Minimization of Two-Level Functions
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- Cory Walton
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1 ¹±oŽ bž± «b ² ««¼ «±
2 Goals: Minimization of Two-Level Functions Minimize cover cardinality Minimize number of literals PLA implementation: Minimize number of rows Minimize number of transistors --> Minimize area and time Karnaugh map: manual minimization f=σm(0,2,3,6,7,8,9,10,13) ab cd f=b d +a c+ac d
3 Quine-McCluskey Method Minimization of Two-Level Functions Exact minimization (global minimum) Generate all prime implicants Start from 0-dimensional cubes (minterms( minterms) Find k-dimensional cubes from (k-1)-dimensional cubes (find cubes that are diffenrent in only one position) Find a minimum prime cover Minimum set of prime implicants covering all the minterms 1 s Minterms m 0 m 2 m 8 m 3 m 6 m 9 m 10 m 7 m ,2 0,8 2,3 2,6 2,10 8,9 8,10 3,7 6,7 9,13 1-Cubes 00x0 x x 0x10 x x * 10x0 0x11 011x 1x01* 0,2,8,10 2,3,6,7 2-Cubes x0x0 * 0x1x *
4 Minimization of Two-Level Functions * 0,2,8,10 essential * 2,3,6,7 8,9 * 9,13 f=m 0,2,8,10 +m 2,3,6,7 +m 9,13 =b d +a c+ac d
5 «µb ª± Minimization of Two-Level Functions x y sw uz vx vy b 0001x1 b x01111 c 00x111 d 10111x e 10x110 f x00110 covering proposition CP=(a+f)(a+c)(b+c)(e+f)(d+e)(b+d) m 6 m 7 m 15 m 38 m 46 m 47 =abe+abdf+acde+bcef+cdfcdf (exponential) select terms with fewest product terms (abe( abe, cdf) ---> then select the term with fewest literals
6 ª«b ª± Minimization of Two-Level Functions select a no yes select b select b no yes... select c... select c If worse, prune the branch
7 Complexity: Minimization of Two-Level Functions Number of minterms: ~ 2 n Number of prime implicants: ~ 3 n /n Very large table Covering problem is NP-complete --> Branch and bound technique Use heuristic minimization Find a minimal cover using iterative improvement. Example: MINI, PRESTO, ESPRESSO
8 MINI S.J.Hong, R.G. Cain, and D.L.Ostapco Ostapco,, "MINI: a heuristic approach for logic minimization," IBM J. of Res.. and Dev., Sep Three processes Expansion: Expand implicants and remove those that are covered Reduction: Reduce implicants to minimal size Reshape: Modify implicants
9 Expansion MINI Iterate on implicants Make implicants as large as possible Remove covered implicants Goal: Minimal prime cover w.r.t. single cube containment Most minimizers use this technique Algorithm: For each implicant { For each care literal { Replace literal by * If (implicant not valid) restore } Remove all implicants covered by expanded implicants }
10 MINI Example * Take Expand * 0 0 ok Expand * * 0 ok Expand * * * not ok Restore * * 0 Remove covered implicants * * Take Expand * 0 1 not ok Restore Expand 0 * 1 not ok Restore Expand 0 0 * ok Remove covered implicants * * *
11 Use heuristics to: MINI Order the implicants --> Place on top of the list those cubes that are hard to merge with other cubes encoding: 1-->01 0-->10 X-->11 null --> 00 C >inner product:1+3+2=6 C C C # of 1s: C4 higher priority hard to merge C3 expand these first C1 C2
12 MINI Order the literals (direction of expansion) --> Choose a permutation for which expansion covers as many cubes as possible If we expand this literal, C1 covers many other cubes --> Choose it C C covers Validity check: Check intersection of expanded implicant with X OFF
13 Reduce MINI Iterate on implicants Reduce implicant size while preserving cover cardinality Goal: Escape from local minima Alternate with expansion Algorithm: For each implicant { For each don't care literal { Replace literal by 1 or 0 If (implicant not valid) restore } } Heuristics: ordering Example: * * 0 can't be reduced 0 0 * can be reduced to 0 0 1
14 Reshape MINI Modify implicants while preserving cover cardinality. Goal: Escape from local minima. A, B : disjoint A and B are different in exactly two parts One different part of A covers the corresponding part of B A=π 1 π 2 π i π j π p B=π 1 π 2 µ i µ j π p π j covers µ j A =π 1 π 2 π i (π j µ j ) π p B =π 1 π 2 (π i µ i ) µ j π p A=1x1 A =111 B=001 B =X01
15 MINI Alternate with expansion and reduce. Example: 1x1 a b c x reshape 111 a b c x x expand a b c x x x01 11x
16 µ² µµ±b p p» ±» ± nb p pš ª Š ª nb p p n ƒpžp «± ««± «o ««««nb Ž± «b b b «««¼ «± ƒ ± «ª µb ± b Ž b» ª µ«µnb bbb ¹ bbb ¹ b ƒ ««µª µnbs{zvp µ µb b± b ± b «b µ
17 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
18 «««± µ µ² µµ±b nbb bb nb b ˆˆ bboo b ± ²»bµ² ««b «± µ µb ± b «² b± ² µ bb s npppn n ms msnpppn m Ÿ ¹ª b «bbrnb«bº bº «bj«o ª ªb«² b «kb ²² µb ± ² bbbbbbbbbbbbbsnb«bsnb«bº«bº«b ² b ²² µb ± b ± ² ² µb ± b ± ² bbbbbbbbbbbbbtnb«bº«bº«b ± µb ± b ²² b ± µb ± b ²² bbbbbbbbbbbbbunb«b ª b ± µ²± «b² ± b b«µb ± bbbbbbb µ b ± b ª bj«o ko ª ªb± ² bbbbbbbbbbbbbvnb«b ª b ± µ²± «b² ± b b«µb µ bbbbbbbbbbbbbbbb ± b ª bj«o ko ª ªb± ² º ² sbbºtimºsºui tbbºtbmºsºui b ± bºtib«µb tbrbtbvbuÿ b ± bºsºuib«µb sbtbrbvbvÿ
19 ± ± µ² µµ±b ± ± b ± b b b ¹p p pb b ²b«µb «b µb b b b ² b¹«ª ± ²± µ bbbbbbbbbj ² k bb nbb«b b²bbr bbbbbbbbbbbbbbbbbbbtnbb«b² b² bbrb± bs bbbbbbbbbbbbbbbbbbbvnbb«b² b² bbu bbbbbbbbbbbbbbbbbbb nbb± ª ¹«µ oo b² s t bbbb ² j jjjb bk jjjb bk s k t k k ª b ± b «««± bµ «µ «µ b²b b b ² b b bmb² b bmb²
20 µ² µµ±b ± ± b ² b± b bµ b± b µb b¹«ªb µ² b ±b b²b«µb µ b± b µb ª b b ± ± µb± b ª b µb«b bbbbb ºkbbbbbbbbsbsbrbtbvbv bbbbbbbbbbbb bbrbsbtbrbvbv bbbbbbbbbbbbbbbbsbsbsbsbvbu bbbbbbbbbbbb²bbºvbb tbtbtbsbvbvÿ bbbbbbbbbbbboo b ºv bbsbsbrbtbvbv bbbbbbbbbbbbbbbbbbbbbsbsbsbtbvbu bbbbbbbbbbbb²bbºv bb tbtbtbrbvbvÿ bb tbtbtbrbvbvÿ bbbbbbbbbbbboo b ºv bbsbsbrbtbvbv bbbbbbbbbbbbbbbbbbbbbbrbsbtbtbvbv
21 ª ± b º² µ«± bbº b bº b º bmbº bº b b º bbbbbb ºkbbbbbbbsbsbrbtbvbv bbrbsbtbrbvbv bbbbsbsbsbsbvbu µ² µµ±b ºv bbsbsbrbtbvbv bbbbbbsbsbsbtbvbu ºv bbsbsbrbtbvbv bbbbbbrbsbtbtbvbv bbbbbbbbbbbbºvb ºv bmbºv b b ºv bbbbbbbbbbbbbsbsbrbsbvbvbbbbmbbbbsbsbrbrbvbv bbbbbbbbbbbbbbsbsbsbsbvbubbbbbbbbbrbsbtbrbvbv b
22 ± ± ± b«µ«µ² µµ±b b«µb ± ± ± b«µ«b«bº bº b«bº bº b brboo bsb µ µbb b brboo bsb µ µbb ª «b± ² µb± b b brboo bs ±b µbª bº bº ± ± ± b µ«b«µb ± ± ± b«µ«b«bº bº b«bº bº b brboo bsb µ µbb b brboo bsb µ µbb ª «b± ² µb± b b bsboo br ±b µbª bº bº m ± ± ± b«µ«b«b ± ± ± b µ«b«b b «± ƒb «± b«µb b bj ± ± ± kb«b«b«µ bj ± ± ± kb«b«b«µb b bj«µ«b± bj«µ«b± µ«kb«b b«µb «µ
23 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
24 ± ² b ± ² «± µ² µµ±b ± ² «± b± b b b b ± bˆb b b º² µµ b µ b ± bˆb b b º² µµ b µ bbbbˆ b bbº bº bjˆº k bmbjˆº k bb«bˆb«µb ± ± ± b«µ«b«bº bº b b bbbbˆ b bbº bº bjˆº k bmbjˆº º k bb«bˆb«µb ± ± ± b µ«b«bº bº bb ±± bb bˆb«µb ± ± ± b«µ«b«bº bº nb ª ˆº º b bˆº oo bˆbbº bˆ bº bˆº bmbº bº bˆº b bbbbbbbbº bˆ bº bˆº bmbº bˆ bº bˆº º bmbº bº bˆ bbbbbbbbbbbbbbbbbbbbbbbbbbº bˆ bº bˆº bmbˆ bˆº º b bbbbbbbbº bˆ bº bˆº bmbˆ bˆº º ˆº bbbbbbbbbbbbbbbbbbbbbbbbbbjº º bmbˆº º kbˆº bbbbbˆ b bbº bº bjˆº k bmbjˆº k bˆº º
25 µ² µµ±b ª bˆb«µb ± ± ± b«µ«b«bº bº n ± ² «± b± bº bº bjˆº k b b bbbbb¾ˆº º ¾b~b¾ˆ¾bj µb¹«ªbtiµb«b o ª ªb²±µ««± k bbbbb b± b µµb «bjº º b«µb ± k ± ² «± b± bjˆº k b b bbbbb¾ˆº ¾bb¾ˆ¾ bbbbb b± b µµb «bjº º b«µb ± k µ«b ± ² «± bbbbbˆ bbºs bbºs jˆºs k mbjˆºs k bbbbbbbbbbºs jˆ jˆºs ºs k mbºt jˆºsºt ºsºt k mbjˆºsºt k bbbbbbbbbppp bºs jˆ jˆºs ºs k mbºt jˆºsºt ºsºt k mbpppbmbº jˆºsºt ºsºt º º k mbjˆºsºt k ºsºt º º bbbbbboo b b b «nb«snpppn nb b ª b±»b «µb «b ± btn ª bˆºsºt ºsºt º º b«µb ± ±»b b ª b ± ² b«µb ²»p j b«µb b b«bˆb b b «b«µb ª b«o ª ªb ± ²± b± b k
26 µ² µµ±b ª±«b± b «µ ª±±µ b «µb ± b ª b µ b b j¹«ªb º«b b± tiµkboo b «µb ª b µ«± b µ b ª b nb ª±±µ b «µ b ª b «b ª b ²² b ±µ b± «b ª b± ª b µb± bˆboo b µb µb»bt µb µb²±µµ«µb µb²±µµ«± ² «± bj ± o k ˆ bjº ˆº mº º ˆº º k bbbbbbjº ˆ º ˆº k jº º ˆº º k b bbbbjº º mjˆº k kjº º mjˆº º k k bbbbbbº bº jˆº º k mº º jˆº k mjˆº k jˆº º k bbbbbbº bº jˆº º k mº º jˆº k bbbbbbj¹ª bº bº rnbº bº jˆº º k mº º jˆº k bjˆº º k bjˆº k jˆº º k ¹ª bº bº snbº bº jˆº º k mº º jˆº k bjˆº k bjˆº k jˆº º k k
27 µ² µµ±b If c F, c F=0 --> c (cfc cf c + c F c )=0 --> c F c =0 F=cF c + c F c = cf c --> F =cf =c +( +(F c ) x1 x2 x3 x4 x F = > c=[ ]=x2 --> F =x2f =x2 +(F x2 ) simpler than F x1 x2 x3 x4 x x2 = F x2
28 µ² µµ±b Procedure COMP1(F) /* Given a cover F of ff = { f, d, r}, r a single output logic function. /* returns R, a cover of the offset r. /* n is the number of input variables. Begin if (row of all 2's) Return (R<-- ) if (F is unate) ) Return (R<--UNATE COMPLEMENT(F)) c<--f 1 /* Initialize with first cube for (j=1,,, n) /* Extract cube c Begin /* representing columns for (i,= 2,..., F ) /* of all 0's or all 1's If (c( j F i j) ) then c j <-- 2 End R<--UNATE COMPLEMENT({c}) /* Complement cube c F<--F c j<--binate SELECT(.F) /* Select the most /* binate variable. R<--R MERGE_WITH_CONTAINMENT(COMP1(F xj ), COMP1(F xj )) /* Complement cofactors /* and merge. Append /* to partial result R Return End
29 ƒ Ž jˆk µ² µµ±b b º bj² r j km² s j kk ¹ª b² r j kb«µbeb± b µb¹«ªbr µb«b o µb«b o ª ªb«² b²±µ««± b² s j kb«µbeb± b µb¹«ªbs µb«b o µb«b o ª ªb«² b²±µ««± Š ƒ (COMP1( (COMP1(F xj ), COMP1(F xj = xj(f xj ) +xj (F xj ) )) xj )) x1 x2 x3 x4 x x2 = F x2 (F x2 most binate p 0 (3)=2, p 1 (3)=2 x2 ) = x3(f x2x3 ) +x3 (F (F x2x3 ) x1 x2 x3 x4 x5 x2x3 = F x2x3 x1 x2 x3 x4 x5 x2x3 = F x2x3
30 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
31 º² µ² µµ±b «b µb«b ª b ± ƒ b µb«b ª b± b± b µ«bµ«¼ p oo b Ž b µb b ± b b ±b ± b± ª b µb b µµ b ±b b ± b»b± ª b µp b ª b b ± ² µb«b² ± µp «b ± µb«b b bj k spb b b bj kb«b ª b± oµ bª µb±»b± b ± bj kb ª µ «µ «µb ª b ± ±¹«b ± ««± n bbbbbbbbj sb b b rkb± bj rb b b sk ± b b ± b b º² p ºkb rstnbb sss oo b ± bsb± b b ± b b º² p b² ± bµ«¼ b»b «««b ª b º b ± nb nb µb ª b ± b b ± p
32 µ² µµ±b tpb b ± µb ª b b b º² b ±b ± b µb»b µb«ª b± oµ b µb²±µµ«p b ª b b ±b ± b ± b µnbµ b b ± b¹«ª º«b ± «µb ¹ b b b± ª b µb«b ª b± oµ p ª b ± µ²± «b ± µb b ± b µb b ««p µb b ««p upb b b µb«b ª b± oµ bª b ±b ± «b± b ± b n µ b ª b ± b ± b º² µ«± pb ª b ± µ²± «b ± b ± b µb b ««p vp ² bµ ²bsnbtnb bub «jskb b ± µ b ««oo b ± jtkb b µb«b ª b± oµ b b ««oo bµ b b ««b ± µb ± b º² µ«± jukb b µb«b ª b± oµ b b ± oo bµ b ± b º² µ«± b µb»b ± µb µb²±µµ«nb ± bbbb beb± b «µ oo b «b ª b ««b ± b ± b± b ª b ± «b «º oo b ob ± ² oo b µ bª «µ «µ
33 µ² µµ±b ± «b «º ± oµ bb srsrs ± oµ b ssslrr tsrsss urssrs vssrrr r r r r when c j ri j c is not expanded for these columns --> {c} R=
34 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
35 µµ «b² «µ µ² µµ±b ± µ µ µb µ µ µb± b ¹±b µb b b b«µb b bµ ªb ª b j nb kbbrnb bb b b n b n «b j nb kbbsnb b «bb b ««nb«b b ««b brnbrn bbbbtbbnb± ª ¹«µ n «b j nb kb btnbb bb btnbb bb d=121 e=102 c=200
36 µ² µµ±b ˆ± b b² «b²nb«b«b ª b ± µ µ µb± bjj± oµ b b oµ ko½² kb oµ ko½² k ¹«ªb²b ± ²»b ± µb²nb²b«µb ± b µµ «p p p essential not essential ª»b b ± µ µ µ p ((on-set dc-set)-{p}) does not cover p even though p is not essential
37 µ² µµ±b ƒbµ b± b µb b ± µb b b²b«b b«b ² bj ± ± b± b b¹p p p ²kb«µb b ± ±»p ±± b ²²±µ b b ± µb² oo b b²bb² ex) bbbbb ²²±µ b 021 C = 102 oo bj b²k ² ² ² s bbbbj b²k ² ² b² ² ²b s jsk jtk s ² ± bjskjtknb b sbj ± ±»k b ² s oo b b²bbj²b ²b mb²ib b ²i kb² b²b b ² b²b oo b b ± µb² p = 201 j µ b ² sb»b µµ ² «± k C p 022 = 122 = 222
38 ± ±»b ± ² «± µ² µµ±b b«b ª b«µb b ±¹b± btiµb«b ª b ± p ˆ µ b«b ª b ± bª µb b ± b± b bsiµb± b briµp ˆ µ b «b ª b µ b ± b b± b «µ b «µb ± «b«b ªb b ± «b«b ªb jt ± tiµ kb± b ª b ± b«µb µµb ª bt p b ± b ± b ª b ± nb µb ± ² b ± ±»b ± b ª ± ± µp b«µb b ± ±»b«b b«b º b b b º i b b ± ± «µp ±± ~b b b º bb b º i bbsn bbb bbº bº º bmbº bº i º i i bbº bº bmbº bº ibbsp bb b ²²±µ b bbsb b b º b bspb ª bspb ª b º bbrb ± bµ± b«² bbbbb ± ««± p bbbbb µ b b º b«µb«² b± bº bº nb¹ b bµ bº bº bbsb¹«ª± bbbbb «b ª b b b º bbrp bbboo b bbº bº º bmbº bº i º i i bbr bbboo b b«µb ± b b ± ±» bbboo b ± ««±
39 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
40 b ± µ² µµ±b ««± b ª b² «b ± b«±b ¹±bµ µ µ b b± b b µµ «b µ µ b b± b b µp ˆ± b b b b«b ª b ± bj± oµ knb«bjj± oµ b b oµ kbo b oµ kbo ½ k b«µb b ± ±»nb ª b«µnb«bjj± oµ b b oµ kbob½ k b oµ kbob½ k ± µb nb ª b b«µb b b bj b b knb µ b b«µb b knb µ b b«µb b µµ «b bj b b kp b kp ƒb b b b«µb² b b«bj oµ b «µb² b b«bj oµ b b k «µb ± b b ± ±»p ««b µb«b b b ±»b p ±»b b µb b ± p ˆ ± b ª bµ b b ² b± b² b b µnb º b ««bµ b b bµ ªb ª b b b b b«µbµ «b b ± p oo b ««b ± b ±
41 µ² µµ±b d d c c b p b p a a, d : relatively essential b, c : partially redundant a a, c : relatively essential b : totally redundant
42 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
43 «± µ² µµ±b «b µb ± b «± b ª b µ b pb b ª b ««b µb ««µ«b²µ ±o «µ bj b± b «µ ª µkp ºkb² jrstsuvnbrtssvvkbbu oo Ž µ b µb b b b ±µ b µp Ž b º² µ«± b ±b ± b«µb «ª ± µp
44 µ² µµ±b ˆ± b b b b «b ª b ± nb ± ² b ª b µ µ b b µ ± ««bjj± oµ bob½ kb b oµ k b oµ k bb bjˆj kk on-{c} {c} dc µbb j bjˆj kk k µb b boo b boo bµ bµ b brboo bµbb bµ bµ m bµ bµ µbb bµ bµ b bj j bjˆj kk kk
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46 µ² µµ±b µbb bj j bjˆj kk bj j bjˆj kk kk b b jj bjˆj kk k kboob»bjtkb b bjˆj kk b b b b b j bjjˆj kk kk k k b b jjˆj jˆj k k k b b jˆj k k j kbb jº j º kbmbº bº j º oo b µ«b ± ² «± kk º kk j kbb jº º bmbº bº º k bbbbb b j jº º kbmb jº º k bbbbbbº bº j º kbmbº bº b j º kboob»bjsk oo b j j kkb b j b jº j º kbmbº bº b j º kk bbº bº º bmbº bº º bº j º kbmbº bº b j º k oo b j kb b j b jº j º kbmbº bº b j º kk ˆ ± bjukb bjvkn j kbb bjº j º kbmbº bº b j º kk j kbb j k b jº j º kbmbº bº b j º kk b jº j º kbmbº bº b j º kkbboob b º bj º bbbbbbbbjuk bbbbbbbbjvk º k
47 ³ b± b±² «± µ spb ± ² µ² µµ±b ± ² b ª b± oµ bj ± ² b± bš b bš k tpb º² º² b ªb«² «b«±b b² «b b ± b ± «² «µ upb µµ «b² «µ º b µµ «b² «µb b² b ª b«b ª b ± i b bµ vpb b ± ˆ«b b ««b«b ± wpb b ªb«² «b ±b b ««b µµ «b«² «xpb btnbvnb bwb «b ±b«² ± ypbž µ µ²»b nb º² nb b«b ± b µ«b b «µ» bµ µµ nb ± «b ª b««± zpb µ² µ b ª b µµ «b² «µb b«±b ª b ± b b b ª Žƒbµ b µbµ² µ b µb²±µµ«
48 Ž µ µ² µ² µµ±b b µb«²»p oo b± b«² p ª b µ b ± b b µb ½ snb tnb pppb ² b ² b «µb ± µµ b b ± p º² b ª b b µb ±b b bµ b± b ¹b² «µp b ƒ b¹«ªbˆb bšnb¹ª bˆb«µb ª bµ b± b± bšnb¹ª bˆb«µb ª bµ b± b± ² «µb bšb«µb ª bµ b± b ¹b² «µp
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50 µ² µ µ² µµ±b b b± b «µb«b ªb p oo b b Žƒb «ºb µbµ² µ b µb²±µµ«p oo b ª b «b ±b b ± b b«² ± b «bbb² ±² «µp b ± b ± b «² b ± ² b b ª b µb ± bµ«b± ² p j ºk rstuvv rssvvu rsrvuu bbboo bt b b«µb b ± bt b± ² p Ž±¹ b± ² b² b± ² b² bµ² µ ± ² b«b ± b ± bµ«b± ² j ºk rstuvv rssvvuboo brssvuu rsrvuu
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