Category:Algebraic normal form (image set)

See also: v:Algebraic normal form
Algebraic normal forms are vertical, and truth tables are horiozontal bit-patterns.
The calculation ANF to TT uses an upper triangular Sierpiński triangle with rows corresponding to ANF.
The calculation ANF from TT uses a lower triangular Sierpiński triangle with columns corresponding to TT.
The sections below illustrate cycles and fixed points in Zhegalkin permutation Ж3.
(It is a Walsh permutation corresponding to the 8×8 lower triangular Sierpiński triangle. It has 120 transpositions and 16 fixed points.)
This permutation is the self-inverse map between integers representing ANFs and truth tables of length 8.
(The pairs of bit-patterns are the same, no matter which is intepreted as ANF or TT.)
cycle (110, 142) |
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0111 0110 little-endian binary = 110 decimal ANF 142 to TT 110 ANF 142 from TT 110 ANF 110 to TT 142 ANF 110 from TT 142 |
cycle (143, 145) |
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1111 0001 little-endian binary = 143 decimal ANF 143 to TT 145 ANF 143 from TT 145 ANF 145 to TT 143 ANF 145 from TT 143 |
cycle (22, 150) |
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0110 1000 little-endian binary = 22 decimal ANF 22 to TT 150 ANF 22 from TT 150 ANF 150 to TT 22 ANF 150 from TT 22 |
fixed point 30 |
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0111 1000 little-endian binary = 30 decimal ANF 30 to TT 30 ANF 30 from TT 30 |
compared to prefects | ||
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AND XAND 8 Walsh functions (left) and their complements (right) The yellow matrix used to convert an ANF to its Boolean function is a Sierpiński triangle. Its rows are variadic ANDs. Instead using a matrix whose rows are variadic biconditionals (XAND) also creates a Boolean function. So every Boolean function can be assigned a linear Boolean function, which shall be called its prefect. The 16 2-ary functions are partitioned into 2⋅4=8 prefectures, each with 2 functions. | ||
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Subcategories
This category has only the following subcategory.
Z
- Zhegalkin index to prefect (image set) (1 P, 3 F)
Pages in category "Algebraic normal form (image set)"
This category contains only the following page.
Media in category "Algebraic normal form (image set)"
The following 16 files are in this category, out of 16 total.
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ANF from 0110 1000.svg 600 × 600; 52 KB
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ANF from 0110 1001.svg 600 × 600; 52 KB
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ANF from 0111 0001.svg 600 × 600; 53 KB
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ANF from 0111 0110.svg 600 × 600; 52 KB
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ANF from 0111 1000.svg 600 × 600; 53 KB
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ANF from 1000 1001.svg 600 × 600; 53 KB
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ANF from 1111 0001.svg 600 × 600; 53 KB
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ANF to 0110 0011.svg 600 × 600; 55 KB
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ANF to 0110 1000.svg 600 × 600; 53 KB
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ANF to 0110 1001.svg 600 × 600; 53 KB
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ANF to 0111 0001.svg 600 × 600; 54 KB
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ANF to 0111 0110.svg 600 × 600; 54 KB
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ANF to 0111 1000.svg 600 × 600; 54 KB
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ANF to 1000 1001.svg 600 × 600; 54 KB
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ANF to 1011 0010.svg 600 × 600; 54 KB
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ANF to 1111 0001.svg 600 × 600; 54 KB