Consequence relations and data science: From Galois mappings to data interpretation

Abstract

The concepts of a consequence relation and operation, though very abstract and theoretical, may be related to specific categories of information systems (i.e. mathematical frontends of data tables); as it has been demonstrated by D. Vakarelov, there exist correspondence between Pawlak information systems and Scott as well as Tarski consequence operations. This line of research goes (via representation) from abstract concepts to data. In this paper we would like to take the opposite direction: from data (via construction) to consequence relations. The main emphasis is laid here not on general categories of consequence relations (e.g. Scott or Tarski ones) but on concrete operators that can be retrieved from information systems (e.g. different examples of Scott consequence). To this end, we employ Galois connections and adjunctions (en masse called Galois mappings) and study the consequence relations that can be built via these maps. The main novelty of our research comes from the investigation of consequence relations induced by adjunctions rather than monotone Galois connections, which have been the main subject of studies so far. Surprisingly, the operations obtained from adjunctions possess a number of counter-intuitive properties, which (in turn) request some intelligible interpretations. And this is our next objective: to make sense of these consequence relations in the context of information processing.