Approximate logic synthesis for error tolerant applications

Transcription

1 Approximate logi synthesis for error tolerant appliations Doohul Shin and Sandeep K. Gupta Eletrial Engineering Department, University of Southern California, Los Angeles, CA 989 {doohuls, Abstrat Error tolerane formally aptures the notion that for a wide variety of appliations inluding audio, video, graphis, and wireless ommuniations a defetive hip that produes erroneous values at its outputs may be aeptable, provided the errors are of ertain types and their severities are within appliation-speified thresholds. All previous researh on error tolerane has foused on identifying suh defetive but aeptable hips during post-fabriation testing to improve yield. In this paper, we explore a ompletely new approah to exploit error tolerane based on the following observation: If ertain deviations from the nominal output values are aeptable, then we an exploit this flexibility during iruit design to redue iruit area and delay as well as to inrease yield. The speifi metri of error tolerane we fous on is error rate, i.e., how often the iruit produes erroneous outputs. We propose a new logi synthesis approah for the new problem of identifying how to exploit a given error rate threshold to maximally redue the area of the synthesized iruit. Experiment results show that for an error rate threshold within %, our approah provides 9.43% literal redutions on average for all the benhmarks that we target. Keywords error tolerane, logi synthesis, approximate logi funtion, funtional yield I. INTRODUCTION As the VLSI fabriation tehnology saling is reahing nano-sale, dramati improvements in most attributes of iruits, espeially delay and yield, provided by saling are beginning to derease. One of the main reasons for this trend is the inrease in non-idealities, suh as defet rates and variations due to the manufaturing proess []. To mitigate the effets of these non-idealities, researhers have proposed onepts of fault tolerane and defet tolerane. However, the disadvantage of these tehniques is that they require additional hardware hene inrease the omplexity of the iruit, when ompared to the original implementation. The onept of error tolerane was proposed to ombat non-idealities without inreasing iruit omplexity [2]. Traditional test tehniques lassify hips as either perfet, i.e., without any error-produing defets, or imperfet, i.e., those that have one or more error-produing defets or variability. In traditional testing, every imperfet hip is disarded. The ore onept of error tolerane is that, for a wide range of appliations inluding image, video, audio, graphis, games, and error-orreting odes for wireless ommuniation, an imperfet hip an still be used, provided the types and severities of errors are within ertain appliation-speified threshold. These imperfet but useable hips (i.e., hips that produe only aeptable errors) are lassified as aeptable hips [3, 4]. Two quantitative metris have been previously proposed to measure severity of errors [3]. Error signifiane (ES) for a set of outputs is defined as the maximum amount by whih the numerial value at the outputs of an imperfet iruit version an deviate from the orresponding value for the perfet version. Error rate (ER) is the perentage of vetors for whih values at a set of outputs deviate from the error free response, during normal operation. Composite metris have also been defined using ER and ES. In [5, 6], faults in iruits used for multimedia appliations are analyzed and it is shown that for a signifiant perentage of faults the output quality degradation is not signifiant. Also in [7] an algorithm has been developed to generate tests that detet unaeptable hips without rejeting aeptable hips, for appliations that use ER as the metri to define aeptability. The main fous of all previous researh on error tolerane was to identify hips during post-fabriated testing that are imperfet but still aeptable. However, [8] we showed that exploiting error tolerane during the design of the iruit an redue iruit s omplexity, i.e., delay and area. Sine redution in iruit omplexity an be translated into yield improvement and redution in fabriation ost, this represents a new way of exploiting error tolerane. In this paper we propose a logi synthesis approah to design iruits that implement approximate versions of the given funtion. We onsider ER as the metri for error tolerane. The appliations for whih the hip is to be used must be analyzed to determine the threshold on error rate. Chips that have error rates smaller than this threshold would then be deemed aeptable. The objetive is to obtain designs that have minimum area (minimum number of literals) for a given error rate threshold. Assuming that every input vetor is equally likely during normal operation, the appliation-speified error rate threshold determines the number of minterms in the are-set of the given funtion for whih the output value an be omplemented. With this observation, we an define the objetive of our algorithm as: Identify minterm omplements that produe an approximate iruit version that has the smallest number of literals for a given error rate threshold. Even though a omposite metri of ES and ER is useful for many appliations, analyzing the eah output s ES for multiple output funtions is beyond the sope of this paper and a subjet of our ongoing researh. First we investigate this problem using exhaustive searh where we enumerate all possible approximate funtions, eah obtained by omplementing a different minterm. This simple experiment showed that signifiant literal redution is possible even when a single minterm of the original funtion is omplemented. However, the main hallenge with exhaustive searh is that its time omplexity inreases exponentially with the number of iruit inputs, and the inrease in time omplexity with given error rate threshold. In this paper, we identify several new properties and integrate these into a new heuristi approah to synthesize approximate iruit versions for

2 higher error rate thresholds. This paper is organized as follows. In the next setion, we show the results of our exhaustive searh experiments. In Setion III, we develop several new properties to redue the number of andidate minterms that an be omplemented to redue the number of literals. In Setion IV, we propose our heuristi approah to identify minterms that an be omplemented to maximally redue the number of literals. Finally in Setion V, we present experimental results for benhmark iruits and report the yield improvements provided by our approah. II. EXHAUSTIVE SEARCH TO FIND OPTIMAL MINTERM COMPLEMENTS A. Example of reduing literals in a iruit by omplementing minterms of the original funtion. Definition ) Minterm omplement threshold (C t ): The maximum number of minterms in the are-set (on-set and off-set) of the Boolean funtion that an be omplemented under the given error rate threshold. For example, for an n input funtion, if the error rate threshold is r, then we are allowed to omplement up to C t = r 2 n minterms in its are-set. x (: omplemented minterms) Figure. Literal redutions obtained by omplementing a minterm. The over in Figure is the minimum sum-of-produt (SOP) over. Suppose C t =. Then we an omplement any one minterm from the original funtion. To redue the number of literals from original minimum over, we hoose minterm x to omplement from to. The minimum over for this simplified funtion is shown in Figure. The omplemented minterm expands the prime impliant (PI) x 4 to, expands xx2x3x4 to x x 4 and removes x. This redues the number of literals by,, and 3, respetively, with respet to the original PIs. Hene, the total literal redution is six for this minterm omplement, sine removing a PI also redues literal in the OR part of SOP representation. A signifiant redution in the number of literals an hene be expeted by seleting to minterm omplements that an expand many original PIs. With the same onstraint of C t =, Figure () shows an example where omplementing a minterm from to redues the x x () number of literals. In this ase PI x an be removed, whih redues five literals from the original minimum over. B. Exhaustive searh to find a minterm omplement for maximum literal redution This sub-setion presents the results of our exhaustive searh to identify the minterm omplements that redue the greatest number of literals for a given C t. Eah possible ombination of C t or fewer minterm omplements produes one approximate version of the funtion. Hene, assuming C t minterms have been omplemented to generate eah approximate funtion, we have C(2 n, C t ) possible approximate funtions, where n is the number of inputs of the funtion. (Note: C(p, q) denotes all ombinations of q items seleted from a given set of p items.) We synthesize eah approximate funtion and ount the number of literals in the synthesized iruit. For example, if C t = (i.e., error rate threshold is /2 n ), first we omplement minterm ( ) to obtain an approximate version of the funtion, synthesize it, and ount the number of literals in the iruit. This experiment is repeated for all single omplements from minterm ( ) to ( ). We use two-level logi synthesis tool ESPRESSO-MV [9] Number of iruit versions to to Number of literals Number of iruit versions to to Number of literals Figure 2. Number of literals in different iruit versions when Ct = : Z9sym.pla, and sym.pla. Figure 2 shows the histograms of number of literals in different iruit versions obtained in this manner for Z9sym.pla (a 9-input, -output funtion), and sym.pla (a -input, -output funtion), whih are benhmark iruits from the two-level synthesis suite. The numbers of literals in the original minimum overs are 6 and 47, respetively. We ahieve 4.8% and 8.5% literal (area) redution for Z9sym.pla and sym.pla, respetively, when we omplement a single minterm that redues the number of literals by the maximum amount, ompared to the minimum over for the orresponding original funtion. One important observation from Figure 2 is that all approximate versions of iruits that have the smallest numbers of literals are obtained from approximate funtions obtained by omplementing a minterm in the off-set (i.e., by a to omplement). This is beause to redue the number of literals by exploiting a to omplement, we have to remove an existing PI. With one to omplement, we an only remove at most one PI in the original minimum over. However, a to omplement an expand many PIs in the original minimum over. Also, if an expanded PI overs all the minterms in another PI, then the PI that has been overed beomes redundant and hene an be removed from the minimum over for approximate funtion. Figure shows the removal of a PI made redundant by a to omplement. A similar reasoning also suggests that to 2

3 omplements often provide greater redutions in number of literals for ases where we are allowed to omplement multiple minterms of the funtion. In Figure 2, we an also observe that several iruit versions have larger numbers of literals than the original minimum over. There are two reasons for this: (i) a to minterm omplement adds a new PI to the original minimum over, and (ii) a to minterm omplement redues the size of PIs that were used in the original minimum over and this neessitates the use of multiple PIs, eah with more literals. Suh an exhaustive searh an only be used for iruits with small number of inputs and to selet a small number of minterms to omplement, sine omplexity grows exponentially with the number of inputs and inrease in omplexity with C t. Hene, we perform exhaustive searh only when C t = or 2 for most of the benhmarks to ompare with the heuristi approah that we desribe in Setion IV. III. DETERMINISTIC PRUNE OF MINTERM COMPLEMENT CANDIDATES In the previous setion, we quantified literal redutions obtained by exhaustively enumerating all possible approximations. From the results, we notie that to omplements are typially more benefiial than to omplements and suggested possible reasons for this phenomenon. In the rest of this paper, we will fous only on to omplements. In this setion, we introdue some useful properties to eliminate to omplement andidates that annot redue the number of literals. Definition 2) Adjaent minterms: Consider two minterms m i and m j. m i and m j are said to be adjaent if the hamming distane between m i and m j is in the sense that m i and m j differ only in one bit. For example, in a four-input funtion, minterms x and xx 4are adjaent to eah other. Definition 3) Minterm luster (M ): A minterm luster is a set of minterms omplemented from the original funtion from to suh that for any two minterms, m i and m j M, there exists a sequene of minterms (m i, m i,, m i2,,m iα, m j ) where m i,, m i2,,m iα belong to M, and every pair of onseutive minterms in the sequene are adjaent. For developing properties, let us denote the set of all on-set minterms in the original funtion by a set M o and the set of PIs newly generated by omplementing the set of minterms M as PI(M ). Also, denote the set of minterms in a prime impliant PI j as M PIj. Property ) Consider a to minterm omplement m f. If M PIδ M o = φ, for all PI suh that m f PI δ, then there exist other approximate versions of the original funtion with fewer literals and smaller sets of minterm omplements that do not inlude the minterm omplement m f. Proof) To over m f in the new minimum over, we need at least one PI δ. Sine M PIδ M o = φ, if we do not omplement minterms that are only overed by PI δ, inluding m f, we an remove PI δ s from the over for the approximate funtion. This leads to a new iruit version with fewer literals and fewer minterm omplements, beause without PI δ, the number of literals redues and we omplement fewer minterms from the original funtion. Hene, under the given minterm omplement threshold, there exists another approximate iruit version with fewer literals and fewer minterm omplements where we do not omplement m f. Above property shows us that if there exists a minterm omplement m f and for all the PI δ s for whih m f PI δ don t overlap with M o, then we do not have to onsider omplementing m f from the original funtion. For example, in Figure 3, it is unneessary to omplement minterm x with x beause when we omplement both minterms, we need an additional PI xx 4to over minterm x. Even though the minimum over for the approximate funtion obtained by omplementing both minterms has fewer literals than the original minimum over, there exists another approximate version of iruit, obtained by omplementing only x, whih has fewer literals and fewer minterm omplements. Corollary ) If M PIδ M o = φ, for all PI δ PI(M ), then the number of literals in the minimum over of approximate funtion obtained by omplementing M is greater than the number of literals in the original minimum over. Proof) For M M o, the original minimum over is still the minimum over for M o beause M PIδ M o = φ, for all PI δ PI(M ). Sine we need additional PIs to over M in addition to the PIs that over M o, the literals for these additional PIs are added to the number of literals in the original minimum over. x x Figure 3. Examples of minterm omplement andidates. Figure 3 shows that omplementing M = { xx2x3x4, x, xx2x3 } annot redue the number of literals from the original minimum over and M requires additional PI to over. Conversely, for a M to redue number of literals from original minimum over, there must exist PI δ PI(M ) suh that M P I δ M o φ (i.e., at least one PI in PI(M ) must overlap with M o ). Corollary 2) Let d be the hamming distane between a to minterm omplement m f and a on-set minterm in M o that is at the minimum distane from m f. For the new approximate funtion obtained by omplementing m f to redue the number of literals in the minimum over, d log 2 (C t +). Proof) Let m be a losest on-set minterm in M o to m f. Suppose we omplement m f to redue literals from the original minimum over. For m f to redue literals from the original minimum over, PI δ suh that m f M PIδ and M PIδ M o φ. The prime impliant PI δ that ontains m f, overlaps with M o and ontains the smallest number of minterm omplements is the PI whih onsists of literals ommon to minterms m f and m. All the minterms in this PI must have the literals ommon to m f and m and the distane from eah minterm in the PI (exept for m f ) to m must be smaller than the distane from m f to m. Sine we assume that m is the on-set minterm in M o that is the losest to m f, all the other minterms exept m in the PI are not in M o. This in turn means all these are off-set minterms for the original 3

4 funtion. Sine to omplement all minterms in the PI exept m requires 2 d - omplements, 2 d - C t. Hene, we see that d log 2 (C t +). Figure 4. Candidate minterm flips when C t = : All the minterms in CM, Minterms that an expand the original PIs. Figure 4 shows the possible minterm omplements when C t =. The shaded off-set minterms are the ones for whih d log 2 (C t +) (note that d is a non-negative integer). Only the off-set minterms among these andidates an redue the number of literals ompared to the original minimum over when C t =. We an try different ombinations of C t or fewer omplements out of these andidates and synthesize eah approximate funtion to obtain the approximate version of iruit that has the minimum number of literals, for the given C t. Assuming we omplement all the possible C t omplements for the approximate funtion with above proprieties, we an only onsider C(CM, C t ) ases to omplement, where CM is the set of offset minterms suh that, for eah minterm, d log 2 (C t +). IV. x APPROXIMATE LOGIC SYNTHESIS BY ASSISTING EXPANSIONS With the observations in Setion II and properties derived in Setion III, we identify the fats that to omplements that are within a partiular hamming distane of the minterms in the original funtion are possible andidate minterms to omplement to redue the number of literals from the original minimum over. However, a searh algorithm that onsiders C(CM, C t ) approximate funtions still has exponential worst-ase run-time omplexity in the number of funtion inputs, n, and a signifiant omplexity with C t. In this setion, we propose a heuristi approah to selet to minterm omplements to find an approximate iruit version that has the minimum number of literals for a given value of C t. Among the minterms in CM, the minterms that we fous on are the minterms that an assist expanding PIs in the original minimum over. The main reason is beause when we omplement only minterms from to that an expand one or more PIs in the original minimum over, the given minimum over of the orresponding approximate funtion is guaranteed to have a smaller number literals than the minimum over for the original funtion. On the other hand, omplementing minterms in CM that annot expand any PI in the original over requires new PIs to over the omplemented minterms whih may inrease the total number of literals. In some ases, omplementing minterms that annot expand any original PI an also redue the total number of literals by produing a different over for the original funtion if minterm omplements are in CM (minterm omplements that an expand one or more original PIs are also in CM). In this ase, the total literal redution is the differene between the inrease in the number of literals due to the new PIs that over minterm omplements and the derease in the number of literals due to the use x of a different over for the original funtion. Figure 4 shows the andidate minterm omplements when we onsider all minterms in CM for C t =, and the andidate minterms that an expand one or more PI in the original over, for the ase where C t =. Typially, the latter set ontains muh fewer andidate minterms. Our heuristi exploits the fat that in most ases, a ombination of minterm omplements that an assist PI expansions an provide an approximate iruit version with the minimum number of literals at given C t. However, there exist some original funtions where minterm omplements that do not expand any PI in the given original funtion minimize the number of literals for a given C t. Based on above observations, we develop proedures to enumerate all the possible expansions for the original PIs and for identifying approximate iruit versions with the minimum number of literals for a given C t. Definition 3) Minterm set to expand original PIs (MSEOP): A set ontaining all off-set minterms that prevent expanding one or mo re PIs in a ertain diretion. Sine a PI an expand by eliminating different literals in the PI, multiple MSEOPs an exist for a PI. For example, in Figure 4, { xx2x3x4}, { xx2x3x4, xx2x3}, and { x x 4, xxxx 2 3 4} are a few different MSEOPs that eah expand (by removing one literal) the PI xx2x4in the original over. We first enumerate all MSEOPs that an expand PIs in the original minimum over to identify the MSEOP that leads to an approximate iruit version for a given C t. Let us denote the ardinality of MSEOP by MSEOP. If MSEOP C t, we store the original PI, the expanded PI, and the number of literals redued by the expansion to a list, L(MSEOP). After searhing all the PIs that an expand in the given C t, the list ontains the information about PIs that an be expanded by omplementing MSEOPs. Figure 5 shows the proedure Generate-list. //Generate-list ( ) begin foreah PI in the original minimum over i, foreah possible expansion of PI i j, if MSEOP ij C t, then add info about PI ij to expansion list L(MSEOP ij ) end Figure 5. Proedure to generate list of MSEOP that an expand PI. //Generate-union-list (MSEOP i ) begin foreah L(MSEOP) j, if MSEOP i MSCOP j C t, Merge L(MSEOP i ) and L(MSEOP i ) into L(MSEOP i MSEOP j ) Generate-union-list (MSEOP i MSEOP j ) end Figure 6. Proedure for generating union list for union MSEOP. Lists from Generate-list onsider MSEOPs that an expand at least one PI in the original minimum over. We an onsider omplementing union of multiple MSEOPs, suh that MSEOP i MSEOP j MSEOP k C t. 4

5 Figure 6 shows the proedure for generating suh an union list. These generated lists ontain information regarding the literal redution when omplementing an MSEOP or union of multiple MSEOPs. However, for aurately estimating the literal redution, we have to examine PIs after expansion. This is beause, in many ases, the expanded PIs an make other PIs redundant and ounting the literals of these redundant PIs an underestimate the total redution in the number of literals. Hene for aurate estimation of literal redution, we develop and use following properties. Let us denote two different PIs (PI and PI 2 ) and a MSEOP for PI, MSEOP, and a MSEOP for PI 2, MSEOP 2. Also denote the number of literals redued by expanding PI using MSEOP by l and PI 2 using MSEOP 2 by l 2. Property 2) If MSEOP = MSEOP 2 and the PIs after expanding PI and PI 2 are not idential, then the number of literals redued by omplementing MSEOP is (l + l 2 ). Property 3) If MSEOP = MSEOP 2 and the PIs after expanding PI and PI 2 are idential, then the number of literals redued by omplementing MSEOP is: l + (all the literals in PI 2 ) +. Property 4) If MSEOP MSEOP 2 and PI = PI 2, then the number of literals redued by omplementing MSEOP ( MSEOP 2 ) = l. Property (2) shows that if the MSEOP is the same for the different PIs, we add the literal redution for the two PIs, if no PI beomes redundant after expansion. In Property (3), the number of redued literals in PIs that beome redundant is added to the total literal redution estimation. Property (4) prevents overestimating the number of literals when multiple MSEOPs exist for the same PI, and one of the MSEOPs is a superset of the other. In suh ase, expanded PI obtained by the superset MSEOP is only the PI that is ounted for omputing literal redution. Our list generating proedure and literal estimating properties provide tight estimation of the lower bound on literal redution by omplementing ertain MSEOP and union of MSEOPs. V. EXPERIMENTAL RESULTS A. Comparison of assisting expansion heuristi and exhaustive searh To show the near-optimality and effiieny of our heuristi, we performed exhaustive searh for six benhmark iruits. The results an be obtained only for C t = and 2 under our experimental environment due to the high time omplexity of the exhaustive searh. The experiments were performed on a 2.66 Ghz quad ore Intel Xeon proessor with 2 GB of main memory. For the sum of runtime for ases, assisting expansion is times faster than exhaustive searh. Also the perentage differene in number of literals between the two approahes is 2.27%, on average. From Table, we see that multiple output benhmarks have greater differene in the numbers of literals between the exhaustive and heuristis than single output benhmarks. This is beause for multiple output funtions, our heuristi partitions the funtion into single output funtions and generates input and output relationship for eah output separately. After partitioning the funtion, we selet minterm omplements that an redue the number of literals for eah single output funtion. The minterm omplements that have the greatest total estimated value of literal redutions for separate output funtions is seleted. However, above heuristi for multiple output funtions does not take into aount multiple output impliants, i.e., impliants that have output value of for more than one output [3]. Table. The numbers of literals and runtimes. Benhmark C t # of literals Runtime(se) Exhaustive Heuristi Exhaustive Heuristi Z9sym (9,, 6) sym (,, 47) rd (7, 3, 93) lip (9, 5, 793) sao2 (, 4, 496) xp (5, 7, 347) (Legend: # of I/Ps, # of O/Ps, number of literals in the original funtion.) B. Literal redutions for benhmark iruits We used the heuristi desribed in Setion IV to identify minterm omplements that maximally redue the number of literals for various values of C t. We synthesized seven benhmark iruits from the twolevel synthesis suite that implement arithmeti funtions sine the idea of error tolerane has been shown to be appliable to datapath funtions. To alulate the number of literals in eah approximate version of the iruit, minterms seleted by our heuristis are omplemented in the original minimum over to obtain an approximate over, and two-level synthesis is performed. For eah iruit, we perform above experiment for C t =, 2, 4, 8. Sine error rate threshold is C t / 2 n for a given C t, where n is number of inputs, idential C t values orresponds to different error rate thresholds for different iruits. Typially, as the number of iruit inputs inreases, larger literal redution is ahieved for the same error rate threshold. This ours beause the same error rate threshold allows exponentially larger number of minterm omplements for a iruit with larger number of inputs. The results in Figure 7 show that, on an average, for an error rate threshold of.2% we ahieve 3.75% literal redution, and for an error rate threshold of % we ahieve 9.43 % literal redution. % redution in literals error rate Figure 7. Literal redution for different error rates. rd73(7/3/93) lip(9/5/793) sao2(/4/496) 5xp(7//347) Z9sym(9//6) sym(//47) t48(6//5233) (legend: #of inputs/#of outputs/number of literals in original funtion.) C. Funtional Yield improvement due to literal redution We use the negative binomial yield model [7] to estimate funtional yield improvement due to iruit area redution that we alulated from the heuristi. Sine we don t have any information 5

6 regarding manufaturing proess, first we assume the yield of the original iruit is.7 and the lustering fator (α) is 4, for all the benhmark iruits. These assumptions enable us to alulate the value of defet density for the manufaturing proess. Funtional yield is then alulated using above defet density and the redued area of approximately synthesized iruits. Sine the iruit size is smaller then original iruit, number of defet in the iruit is smaller then original iruit that we an expet the funtional yield improvement of the iruit. Table 2 shows the result of the experiment for different benhmarks. On average, we obtain 3.26% yield improvement with % error rate threshold and 4.35% yield improvement with 2% error rate threshold. Table 2. Yield improvement assuming iruit area is proportional to number of literals. Benhmarks Original Error rate =. Error rate =.2 Approximate % improved Approximate % improved Z9sym sym lip rd sao t xp Yield improvement obtained by approximate synthesis grows with the defet density. In addition to assuming original yield as.7 for above experiment, we also realulate yield improvements by assuming the original yield values as being.9,.5,.3, and. and measure the yield improvements with approximate logi synthesis. The derease in original yield represents the inrease in defet density. The results show that the % improvement in yield inrease drastially as original yield dereases. This means that for the high defet density, we an expet huge % yield improvements using approximate logi synthesis. Figure 8 shows the yield improvement for different values of yield for the original design. Sine the main objetive of error-tolerane tehnique is to ombat yield derease due to inreasing defet densities in future nano-sale fabriation proesses, these urves learly show that as the problem of yield derease beome more serious, our tehnique beomes more useful. % improvement in yield Yield for original design % improvement in yield Yield for original design Figure 8. Yield improvement: Z9sym and sym for different original yield values (error rate threshold =.). VI. CONCLUSIONS In this paper, we present a new logi synthesis approah for error tolerant appliations. The major ontributions of this paper are (i) Proposing the idea of using error tolerane threshold during the design phase and demonstrating that it an provide dramati improvements in hip ost and yield. On average, a.2% error rate threshold an redue the number of literals by more than 3.75% and a % error rate threshold an redue the number of literal by 9.43%. (ii) Derivation of properties that eliminate large number of to omplements that annot redue the number of literals for a given original minimum over. (iii) Development of a heuristi approah that is pratially useable for larger iruits and high values of C t and demonstration of the fat that our heuristi is near-optimal for all ases that we have tried. (iv) Calulation of the yield improvement using our tehnique and the demonstration that yield improvement beomes more signifiant as the original yield dereases. This work foused on synthesis of approximate two-level iruits starting with a given logi funtion. Sine most of the well known multi-level synthesis heuristis fatorize the funtion using boolean or algebrai division [, 2], for some ases, a small number of literals in the two-level SOP representation does not mean a small number of literals in the fatored form or small area after tehnology mapping in multi-level synthesis. To find minterm omplements that an redue area and delay of multi-level iruits is a subjet of our ongoing researh. We are also developing methods to maximally simplify a given implementation of a iruit for implementation at a given value of C t. VII. REFERENCES [] International Tehnology Roadmap for Semiondutors (ITRS) 23 [Online]. [2] M. A. Breuer and S. K. Gupta, Intelligible testing, In Pro Int l Workshop on Miroproessor Test and Verifiation, 999. [3] M. A. Breuer, Intelligible test tehniques to support error tolerane, In Pro. Asian Test Symposium, 24, pp [4] M. A. Breuer, S. K. Gupta, and T. M. Mak, Defet and errortolerane in the presene of massive numbers of defets, IEEE Design and Test Magazine, 2, pp , May 24. [5] I. S. Chong and A. Ortega, Hardware testing for error tolerant multimedia ompression based on linear transforms, In Pro. Defet and Fault Tolerane Conferene, 25, pp [6] H. Chung and A. Ortega, Analysis and testing for error tolerant motion estimation, In Pro. 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