SMB algebras II: On the Constraint Satisfaction Problem over Semilattices of Mal'cev Blocks
This work addresses the CSP for theoretical computer scientists, but it is incremental as it reproves known results and focuses on similarities between existing proofs.
The paper tackles the Constraint Satisfaction Problem (CSP) by defining SMB algebras and proves that all such algebras induce tractable CSP templates, a result previously established by Bulatov, while comparing two general proofs of the CSP Dichotomy.
We define a class of algebras, the semilattices of Mal'cev blocks (for short, SMB algebras). In a nutshell, these algebras are semilattices in which each element gets blown up into a Mal'cev algebra. We publish for the first time our old proofs that some SMB algebras induce tractable templates of the reprove that the Constraint Satisfaction Problem. Next, we reprove that, in fact, all SMB algebras induce tractable templates of the Constraint Satisfaction Problem, a result already proved by A. Bulatov. Also, we compare the two general proofs of the CSP Dichotomy and prove they are more similar than initially thought when they are applied to SMB algebras. This paper is the second in the series of papers investigating the SMB algebras and it is a precursor to our further research on the similarities between the proofs of the Dichotomy Theorem.