Validation of Equivalence Structure Incremental Search

Equivalence structure (ES) extraction focuses on multidimensional temporal patterns that appear in a multidimensional sequence and in a different multidimensional sequence. The input of the task is a set of sequences and the output is a set of ESs. An ES is a set of $K$-tuples comprising elements of $K$ IDs to specify $K$ sequences, and it shows which $K$-dimensional sequences composed of $K$ sequences specified by its $K$-tuples are considered equivalent. The standard for determining whether $K$-dimensional sequences are equivalent is based on the subsequences of the $K$-dimensional sequences. ES extraction can be used for multidimensional temporal feature extraction, as well as preprocessing for transfer learning or imitation learning. A method called ES incremental search (ESIS) was recently proposed, which is much faster than brute-force search, but the proofs necessary to derive it had not been sufficient. Therefore, it has been unclear why ESIS worked and can be reliable. This paper presents proofs to validate ESIS, as well as a property of the solution of ESIS that could be useful for developing a faster method.