Trougaone norme prioriteta i njihova primena na modeliranje ispunjenja fazi ograničenja

Abstract

<p>Doktorska disertacija pripada oblasti fazi skupova, inteligentnih sistema i baza podataka.<br />Osnovni cilj ove teze je koriˇs&acute;cenje konkretnih rezultate iz oblasti fazi skupova<br />i PFCSP sistema u okviru fazi relacionih baza podataka. U skladu sa tim implementirana<br />je aplikacija koja ilustruje ove primene.<br />U doktorskoj disertaciji su obrad&macr;ene t-norme prioriteta, jer imaju znaˇcajnu ulogu u<br />PFCSP sistemima. T-norme prioriteta su ustvari Schur-konkavne t-norme. Originalni<br />doprinos autora je karakterizacija jedne klase konkavnih t-normi preko nivoskih linija<br />(teorema 2.3).<br />Od inteligentnih sistema prouˇcavani su PFCSP sistemi i specijalne fazi relacione<br />baze podataka. PFCSP (engleski Priority Fuzzy Constraint Satisfaction Problem) su<br />sistemi koji se koriste za odluˇcivanje kada ograniˇcenja mogu imati fazi vrednosti i<br />razliˇcite prioritete. Autor je uveo novi tip PFCSP sistema koji definiˇsu stroˇziji koncept<br />prioriteta.<br />Svaki model zasnovan na klasiˇcnoj logici pretpostavlja dihotomiju, med&macr;utim u realnim<br />sistemima ˇcesto se ne vidi granica izmed&macr;u dve alternative. Ukratko, fazi logika<br />predstavlja fleksibilan sistem koji se koristi kao semantiˇcka osnova za prirodne jezike.<br />Preslikavanja iz jediniˇcnog kvadrata na jediniˇcni interval koja su monotona, komutativna,<br />asocijativna i imaju jediniˇcni element 1 nazivaju se t-norme. Uopˇstenje<br />operatora konjukcije u fazi logici su t-norme.<br />Schur-konkavnost je specijalna vrsta konkavnosti. Za funkciju dve promenjive<br />f kaˇzemo da je Schur-konkavna ako za svako x, y, u, v takvo da x + y = u + v,<br />x &le; u &le; v &le; y vaˇzi: f(u, v) &ge; f(x, y). One se primenjuju u PFCSP sistemima kao<br />operatori koji agregiraju lokalne mere zadovoljenja ograniˇcenja u globalnu meru.<br />CSP (eng. Constraint satisfaction problems) sistemi koriˇs&acute;ceni su za apstrakciju<br />praktiˇcnih problema. Fazifikacijom CSP sistema dobijamo FCSP (eng. Fuzzy constraint satisfaction problems) sisteme kod kojih su ograniˇcenja modelirana kao fazi</p><p>podskupovi domena ili Dekartovog proizvoda nekog podskupa skupa domena. Cˇ esto<br />se deˇsava da ograniˇcenja nemaju isti prioritet, odnosno da je jedno malo &rdquo;vaˇznije od<br />drugog&rdquo;. Zbog toga je prioritet aksiomatski uveden u FSCP sisteme i dobijeni su<br />PFCSP sistemi.<br />Razvojem raˇcunara sredinom devedesetih godina relacioni model postaje standard<br />u bazama podataka. Jedan od nedostataka ovog modela bio je ˇsto nije omogu&acute;cavao<br />neodred&macr;ene i neegzaktne vrednosti podataka. Koriˇs&acute;cenje fazi skupova i fazi logike za<br />proˇsirenje relacionog modela jedan je od naˇcina za ispravljanje ovog nedostatka, ˇsto<br />rezultuje dobijanjem fazi relacionih baza podataka. Oslanjaju&acute;ci se na PFCSP sisteme<br />proˇsiruje se i SQL, ˇsto rezultira PFSQL-om jezikom za postavljanje upita nad fazi<br />bazom podataka kod kojih se pojavljuju uslovi sa razliˇcitim prioritetom.</p>

<p>The scientific fields of this thesis are fuzzy sets, intelligent systems and databases.<br />The goal of the thesis is to use results obtained in the field of fuzzy sets in PFCSP<br />systems and databases. According to that, a software package has been implemented.<br />In the PhD thesis priority t-norms have been studied, since they play an important<br />role in PFCSP systems. Priority t-norms are actually Schur-concave t-norms. The<br />original contribution of the author is the characterization of a class of concave t-norms<br />by their level lines.<br />From the field of intelligent systems PFCSP systems have been studied. PFCSP<br />(Priority Fuzzy Constraint Satisfaction Problem) are used for decision making when<br />the constraints are fuzzy with different priorities. Author had found a new type of<br />PFCSP systems which define a stricter concept of priority.<br />Each model based on classical logic imposes dichotomy. However, the difference<br />between two alternatives is not always clear. Fuzzy logic represents a flexible system<br />used as a semantic base for natural languages.<br />Mappings form the unit square to the unit intervals that are monotone, symmetric,<br />associative and 1 is their unit element are called t-norms. They are a generalization of<br />the conjunction operator from classical logic.<br />Schur-concavity is a special type of concavity. If for a function the following holds<br />for each x, y, u, v such that x+y = u+v, x &le; u &le; v &le; y then f(u, v) &ge; f(x, y), the<br />function is called Schur-concave. They are used in PFCSP systems as operators that<br />aggregate local satisfaction degrees in order to obtain the global satisfaction degree.</p><p>CSP (Constraint satisfaction problems) are used to model practical problems. Fuzzyfication<br />of CSP leads to FCSP (Fuzzy constraint satisfaction problems) systems which<br />allow fuzzy values constraint satisfaction. It is common that the constraints do not<br />have the same priority, thus priority is axiomatically induced into FCSP resulting in<br />PFCSP.<br />With the development of computers in the mid nineties relational model became<br />the standard in the filed of databases. One of its flaws is the lack of representation for<br />incomplete and imprecise information. The use if fuzzy theory in order to expand the<br />relational model is a way to correct this flaw, resulting in fuzzy relational databases.<br />Relying of PFCSP systems SQL is also expanded resulting in PFSQL language which<br />allows different priorities of query conditions.</p>