Some topics in Discrete Mathematics

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Some topics in Discrete Mathematics Fairouz Tchier, ...∗ Mathematics department, King Saud University, Riyadh 11495, Saudi Arabia .. .

Abstract The importance of relations is almost self-evident. Science is, in a sense, the discovery of relations between observables. Zadeh has shown the study of relations to be equivalent to the general study of systems (a system is a relation between an input space and an output space). Goguen, J.A. [6]Goguen67

Key words: Relational algebra, nondeterministic semantics, demonic calculus, fuzzy calculus, demonic calculus. 2010 Mathematics Subject Classification: 18C10 ,18C50, 68Q55 ,68Q65,03B70,06B35 §1. Introduction The calculus of relations has been an important component of the development of logic and discrete mathematics since the middle of the nineteenth century. George Boole, in his ”Mathematical Analysis of Logic” [2]Boole47, initiated the treatment of logic as part of mathematics, specifically as part of algebra. Quite the opposite conviction was put forward early this century by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica [?]Whitehead10: that mathematics was essentially grounded in logic. The logic is developed in two streams. On the one hand algebraic logic, in which the calculus of relations played a particularly prominent part, was taken up from Boole by Charles Sanders Peirce, who wished to do for the ”calculus of relatives” what Boole had done for the calculus of sets, Peirce’s work was in turn taken up by Schr¨oder in ”Algebra und Logik der Relative” [10]Schroder95. Schr¨ oder’s work, however, lay dormant for more than 40 years, until revived by Alfred Tarski in his seminal paper ”On the calculus of binary relations” [11]Tarski41. Tarski’s paper is still often referred to as the best introduction to the calculus of relations. It gave rise to a whole field of study, that of relation algebras, and more generally Boolean algebras with operators. This important work defined much of the subsequent development of logic in the 20th century, completely eclipsing for some time the development of algebraic logic. In this stream of development, relational calculus and relational methods appear with the development of universal algebra in the 1930’s, and again with model theory from the ∗

Fairouz Tchier

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1950’s onwards. In so far as these disciplines in turn overlapped with the development of category theory, relational methods sometimes appear in this context as well. It is fair to say that the role of the calculus of relations in the interaction between algebra and logic is by now well understood and appreciated, and that relational methods are part of the toolbox of the mathematician and the logician. The main advantages of the relational formalization are uniformity and modularity. Actually, once problems in these fields are formalized in terms of relational calculus, these problems can be considered by using formulae of relations, that is, we need only calculus of relations in order to solve the problems. This makes investigation on these fields easier. Demonic relational calculus In the context of software development, one important approach is that of developing programs from specifications by stepwise refinement, see, e.g. [1, 13, 15]Back81a, Tchier2002a, Tchier2003. One point of view is that a specification is a relation constraining the input-output (respectively, argument-result) behaviour of programs. Inspired by interpretation of relations as non-deterministic programs, demonic, angelic and erratic variants of relational operations have been studied. The demonic interpretation of non-deterministic turns out to closely correspond to the concept of under-specification, and therefore the demonic operations are used in most refinement calculus. Fuzzy Calculus Fuzzy ”As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics.” [?]Zadeh73 Let us consider characteristic features of real-world systems again: real situations are very often uncertain or vague in a number of ways. Due to lack of information the future state of the system might not be known completely. This type of uncertainty has long been handled appropriately by probability theory and statistics. This Kolmogoroff-type probability is essentially frequentistic and bases on settheoretic considerations. Koopman’s probability refers to the truth of statement and therefore bases on logic. On both types of probabilistic approaches it is assumed, however, that the events (elements of sets) or the statements, respectively, are well defined. We shall call this type of uncertainty or vagueness stochastic uncertainty by contrast to the vagueness concerning the description of the semantic meaning of the events, phenomena or statements themselves, which we shall call fuzziness. Fuzziness can be found in many areas of daily life, such as in engineering, in medicine, in meteorology, in manufacturing [8]Mamdani81; and others. The first publications in fuzzy set theory by Zadeh [17]Zadeh65 and Goguen [6, 7]Goguen67,Goguen69 show the intention of the authors to generalize the classical notion of a set. Zadeh writes:”The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of mathematics and computer science (pattern classification and information processing). Fuzzy set theory provides a strict mathematical framework ( there is nothing fuzzy about fuzzy set theory) in which vague conceptual phenomena can be precisely and rigorously studied. It can also be considered as a modeling language well suited for situations in which fuzzy relations, criteria, and phenomena exist. It will mean different things, depending on the application area and the way it is measured. In the meantime, numerous authors have contributed to this theory. In 1984 as many as 4000 publications may already exist. The specialization of those publications conceivably increases, making it more and more difficult for new comers to this area. Roughly speaking, fuzzy set is a formal theory which, when maturing, became more sophisticated and specified and was enlarged by original ideas and concepts as well as by ”embracing” classical mathematical areas such as algebra, graph theory, topology, and so on by generalizing (fuzzifying) them. As a very powerful modeling language, that can cope with a large fraction of uncertainties of real-life situations. There are countless applications for fuzzy logic. In fact, some claim that fuzzy logic is the encompassing theory over all types of logic. The items in this list are more common applications that one may encounter in everyday life, for example Temperature control (heating/cooling), Medical diagnoses, Predicting travel time, auto-Focus on a camera, predicting genetic traits and bus Time Tables.

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[1] ok Back, R. J. R. : On correct refinement of programs . J. Comput. System Sci., 23(1):49–68, 1981. [2] Boole, G..: The Mathematical Analysis of Logic, Being an Essay Toward a Calculus of Deductive Reasoning. Macmillan, Cambridge, 1847. [3] Desharnais, J. B. M¨ oller, and F. Tchier. Kleene under a demonic star. 8th International Conference on Algebraic Methodology And Software Technology (AMAST 2000), May 2000, Iowa City, Iowa, USA, Lecture Notes in Computer Science, Vol. 1816, pages 355–370, Springer-Verlag, 2000. [4] Desharnais, J., Belkhiter, N., Ben Mohamed Sghaier, S., Tchier, F., Jaoua, A., Mili, A. and Zaguia, N.: Embedding a Demonic Semilattice in a Relation Algebra. Theoretical Computer Science, 149(2):333–360, 1995. [5] Dijkstra, E. W. and Feijn, W. : Predicate calculus and program semantics. Addison-Wesley, 1988. [6] Goguen, J. A. . : L-fuzzy sets. JMAA 18, 145-174. [7] Goguen, J. A. . : The logic of inexact concepts. Synthese 19, 325-373. [8] Mamdani, E. H. : ”Advances in the Linguistic Syntesis of Fuzzy Controllers In Fuzzy Reasoning and Its Applications”, Academic Press, London (1981). [9] Schmidt, G. and Str¨ ohlein, T.: Relations and Graphs. EATCS Monographs in Computer Science. SpringerVerlag, Berlin, 1993. [10] Str¨ ohlein, E.: Vorlesungen u ¨ber die Algebra der Logik(exacte Logik). Teubner, Leipzig, 1895.Vol.3, Algebra und Logik der Relative, part I,2nd edition published by Chelsea, 1966. [11] Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 3, 1941, 73–89. [12] Tchier, F.: S´emantiques relationnelles d´emoniaques et v´erification de boucles non d´eterministes. Theses of doctorat, D´epartement de Math´ematiques et de statistique, Universit´e Laval, Canada, 1996. [13] Tchier, F.: Demonic semantics by monotypes. International Arab conference on Information Technology (Acit2002),University of Qatar, Qatar, 16-19 December 2002. [14] Tchier, F.: Demonic relational semantics of compound diagrams. In: Jules Desharnais, Marc Frappier and Wendy MacCaull, editors. Relational Methods in computer Science: The Qu´ebec seminar, pages 117-140, Methods Publishers 2002. [15] Tchier, F.: While loop demonic relational semantics monotype/residual style. 2003 International Conference on Software Engineering Research and Practice (SERP03), Las Vegas, Nevada, USA, 23-26, June 2003. [16] Tchier, F..: Demonic Semantics: using monotypes and residuals. IJMMS 2004:3 (2004) 135-160. (International Journal of Mathematics and Mathematical Sciences) [17] Zadeh, L. A. .: Fuzzy Sets. Inform and Control 8 1965, 338–353. [18] Zimmermann, H. J. .: Fuzzy Set Theory And its Applications, Second edition. Kluwer Academic Publishers, Boston, Dordrecht, London, 1990.

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