🤖 AI Summary
This work investigates the effective decomposition of higher-arity relations over finite domains into binary relations, with applications in constraint satisfaction problems, clone theory, and relational databases. By interpreting relations as partially defined graphs of multi-valued functions and leveraging functional completeness and function decomposition techniques from many-valued logic, the authors first reduce arbitrary high-arity relations to ternary ones. Subsequently, they achieve full binarization through a transformation that replaces disjunctions with existential quantifiers. This approach yields an elementary and computationally efficient constructive proof, unifying and rederiving Pierce’s reduction theorem for finite domains. Moreover, it demonstrates that the graph of any Sheffer function suffices to generate all relations, thereby establishing a computable theoretical foundation for the aforementioned fields.
📝 Abstract
We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued 'functions'. The 'functions' are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce's reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there.