Manuscript Oai119-A@1 · Oai119-A.tex · section · fdff7292

Exhaustive classification of the sixteen functions

Stages 3 and 6: the complete finite certificate › Exhaustive classification of the sixteen functions · lines 813–865

Oai119-A@1#eq:posterior

The complete representative calculation is:

Representative	$m$	$f (v_{1})$	$f (v_{2})$	$f (v_{3})$	$f (v_{4})$
$1$	$1$	$1$	$1$	$1$	$1$
$x_{1}$	$0$	$1$	$1$	$- 1$	$- 1$
$x_{1} x_{2}$	$0$	$1$	$- 1$	$- 1$	$1$
$q$	$1/2$	$- 1$	$1$	$1$	$1$

Representative	$T_{ρ_{0}} f (v_{1})$	$T_{ρ_{0}} f (v_{2})$	$T_{ρ_{0}} f (v_{3})$	$T_{ρ_{0}} f (v_{4})$
$1$	$1$	$1$	$1$	$1$
$x_{1}$	$1/2$	$1/2$	$- 1/2$	$- 1/2$
$x_{1} x_{2}$	$1/4$	$- 1/4$	$- 1/4$	$1/4$
$q$	$- 1/8$	$5/8$	$5/8$	$7/8$

These entries follow by multiplication by (eq:fourmatrix); no dimension induction is being used.

For the biased representative, the full three-atom posterior table is

d = T_{ρ_{0}} q (Y) P (d) P (q (X) = 1 ∣ Y) H (d) - 1/8 1/4 7/16 37 /8 5/8 1/2 13/16 39 /8 7/8 1/4 15/16 15 /8 .

This is a posterior law for this function, not a solution of the unrelated joining optimization (eq:joiningproblem).

Displayed formulas in this passage

Oai119-A@1#eq:posterior · line 853
$d = T_{ρ_{0}} q (Y) P (d) P (q (X) = 1 ∣ Y) H (d) - 1/8 1/4 7/16 37 /8 5/8 1/2 13/16 39 /8 7/8 1/4 15/16 15 /8 .$