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  1. Kerkko Luosto (2012). On Vectorizations of Unary Generalized Quantifiers. Archive for Mathematical Logic 51 (3-4):241-255.
    Vectorization of a class of structures is a natural notion in finite model theory. Roughly speaking, vectorizations allow tuples to be treated similarly to elements of structures. The importance of vectorizations is highlighted by the fact that if the complexity class PTIME corresponds to a logic with reasonable syntax, then it corresponds to a logic generated via vectorizations by a single generalized quantifier (Dawar in J Log Comput 5(2):213–226, 1995). It is somewhat surprising, then, that there have been few systematic (...)
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  2. Kerkko Luosto (2000). Hierarchies of Monadic Generalized Quantifiers. Journal of Symbolic Logic 65 (3):1241-1263.
    A combinatorial criterium is given when a monadic quantifier is expressible by means of universe-independent monadic quantifiers of width n. It is proved that the corresponding hierarchy does not collapse. As an application, it is shown that the second resumption (or vectorization) of the Hartig quantifier is not definable by monadic quantifiers. The techniques rely on Ramsey theory.
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  3. Lauri Hella, Phokion G. Kolaitis & Kerkko Luosto (1997). How to Define a Linear Order on Finite Models. Annals of Pure and Applied Logic 87 (3):241-267.
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  4. Lauri Hella, Phokion G. Kolaitis & Kerkko Luosto (1996). Almost Everywhere Equivalence of Logics in Finite Model Theory. Bulletin of Symbolic Logic 2 (4):422-443.
    We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (C)=1 (...)
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  5. Lauri Hella, Kerkko Luosto & Jouko Väänänen (1996). The Hierarchy Theorem for Generalized Quantifiers. Journal of Symbolic Logic 61 (3):802-817.
    The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...)
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  6. Lauri Hella & Kerkko Luosto (1995). Finite Generation Problem and N-Ary Quantifiers. In. In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers. 63--104.
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  7. Lauri Hella & Kerkko Luosto (1992). The Beth-Closure of L(Qα) is Not Finitely Generated. Journal of Symbolic Logic 57 (2):442 - 448.
    We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).
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  8. Lauri Hella & Kerkko Luosto (1992). The Beth-Closure of $Mathscr{L}(Q_alpha)$ is Not Finitely Generated. Journal of Symbolic Logic 57 (2):442-448.
    We prove that if $\aleph_\alpha$ is uncountable and regular, then the Beth-closure of $\mathscr{L}_{\omega\omega}(Q_\alpha)$ is not a sublogic of $\mathscr{L}_{\infty\omega}(\mathbf{Q}_n)$, where $\mathbf{Q}_n$ is the class of all $n$-ary generalized quantifiers. In particular, $B(\mathscr{L}_{\omega\omega}(Q_\alpha))$ is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set $\mathbf{Q}$ of Lindstrom quantifiers such that $B(\mathscr{L}_{\omega\omega}(Q_\alpha)) \leq \mathscr{L}_{\omega\omega}(\mathbf{Q})$.
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