{'fraction_non_alphanumeric': 0.014012738853503185, 'fraction_numerical': 0.0, 'mean_word_length': 5.238095238095238, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Indexed rough approximations and generalized possibility theory

Indexed rough approximations that generalize the fuzzy rough sets are proposed. A family of indexed relations between objects with the set of indices being a lattice is considered. Relations in the family are ordered by the inclusion, and moreover the ordering is assumed to be consistent with the ordering of the lattice. Thus, a collection of rough approximations, each of which is induced from a relation in the family, is obtained. The corresponding modal operators with the indices are defined; the completeness between the axiomatic system and the Kripke model which is the above collection of rough approximations is proved. A generalized possibility theory is derived from the collection of rough approximations.