Abstract
We introduce the definition of unary algebra as well as subclasses of it called k-valued, strongly-k-valued and strongly-k-generated. Then we proceed with the simplification algorithm that transforms each system of equations into a more regular one at the expense of adding some definable constraints. Finally we give computational complexity characterization of SysTermSat over three-element unary algebras that depends on width of a special preorder constructed from given algebra.
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Notes
- 1.
We use here (and in the rest of the text) polynomial-time many-one reductions also known as polynomial transformations.
- 2.
The relation defined by Positive-1-in-3-Sat is \(\left\{ (1,0,0),(0,1,0),(0,0,1)\right\} \) and it is not closed under any of the operations listed in Fact 1.15.
- 3.
We will use only \(f_1,f_2\) and \(f_3\) for which the situation \(a_1=1,b_1=0\) is symmetric.
- 4.
Obviously \(X\Leftrightarrow Y\) is a pair of 2-Sat clauses: \(\lnot X \vee Y\) and \(X \vee \lnot Y\).
References
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Broniek, P. (2015). Unary Algebras. In: Computational Complexity of Solving Equation Systems. SpringerBriefs in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-21750-5_2
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DOI: https://doi.org/10.1007/978-3-319-21750-5_2
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