"... In PAGE 27: ... As a consequence the CAG is empty and a default alignment is given for them. In this case, and as shown in Table7 , a two-dimensional partition is selected for array A, with rows cyclically Table 6: Summary of the data distribution strategies for the arrays in jacobi. Array name Size PARADIGM ALIGN PARADIGM DISTRIBUTE DDT ALIGN DDT DISTRIBUTE A, B 514, 514 i, j BLOCK, BLOCK i, j... In PAGE 28: ...ptimization described in Section 2.2.2 is enabled in the CAG control panel, these two edges also appear in the CAG and the same alignment is obtained. Table7 : Summary of the data distribution strategies for the arrays in dgefa. Array name Size PARADIGM ALIGN PARADIGM DISTRIBUTE DDT ALIGN DDT DISTRIBUTE A 512, 512 i, j CYCLIC, CYCLIC i, j CYCLIC(2), CYCLIC(8) IPVT 512 i, 1 BLOCK i, 1 CYCLIC(2) Table 8: Summary of the data distribution strategies for the arrays in olda.... In PAGE 52: ...A.7 OCEAN Table7 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread acac 11 2 0 0 0 0 9 csr 9 5 0 0 0 0 4 dealse 0 0 0 0 0 0 0 ftrvmt - - - - - - - in 1 0 0 0 0 0 1 out 0 0 0 0 0 0 0 rcs 9 5 0 0 0 0 4 scsc 17 4 0 0 0 2 11 Table 7.... In PAGE 52: ...7 OCEAN Table 7.i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread acac 11 2 0 0 0 0 9 csr 9 5 0 0 0 0 4 dealse 0 0 0 0 0 0 0 ftrvmt - - - - - - - in 1 0 0 0 0 0 1 out 0 0 0 0 0 0 0 rcs 9 5 0 0 0 0 4 scsc 17 4 0 0 0 2 11 Table7 .ii: General Communication Routines routine general Decomp acac 0 0 csr 0 0 dealse 0 0 ftrvmt - - in 0 0 out 0 0 rcs 0 0 scsc 0 0 Table 7.... In PAGE 52: ...i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread acac 11 2 0 0 0 0 9 csr 9 5 0 0 0 0 4 dealse 0 0 0 0 0 0 0 ftrvmt - - - - - - - in 1 0 0 0 0 0 1 out 0 0 0 0 0 0 0 rcs 9 5 0 0 0 0 4 scsc 17 4 0 0 0 2 11 Table 7.ii: General Communication Routines routine general Decomp acac 0 0 csr 0 0 dealse 0 0 ftrvmt - - in 0 0 out 0 0 rcs 0 0 scsc 0 0 Table7 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns acac 16 11 0 4 0 2 csr 16 9 0 7 0 0 dealse 0 0 0 0 0 0 ftrvmt - - - - - - in 1 1 0 0 0 0 out 0 0 0 0 0 0 rcs 16 9 0 7 0 0 scsc 21 17 0 4 0... In PAGE 108: ... NAME ALIGN SHIFT DISTRIBUTE CYCLIC filter a19 a19 a19 fct a19 artdif a19 trans1 a19 a19 a19 trans2 a19 a19 a19 ismax a19 (1) a19 a19 (2), (3) t1 a19 t2 Table 6: Communication Costs for different distributions. NAME 16, 1 8, 2 4, 4 2, 8 1, 16 filter 779 1413 2242 4191 9051 fct 2134 1259 1738 4134 10408 artdif 410 437 660 1336 2890 trans1 0 0 0 0 0 trans2 0 0 0 0 0 ismax 0 t1 40 32 61 159 412 t2 656 259 108 73 104 NAME no_shict shift artdif 410 410 t1 32 32 t2 73 73 Table7 : Applicability of techniques described in Section 4. NAME PRIVAT DEPEND REDUCT SPARSE CROSS EQUIV OTHER filter fct a19 a19 a artdif a19 a19 trans1 trans2 ismax... In PAGE 109: .... Some arrays not declared but used within the routine body. ParaScope considers them function calls. No ALIGN directive is generated for them. t1 a19 t2 a19 Table7 : Applicability of techniques described in Section 4. NAME PRIVAT DEPEND REDUCT SPARSE CROSS EQUIV... ..."

"... In PAGE 29: ... ... 20 continue 30 continue Figure 14: CAG for the olda routine in the TRFD benchmark. In this case, the data partitioning strategy obtained is shown in Table9 . Observe that now, the two vectors are aligned with the first dimension of array V and the same alignment as before between arrays XRSIQ and V.... In PAGE 29: ....8.4 DFLUX DFLUX and EFLUX are two routines taken from the FLO52 program of the Perfect Club. For the first one, the alignment and distribution functions for the all the arrays (as determined by Table9 : Summary of the data distribution strategies for the arrays in olda when expression substitution and induction variable detection are enabled. Array name Size DDT ALIGN DDT DISTRIBUTE V 32, 32 i, j CYCLIC, CYCLIC XRSIQ 32, 32 j, i XIJ 32 i, 1 XRSPQ 278784 i, 1 XRSIQ1 XRSIQ2 XRSPQ1 V1 V2 XIJ1 1 4 XRSIQ1 XRSIQ2 XRSPQ1 V1 V2 XIJ1 1 4 1 1 (a)... In PAGE 54: ...17 0.17 udag - - - Table9 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread ffa99 236 17 0 89 0 0 130 ffs99 240 17 0 89 0 0 134 fixtet 20 7 0 4 7 2 0 fl22 6 4 0 0 2 0 0 gfidi - - - - - - - matvec 0 0 0 0 0 0 0 msu22 10 6 0 0 4 0 0 psu22 10 6 0 0 4 0... In PAGE 55: ...sumpls 0 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 0 Table9 .ii: General Communication Routines routine general Decomp ffa99 0 0 ffs99 0 0 fixtet 0 0 fl22 0 0 gfidi - - matvec 0 0 msu22 0 0 psu22 0 0 sumpls 0 0 uvglob 0 0 Table 9.... In PAGE 55: ...sumpls 0 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 0 Table 9.ii: General Communication Routines routine general Decomp ffa99 0 0 ffs99 0 0 fixtet 0 0 fl22 0 0 gfidi - - matvec 0 0 msu22 0 0 psu22 0 0 sumpls 0 0 uvglob 0 0 Table9 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns ffa99 264 236 0 28 0 0 ffs99 270 240 0 30 0 0 fixtet 20 20 0 0 0 0 fl22 10 6 0 4 0 0 gfii - - - - - - matvec 0 0 0 0 0 0 msu22 18 10 0 8 0 0 psu22 18 10 0 8 0 0 sumpls 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 Table 9.... In PAGE 55: ... Table9 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi... In PAGE 56: ...i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread bstate 0 0 0 0 0 0 0 comcov 9 7 0 0 0 0 2 copytr 5 0 5 0 0 0 0 extend - - - - - - - fitpar 10 8 0 0 0 0 2 fptrak 16 9 1 0 2 2 2 inv4 1 1 0 0 0 0 0 nlfilt 17 7 2 0 4 3 1 profyl 0 0 0 0 0 0 0 Table 10.ii: General Communication Routines routine general Decomp bstate 0 0 comcov 0 0 copytr 0 0 extend - - fitpar 0 0 fptrak 5 2 inv4 0 0 Table9 .iv: Inter-component Alignment Execution Times.... In PAGE 110: ...2 SU2COR Table 8: Analysis of alignments and distributions. NAME ALIGN SHIFT DISTRIBUTE CYCLIC sweep a19 a19 a19 matmat a19 a19 a19 int2v a19 a19 a19 trngv a19 a19 (3) bespol a19 a19 a19 int4v a19 a19 a19 loops a19 a19 a19 perm a19 a19 a19 Table9 : Communication Costs for different distributions. NAME 16, 1 8, 2 4, 4 2, 8 1, 16 matmat 0 8192 24576 - - int2v 1540 2563 4610 - - trngv 1707 bespol 8008 9009 10010 12012 - int4v 1028 2307 4866 - - perm 4096 4608 5120 6144 - sweep loops 16,1,1,1 0 22128128 8,2,1,1 768 13832384 4,4,1,1 2304 11069952 2,8,1,1 - 13841600 1,16,1,1 - 22149632 8,1,2,1 16896 22741504 1,8,2,1 - 23452672 4,1,4,1 34816 24385024 1,4,4,1 148480 24998400 2,1,8,1 71680 28702208 1,2,8,1 149504 29111808 1,1,16,1 147456 17268736 8,1,1,2 16896 22741504 1,8,1,2 - 23452672 1,1,8,2 98304 16973824 4,1,1,4 34816 24385024 1,4,1,4 148480 24998400 1,1,4,4 81920... ..."

"... In PAGE 29: ... ... 20 continue 30 continue Figure 14: CAG for the olda routine in the TRFD benchmark. In this case, the data partitioning strategy obtained is shown in Table9 . Observe that now, the two vectors are aligned with the first dimension of array V and the same alignment as before between arrays XRSIQ and V.... In PAGE 29: ....8.4 DFLUX DFLUX and EFLUX are two routines taken from the FLO52 program of the Perfect Club. For the first one, the alignment and distribution functions for the all the arrays (as determined by Table9 : Summary of the data distribution strategies for the arrays in olda when expression substitution and induction variable detection are enabled. Array name Size DDT ALIGN DDT DISTRIBUTE V 32, 32 i, j CYCLIC, CYCLIC XRSIQ 32, 32 j, i XIJ 32 i, 1 XRSPQ 278784 i, 1 XRSIQ1 XRSIQ2 XRSPQ1 V1 V2 XIJ1 1 4 XRSIQ1 XRSIQ2 XRSPQ1 V1 V2 XIJ1 1 4 1 1 (a)... In PAGE 54: ...17 0.17 udag - - - Table9 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread ffa99 236 17 0 89 0 0 130 ffs99 240 17 0 89 0 0 134 fixtet 20 7 0 4 7 2 0 fl22 6 4 0 0 2 0 0 gfidi - - - - - - - matvec 0 0 0 0 0 0 0 msu22 10 6 0 0 4 0 0 psu22 10 6 0 0 4 0... In PAGE 55: ...sumpls 0 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 0 Table9 .ii: General Communication Routines routine general Decomp ffa99 0 0 ffs99 0 0 fixtet 0 0 fl22 0 0 gfidi - - matvec 0 0 msu22 0 0 psu22 0 0 sumpls 0 0 uvglob 0 0 Table 9.... In PAGE 55: ...sumpls 0 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 0 Table 9.ii: General Communication Routines routine general Decomp ffa99 0 0 ffs99 0 0 fixtet 0 0 fl22 0 0 gfidi - - matvec 0 0 msu22 0 0 psu22 0 0 sumpls 0 0 uvglob 0 0 Table9 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns ffa99 264 236 0 28 0 0 ffs99 270 240 0 30 0 0 fixtet 20 20 0 0 0 0 fl22 10 6 0 4 0 0 gfii - - - - - - matvec 0 0 0 0 0 0 msu22 18 10 0 8 0 0 psu22 18 10 0 8 0 0 sumpls 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 Table 9.... In PAGE 55: ...ii: General Communication Routines routine general Decomp ffa99 0 0 ffs99 0 0 fixtet 0 0 fl22 0 0 gfidi - - matvec 0 0 msu22 0 0 psu22 0 0 sumpls 0 0 uvglob 0 0 Table 9.iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns ffa99 264 236 0 28 0 0 ffs99 270 240 0 30 0 0 fixtet 20 20 0 0 0 0 fl22 10 6 0 4 0 0 gfii - - - - - - matvec 0 0 0 0 0 0 msu22 18 10 0 8 0 0 psu22 18 10 0 8 0 0 sumpls 0 0 0 0 0 0 uvglob 0 0 0 0 0 0 Table9 .iv: Inter-component Alignment Execution Times.... In PAGE 55: ...01 - ffs99 - 0.01 - Table9 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi... In PAGE 110: ...2 SU2COR Table 8: Analysis of alignments and distributions. NAME ALIGN SHIFT DISTRIBUTE CYCLIC sweep a19 a19 a19 matmat a19 a19 a19 int2v a19 a19 a19 trngv a19 a19 (3) bespol a19 a19 a19 int4v a19 a19 a19 loops a19 a19 a19 perm a19 a19 a19 Table9 : Communication Costs for different distributions. NAME 16, 1 8, 2 4, 4 2, 8 1, 16 matmat 0 8192 24576 - - int2v 1540 2563 4610 - - trngv 1707 bespol 8008 9009 10010 12012 - int4v 1028 2307 4866 - - perm 4096 4608 5120 6144 - sweep loops 16,1,1,1 0 22128128 8,2,1,1 768 13832384 4,4,1,1 2304 11069952 2,8,1,1 - 13841600 1,16,1,1 - 22149632 8,1,2,1 16896 22741504 1,8,2,1 - 23452672 4,1,4,1 34816 24385024 1,4,4,1 148480 24998400 2,1,8,1 71680 28702208 1,2,8,1 149504 29111808 1,1,16,1 147456 17268736 8,1,1,2 16896 22741504 1,8,1,2 - 23452672 1,1,8,2 98304 16973824 4,1,1,4 34816 24385024 1,4,1,4 148480 24998400 1,1,4,4 81920... ..."

"... In PAGE 12: ...Overview of the Constraints and the Adaptability Space As illustrated by Table3 , all the three frameworks require interaction and composability constraints to guarantee correct behavior. These constraints are the boundaries of the adaptability space of each... ..."

"... In PAGE 23: ..... A(ck, j) lt;- B(2, j) which correspond to several COPY can be aggregated as A(2, j) lt;- B(2, 3) which corresponds to a single SPREAD if k is large. Table5 shows the different options implemented in DDT for this optimization and conditions for being applied (n is the number of times the pattern appear, Np is the number of processors and dimp the size of the target in the dimension where the communication takes place). Other aggregations are possible but not profitable with the communication costs defined in Section 2.... In PAGE 23: ...5% of the reference patterns. Table5 : Aggregating Patterns options and profitability original patterns resulting pattern condition n COPY, same source SPREAD n gt; log Np n SPREAD MULTISPREAD n gt; dimp/log Np n REDUCE MULTISPREAD... In PAGE 48: ...31 1.55 psmoo - - - Table5 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread correc 4 1 0 0 0 0 3 cshift 1 1 0 0 0 0 0 interf 173 53 0 0 120 0 0 intraf 21 6 0 0 0 0 15 poteng 16 7 0 0 0 0 9 predic 1 1 0 0 0 0 0 Table 4.... In PAGE 49: ...A.6 MG3D Table5 .ii: General Communication Routines routine general Decomp correc 0 0 cshift 0 0 interf 0 0 intraf 0 0 poteng 0 0 predic 0 0 Table 5.... In PAGE 49: ...6 MG3D Table 5.ii: General Communication Routines routine general Decomp correc 0 0 cshift 0 0 interf 0 0 intraf 0 0 poteng 0 0 predic 0 0 Table5 .iii: Reference Pattern Optimizations... In PAGE 108: ...A.1 HYDRO2D Table5 : Analysis of alignments and distributions. NAME ALIGN SHIFT DISTRIBUTE CYCLIC filter a19 a19 a19 fct a19 artdif a19 trans1 a19 a19 a19 trans2 a19 a19 a19 ismax a19 (1) a19 a19 (2), (3) t1 a19 t2 Table 6: Communication Costs for different distributions.... ..."

"... In PAGE 27: ....8.1 JACOBI This is a simplified version of the Jacobi program. In this program, there are two assignment statements involving arrays A and B and they suggest the same alignment for the arrays, as shown in Table6 . The analysis of communication costs suggests a two-dimensional partitioning of the arrays, where both the rows and the columns are distributed on 4 processors each.... In PAGE 27: ... That leads to 4 shift communication routines, each with a message size of n/4 words (n=514). As shown in Table6 , [Gupt92] also suggests the same data partitioning strategy. However, he considers a more realistic model for the transfer time (according to the transfer times in the iPSC/2 machine).... In PAGE 27: ... As a consequence the CAG is empty and a default alignment is given for them. In this case, and as shown in Table 7, a two-dimensional partition is selected for array A, with rows cyclically Table6 : Summary of the data distribution strategies for the arrays in jacobi. Array name Size PARADIGM ALIGN PARADIGM DISTRIBUTE DDT ALIGN DDT DISTRIBUTE A, B 514, 514 i, j BLOCK, BLOCK i, j... In PAGE 49: ...01 0.01 predic - - - Table6 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread cfft9x 4 0 0 0 0 0 4 cpass 0 0 0 0 0 0... In PAGE 50: ...cpassm 0 0 0 0 0 0 0 difx 8 6 0 2 0 0 0 dify 8 6 0 2 0 0 0 ffb 2 2 0 0 0 0 0 fff 1 1 0 0 0 0 0 fft991 1 0 0 1 0 0 0 march 10 6 4 0 0 0 0 migrat 12 11 0 0 0 1 0 mul 6 6 0 0 0 0 0 rfft 3 1 0 0 0 0 2 rpass 504 0 0 0 0 0 504 rpassm 405 0 0 0 0 0 405 Table6 .ii: General Communication Routines routine general Decomp cfft9x 0 0 cpass 0 0 cpassm 0 0 difx 4 0 dify 4 0 ffb 0 0 fff 0 0 fft991 0 0 march 4 6 migrat 0 12 mul 0 0 rfft 0 0 rpass 0 0 rpassm 0 0 Table 6.... In PAGE 50: ...cpassm 0 0 0 0 0 0 0 difx 8 6 0 2 0 0 0 dify 8 6 0 2 0 0 0 ffb 2 2 0 0 0 0 0 fff 1 1 0 0 0 0 0 fft991 1 0 0 1 0 0 0 march 10 6 4 0 0 0 0 migrat 12 11 0 0 0 1 0 mul 6 6 0 0 0 0 0 rfft 3 1 0 0 0 0 2 rpass 504 0 0 0 0 0 504 rpassm 405 0 0 0 0 0 405 Table 6.ii: General Communication Routines routine general Decomp cfft9x 0 0 cpass 0 0 cpassm 0 0 difx 4 0 dify 4 0 ffb 0 0 fff 0 0 fft991 0 0 march 4 6 migrat 0 12 mul 0 0 rfft 0 0 rpass 0 0 rpassm 0 0 Table6 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns cfft9x 4 4 0 0 0 0 cpass 0 0 0 0 0 0 Table 6.... In PAGE 50: ...ii: General Communication Routines routine general Decomp cfft9x 0 0 cpass 0 0 cpassm 0 0 difx 4 0 dify 4 0 ffb 0 0 fff 0 0 fft991 0 0 march 4 6 migrat 0 12 mul 0 0 rfft 0 0 rpass 0 0 rpassm 0 0 Table 6.iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns cfft9x 4 4 0 0 0 0 cpass 0 0 0 0 0 0 Table6 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi... In PAGE 51: ...07 rfft - - - rpass - - 0.01 rpassm - - - Table6 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating... In PAGE 108: ...1 HYDRO2D Table 5: Analysis of alignments and distributions. NAME ALIGN SHIFT DISTRIBUTE CYCLIC filter a19 a19 a19 fct a19 artdif a19 trans1 a19 a19 a19 trans2 a19 a19 a19 ismax a19 (1) a19 a19 (2), (3) t1 a19 t2 Table6 : Communication Costs for different distributions. NAME 16, 1 8, 2 4, 4 2, 8 1, 16 filter 779 1413 2242 4191 9051 fct 2134 1259 1738 4134 10408 artdif 410 437 660 1336 2890 trans1 0 0 0 0 0 trans2 0 0 0 0 0 ismax 0 t1 40 32 61 159 412 t2 656 259 108 73 104 NAME no_shict shift artdif 410 410 t1 32 32 t2 73 73 Table 7: Applicability of techniques described in Section 4.... ..."

"... In PAGE 30: ... In this case, the second dimension would have allocated less processors than Table 10: Summary of the data distribution strategies for the arrays in dflux. Array name Size PARADIGM ALIGN PARADIGM DISTRIBUTE DDT ALIGN DDT DISTRIBUTE FS, FW 193, 34, 4 i, j, k BLOCK, :, : i, j, k BLOCK(8), BLOCK(2), : DW 194, 34, 4 i, j, k i, j, k W 194, 34, 4 i, j, k i-1, j, k DP 195, 35 i, j, 1 i, j, 1 RADJ 194, 34 i, j, 1 i, j, 1 DTL, P, RADI, VOL 194, 34 i, j, 1 i, j, 1 EP, DIS4, DIS2 193, 33 i, j, 1 i, j, 1 Table11 : Estimated communication costs for the different processor-allocation strategies evaluated by DDT. permutation estimated cost permutation estimated cost 16 x 1 x 1 1470 1 x 8 x 2 10140 8 x 2 x 1 1393 4 x 1 x 4 21105 4 x 4 x 1 1683 1 x 4 x 4 22815 2 x 8 x 1 2798 2 x 1 x 8 55072 1 x 16 x 1 5265 1 x 2 x 8 56452 8 x 1 x 2 7735 1 x 1 x 16... In PAGE 58: ...A.11 TRFD Table11 .i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread intgrl 0 0 0 0 0 0 0 olda 14 6 0 0 0 2 6 Table 11.... In PAGE 58: ...11 TRFD Table 11.i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread intgrl 0 0 0 0 0 0 0 olda 14 6 0 0 0 2 6 Table11 .ii: General Communication Routines routine general Decomp intgrl 0 0 olda 6 0 Table 11.... In PAGE 58: ...i: Simple Communication Routines routine simple LMA Copy Shift Spread Reduce Multi Spread intgrl 0 0 0 0 0 0 0 olda 14 6 0 0 0 2 6 Table 11.ii: General Communication Routines routine general Decomp intgrl 0 0 olda 6 0 Table11 .iii: Reference Pattern Optimizations routine before After Constant Propagation Combining Identical Combining Subset Aggregating Patterns intgrl 0 0 0 0 0 0 olda 20 20 0 0 0 0 Table 11.... In PAGE 112: ...A.3 MDLJDP2 and MDLJSP2 Table11 : Analysis of alignments and distribtuions. NAME ALIGN SHIFT DISTRIBUTE CYCLIC jloopu a19 a19 a19 (3) jloopb a19 a19 a19 (3) postfr a19 a19 a19 (3) prefor a19 a19 a19 (3) Table 12: Communication Costsfor different distributions.... ..."

"... In PAGE 55: ... In counting the P- and NP-equivalence classes, only permutations and negations of the k de ning variables are considered. Table4 collates the data from Table 3 to indicate the number of CB2-representable functions as a function of the size of the target concept k. The quantity g(k) is the number of functions dependent on exactly k bits of the representation that are CB2-representable with respect to a xed target concept t MN;k.... In PAGE 55: ...Table 4 also lists jHCB2 Mk;kj, which is the number of functions that are CB2-representable with respect to any target concept de ned on k out of k literals (all negations included). Clearly jHCB2 Mk;kj is a sum over the sizes of the NP-equivalence classes in Table 3, or alternatively: jHCB2 Mk;kj = k X k0=0 2k0 k k0 g(k0) (133) The nal column in Table4 also lists for comparison the total number of functions de nable on a k-dimensional example space, jBkj = 22k. Hence combining the result of Corollary 5.... In PAGE 55: ... Hence combining the result of Corollary 5.26 with the values of jHtj listed in Table4 , the following upper bounds on sample complexity may be stated: 8N 1 mCB2(MN;1; ; ) 1 loge 2 lt; 1 + 1 loge 1 (134)... In PAGE 57: ...g(k) jHtj (t MN;k) jHCB2 Mk;kj jBkj = 22k 0 2 2 2 2 1 1 3 4 4 2 1 5 10 16 3 5 13 60 256 4 30 62 674 65,536 5 603 820 22,148 4,294,967,296 Table4 : Numbers of CB2-representable functions 0 0.2 0.... ..."