Numerical characterisation of n-cube subset partitioning

We consider the problem of query based algorithmic identification/recognition of monotone Boolean functions, as well as of binary functions defined on multi-valued discrete grids. Hansel’s chain-split technique of $n$-cubes is a well known effective tool of monotone Boolean recognition. An extension by Alekseev is already applied to the grid case. The practical monotone recognition on $n$-cubes is provided by the so called chain-computation algorithms that is not extended to the case of multi-valued grids. We propose a novel split construction based on partitioning the grid into sub-grids and into discrete structures that are isomorphic to binary cubes. Monotonicity in a multi-valued grid implies monotonicity in all induced binary cubes and in multi-valued sub-grids. Applying Hansel’s technique for identification of monotone Boolean functions on all appearing binary cubes, and Alekseev’s algorithm on all sub-grids leads to different scenarios of reconstruction of monotone functions. On one hand such partitioning technique makes parallel recognition possible, on the other hand — the method can be used in practical identification algorithms due to simple structures and easily calculable quantities appearing after the partition to the $n$-cubes. Complexity issues of considered algorithms were also elaborated.