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Marx, Maarten; Venema, Yde
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After seeing so many multidimensional modal logics, the reader might wonder if not every modal logic is (or can be seen as) a multidimensional one. The answer is in some sense positive. We will show that for every modal similarity type S, the basic derivation system K_{S} is strongly sound and complete with respect to a multidimensional class of frames. The dimension and the interpretation of the modalities depends only on the type S.
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Marx, Maarten; Venema, Yde
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This chapter contains a technical introduction to the world of multidimensional modal logics. We will treat some relatively simple logics with a twodimensional semantics. In section 2.1, we introduce the family of modal operators we are going to study, with their twodimensional semantics. In sections 2.2 and 2.3, we study twodimensional modal logic with unary operators. These sections can be seen as an appetizer for the αdimensional case which is treated in chapter 5. Section 2.4 deals with the modal logic of composition. This section is an introduction to chapter 3, which is completely devoted to logics with composition as their main connective. Section 2.5, finally, is about twodimensional tense logic, a subject which is taken up again in chapter 4. We conclude this chapter with some historical notes on the logics described here.
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Marx, Maarten; Venema, Yde
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In other chapters we have seen or will see how logics from various origins can be treated in the unifying framework of multidimensional modal logic. In this chapter we will apply the same methodology to a very familiar logical system, namely that of firstorder logic itself. Our modal analysis of the semantics of firstorder logic naturally leads to the multidimensional perspective, and within this approach we will encounter some unexpected mysteries such as decidable versions of the predicate calculus.
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By
Marx, Maarten; Venema, Yde
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1 Citations
We start with informally defining the subject matter of this book: multidimensional modal logic (MDML). First let us briefly consider what we understand by the notion of “modal logic”. The last decade has seen a development in modal logic towards a more abstract and technical approach. In this perspective of what one might call abstract modal logic, arbitrary relational structures can be seen as models for an (extended) modal language: any relation is a potential accessibility relation of some suitably defined modal operator. As the essentially modal aspect of the framework one could point out that the mechanism for evaluating formulas forces certain moves along the accessibility relations. Thus, for instance quantification over a model is restricted to an “accessible” part of the structure.
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By
Marx, Maarten; Venema, Yde
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In this chapter we show how one can treat temporal logics of periods in the style of twodimensional modal logic, looking at intervals as points in a plane. In this section we first give a general introduction to periodbased temporal logics, and then proceed with a more technical part motivating the treatment of interval logic as twodimensional modal logic.
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By
Marx, Maarten; Venema, Yde
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In this chapter, we continue the study of twodimensional frames. Here we look at these frames from a slightly different perspective, namely apart from taking the states in the twodimensional frames to be just pairs (u,v), we will view them as arrows leading from u to v. We will study a similarity type which, interpreted on squares, is very expressive. This similarity type consists of the following three modalities, a dyadic ○ a monadic ⊗ and a constant ι δ. All of these were discussed before; we recall their definitions on squares,
$$\begin{array}{*{20}{l}}
{\mathfrak{M}, (u,\upsilon ) \Vdash \varphi \circ \psi }&{\mathop \Leftrightarrow \limits^{def} }&{(\exists w) : \mathfrak{M}, (u,w) \Vdash \varphi \& \mathfrak{M},(w,\upsilon ) \Vdash \psi } \\
{\mathfrak{M},(u,\upsilon ) \Vdash \otimes \varphi }&{\mathop \Leftrightarrow \limits^{def} }&{\mathfrak{M},(\upsilon ,u) \Vdash \varphi } \\
{\mathfrak{M},(u,\upsilon ) \Vdash \iota \delta }&{\mathop \Leftrightarrow \limits^{def} }&{u = \upsilon .}
\end{array}$$
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By
Marx, Maarten
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1 Citations
The complexity of the satisfiability problems of various arrow logics and cylindric modal logics is determined. As is well known, relativising these logics makes them decidable. There are several parameters that can be set in such a relativisation. We focus on the following three: the number of variables involved, the similarity type and the kind of relativised models considered. The complexity analysis shows the importance and relevance of these parameters.
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By
Schuth, Anne; Marx, Maarten
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2 Citations
We introduce two metrics aimed at evaluating systems that select facetvalues for a faceted search interface. Facetvalues are the values of metadata fields in semistructured data and are commonly used to refine queries. It is often the case that there are more facetvalues than can be displayed to a user and thus a selection has to be made. Our metrics evaluate these selections based on binary relevant assessments for the documents in a collection. Both our metrics are based on Normalized Discounted Cumulated Gain, an often used Information Retrieval metric.
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By
Marx, Maarten; Mikulás, Szabolcs
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3 Citations
We consider the problem of the product finite model property for binary products of modal logics. First we give a new proof for the product finite model property of the logic of products of Kripke frames, a result due to Shehtman. Then we modify the proof to obtain the same result for logics of products of Kripke frames satisfying any combination of seriality, reflexivity and symmetry. We do not consider the transitivity condition in isolation because it leads to infinity axioms when taking products.
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Azarbonyad, Hosein; Saan, Ferron; Dehghani, Mostafa; Marx, Maarten; Kamps, Jaap
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1 Citations
Text interestingness is a measure of assessing the quality of documents from users’ perspective which shows their willingness to read a document. Different approaches are proposed for measuring the interestingness of texts. Most of these approaches suppose that interesting texts are also topically diverse and estimate interestingness using topical diversity. In this paper, we investigate the relation between interestingness and topical diversity. We do this on the Dutch and Canadian parliamentary proceedings. We apply an existing measure of interestingness, which is based on structural properties of the proceedings (eg, how much interaction there is between speakers in a debate). We then compute the correlation between this measure of interestingness and topical diversity.
Our main findings are that in general there is a relatively low correlation between interestingness and topical diversity; that there are two extreme categories of documents: highly interesting, but hardly diverse (focused interesting documents) and highly diverse but not interesting documents. When we remove these two extreme types of documents there is a positive correlation between interestingness and diversity.
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By
Bezhanishvili, Nick; Marx, Maarten
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3 Citations
We show that every proper normal extension of the bimodal system S5^{2} has the polysize model property. In fact, to every proper normal extension L of S5^{2} corresponds a natural number b(L)  the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every Lconsistent formula ϕ is satisfiable on an Lframe whose universe is bounded by P(ϕ), where ϕ denotes the number of subformulas of ϕ. It is shown that this bound is optimal.
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By
Hoogland, Eva; Marx, Maarten
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7 Citations
The guarded fragment (GF) was introduced by Andréka, van Benthem and Németi as a fragment of first order logic which combines a great expressive power with nice, modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. Slightly generalizing the admissible relativizations yields the packed fragment (PF). In this paper we investigate interpolation and definability in these fragments. We first show that the interpolation property of first order logic fails in restriction to GF and PF. However, each of these fragments turns out to have an alternative interpolation property that closely resembles the interpolation property usually studied in modal logic. These results are strong enough to entail the Beth definability property for GF and PF. Even better, every guarded or packed finite variable fragment has the Beth property. For interpolation, we characterize exactly which finite variable fragments of GF and PF enjoy this property.
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By
Dehghani, Mostafa; Azarbonyad, Hosein; Kamps, Jaap; Marx, Maarten
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2 Citations
There is an increasing volume of semantically annotated data available, in particular due to the emerging use of knowledge bases to annotate or classify dynamic data on the web. This is challenging as these knowledge bases have a dynamic hierarchical or graph structure demanding robustness against changes in the data structure over time. In general, this requires us to develop appropriate models for the hierarchical classes that capture all, and only, the essential solid features of the classes which remain valid even as the structure changes. We propose hierarchical significant words language models of textual objects in the intermediate levels of hierarchies as robust models for hierarchical classification by taking the hierarchical relations into consideration. We conduct extensive experiments on richly annotated parliamentary proceedings linking every speech to the respective speaker, their political party, and their role in the parliament. Our main findings are the following. First, we define hierarchical significant words language models as an iterative estimation process across the hierarchy, resulting in tiny models capturing only well grounded text features at each level. Second, we apply the resulting models to party membership and party position classification across time periods, where the structure of the parliament changes, and see the models dramatically better transfer across time periods, relative to the baselines.
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