whch was publshed n 98 [Wll 8]. The Wllams Brown model uses fault coverage and the ntal yeld before test patterns have been appled to predct the fnal defectve part level accordng to the followng formula. DL = Y FC Here, Y s the manufacturng yeld and FC s the fault coverage of the test pattern set appled generally the stuck-at fault coverage. (In ths model and most other defectve part level models, the predcted value of the DL s a fracton whch can obvously be converted to parts per mllon by multplyng by 0 6.) However, the sngle stuck-at fault model (and n fact any sngle fault model) cannot represent the effects of all possble defects. Thus, stuck-at fault test sets may not detect an adequate number of the untargeted defects [Mll 88], [Max 92], [Ma 95], [Fran 95], [Butl 9]. Ths s true even though the Wllams Brown model predcts that, at 00% fault coverage, the defectve part level wll be zero. In fact, as fault coverages approach 00%, the tests targeted at the remanng stuck-at faults are based n favor of detectng those faults at the expense of the remanng defects [Wang 95a], [Wang 95b], [Wang 96]. Furthermore, test pattern sets wth dentcal fault coverages may have very dfferent defect coverages and thus produce very dfferent defectve part levels [Kapu 92], [Max 9], [Park 94]. In fact, as fault coverages approach 00%, the standard devaton of the defect coverages of the test sets ncreases makng fault coverage an naccurate metrc for predctng defect coverage and defect level [Park 94]. However, whle stuck-at faults do not capture the behavor of all possble defects, there s a common requrement for detectng any defect, regardless of type: the ste where the defect occurs must be observed at an output. In other words, the erroneous value at the ste where the defect occurs must be propagated through the crcut logc to a prmary output. In contrast, the exctaton requrements vary from one defect type to another. (Exctaton refers to the need to create a dfference n logc value between those present at the defect ste n the defectve and non-defectve crcuts.) Both exctaton and observaton must occur smultaneously for the defect to be detected. In addton, just as stuck-at faults may be fortutously detected by test patterns targetng other stuck-at faults, a defect not well-modeled by a stuck-at fault may be fortutously detected by a test pattern that targets that fault f the ste where the defect occurs s observed and the defect happens to be excted. In fact, t has been found that as a ste s observed more tmes, the probablty of an undetected defect never havng been smultaneously excted decreases sgnfcantly. Ths analyss of the commonalty and dfferences among exctaton and observaton requrements of defects lead to the DO-RE-ME (Determnstc Observaton, Random Exctaton, and MPG-D Defectve Part Level Estmaton) test pattern generaton method [Grm 99]. When the DO-RE-ME method s used, emphass s placed on observng every crcut ste, especally those that are dffcult to observe, as many tmes as possble whle randomly exctng whatever defects may occur at those stes. In addton, the MPG-D
defectve part level model s used to predct the defectve part level of the resultng test set and to choose among possble subsets f too many vectors are ntally generated to ft n the tester memory. Unlke defectve part level models that predct the defectve part level based upon smple fault coverage, the MPG-D model predcts the defectve part level based upon the number of observatons of dfferent crcut stes or faults and has been shown to be more accurate, especally at very hgh fault coverages [Dwor 0]. The observaton data requred for the MPG-D defectve part level model can be obtaned from a fault dctonary. Intally, all crcut stes are assgned a (usually equal) contrbuton to the overall defectve part level. Thus, the defect level contrbuton of ste before any test patterns have been appled s: Yeld DL (0) = # of stes The defect level contrbuton of each ste then changes as patterns are appled based upon observaton counts of those stes. The probablty of exctng an undetected defect at a ste gven that that ste s observed has been studed [Dwor 00] and shown to be a decayng exponental functon of the number of tmes that ste has been observed prevously and a tme constant τ: P excte = e obs # obs τ Ths makes ntutve sense. Consder Fgure. Here the boxes represent all test patterns (nput combnatons) that wll observe a gven ste. Each of the ovals wthn the boxes represent the test patterns that wll detect a correspondng undetected defect. In other words, those test patterns wll excte that defect whle t s observed. The left rectangle represents the test spaces for undetected defects before a test pattern has been appled. A large porton of the box s covered, ndcatng that the smultaneous exctaton of at least one undetected defect gven that ths ste s observed s hghly lkely the frst tme t s observed. Now assume that the frst test pattern to observe ths ste s located at the pont ndcated by the star. If ths test pattern s chosen, then several of the defects wll be detected and therefore do not appear n the box on the rght. The probablty of exctng at least one undetected defect gven that the ste s observed s now consderably lower. Ths nformaton s used to calculate the change n defectve part level contrbuton of each ste as a result of whether or not t has been observed by a pattern accordng to the followng formula: ste DL ( n ) * ( A* Pexcte obs ), f ste was observed by pattern n = 0, otherwse Here, the constant A represents the fracton of defectve part level contrbuton that wll be removed from the ste gven that at least one undetected defect s excted and observed. Other equatons are used to calculate addtonal changes n defectve part level contrbuton due to the sharng of defects among crcut stes, gvng a resultng value of share. Then a new value for each ste s defectve part level contrbuton after pattern n has been appled s calculated accordng to the followng equaton: DL ( n) = DL ( n ) ste share before The overall defectve part level s calculated by summng the defectve part level contrbutons of every ste. Total _ DL( n) = # of stes = after Test pattern appled Fgure : Test spaces of undetected defects gven that ste s observed before and after the test pattern s appled DL ( n)
However, whle the defectve part level obtaned by a gven test pattern set s very valuable nformaton for both the ntegrated crcut manufacturer and the customer, another valuable metrc to consder would be the mean tme to falure of a defectve part. MTTF s mportant precsely because t gves an estmate of how soon the defect wll affect the operaton of a crcut. It allows for a quanttatve analyss of whether defectve part levels obtaned wth current test sets are low enough for a gven applcaton. In addton, t would allow the probablty that the frst error that occurs durng crcut operaton wll be due to a manufacturng defect to be compared to the probablty that that error wll be due to ether a software error or early lfe falure. 3. Evaluaton of the Correlaton between MTTF and Defectve Part Level In order to determne f a correlaton exsts between MTTF and the defectve part level, we tested three of the ISCAS85 [Brgl 85] benchmark crcuts and one of the ISCAS89 [Brgl 89] benchmark crcuts. Usng Verlog, we nserted hgh mpedance ponts nto each nput wre and each nternal wre of the crcut such that only a sngle hgh mpedance pont was nserted for any nstantaton of the crcut. We then appled ATPG test patterns to determne when these smulated defects, or surrogates, would be detected durng testng, and we used random nput patterns to smulate the behavor of the crcut under normal operatng condtons. Thus, we determned how many patterns a crcut contanng a surrogate took to fal, (MTTF), under normal operatng condtons. We dscovered that the correlaton between when surrogates are detected durng testng and how quckly they are detected durng normal operatons depends greatly on the permeablty of the crcut (how easy crcut stes are to observe) and whether the crcut s sequental or combnatonal n nature. 800 700 600 500 400 300 200 00 0 to Detect Falure vs. Percentage of Tme Defect was Detected Durng ATPG Testng 0 0.2 0.4 0.6 0.8.2 Percent Defect Detecton Fgure 2: MTTF vs. Percent Defect Detecton for C880 Specfcally, we found that for C880, a hghly permeable crcut, no correlaton exsts between when surrogates are detected durng ATPG and when they are detected durng normal operaton. The data for C880 s shown n Fgure 2. For nearly all of the 46 surrogates shown n Fgure 2, the tme the resultng crcut takes fal s very small on the order of 0 nput vectors. In contrast, for sem-permeable crcuts, there s a greater correlaton between MTTF and the defectve part level. 40 35 30 25 20 5 0 5 to Detect Falure vs. Percentage of Tme Defect was Detected Durng ATPG Testng 0 0 0.2 0.4 0.6 0.8.2 Percent Defect Detecton Fgure 3: MTTF vs. Percent Defect Detecton for C499 C499, shown n Fgure 3, s an example of a sem-permeable crcut. As shown by the trendlne, MTTF decreases as the percent defect detecton ncreases. In other words, those surrogates detected early durng ATPG tend to lead to smaller values of MTTF under normal crcut operatons. The ponts at 0.2 and 0.4 defect detecton are clustered around 5 nput vectors and 32 nput vectors. Ths may be due to dfferent path utlzatons for ATPG and normal operatons.
6 4 2 0 8 6 4 2 to Detect Falure vs. Percentage of Tme Defect was Detected Durng ATPG Testng 0 0 0.2 0.4 0.6 0.8.2 Percent Defect Detecton Fgure 4: MTTF vs. Percent Defect Detecton for C432 C432, shown n Fgure 4, s another sempermeable crcut. The correlaton s even stronger n ths crcut than n the prevous crcut. Thus, the correlaton between when surrogates are detected durng ATPG and when they are detected durng normal operaton n sem-permeable crcuts appears to be stronger than n hghly permeable crcuts. Ths seems to ndcate that MTTF wll be more dffcult to predct for defects remanng n hghly permeable crcuts or hghly permeable areas of crcuts. Fortunately, hghly permeable areas n a crcut are usually not the problem spots n the crcut. However, whle a correlaton exsts for combnatonal crcuts, partcularly those that are less permeable, the correlaton n sequental crcuts such as S27 descrbed below s much weaker. 8 7 6 5 4 3 2 to Detect Falure vs. Percentage of Tme Defect was Detected Durng ATPG Testng 0 0 0.2 0.4 0.6 0.8.2 Percent Defect Detecton Fgure 5: MTTF vs. Percent Defect Detecton for S27 One tem to notce about the data on S27, shown n Fgure 5, s the ponts n the upper rght corner. These ponts correspond to a hgh probablty of beng found n ATPG testng, but a low probablty of beng found durng normal operaton. These ponts correspond to surrogates that are located at the nputs nto two of the D flpflops n the crcut. Durng scan, these two ponts are pseudo-outputs of the crcut and defects here are easly observable. However durng normal operatons, they are not easly observable because they have a chance of beng masked by other gates n the crcut. Thus, these ponts llustrate the skew we expect n sequental crcuts when tested usng scan. Another tem to notce s breadth of dstrbuton of the data ponts for S27. The data ponts for the combnatonal crcuts were generally focused n narrow columns. The same s not true for the sequental crcut. Instead, the data ponts are scattered. We expect that ths wll lead to greater uncertanty n our mathematcal model of the relatonshp between MTTF and defectve part level. Thus, from ths data, we can conclude that a smple mappng between defectve part level and MTTF may not be adequate, especally for sequental crcuts, because of the mportance that the probablty of observaton of the defect stes durng normal operaton appears to have on how quckly a defectve crcut wll fal. Specfcally, t was shown that for the hghly permeable C880 crcut (n whch nearly all of the crcut stes were hghly observable) falure occurred very quckly durng normal crcut operaton regardless of where n the crcut the surrogate was located. In addton, the change n observaton probabltes that occurred for the nputs to D flp-flops that would have been contaned n the scan chan n S27 dramatcally affected when those surrogates would be detected durng normal crcut operaton vs. durng ATPG. Thus, ths ndcates that an accurate estmator of MTTF wll need to nclude nformaton on the probablty of observaton of dfferent crcut stes durng normal operaton n addton to the lkelhood that each of those stes are lkely to stll contan defects. Fortunately, data collected whle calculatng the defectve part level usng the MPG-D model ntroduced earler may prove useful n predctng MTTF. 4. Extendng MPG-D to Predct Mean Tme to Falure Obvously, the underlyng requrements for detectng a defect durng testng are exactly those requrements that must be met for a defect to cause ncorrect behavor durng normal operaton.
Specfcally, the defect must be both excted and observed for ths to occur. Recall that the probablty of exctng an undetected defect gven that a ste s observed s modeled as a decayng exponental n MPG-D. Thus, after test patterns have been appled, many of the IC s contanng easy-to-detect defects have already been dentfed and removed from consderaton. The remanng defectve parts are most lkely to contan those defects that are harder to excte and thus detect. It s how quckly these defects cause observed errors durng normal operaton that wll determne our value of MTTF. Ideally, when calculatng MTTF, we would assume that only one defect occurs on any gven defectve chp and would consder the weghted average of the test set szes for each of the remanng potental defects (as depcted n Fgure ) based upon probablty of occurrence of that defect whle calculatng the probablty of exctng and defect whle observng the ste. However, detaled nformaton of the remanng defect types and ther correspondng test spaces s unknown. Thus, we propose to use the probablty of exctng an undetected defect gven that the ste s observed calculated accordng to the MPG-D formula P excte = e obs # obs τ (where #obs s equal to the number of tmes that ste was observed durng ATPG) as an upper bound on the probablty of exctng an undetected defect at a ste gven that that ste s observed durng normal operaton. We wll use ths value n our MTTF calculatons. The other requrement that must be satsfed s observaton of the ste where the defect occurs. However, the observaton probablty used should be the observaton probablty of each ste under the condtons of normal operaton,.e. under normal applcatons. In the absence of applcaton data, probablty of observaton data for random vectors may be used. The probablty that the value of a crcut ste n a sngle clock cycle (when the defect was excted) would affect the outputs n ether that clock cycle or n a subsequent clock cycle would need to be determned. If t generally takes many clock cycles for that value to affect the output, then the average number of clock cycles needed should also be collected so that ths can be factored nto MTTF calculatons. In ether case, some sort of smulaton wll lkely need to be done to collect ths data. Once the probablty of observaton under normal operatng condtons and the probablty of exctaton of an undetected defect gven that the ste s observed and gven that t was observed for a certan number of tmes n testng have been obtaned for every ste, we multply the two values together. Ths wll gve us the probablty of detecton of an undetected defect at each crcut ste gven that that ste s where the defect occurs. Thus, ( P )( P ) P = obs normal operaton exc obs Thus, the probablty of detectng a defect that occurs at ste for the frst tme wth the frst pattern s P. Smlarly, the probablty of detectng a defect that occurs at ste for the frst tme wth the second pattern (assumng ndependence between patterns) s P ( P ). If we extend ths, then the probablty of detectng a defect that occurs at ste for the frst tme wth the nth pattern s n P ( ) P. We can use ths to fnd the average number of patterns that wll be appled (number of clock cycles) before the defect at ste causes an error n normal crcut operaton n np ( P ) = n= P Thus, as expected, the average number of clock cycles that we can expect to pass before the defect at ste s detected s nversely proportonal to the probablty of that defected beng detected. If many addtonal cycles are expected to be needed before an error wll appear at an output, these can be added to our expected value at ths pont. However, each ste s not equally lkely to be the ste where the defect occurs. Stes that were observed many tmes durng testng are much less lkely to contan the undetected defect than stes that were observed few tmes, f at all. Thus, when we fnd the average patterns to detecton for the entre crcut, we wll need to take a weghted average where the weghts are based upon each ste s lkelhood of contanng the defect. For ths, we can use the DL contrbuton of every ste calculated by MPG-D. DL ( n) = DL ( n ) ste share where n s equal to the number of test patterns appled durng ATPG testng. Thus, we can calculate the expected number of clock cycles before falure for a crcut gven that the crcut s defectve and has been tested wth a
manufacturng test pattern set of gven characterstcs as: number of = stes DL DL P Obvously, we can convert ths to tme and thus MTTF usng the clock speed. We may also fnd that we want to take stes wth ncredbly low DL contrbutons and remove them from consderaton f we are farly confdent that no defects could reasonably occur there. Ths could reduce the smulaton tme requred for determnng the probablty of observaton values durng normal operaton. 5. Future Work In order to better understand the correlaton between the defectve part level and MTTF, more smulatons wll be conducted to collect data on varous crcuts for analyss and comparson. In addton to the hgh mpedance ponts that are currently modeled for smulaton, surrogates such as brdgng surrogates wll be ntroduced n future smulatons. More complcated surrogates such as delays and couplng effects n sequental crcuts are also beng consdered for future nvestgaton. These surrogate smulatons wll be used to provde addtonal understandng of the correlaton between how quckly surrogates are detected durng testng and MTTF. Data collected from these smulatons wll also be used to estmate defectve part levels and to verfy and refne the MTTF model descrbed n the prevous secton. In addton, we ntend to gather data from these smulatons to try to quantfy the uncertanty n our MTTF predctons. Intutvely, as the probablty of detectng a defect decreases, precsely when that defect can be expected to be detected becomes less certan. Furthermore, n P ( ) P has a geometrc dstrbuton wth 2 varance nversely proportonal to P [Ash 93]. Thus, future work wll contan more detaled analyss of the precson wth whch MTTF can be predcted. Unfortunately, whle smulaton data s very useful, t s not perfect. It has the lmtatons of beng tme consumng and thus lmts the number of clock cycles that can be smulated. It also restrcts the defect types studed to those that are modeled as surrogates n the smulaton. Thus, an even better understandng of MTTF can be accomplshed f experments are done usng actual manufactured ntegrated crcuts n hardware. Thus, after usng surrogate smulaton to gve us prelmnary nformaton about the relatonshp between test patterns appled, defectve part level, and MTTF, we wll contnue our expermentaton usng actual hardware. We wll run parts that were dentfed to be defectve durng manufacture testng n parallel wth a seres of gold standard chps that have been thoroughly tested and are assumed to be non-defectve. By comparng the outputs of the defectve and gold standard chps, we can obtan actual MTTF results. These results can then be analyzed wth respect to when those parts faled manufacture testng and what the defectve part level of the chps would have been had testng been stopped before that defectve crcut was dentfed. Thus, the data obtaned from these hardware experments wll be compared to smulaton results and used to further refne our models whle possbly trggerng more nterestng questons to be nvestgated. 6. Concluson The ablty to predct MTTF would be a valuable asset for both ndustry and consumers. It could change the way that ndustry sets the maxmum allowed defectve part level for each product and lead to better use of testng resources. If a company knew the estmated MTTF that corresponds to the defectve part level they attan after testng has been completed, t could use that nformaton to adjust ts warranty perod or to relax the defectve part level requrement, turnng products that would have been unnecessarly elmnated nto proftable tems. In ths paper we have descrbed our ntal work wth respect to predctng MTTF. We have shown that observablty of crcut stes durng normal operaton has great nfluence on MTTF values for defects that occur at those stes. Thus, the probablty of observaton of crcut stes durng normal operaton wll be an mportant component of any accurate model for MTTF. Accordngly, we have ntroduced a prelmnary model whch attempts to relate. the observablty of crcut stes durng normal crcut operaton 2. the probablty of exctng an undetected defect at one of those crcut stes gven that t s observed and gven that a certan number of observatons of that ste occurred durng manufacture testng, and 3. the probablty that a defect remans at that ste
to MTTF. However, ths model needs to be further evaluated usng both surrogate smulaton and experments run usng actual defectve crcuts. Developng a model to predct MTTF s not a smple task. The problem s qute complcated, and there are stll many aspects we have yet to explore. However, our prelmnary results have gven us valuable nformaton and drecton n formulatng an ntal model, and we are confdent that further expermentaton wll lead to even more valuable nsghts. Thus, we beleve that ths journey to learn more about MTTF wll be rewardng and wll ultmately provde valuable knowledge for those n ndustry and those who buy ther products. References [Eldr 59] R.D. Eldred, Test routnes based on symbolc logc statements. Journal of the ACM, Vol. 6, no., 959, pp. 33-36. [Wll 8] T. W. Wllams and N. C. Brown, Defect level as a functon of fault coverage, IEEE Trans. on Computers, Vol. C-30, no. 2, 98, pp. 987-988. [Mll 88] S.D. Mllman and E.J. McClusky, Detectng brdgng faults wth stuck-at test sets, Proc. Internatonal Test Conference, 988, pp. 773-783. [Max 92] P.C. Maxwell, R. C. Aken, V. Johansen, and I. Chang, The effectveness of I DDQ, functonal and scan tests: how many fault coverages do we need? Proc. Internatonal Test Conference, 992, pp. 68-77. [Ma 95] S. C. Ma, P. Franco, and E. J. McCluskey, An expermental chp to evaluate test technques: experment results, Proc. Internatonal Test Conference, 995, pp. 663-672. [Fran 95] P. Franco, W. D. Farwell, R. L. Stokes, and E. J. McCluskey, An expermental chp to evaluate test technques: chp and expermental desgn, Proc. Internatonal Test Conference, 995, pp. 653-662. [Butl 9] K. M. Butler and M. R. Mercer, Quantfyng non-target defect detecton by target fault test sets, Proc. European Test Conference, Munch Germany, Aprl 0-2, 99, pp. 9-00. [Wang 95a] L-C. Wang, M. R. Mercer, and T. W. Wllams, On effcently and relably achevng low defectve part levels, Proc. Internatonal Test Conference, 995, pp. 66-625. [Wang 95b] L-C. Wang, M. R. Mercer, T. Wllams, S.W. Kao, On the declne of testng effcency as fault coverage approaches 00%, Proc. VLSI Test Symposum, Prnceton, NJ, 995, 74-83. [Wang 96] L-C. Wang, M. R. Mercer, T. Wllams, A better ATPG algorthm and ts desgn prncples, Proc. Internatonal Conference on Computer Desgn, 996, pp. 248-252. [Kapu 92] R. Kapur, J. Park, M. R. Mercer, All tests for a fault are not equally valuable for defect detecton, Proc. Internatonal Test Conference, 992, pp. 762-769. [Max 9] P. C. Maxwell, R. C. Atken, V. Johansen, and I. Chang, The effect of dfferent test sets on qualty level predcton: when s 80% better than 90%? Proc. Internatonal Test Conference, 99, pp. 358-364. [Park 94] J. Park, M. Navar, R. Kapur, M. R. Mercer, and T. W. Wllams, Lmtatons n predctng defect level based on stuck-at fault coverage, Proc. VLSI Test Symposum, 994, pp. 86-9. [Grm 99] M. R. Grmala, S. Lee, J. Dworak, K. M. Butler, F. Stewart, H. Balachandran, B. Houchns, V. Mathur, J. Park, L-C. Wang, and M. R. Mercer, REDO Random Exctaton and Determnstc Observaton frst commercal experment, Proc. VLSI Test Symposum, 999, pp. 268-274. [Dwor 0] J. Dworak, J. Wcker, S. Lee, M. R. Grmala, K. M. Butler, B. Stewart, L-C. Wang, and M. R. Mercer, Defect-Orented Testng and Defectve-Part-Level Predcton, IEEE Desgn and Test of Computers, January- February, 200, Vol. 8, No., pp. 3-4. [Dwor 00] J. Dworak, M. R. Grmala, S. Lee, L-C. Wang, and M. R. Mercer, Enhanced DO-RE- ME Based Defect Level Predcton Usng Defect Ste Aggregaton MPG-D, Proceedngs of the 2000 Internatonal Test Conference, Atlantc Cty, NJ, October 3-5, 2000, pp. 930-939. [Brgl 85] Brglez, F. and Fujwara, A Neutral Netlst of 0 Combnaton Benchmark Crcuts and a Target Translator n FORTRAN. Proc. Int. Symp. Crcuts Syst., 985. [Brgl 89] F. Brglez, D. Bryan, and K. Kozmnsk, Combnatonal Profles of Sequental Benchmark Crcuts Proc. Int. Symp. Crcuts Syst., 989. [Ash 93] C. Ash, The Probablty Tutorng Book: An Intutve Course for Engneers and Scentsts (and everyone else!), IEEE Press, 993.