A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Ristikkotorniprojektin jälkiseuranta sekä valmistus-, kuljetus- ja asennusnäkökohdat massiivisen tornin suunnittelussa

Tässä työssä tutkittiin jälkiseurantana putkipalkkirakenteisen ristikkotornin rakennusprojektia. Tutkimuksen kohteena oli suomalaiselle öljynjalostamolle vuonna 1992 rakennettu vapaastiseisova 105 m korkea soihtutorni. Työn tarkoituksena oli tutkia teräsrakennekokonaisuutta toisaalta suunnittelun ja lujuuslaskennan, toisaalta valmistuksen, kuljetuksen ja asennuksen kannalta. Lisäksi työssä etsittiin soihtutornin vaihtoehtoisia rakenneratkaisuja. Esisuunnittelun tuloksena syntyneitä rakenteita verrattiin vuonna 1992 toteutettuun rakenteeseen. Toteutetun ristikkomaston lujuusominaisuuksia arvioitiin käymällä läpi tehdyt lujuuslaskelmat ja selvittämällä mm. rakenteen eri kohdista mitoituksen tasaisuus. Havaittiin, että rakenteen lujuuden käyttöaste vaihteli 0,7:n ja 0,9:n välillä 72 m:n korkeuteen asti. Tästä ylöspäin käyttöasteet olivat pienempiä. Konepajavalmistuksen, kuljetusten ja asennustöiden onnistumista tutkittiin haastattelemalla sekä konepajan että tilaajan edustajia. Ylöspäin kapeneva rakenne osoittautui lujuusopillisesti järkeväksi, mutta valmistusteknisesti vanhanaikaiseksi. Massiivisten ristikkotornien kuljetus- ja nostomahdollisuuksia tutkittiin selvittämällä lainsäädännön asettamat rajoitukset ja käsikirjojen antamat ohjeet. Lisäksi haastateltiin erikoiskuljetuksia ja -nostoja suorittavan yrityksen edustajaa. Uusien rakenneratkaisujen suunnittelussa korostettiin pitkälle vietyä osavalmistusta sekä kuljetus- ja nostomahdollisuuksia. Uusista tornivaihtoehdoista tutkittiin tarkemmin massiivisista pyöreistä putkipalkeista tehty torni, joka on rakenteeltaan Vierendeel-kehän sovellutus. Rakenne on 62 % painavampi kuin vuonna 1992 toteutettu soihtutorni, mutta silti 9 % halvempi. Uuden rakenteen valmistusystävällisyys on merkittävästi parempi kuin vanhan.

Development of an HTR-10 model in the Serpent reactor physics code

The use of exact coordinates of pebbles and fuel particles of pebble bed reactor modelling becoming possible in Monte Carlo reactor physics calculations is an important development step. This allows exact modelling of pebble bed reactors with realistic pebble beds without the placing of pebbles in regular lattices. In this study the multiplication coefficient of the HTR-10 pebble bed reactor is calculated with the Serpent reactor physics code and, using this multiplication coefficient, the amount of pebbles required for the critical load of the reactor. The multiplication coefficient is calculated using pebble beds produced with the discrete element method and three different material libraries in order to compare the results. The received results are lower than those from measured at the experimental reactor and somewhat lower than those gained with other codes in earlier studies.