Pawlak: rough sets

In January 1981 Zdzisław Pawlak of the Institute of Computer Science of the Polish Academy of Sciences issued in Warsaw the report 'Classification of objects by means of attributes', with the upper and lower approximations and an appendix 'Rough sets', in February the report 'Rough sets. Basic notions', and in October 1982 published the paper 'Rough Sets', on approximate operations on sets, approximate equality and inclusion. The author presents the approach as an alternative to fuzzy set theory and tolerance theory and as a mathematical foundation for artificial intelligence: classification, inductive reasoning, pattern recognition.

ICS PAS Report 429, 'Classification of objects by means of attributes. An approach to inductive inference', has twenty numbered pages, printed as a manuscript in 700 copies; the cover reads 'January 1981, Warszawa', and the imprint says it was sent to print in January 1981. Its departure point is the information system of Marek and Pawlak (1976), here called a knowledge representation system. In section 2 the smallest composed set containing Y is called the upper approximation of Y and the greatest composed set contained in Y its lower approximation, and the accuracy of an approximation is the size of Y divided by the size of its upper approximation; reducts of the set of attributes, a classification algorithm and an example from medical diagnosis, without patient data, follow. The abstract says the approach leads to a new formulation of fuzzy sets, 'called here the rough sets'; a footnote names 'surely belongs' and 'possibly belongs' as a basis for a theory of rough sets, to be discussed elsewhere; and the appendix 'Rough sets' (pp. 18-19) gives nine axioms for the approximations, calls this an alternative approach to Zadeh's fuzzy set theory and notes that the upper and lower approximations satisfy the axioms of topological closure and interior. Report 431, a month later, has twelve numbered pages, printed as a manuscript in 700 copies; the cover reads 'February 1981, Warszawa', and the imprint says it was sent to print in February 1981. Its approximation space is a set with an equivalence relation. The least composed set containing X is called the best upper approximation of X, the greatest composed set contained in X the best lower approximation, and their difference the boundary. Membership is strong or weak: 'surely belongs' and 'possibly belongs'. Rough equality and rough subsets follow. The report gives no separate definition of a rough set and does not call itself the first: its introduction proposes 'another approach' beside Zadeh's fuzzy sets and Zeeman's tolerance spaces and cites Pawlak's earlier Report 429. That rough sets were introduced in Report 431 is what Pawlak himself says in the first sentence of the 1982 paper; these are the author's words, not an outside witness of priority, and the name and both approximations already stand in Report 429. The paper was received in June 1981 and revised in September 1982. Its first page lists among the key words artificial intelligence, learning algorithms and pattern recognition; on the second the author says he aims at mathematical foundations for artificial intelligence rather than a new set theory, and that he was inspired by Michalski's results on automatic classification. In 2002 Pawlak wrote that the theory overlaps with fuzzy sets and Bayesian inference but is an independent, complementary, not competing discipline. What the record does not claim: any application or measurement, since the paper only outlines applications and Report 429 describes diagnosis without data; that either report is where the idea first appeared, since the record says only what each document prints; the day either report came out, since they print only months.