Subshifts, MSO Logic, and Collapsing Hierarchies

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014, published by Springe

The Monadic Quantifier Alternation Hierarchy over Grids and Pictures

The subject of this paper is monadic second-order logic over two-dimensional grids. We give a game-theoretical proof for the strictness of the monadic second-order quantifier alternation hierarchy over grids. Additionally, we can show that monadic secondorder logic over coloured grids is expressive enough to define complete problems for each level of the polynomial time hierarchy. 1 Introduction Grids are "finite graphs with two edge relations" whose elements are arranged as the elements of a matrix. The two edge relations are successor relations connecting each element with the proper element in the following row, respectively column. Pictures are coloured grids, i.e. grids with some additional unary relations, and can be viewed as "two-dimensional words". Monadic second-order logic is the fragment of second-order logic in which second-order quantifiers may only range over monadic predicates, i.e. over sets. \Sigma 1 k is the set of all secondorder formulas having a prefix of ..

Modal Fragments of Second-Order Logic

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 exactly captures the expressivity of second-order logic