Zdzislaw Pawlak [pron. ZDHI-suav PAH-vlak] was born on November 10, 1926, in Lódz, Poland, and died on April 7, 2006, in Warsaw. He obtained his doctorate in technical sciences, in 1958, and his D.Sc. in mathematics, in 1963, from the Mathematical Institute of the Polish Academy of Sciences (PAS). Between 1951-57 he worked on the team designing the first Polish digital computer (Mathematical Institute PAS). Next, between 1957-59, he led the team at Warsaw University of Technology, to design a digital computer based on the “-2†number system. Then, he worked at various institutes (Mathematical Institute, Institute of Computer Science, and Institute of Theoretical and Applied Informatics) of the PAS until his death in 2006, and served as director of the Institute of Computer Science of the Warsaw University of Technology (1989-96).
Major Contribution: Invention of the rough set theory in 1981 [1,2].
It must be noticed that Z. Pawlak made a multitude of other contributions to computing, for example, in addition to his contributions to hardware design [3-5], he proposed a new class of parenthesis-free notations, which is a generalization of the Polish Notation [6,7], presented a formal model of a computer, known as the Pawlak machine [8,9] (different from Turing machines and from von Neumann architecture), and created the first mathematical model of a genetic code [10]).
Basic Questions
What is a rough set? Briefly, a rough set is a set with vague boundaries. By comparison, a regular set has sharp boundaries, that is, elements on the boundary either belong to the set or do not belong to it. For a fuzzy set, elements on its boundaries may belong to a set with a certain degree, which is characterized by a number from the interval (0, 1). In contrast to that, set membership of elements on the boundary of a rough set in undetermined, that is, it is not known whether they belong to the set or not, and to what degree. A rough set is characterized by its lower approximation, that is, a collection of elements which belong to the set for sure, and its upper approximation, that is a collection of elements which may belong to a set. A boundary of a rough set is defined as a difference between the upper approximation and the lower approximation.
The difference between rough sets and fuzzy sets has been described by Pawlak himself [11]. There are good tutorials on rough sets available on the Internet, some of them linked (with permission from the authors) to the Resources button of this webpage.
What is fascinating about the work of Zdzislaw Pawlak, is that it had such a broad scope. One specific topic from his numerous publications especially attracted our interest: What is a “-2†number system? It is one of the negative base number systems. One can learn a lot on the negative base number systems from a very interesting tutorial presented by W.J. Gilbert and R.J. Green [12]. Formally, negative bases are as good for representing numbers as traditional systems we are familiar with, that is, a positive 10 (decimal) and positive 2 (binary) numeration systems. For example, -326 expressed in a positive decimal notation is represented by 1734 in the negative decimal notation, which is just an abbreviation of the full formula: 1(-10)^3 + 7(-10)^2 + 3(-10)^1 + 4(-10)^0. However, it is interesting to learn: How does one come up with the individual digits of 1734? Roughly speaking, that happens the same way as converting regular decimal numbers to the binary representation, except that in the conversion algorithm one divides the positive decimal number by negative 10, instead of positive 2. As an example, here are all the steps to convert -326 expressed in positive decimal notation to its negative decimal representation (remember that for the proper conversion a remainder in each step has to be always positive, so adjusting the dividend is sometimes necessary):
- dividing -326 by -10 results in 33, remainder 4, which we hold as a number of units of the new representation
- dividing 33, obtained from previous step, by -10 results in -3, remainder 3, which we hold as a number of tens of the new representation
- dividing -3, obtained from previous step, by -10 results in 1, remainder 7, which we hold as a number of hundreds of the new representation
- the final result, 1, is held as the number of thousands of the new epresentation.
That way, we obtained 1734 as a negative decimal representation of a number -326 expressed in positive decimal notation. It must be noted that negative numeration systems have an advantage over the positive ones in that there is no need for extra representation of a sign, because the sign is embedded in the number itself. Thanks to this property, a problem with negative 0 and positive 0 is avoided.
Significance
Rough set theory, since its introduction in 1982 [1,2], has developed into a comprehensive discipline. It deals with uncertainty and decision making under circumstances with insufficient information, so it is applicable to all kinds of environments and has been used in a wide spectrum of applications, from high tech industries [13] and NASA space program [14], to medicine [15] and economics [16,17]. About three thousand papers on rough sets and their applications have been published world wide. Many international workshops and conferences on rough sets have been held in recent years. Over two dozen computer software tools have been developed for rough set calculations. Its recent extensions include, naming a few by Pawlak himself, the analogy with Bayes’ theorem [18], decision making with graphs [19], application to conflict analysis, and a new view of set theory [20]. Most of the respective information on rough sets is available from the website of the International Rough Sets Society.
Findings
The historical contribution of this website is in bringing, for the first time to the general audience, pieces of information on Zdzislaw Pawlak that have been completely unknown by the general public. This includes his unique practical contributions to the early computer development [3-5], as well as original theoretical results: extension of the parenthesis-free notation [6,7], a new concept of a description of a computer [8,9], and a formal mathematical model of genetic codes [10], with a selection of respective early references to his works [22-25].
References [show]
[1] Z. Pawlak, Rough Sets, International Journal of Information and Computer Sciences, Vol. 11, No. 5, pp. 341-356, 1982
[2] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991
[3] Z. Pawlak, Flip-Flop as a Generator of Random Binary Digits, Mathematical Tables and Other Aids to Computation, Vol. 10, No. 53, pp. 28-30, January 1956
[4] Z. Pawlak, An Electronic Digital Computer Based on the “-2†System, Bulletin de l’Academie Polonaise de Sciences, Ser. Sci. Tech., Vol. VII, No. 12, pp. 713-721, 1959
[5] Z. Pawlak, The Organization of a Digital Computer Based on the “-2†System, Bulletin de l’Academie Polonaise de Sciences, Ser. Sci. Tech., Vol. VIII, pp. 252-258, 1960
[6] Z. Pawlak, New Method of Parenthesis-free Notation of Formulae, Bulletin de l'Academie polonaise des sciences. Serie des sciences mathematiques, astronomiques, et physiques, Vol. 8, pp. 197-199, 1960
[7] Z. Pawlak, New Class of Mathematical Languages and Organization of Addressless Computers, pp. 227-238, Colloquium on the Foundations of Mathematics, Mathematical Machines and Their Applications, L. Kalmar (Ed.), Akademiai Kiado, Budapest, 1965
[8] Z. Pawlak, On the Notion of a Computer, pp. 255-267, Logic, Methodology and Philosophy of Science III, B. van Rootselaar, J.F. Staal (Eds.), North-Holland, Amsterdam, 1968.
[9] Z. Pawlak, Maszyny programowane (Stored Program Computers, in Polish), Algorytmy, Vol. 5, No. 10, pp. 5-19, 1969
[10] Z. Pawlak, Gramatyka i matematyka (Grammar and Mathematics, in Polish), PZWS, Warsaw, 1965
[11] Z. Pawlak, Rough Sets and Fuzzy Sets, Fuzzy Sets and Systems, Vol. 17, pp. 99-102, 1985
[12] W.J. Gilbert, R.J. Green, Negative Based Number Systems, Mathematical Magazine, Vol. 52, No. 4, pp. 240-244, September 1979.
[13] A. Mrozek, Rough Sets in Computer Implementation of Rule-based Control of Industrial Processes, Intelligent Decision Support: Handbook of Applications and Advances of the Rough Set Theory, S.-Y. Huang (Ed.), Kluwer Academic Publishers, Dordrecht, 1992, pp. 19-31
[14] Z.M. Wojcik, Detecting Spots for NASA Space Program Using Rough Sets, Rough Sets and Current Trends in Computing, W. Ziarko, Y. Tao (Eds.), Springer-Verlag, 2001, pp. 569-576
[15] K. Slowinski, R. Slowinski, J. Stefanowski, Rough Sets Approach to Analysis of Data from Peritoneal Lavage in Acute Pancreatitis, Medical Informatics, Vol. 13, pp. 143-159, 1989
[16] F.E.H. Tay, L. Shen, Economic and Financial Prediction Using Rough Sets Model, European Journal of Operational Research, Vol. 141, pp. 641-659, 2002
[17] A.I. Dimitras, R. Slowinski, R. Susmaga, C. Zopounidis, Business Failure Prediction Using Rough Sets, European Journal of Operational Research, Vol. 114, pp. 263-280, 1999.
[18] Z. Pawlak, Rough Sets, Decision Algorithms and Bayes’ Theorem, European Journal of Operational Research, Vol. 136, pp. 181-189, 2002
[19] Z. Pawlak, Data Analysis and Flow Graphs, Journal of Telecommunications and Information Technology, No. 3, pp. 1-5, 2004
[20] Z. Pawlak, Some Remarks on Conflict Analysis, European Journal of Operational Research, Vol. 166, pp. 649-654, 2005
[21] Z. Pawlak, Orthodox and Non-orthodox Sets: Some Philosophical Remarks, Foundations of Computing and Decision Sciences, Vol. 30, No. 2, pp. 133-140, 2005
[22] M. Novotny, On Some Problems Concerning Pawlak’s Machines, Proc. 4th Symp. Mathematical Foundations of Computer Science, Marianske Lazne, Czechoslovakia, Sept. 1-5, 1975, J. Becvar (Ed.), Springer-Verlag, Heidelberg, 1975, pp. 88-100
[23] J. Fried, Simulations of Pawlak Machines as Fuzzy Morphisms of Partial Algebras, Commentationes Mathematicae Universitatis Carolinae, Vol. 18, No. 2, pp. 343-350, 1977
[24] W.J. Meyers, Linear Representation of Tree Structure: A Mathematical Theory of Parethesis-Free Notation, pp. 50-62, Proc. 3rd Ann. ACM Symposium on Theory of Computing, 1971
[25] S. Marcus, Linguistic Structures and Generative Devices in Molecular Genetics, Cahiers de Linguistique Theorique et Appliquee, Vol. 11, No. 1, pp. 77-104, 1974
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