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Internalizing Labelled Deduction
 Journal of Logic and Computation
, 2000
"... This paper shows how to internalize the Kripke satisfaction denition using the basic hybrid language, and explores the proof theoretic consequences of doing so. As we shall see, the basic hybrid language enables us to transfer classic Gabbaystyle labelled deduction methods from the metalanguage to ..."
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Cited by 74 (20 self)
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This paper shows how to internalize the Kripke satisfaction denition using the basic hybrid language, and explores the proof theoretic consequences of doing so. As we shall see, the basic hybrid language enables us to transfer classic Gabbaystyle labelled deduction methods from the metalanguage to the object language, and to handle labelling discipline logically. This internalized approach to labelled deduction links neatly with the Gabbaystyle rules now widely used in modal Hilbertsystems, enables completeness results for a wide range of rstorder denable frame classes to be obtained automatically, and extends to many richer languages. The paper discusses related work by Jerry Seligman and Miroslava Tzakova and concludes with some reections on the status of labelling in modal logic. 1 Introduction Modern modal logic revolves around the Kripke satisfaction relation: M;w ': This says that the model M satises (or forces, or supports) the modal formula ' at the state w in M....
The Computational Complexity of Hybrid Temporal Logics
 Logic Journal of the IGPL
, 2000
"... In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specic times possible, they remove the most serious obstac ..."
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Cited by 55 (11 self)
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In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specic times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics. In this paper we analyze the complexity of the satisability problem of a number of hybrid temporal logics: the basic hybrid language over transitive frames; nominal tense logic over transitive frames, strict total orders, and transitive trees; nominal Until logic; and referential interval logic. We discuss the eects of including nominals, the @ operator, the somewhere modality E, and the dierence operator D. Adding nominals to tense logic leads for several frame{classes to an increase in complexity of the satisability pro...
Normal Multimodal Logics With Interaction Axioms: A Tableau Calculus and Some (Un)Decidability Results
, 2000
"... In this paper we present a prefixed analytic tableau calculus for a wide class of normal multimodal logics; the calculus can deal in a uniform way with any logic in this class. To achieve this goal, we use a prefixed tableau calculus a la Fitting, where we explicitly represent accessibility relation ..."
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Cited by 18 (8 self)
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In this paper we present a prefixed analytic tableau calculus for a wide class of normal multimodal logics; the calculus can deal in a uniform way with any logic in this class. To achieve this goal, we use a prefixed tableau calculus a la Fitting, where we explicitly represent accessibility relations between worlds by means of a graph and we use the characterizing axioms as rewriting rules. Such rules create new paths among worlds in the countermodel construction. The prefixed tableau method is, then, used to prove (un)decidability results about certain classes of multimodal logics.
Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?
, 1999
"... . We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule ..."
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Cited by 15 (4 self)
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. We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cutformulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...
Labelled Modal Logics: Quantifiers
, 1998
"... . In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logic ..."
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Cited by 15 (2 self)
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. In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework. 1 Introduction Motivation Modal logic is an active area of research in computer science and artificial intelligence: a large number of modal logics have been studied and new ones are frequently proposed. Each new log...
Bringing them all Together
, 2001
"... this paper, Jerry Seligman takes us on an interesting journey. The satisfaction denition of most modal operators is specied in terms of rstorder conditions. Hence we can always obtain a complete calculus for the basic logic characterizing any collection of such operators by appealing to a calculus ..."
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Cited by 14 (0 self)
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this paper, Jerry Seligman takes us on an interesting journey. The satisfaction denition of most modal operators is specied in terms of rstorder conditions. Hence we can always obtain a complete calculus for the basic logic characterizing any collection of such operators by appealing to a calculus which is complete for the full rstorder language. Seligman shows here that by making use of the expressiveness provided by the hybrid apparatus, we can, step by step, transform a rstorder sequent calculus into an internalized sequent calculus specically tailored for a particular hybrid fragment
Fibring Labelled Deduction Systems
 Journal of Logic and Computation
, 2002
"... We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial ..."
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Cited by 13 (9 self)
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We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.
A New Method for Bounding the Complexity of Modal Logics
, 1997
"... . We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility r ..."
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Cited by 12 (2 self)
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. We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with bounded space requirements. As examples we give O(n log n) space procedures for the modal logics K and T. 1 Introduction We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as cutfree labelled sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with space requirements that are easily bounded. As examples we give O(n log n) space decision procedures f...
Natural Deduction for NonClassical Logics
, 1996
"... We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke m ..."
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Cited by 11 (3 self)
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We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.
Representation, Reasoning, and Relational Structures: a Hybrid Logic Manifesto
 Logic Journal of IGPL
, 2000
"... This paper is about the good side of modal logic, the bad side of modal logic, and how hybrid logic takes the good and xes the bad. In essence, modal logic is a simple formalism for working with relational structures (or multigraphs) . But modal logic has no mechanism for referring to or reasoning ..."
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Cited by 11 (1 self)
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This paper is about the good side of modal logic, the bad side of modal logic, and how hybrid logic takes the good and xes the bad. In essence, modal logic is a simple formalism for working with relational structures (or multigraphs) . But modal logic has no mechanism for referring to or reasoning about the individual nodes in such structures, and this lessens its eectiveness as a representation formalism. In their simplest form, hybrid logics are upgraded modal logics in which reference to individual nodes is possible. But hybrid logic is a rather unusual modal upgrade. It pushes one simple idea as far as it will go: represent all information as formulas. This turns out to be the key needed to draw together a surprisingly diverse range of work (for example, feature logic, description logic and labelled deduction) . Moreover, it displays a number of knowledge representation issues in a new light, notably the importance of sorting. Keywords: Labelled deduction, description logic, f...