Updated index.rst

index.rst

@@ -29,6 +29,15 @@ NeuronRain repositories are in:
 
         (*) NeuronRain Antariksh -  https://gitlab.com/shrinivaasanka - Drone development
 
+1402. NeuronRain - ReadTheDocs URLs:
+------------------------------------
+        1402.1 NeuronRain Docuementation - unified - https://neuronrain-documentation.readthedocs.io/en/latest/ 
+        1402.2 NeuronRain Theory Drafts - https://acadpdrafts.readthedocs.io/en/latest/ 
+	1402.3 NeuronRain AstroInfer - https://astroinfer.readthedocs.io/en/latest/
+	1402.4 NeuronRain USBmd64 - https://usb-md64-github-code.readthedocs.io/en/latest/
+	1402.5 NeuronRain VIRGO64 - https://virgo64-linux-github-code.readthedocs.io/en/latest/
+	1402.6 NeuronRain KingCobra64 - https://kingcobra64-github-code.readthedocs.io/en/latest/
+
 1331. NeuronRain Documentation Repositories:
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         (*) https://github.com/shrinivaasanka/Krishna_iResearch_DoxygenDocs
@@ -334,7 +343,7 @@ isolate the meanings of common pictogram B in three ways by 1) Set intersection
 	6. Integer Partitions and Tabulation Hashing are isomorphic e.g partition of an integer 21 as 5+2+3+4+5+2 and Hash table of 21 values partitioned by keys on bucket chains of sizes 5,2,3,4,5,2 are bijective. Both Set Partitons and Hash tables are exact covers quantified by Bell Numbers/Stirling Numbers. Partitions/Hashing is a special case of Multiple Agent Resource Allocation problem. Thus hash tables and partitions create complementary sets defined by Diophantine equations. Pareto Efficient resource allocation by Multi Agent Graph Coloring - coloring partition of vertices of a graph - finds importance in GIS and Urban Sprawl analytics, Resource Scheduling in Operating Systems (allocating processors to processes), Resource allocation in People Analytics (allocating scarce resources - jobs, education - to people) by a Social welfare function e.g Envy-Free, Pareto efficient Multi Agent Graph Fair Coloring of Social Networks to identify communities, allocate resources to communities of social networks in proportion to size of each community.
 	7. Ramsey Coloring and Complementation are equivalent. Ramsey coloring and Complement Diophantines can quantify intrinsic merit of texts.
 	8. Graph representation of Texts and Lambda Function Composition are Formal Language and Algorithmic Graph Theory Models e.g parenthesization of a sentence creates a Lambda Function Composition Tree of Part-of-Speech.
-	9. Majority Function - Voter SAT is a Boolean Function Composition Problem and is related to an open problem - KRW conjecture - and hardness of this composition is related to another open problem - P Vs NP and Knot Theory. Theoretical Electronic Voting Machine (which is a LSH/set partition for multipartisan election) for two candidates is the familiar Boolean Majority Circuit whose leaves are the binary voters (and their VoterSATs in Majority+VoterSAT circuit composition). Pseudorandom shuffle of leaves of Boolean majority circuit simulates paper ballot which elides chronology. Pseudodrandomly shuffled Electorate Leaves of the Boolean Majority Circuit are thus Ramsey 2-colored (e.g Red-Candidate0, Blue-Candidate1) by the candidate indices voted for. Pseudorandom shuffle and Ramsey coloring are at loggerheads - arithmetic progression order arises in pseudorandomly shuffled bichromatic electorate disorder and voters of same candidate are equally spaced out which facilitates approximate inference of voting pattern. Hardness of inversion in the context of boolean majority is tantamount to difficulty in unravelling the voters who voted in favour of a candidate - voters_for(candidate) - pseudorandom shuffle of leaves of boolean majority circuit must minimize arithmetic progressions emergence which amplifies hardness of the function voters_for(candidate). Another instance of order emergence from disorder is the group of half-turn moves of Rubik's Cube, Cayley graph of which has been shown to have a diameter of 20 - https://tomas.rokicki.com/rubik20.pdf, https://www.cube20.org/ - "... In group theory language, the problem we solve is to determine the diameter, i.e., maximum edge-distance between vertices, of the HTM-associated Cayley graph of the Rubik’s Cube group. As summarized in the next section, many researchers have found increasingly tight upper and lower bounds for the HTM diameter of the cube. The present work explains the computational aspects of our proof that it equals 20..." - In other words solution can be reached from any of the 43,252,003,274,489,856,000 positions of Rubik's cube within 20 moves (God's number). God's number of 20 is tight (both lower and upper). If colors of Rubik's cube are replaced by integers,every configuration (vertex of Cayley graph) of Rubik's cube corresponds to a pseudorandom integer (disorder) eventually converging to a monochromatic face solution (order - face of same integers) within 20 moves. Cayley graph is edge colored by generating set of Rubik's cube group. Cayley graph of a Rubik's cube colored by symbols from a formal language leads to a Rubik's cube version of edit distance which is surprisingly computable in O(n^2/logn) which is sublinear while conventional edit distance is quadratic and subquadratic edit distance algorithm implies SETH is false. 
+	9. Majority Function - Voter SAT is a Boolean Function Composition Problem and is related to an open problem - KRW conjecture - and hardness of this composition is related to another open problem - P Vs NP and Knot Theory. Theoretical Electronic Voting Machine (which is a LSH/set partition for multipartisan election) for two candidates is the familiar Boolean Majority Circuit whose leaves are the binary voters (and their VoterSATs in Majority+VoterSAT circuit composition). Pseudorandom shuffle of leaves of Boolean majority circuit simulates paper ballot which elides chronology. Pseudodrandomly shuffled Electorate Leaves of the Boolean Majority Circuit are thus Ramsey 2-colored (e.g Red-Candidate0, Blue-Candidate1) by the candidate indices voted for. Pseudorandom shuffle and Ramsey coloring are at loggerheads - arithmetic progression order arises in pseudorandomly shuffled bichromatic electorate disorder and voters of same candidate are equally spaced out which facilitates approximate inference of voting pattern. Hardness of inversion in the context of boolean majority is tantamount to difficulty in unravelling the voters who voted in favour of a candidate - voters_for(candidate) - pseudorandom shuffle of leaves of boolean majority circuit must minimize arithmetic progressions emergence which amplifies hardness of the function voters_for(candidate). Another instance of order emergence from disorder is the group of half-turn moves of Rubik's Cube, Cayley graph of which has been shown to have a diameter of 20 - https://tomas.rokicki.com/rubik20.pdf, https://www.cube20.org/ - "... In group theory language, the problem we solve is to determine the diameter, i.e., maximum edge-distance between vertices, of the HTM-associated Cayley graph of the Rubik’s Cube group. As summarized in the next section, many researchers have found increasingly tight upper and lower bounds for the HTM diameter of the cube. The present work explains the computational aspects of our proof that it equals 20..." - In other words solution can be reached from any of the 43,252,003,274,489,856,000 positions of Rubik's cube within 20 moves (God's number). God's number of 20 is tight (both lower and upper). If colors of Rubik's cube are replaced by integers,every configuration (vertex of Cayley graph) of Rubik's cube corresponds to a pseudorandom integer (disorder) eventually converging to a monochromatic face solution (order - face of same integers) within 20 moves. Cayley graph is edge colored by generating set of Rubik's cube group. Cayley graph of a Rubik's cube colored by symbols from a formal language leads to a Rubik's cube version of edit distance which is surprisingly computable in O(n^2/logn) which is sublinear while conventional edit distance is quadratic and subquadratic edit distance algorithm implies SETH is false. N*N*N Rubik's cube in its original version supports only 6 colors which requires a binary encoding of natural language vocabulary (ASCII or Unicode) for coloring faces (white,red,blue,orange,green,yellow have to be replaced by letters from a natural language). Tuttminx (shaped as Buckminsterfullerene - Carbon 60 - https://en.wikipedia.org/wiki/Buckminsterfullerene) - https://en.wikipedia.org/wiki/Tuttminx - an advanced version of Rubik's cube supports 32 colored faces (sufficient to encode English or any Latin derived language directly without binary representation) and 150 moveable pieces compared to 20 pieces in Rubik's cube. Number of possible configurations (or number of possible words and vertices on Cayley graph) in Tuttminx is 1.2325 * 10^204. An unusual consequence of God's number arises when Rubik's cube faces are binary encoded: As a solution can be reached from any configuration in 20 moves (diameter of Cayley graph), distance between two configurations (boolean strings encoded on faces of Rubik's cube) is upperbounded by 20 and any two such binary string configurations x and y are correlated (or every bit of x is flipped by noise to y) placing an upperlimit on probability of per bit flip (naive noise probability bound: < 1/20) - https://booleanzoo.weizmann.ac.il/index.php/Noise_sensitivity. 
 	10. Majority Versus Non-Majority Social Choice comparison arises from Condorcet Jury Theorem (recent proof of Condorcet Jury Theorem in the context of Strength of Weak Learnability - Majority Voting in Learning theory - AdaBoost Ensemble Classifier - https://arxiv.org/pdf/2002.03153.pdf) and Margulis-Russo Threshold phenomenon in Boolean Social Choice i.e how individual decision correctness affects group decision correctness. Equating the two social choices has enormous implications for Complexity theory because all complexity classes are subsumed by Majority-VoterSAT boolean function composition. Depth-2 majority (Majority+Majority composition) social choice function - boolean and non-boolean - is an instance of Axiom of Choice (AOC) stated as "for any collection of nonempty sets X, there exists a function f such that f(A) is in A, for all A in X". Depth-2 majority (both boolean and non-boolean voters set-partition induced by candidate voted for), which is the conventional democracy, chooses one element per constituency electorate set A of set of constituencies X in the leaves, at Depth-1.
 
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