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Tampere University

1. Kuusisto, Antti. Modal Fragments of Second-Order Logic .

Degree: 2011, Tampere University

Formaalin logiikan tutkimuskohteina ovat erilaiset muodolliset systeemit eli logiikat, joiden avulla voidaan mm. mekanisoida monenlaisia päättelyprosesseja. Eräs modernin formaalin logiikan keskeisistä tutkimusaiheista on modaalilogiikka, jossa perinteisempää logiikkaa laajennetaan nk. modaliteeteilla. Modaliteettien avulla voidaan luoda mitä erilaisimpia formaaleja systeemejä. Modaalilogiikalla onkin huomattava määrä sovelluksia aina tietojenkäsittelytieteestä ja matematiikan sekä fysiikan perusteista filosofiaan ja kielitieteisiin. Väitöskirja keskittyy modaalilogiikan nk. malliteoriaan. Tutkielmassa luokitellaan erilaisia formaalin logiikan systeemejä perustuen siihen, millaisia ominaisuuksia kyseisten systeemien avulla voidaan ilmaista. Mitä korkeampi ilmaisuvoima formaalilla järjestelmällä on, sitä hitaampaa on järjestelmän avulla suoritettava tietokoneellistettu päättely. Tutkielma käsittelee useita modaalilogiikan systeemejä; painopiste on erittäin korkean ilmaisuvoiman omaavien logiikoiden teoriassa. Tarkastelun kohteena olevat kysymykset liittyvät suoraan muuhun modaalilogiikan alan matemaattiseen tutkimukseen. Tutkielmassa mm. esitetään ratkaisu vuodesta 1983 avoinna olleeseen tekniseen kysymykseen koskien nk. toisen kertaluvun propositionaalisen modaalilogiikan alternaatiohierarkiaa.; In this thesis we investigate various fragments of second-order logic that arise naturally in considerations related to modal logic. The focus is on questions related to expressive power. The results in the thesis are reported in four independent but related chapters (Chapters 2, 3, 4 and 5). In Chapter 2 we study second-order propositional modal logic, which is the system obtained by extending ordinary modal logic with second-order quantification of proposition symbols. We show that the alternation hierarchy of this logic is infinite, thereby solving an open problem from the related literature. In Chapter 3 we investigate the expressivity of a range of modal logics extended with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the chapter is that the resulting extension of (a version of) Boolean modal logic can be effectively translated into existential monadic second-order logic. As a corollary we obtain decidability results for multimodal logics over various classes of frames with built-in relations. In Chapter 4 we study the equality-free fragment of existential second-order logic with second-order quantification of function symbols. We show that over directed graphs, the expressivity of the fragment is incomparable with that of first-order logic. We also show that over finite models with a unary relational vocabulary, the fragment is weaker in expressivity than first-order logic. In Chapter 5 we study the extension of polyadic modal logic with unrestricted quantification of accessibility relations and proposition symbols. We obtain a range of results related to various natural fragments of the system. Finally, we establish that this extension of modal logic…

Subjects/Keywords: matematiikka; matemaattinen logiikka; modaalilogiikka; toisen kertaluvun logiikka; malliteoria; mathematics; mathematical logic; modal logic; second-order logic; model theory

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APA (6th Edition):

Kuusisto, A. (2011). Modal Fragments of Second-Order Logic . (Doctoral Dissertation). Tampere University. Retrieved from http://tampub.uta.fi/handle/10024/66794

Chicago Manual of Style (16th Edition):

Kuusisto, Antti. “Modal Fragments of Second-Order Logic .” 2011. Doctoral Dissertation, Tampere University. Accessed July 17, 2019. http://tampub.uta.fi/handle/10024/66794.

MLA Handbook (7th Edition):

Kuusisto, Antti. “Modal Fragments of Second-Order Logic .” 2011. Web. 17 Jul 2019.

Vancouver:

Kuusisto A. Modal Fragments of Second-Order Logic . [Internet] [Doctoral dissertation]. Tampere University; 2011. [cited 2019 Jul 17]. Available from: http://tampub.uta.fi/handle/10024/66794.

Council of Science Editors:

Kuusisto A. Modal Fragments of Second-Order Logic . [Doctoral Dissertation]. Tampere University; 2011. Available from: http://tampub.uta.fi/handle/10024/66794

.