6. Efficiency

Y c c!

198 From Natural to Artificial Intelligence - Algorithms and Applications

Table 4. Steps II, I, and III for multiplication up to 3 � 3 ¼ 9.

Table 5. Steps II, I, and III by output patterns.

is given in the left column of Table 4.

5.3.3. Step III

p cð Þp cð Þ! <sup>¼</sup> <sup>7</sup>!

1 <sup>1</sup>! � ð Þ <sup>2</sup>! <sup>3</sup>

Step II Step I Step III

bijective functions consistent with the count distribution to choose from in Step II. One of these

The outcome of Steps I + II completely fixes the necessary permutation for Step III. However, we do not have to deal with 2<sup>n</sup> values, but only <sup>∣</sup>Imð Þ<sup>f</sup> <sup>∣</sup> are relevant, in our example 7 instead of

0xFEDCBA9876543210 <sup>0001</sup> 0x0123456789ABCDEF <sup>0011</sup> 0x2AE879D <sup>0001</sup> 0xEFDCAB9867452301 <sup>1010</sup> 0x30127654A99AEFDC <sup>1101</sup> 0x3BF869D <sup>0010</sup> 0xC57E03124D6F8BA9 <sup>0100</sup> 0xADCBE7817557B218 <sup>0110</sup> 0x3BDA49F <sup>0100</sup> 0x817A4352096BCFED <sup>0010</sup> 0xCBAF85E55335F05E <sup>0111</sup> 0x3F9E0DB <sup>1010</sup> 0xA1586370294BEDCF <sup>0001</sup> 0xAED8E29222228729 0xB51CA73 <sup>0101</sup> 0xB0487261395AFDCE <sup>1010</sup> 0xE048F62 <sup>0010</sup> 0x3A62D849B1F057EC <sup>0100</sup> 0xC26AD40 <sup>0101</sup> 0x3E629C0DF5B417A8 <sup>0010</sup> 0x837F915 <sup>0010</sup> 0x3C409E2FD7B6158A <sup>0100</sup> 0xA35D917 <sup>0101</sup> 0x3804DA2B97F651CE <sup>1010</sup> 0xF209D46 <sup>1000</sup> 0xB2A670831D54F9EC <sup>0001</sup> 0x7A81546 <sup>0100</sup> 0xA3B761820D45E9FC <sup>1000</sup> 0x7EC5106 <sup>1010</sup> 0x2B3F690A854DE17C <sup>0100</sup> 0xDCEF9A4 <sup>0101</sup> 0x2F3B6D4EC109A578 0x98BADF1 <sup>1000</sup>

IN Step II Step I Step III IN Step II Step I Step III 0010 5 2 0 0110 1 E 2 0001 7 2 0 1001 4 E 2 0000 8 2 0 1101 3 8 3 0100 9 2 0 0111 C 8 3 0011 A 2 0 1010 D 7 4 1100 B 2 0 1011 6 9 6 1000 E 2 0 1110 F 9 6 0101 0 A 1 1111 2 D 9

<sup>3</sup>! � ð Þ <sup>1</sup>! <sup>3</sup>

3! ¼ 1451520

0x1032579 <sup>0001</sup>

0x0123469

In the case of unfair bijective functions, any subset of cells may fire simultaneously, provided that no adjacent cells are contained in the set.

We define local efficiency of a bijective update sequence by two properties:


Global efficiency—which shortest update sequence generates a certain function/permutation is beyond the scope of this paper (it is dealt with implicitly by brute force in a breadth-first manner).
