SVIBOR - Project code: 1-01-315

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Project code: 1-01-315

THE NUMBER OF MONOTONE FUNCTIONS OF TWO-VALUED LOGIC

Main researcher: VUKOVIĆ, ANTE (53861)

Assistants

Duration from: 01/01/95.

Papers on project (total): 0

Institution name: Fakultet elektrotehnike,strojarstva i brodogradnje, Split (23)
Department/Institute: FESB - Department of Mathematics and Physics Physics Department
Address: Ruđera Boškovića b.b.
City: 21000 - Split, Croatia

Communication: Phone: 385 (0)21563777; Fax: 385 (0)21563877

Summary: An outstanding problem in the theory of multivalued logic is the so-called completness. This problem is connected with functions on ordered sets. Let E_k be such a set, where k is a number of elements of E_k. Then, the so-called n-th Cartesian product Ečn_k is defined in natural way; Ečn_k is again an ordered set. Moreover, MONOTONE functions f(x_1,...,x_n) of n variables from the set E_k are defined. All monotone functions f(x_1,...,x_n) form one of the so-called PRECOMPLETE CLOSED classes, which is crucial for k-valued logic completness. Many authors tried to determine the number M=Psi_E_k(n) of monotone functions of n variables and the asymptotics for Psi_E_k(n) when n aims to infinite. The problems was partially solved about thirty years ago for a two-member set E_2, giving answers to a number of problems in the two-valued logic. For k-valued logic there are estimates, which in the last twenty years was not possible to improve. In 1974., the russian mathematician V.B. Alekseev improved shightly an old result according to which log Psi_E_k(n) behaves kčn/nč0.5. His methods differ from methods used by the head of this project in multiple-valued logic stueies; the last also give some estimates. But, as this theory evolve slowly, the subscribed belives that the field deserves permanent watch, especially because of increasing importance of these topics for applications. Applications encourage the endeavor to throw more light on these problems. Having no round-up and far-developed theory at disposal, applications use approximate and experiene methods. In that, more and more appears the connection between the multiple-valued logic and so-called fuzzy-decision theories. The head of the project belives that this connection revives the interest for multiple_valued logics, and that using various approaches, inclusive his own, the results must be positive. In favour of the above stataments we can quote the fact, that in two most spread mathematical informative journals, in the Notices and in the Bulletin of the American Mathematical Society (AMS), which annually review more then hundred books and publish as much articles, in the last few years there are almost no reports on multiple-valued logics. Only one book has been reviewed in the Bulletin AMS (J.R. Buchi: Finite Automata, 1988.) which touches these problems. But, in the January number of the Notices AMS for 1993. (p.15) there is an interesting review on the by L.Wos's books on Automated Reasoning, which refreshes the whole subject, and in the February number for 1995. of the same journal (p.297) there is a call for papers for the new journal: Multiple-Valued Logic (Gordon and Breach publ., Newark, NJ, USA; Editors: Dan A. Simovici and Ivan Stojmenovic). Let us the four fields of interest of the new journal, as they are announced: 1. Mathematical and Engeneering Aspects of Multiple-Valued Logic (MVL); 2. MVL and Automated Reasoning; 3. Computer Science and MVL; 4. Theoretical and Practical Aspectd of Fuzzy Logic and Philosophical Aspects of MVL. Therefore, the subscribed strongly belives that research on this topics must be continued. Also, it is indispensable to attract new researchers to the project, regardless the fact that results at this moment have not a definite form. The subscribed proposes a financial support for at least one institution in Croatia with the aim to acquire the new MVL journal (there are no possibilityto exchange it for Glasnik Matematicki). There is an unpleasent possibillity that in the field, where we have results and current research, in few years there will be no research, and that once and again we shall find us at the verz begining.

Other information about the project.


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Last update: 09/25/95
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