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The Categorial FineStructure of Natural Language
, 2003
"... Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, i ..."
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Cited by 3 (1 self)
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Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, in a light examplebased manner, where this elegant logical paradigm stands when confronted with the wear and tear of reality. Starting from a brief history of the Lambek tradition since the 1980s, we discuss three main issues: (a) the fit of the lambda calculus engine to characteristic semantic structures in natural language, (b) the coexistence of the original typetheoretic and more recent modal interpretations of categorial logics, and (c) the place of categorial grammars in the complex total architecture of natural language, which involves  amongst others  mixtures of interpretation and inference.
Expressing and Verifying Temporal and Structural Properties of Mobile Agents
, 2006
"... Logics for expressing properties of Petri hypernets, a visual formalism for modelling mobile agents, are proposed. Two classes of properties are of interest—the temporal evolution of agents and their structural correlation. In particular, we investigate how the classes can be combined into a logic ..."
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Logics for expressing properties of Petri hypernets, a visual formalism for modelling mobile agents, are proposed. Two classes of properties are of interest—the temporal evolution of agents and their structural correlation. In particular, we investigate how the classes can be combined into a logic capable of expressing the dynamic evolution of the structural correlation. The problem of model checking properties of a class of the logic on Petri hypernets is shown to be PSPACEcomplete.
On the Mosaic Method for ManyDimensional Modal Logics: A Case Study Combining Tense and Modal Operators
"... Abstract. We present an extension of the mosaic method aimed at capturing manydimensional modal logics. As a proofofconcept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal ” S5like modality. We show that the existence of a model for a g ..."
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Abstract. We present an extension of the mosaic method aimed at capturing manydimensional modal logics. As a proofofconcept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal ” S5like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness for the corresponding Hilbertstyle axiomatization, but also in the development of a mosaicbased tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branchingtime logics, and present for them some partial results, such as a nonanalytic version of the tableau system. 1
The Complexity of Decomposing Modal and FirstOrder Theories
"... We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far the best kn ..."
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We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far the best known lower bound is NEXPhardness shown by Marx and Mikulás in 2001. Our first main result closes this complexity gap: We show that satisfiability inK2 is nonelementary. More precisely, we prove that it is kNEXPcomplete, where k is the switching depth (the minimal modal rank among the two dimensions) of the input formula, hereby solving a conjecture of Marx and Mikulás. Using our lowerbound technique allows us to derive also nonelementary lower bounds for the twodimensional modal logics K4 ×K and S52 ×K for which only elementary lower bounds were previously known. Moreover, we apply our technique to prove nonelementary lower bounds for the sizes of FefermanVaught decompositions with respect to product for any decomposable logic that is at least as expressive as unimodal K, generalizing a recent result by the first author and Lin. For the threevariable fragment FO3 of firstorder logic, we obtain the following immediate corollaries: (i) the size of FefermanVaught decompositions with respect to disjoint sum are inherently nonelementary and (ii) equivalent formulas in Gaifman normal form are inherently nonelementary.
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"... Abstract Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper con ..."
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Abstract Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, in a light examplebased manner, where this elegant logical paradigm stands when confronted with the wear and tear of reality. Starting from a brief history of the Lambek tradition since the 1980s, we discuss three main issues: (a) the fit of the lambda calculus engine to characteristic semantic structures in natural language, (b) the coexistence of the original typetheoretic and more recent modal interpretations of categorial logics, and (c) the place of categorial grammars in the complex total architecture of natural language, which involves amongst others mixtures of interpretation and inference. 1 From Montague Grammar to Categorial Grammar Logic and linguistics have had lively connections from Antiquity right until today (GAMUT 1991). A recurrent theme in this history is the categorial structure of language and ontology, from Aristotle's grammatical categories to Russell's theory of types in the foundations of mathematics. Further bridges were thrown as logic and
Finite Satisfiability of Modal Logic over Horn Definable Classes of Frames
"... Modal logic plays an important role in various areas of computer science, including verification and knowledge representation. In many practical applications it is natural to consider some restrictions of classes of admissible frames. Traditionally classes of frames are defined by modal axioms. Howe ..."
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Modal logic plays an important role in various areas of computer science, including verification and knowledge representation. In many practical applications it is natural to consider some restrictions of classes of admissible frames. Traditionally classes of frames are defined by modal axioms. However, many important classes of frames, e.g. the class of transitive frames or the class of Euclidean frames, can be defined in a more natural way by firstorder formulas. In a recent paper it was proved that the satisfiability problem for modal logic over the class of frames defined by a universally quantified, firstorder Horn formula is decidable. In this paper we show that also the finite satisfiability problem for modal logic over such classes is decidable. Keywords: modal logic, decidability, finite satisfiability 1
On Modal Logics of Hamming Spaces
"... With a set S of words in an alphabet A we associate the frame (S, H), where sHt iff s and t are words of the same length and h(s, t) = 1 for the Hamming distance h. We investigate some unimodal logics of these frames. We show that if the length of words n is fixed and finite, the logics are closely ..."
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With a set S of words in an alphabet A we associate the frame (S, H), where sHt iff s and t are words of the same length and h(s, t) = 1 for the Hamming distance h. We investigate some unimodal logics of these frames. We show that if the length of words n is fixed and finite, the logics are closely related to manydimensional products S5 n, so in many cases they are undecidable and not finitely axiomatizable. The relation H can be extended to infinite sequences. In this case we prove some completeness theorems characterizing the wellknown modal logics DB and TB in terms of the Hamming distance.