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Hybrid Logics: Characterization, Interpolation and Complexity
 Journal of Symbolic Logic
, 1999
"... Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We sho ..."
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Cited by 101 (35 self)
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Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of EhrenfeuchtFrasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rstorder logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...
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 Logic of Correct Description
 Advances in Intensional Logic
, 1995
"... Austin's theory of truth is formulated in terms of a relation of correct description holding between a sentence and a situation. A recursive denition of correct description is provided for rstorder languages containing terms denoting situations and a predicate denoting correctdescription. We exam ..."
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Cited by 23 (1 self)
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Austin's theory of truth is formulated in terms of a relation of correct description holding between a sentence and a situation. A recursive denition of correct description is provided for rstorder languages containing terms denoting situations and a predicate denoting correctdescription. We examine a very strong logic of situations, by restricting our attention to situated consequence between descriptions of omniscient situations, arguing that weaker logics may be obtained using standard methods. Rules of natural deduction for the logic are introduced by way of examples of natural reasoning using spatial indexicals. Finally, a Gentzenstyle sequent calculus is oered. Keywords: Austin, truth, correct description, situation, indexical, natural deduction, sequent calculus. 1 Truth and Correct Description A basic tenet of situation semantics ([BP83, BE87, Bar89]), deriving from Austin's theory of truth ([Aus50]), is that every statement is about a situation. To make a statement by...
Hybrid Completeness
 Logic Journal of the IGPL
, 1998
"... In this paper we discuss two hybrid languages, L(8) and L(#), and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it ..."
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Cited by 13 (7 self)
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In this paper we discuss two hybrid languages, L(8) and L(#), and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it is possible to `label' states by using 8 and # to bind special state variables. This paper explores the consequences of hybridization for completeness. As we shall show, the challenge is to blend the modal idea of canonical models with the classical idea of witnessed maximal consistent sets. The languages L(8) and L(#) provide us with two extreme examples of the issues involved. In the case of L(8), we can combine these ideas relatively straightforwardly with the aid of analogs of the Barcan axioms coupled with a modal theory of labeling . In the case of L(#), on the other hand, although we can still formulate a theory of labeling, the Barcan analogs are not valid. We show how to o...
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...
Internalization: The Case of Hybrid Logics
, 2001
"... A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unattractive features are removed ..."
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Cited by 7 (0 self)
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A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unattractive features are removed onebyone, until the final vestiges of the metalanguage can be set aside to reveal a fully internalized calculus. The techniques are quite general and can be applied to a wide range of hybrid and modal logics.
Hybrid Logic is the Bounded Fragment of First Order Logic
, 1999
"... Hybrid languages are extended modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT99]) has focussed on a more constrained system called H(#; @). The purpose of the present paper is ..."
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Cited by 6 (0 self)
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Hybrid languages are extended modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT99]) has focussed on a more constrained system called H(#; @). The purpose of the present paper is to show in detail that H(#; @) is a modally natural system. We study its expressivity, and provide both model theoretic characterizations (via a restricted notion of EhrenfeuchtFraisse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result is that H(#; @) corresponds precisely to the firstorder fragment which is invariant for generated submodels.
Situated Consequence in Elementary Situation Theory
, 1994
"... this paper to do so. Nonetheless, it seems fairly clear that they have more of a pragmatic character than descriptive conventions. For example, a fairly universal demonstrative convention might be that people make true statements most of the time. This convention, if adhered to, can be enough to ide ..."
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Cited by 5 (2 self)
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this paper to do so. Nonetheless, it seems fairly clear that they have more of a pragmatic character than descriptive conventions. For example, a fairly universal demonstrative convention might be that people make true statements most of the time. This convention, if adhered to, can be enough to identify which situations are being described. Nonlinguistic gestures and the mutual beliefs of speaker and hearer may also be relevant.
Logic Journal of the IGPL 8:3
, 2000
"... Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm ..."
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Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is wellsuited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSpace.We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decis...