Quranic inheritance

The Islamic rules of inheritance are primarily laid down in the Quran 4:11-12 and 4:176. Three verses that compress a great number of inheritance cases - at least 512 as we will see in the following exposition. A fascinating example of algorithmic compression.

First, the sources:

• The verses in Arabic (GIF images) and Yusuf Ali English translation at Sacred Texts: 4:11-12 and 4:176.
• The verses in various English translations at Islam Online.

Second, existing analyses:

• An introduction: The Islamic laws of inheritance, by Dr. Abid Hussain.
• The controversy of consistency and completeness: a critical exposition and its response.

Third, existing implementations:

• Inheritance calculation site at the Saudi Ministry of Islamic Affairs.

The aim is to prove the consistency of the theme, in a Mathematica notebook for example.

ANALYSIS

The legal heirs to the deceased subject, as mentioned in the two verses, constitute the atomic family relationships as in the picture below: {{files/atomicfamily.png}}

For each relationship, we can have male or female members. So in total, we get the following roles: male parent (mp), female parent (fp), male spouse (mi), female spouse (fi) (one of them being the deceased subject), male child (mc), female child (fc), male sibling (ms), female sibling (fs). The inheritance rules given in the verses each apply for some combination of the presence of those members, so we can encode all possible cases of inheritance by combining the possibilities for each role. The possibilities for each role are those explicitly mentioned in the verses:

• mp, fp, mi, fi - 1 bit: absent (0) or present (1)
• mc - 1 bit: none (0) or any number (1)
• fc - 2 bits: none (00), one (01), two (10), more than two (11)
• ms+fs - 2 bits: none (00), one (01), two (10), more than two (11) [NOTE: the verses 4:11-12 do not differentiate between male and female siblings]

So we end up with a 9-bit word, or 2^9 = 512 cases. Our binary representation will look like the following: {{files/familybits.png}}

For example, a deceased male subject leaving behind a mother, a wife, two male children and a sister would be represented as 010110001.