The splitting technique in monotone recognition

Notes on Monotone Recognition in Multi-Valued Grids

“…In this section we introduce the cube-splitting technique of recognition of monotone binary functions [6]. Two homogeneous areas inside the 𝛯𝛯 𝑚𝑚+1 𝑛𝑛 are defined in the following way: upper homogeneous area 𝐻𝐻 � , -this is the set of all "upper" elements of 𝛯𝛯 𝑚𝑚+1 𝑛𝑛 , i.e., elements with all-coordinate values ≥ 𝑚𝑚/2; lower homogeneous area 𝐻𝐻 � , -this is the set of all "lower" elements, i.e., elements with all-coordinate values ≤ 𝑚𝑚/2.…”

“…(2) The "cube-splitting" of 𝛯𝛯 𝑚𝑚+1 𝑛𝑛 keeps the monotonicity property in the following way: let 𝐹𝐹 be a monotone binary function defined on 𝛯𝛯 𝑚𝑚+1 𝑛𝑛 , then either 𝑁𝑁 𝐹𝐹 ∩ ℰ 𝑖𝑖 is empty, or it satisfies the binary monotonicity property, i.e., for arbitrary vertex 𝑎𝑎 of 𝑁𝑁 𝐹𝐹 ∩ ℰ 𝑖𝑖 , all vertices of ℰ 𝑖𝑖 greater than 𝑎𝑎, also belong to 𝑁𝑁 𝐹𝐹 ∩ ℰ 𝑖𝑖 (for 𝑖𝑖 = 1, ⋯ , 𝑛𝑛). By integrating ( 1) and ( 2), a novel monotone recognition method has been proposed in [6].…”

“…A recent novel method of the monotone recognition based on a partitioning of the grid into discrete structures isomorphic to binary cubes (called "cube-split" technique) is proposed in [6], and two algorithms /algorithmic schemes/ solving this problem are also introduced. This paper provides implementation details of these algorithms, as well as focuses on the recognition of monotone binary functions with a small number of units.…”

Notes on Reconstruction of Shadow Minimization Boolean Functions

“…MBFs are used to encode extremely important constructions in various combinatorial optimizations: they provide a natural way to describe compatible subsets of sets of finite constraints. A number of applications (e.g., wireless sensor networks, dead-end tests of tables, data mining [1,2]) use optimization with MBF, where MBFs are represented by constructions such as chains and anti-chains [3] in hypercubes. Other similar applications with MBF can be added to this list [4,5,6].…”

“…There are a number of known effective tools and methods for analyzing MBFs, and new approaches are constantly being sought, investigated, and applied. Well-known open problems in this area includes the reconstruction problem of bounded classes of Boolean functions, with randomization of queries and functions, and the use of cube-splitting and chain-splitting technique of the Boolean domain [7,1].…”