Lindströmov teorem za modalnu logiku

master's thesis

• Author: Jakov Krunić
• Mentor: Mladen Vuković (mentor)
• Mentor: Tin Perkov (sumentor)
• Committee member: Mladen Vuković (predsjednik povjerenstva)
• Committee member: Tin Perkov (član povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Lindströmovi teoremi karakteriziraju logiku prvog reda i predstavljaju temelj apstraktne teorije modela. Oni osiguravaju posebno mjesto logici prvog reda među svim logičkim sustavima. U ovom diplomskog radu prikazuju se analogni rezultati u kontekstu modalnih logika. Najvažniji pojam koji proučavamo jest apstraktna modalna logika. Glavni cilj ovog diplomskog rada je dokazati analogon Lindströmovog teorema o karakterizaciji modalnih logika. Rad je podijeljen u dva poglavlja. Cilj prvog poglavlja je opisati neke modalne sisteme. Najprije se bavimo sintaksom i semantikom modalne logike. Zatim uvodimo pojam osnovne modalne logike. Proučavamo generirane podmodele i bisimulacije te navodimo njihova svojstva. Nakon toga opisujemo dvije poznate modalne logike: propozicionalnu dinamičku logiku i linearnu temporalnu logiku. U drugom poglavlju definiramo pojmove apstraktne modalne logike i proširenja osnovne modalne logike. Zatim uvodimo apstraktne modalne logike konačnog stupnja. Detaljno dokazujemo Lindströmov teorem za modalnu logiku koji kaže da je osnovna modalna logika najjača apstraktna modalna logika konačnog stupnja. Na kraju navodimo i varijantu Lindströmovog teorema za modalnu logiku koristeći pojmove relativizacije i kompaktnosti..

Abstract (english)

Lindström's theorems characterize first-order logic and form the basis of abstract model theory. They provide a special place for first-order logic among all logical systems. This thesis presents analogous results in the context of modal logics. The most important concept we study is abstract modal logic. The main goal of this thesis is to prove an analogue of Lindström's theorem on the characterization of modal logics. The paper is divided into two chapters. The aim of the first chapter is to describe some modal systems. We first deal with the syntax and semantics of modal logic. Then we introduce the notion of basic modal logic. We study the generated submodels and bisimulations and list their properties. We then describe two known modal logics: propositional dynamic logic and linear temporal logic. In the second chapter, we define the concepts of abstract modal logic and extensions of basic modal logic. We then introduce abstract modal logics of finite degree. We prove in detail Lindström's theorem for modal logic which says that basic modal logic is the strongest abstract modal logic of finite degree. Finally, we list a variant of Lindström's theorem for modal logic using the terms relativization and compactness.

• Keywords: apstraktna teorija modela; apstraktna modalna logika
• Keywords (english): abstract model theory; abstract modal logic
• Language: croatian
• URN:NBN: urn:nbn:hr:217:499656
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-11-25 13:53:36