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I am trying to find a generalization to Post's Completeness Criterion for multi-valued logic. The Criteria for functional completeness in Boolean logic for any arity is found in Post’s thesis. Two papers which explain his completeness criteria better (at least for me) are:

1) Easy to understand yet must log into Scribd unfortunately. https://www.scribd.com/document/255983069/Posts-Criterion

2) A little more difficult to follow but more rigorous. Pdf is readily available: https://sites.ualberta.ca/~francisp/Phil428/Phil428.11/PostPellMartin.pdf

However, neither Post nor these papers provide a generalized approach to extend the concept of completeness beyond Boolean logic.

I did find some literature on the subject but to be honest it is not for the faint-hearted.

One paper I did find was by Graham from 1967 however it has been difficult for me to wrap my mind around: http://www.math.ucsd.edu/~ronspubs/67_02_truth.pdf

It seems as if the book “Computer Science and Multiple-Valued Logic” by North-Holland does provide a rigorous theorem on page 161 which I will cite here. That chapter can be found here

3.11 Theorem. Each maximal class in $O_k$ is of the form Pol $\rho$ where $\rho$ is one of the following relations on k:

  1. Every partial order on k with a least and greatest element.
  2. Every relation $\{|x\in k\}$ where s is a permutation of k with k/p cycles of the same prime length p.
  3. Every quaternary relation $\{\in k^4|a_0+’ a_1 = a_2 +’ a_3\}$ where $$ is a p-elementary abelian group (p prime).
  4. Every non-trivial equivalence relation on k.
  5. Every central relation on k.
  6. Every relation $\lambda_T$ determined by an h-regular family $T$ of equivalence relations on k (h>2).

A set is complete if and only if for every relation described under 1.-6. above there exists an $f \in X$ not preserving $\rho$.

I am trying to figure out how to even code this up, preferably in Matlab, I have coded up the Boolean solution, or $O_2^{(n)}$ but I am not sure how to proceed given this Theorem. If it would help, I can supply the code for the Boolean case once I have it tidied up.

Any help and thoughts on the matter would be greatly appreciated!

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