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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 108 (37 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...
A roadmap on complexity for hybrid logics
 Computer Science Logic, number 1683 in LNCS
, 1999
"... Abstract. Hybrid languages are extended modal languages which can refer to (or even quantify over) states. Such languages are better behaved proof theoretically than ordinary modal languages for they internalize the apparatus of labeled deduction. Moreover, they arise naturally in a variety of appli ..."
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Cited by 98 (17 self)
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Abstract. Hybrid languages are extended modal languages which can refer to (or even quantify over) states. Such languages are better behaved proof theoretically than ordinary modal languages for they internalize the apparatus of labeled deduction. Moreover, they arise naturally in a variety of applications, including description logic and temporal reasoning. Thus it would be useful to have a map of their complexitytheoretic properties, and this paper provides one. Our work falls into two parts. We first examine the basic hybrid language and its multimodal and tense logical cousins. We show that the basic hybrid language (and indeed, multimodal hybrid languages) are no more complex than ordinary unimodal logic: all have pspacecomplete Ksatisfiability problems. We then show that adding even one nominal to tense logic raises complexity from pspace to exptime. In the second part we turn to stronger hybrid languages in which it is possible to bind nominals. We prove a general expressivity result showing that even the weak form of binding offered by the ↓ operator easily leads to undecidability.
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 62 (13 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...
Hybrid Logics
"... This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur ..."
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Cited by 58 (18 self)
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This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur
Hybrid languages and temporal logic
 Logic J. IGPL
, 1999
"... Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our ..."
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Cited by 44 (16 self)
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Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the rst technical, the second conceptual. First, we showthathybridization gives rise to wellbehaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full rstorder expressive strength, is demonstrated for a weaker local language capable of de ning the Until operator � we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. 1
Termination for hybrid tableaus
 Journal of Logic and Computation
"... Abstract. This article extends and improves work on tableaubased decision methods for hybrid logic by Bolander and Braüner [5]. Their paper gives tableaubased decision procedures for basic hybrid logic (with unary modalities) and the basic logic extended with the global modality. All their proof p ..."
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Abstract. This article extends and improves work on tableaubased decision methods for hybrid logic by Bolander and Braüner [5]. Their paper gives tableaubased decision procedures for basic hybrid logic (with unary modalities) and the basic logic extended with the global modality. All their proof procedures make use of loopchecks to ensure termination. Here we take a closer look at termination for hybrid tableaus. We cover both types of system used in hybrid logic: prefixed tableaus and internalised tableaus. We first treat prefixed tableaus. We prove a termination result for the basic language (with nary operators) that does not involve loopchecks. We then successively add the global modality and nary inverse modalities, show why various different types of loopcheck are required in these cases, and then reprove termination. Following this we consider internalised tableaus. At first sight, such systems seem to be more complex. However we define a internalised system which terminates without loopchecks. It is simpler than previously known internalised systems (all of which require loopchecks to terminate) and simpler than our prefix systems (no nonlocal side conditions on rules are required).
Resolution in Modal, Description and Hybrid Logic
, 2002
"... We provide a resolutionbased proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybr ..."
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Cited by 25 (9 self)
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We provide a resolutionbased proof procedure for modal, description and hybrid logic that improves on previous proposals in important ways. It avoids translations into large undecidable logics, and works directly on modal, description or hybrid logic formulas instead. In addition, by using the hybrid machinery it avoids the complexities of earlier propositional resolutionbased methods for modal logic. It combines ideas from the method of pre xes used in tableaux, and resolution ideas in such a way that some of the heuristics and optimizations devised in either eld are applicable.
Tableaubased decision procedures for hybrid logic
 Journal of Logic and Computation
, 2005
"... Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is wellknown that various hybrid logics without binders are decidable, but decision procedures are usually not based on tableau systems, a kind of formal proof procedure that lends itself towards ..."
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Cited by 23 (4 self)
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Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is wellknown that various hybrid logics without binders are decidable, but decision procedures are usually not based on tableau systems, a kind of formal proof procedure that lends itself towards computer implementation. In this paper we give four different tableaubased decision procedures for a very expressive hybrid logic including the universal modality; three of the procedures are based on different tableau systems, and one procedure is based on a Gentzen system. The decision procedures make use of socalled loopchecks which is a technique standardly used in connection with tableau systems for other logics, namely prefixed tableau systems for transitive modal logics, as well as prefixed tableau systems for certain description logics. The loopchecks used in our four decision procedures are similar, but the four proof systems on which the procedures are based constitute a spectrum of different systems: prefixed and internalized systems, tableau and Gentzen systems.
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 22 (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...
Pure extensions, proof rules and hybrid axiomatics
 Preliminary proceedings of Advances in Modal Logic (AiML 2004
, 2004
"... We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language ..."
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Cited by 18 (6 self)
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We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language