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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 105 (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 93 (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.
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 77 (21 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 57 (12 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 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 40 (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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Cited by 25 (2 self)
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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).
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 22 (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.
Cutfree Display Calculi for Nominal Tense Logics
 Conference on Tableaux Calculi and Related Methods (TABLEAUX
, 1998
"... . We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Krac ..."
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Cited by 17 (7 self)
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. We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)(C7). Finally, we show a weak Sahlqviststyle theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cutfree display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...
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 restrict ..."
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Cited by 16 (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...
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 15 (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