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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....
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 34 (10 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
A Logic of Reachable Patterns in Linked DataStructures
, 2007
"... We define a new decidable logic for expressing and checking invariants of programs that manipulate dynamicallyallocated objects via pointers and destructive pointer updates. The main feature of this logic is the ability to limit the neighborhood of a node that is reachable via a regular expression ..."
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Cited by 25 (3 self)
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We define a new decidable logic for expressing and checking invariants of programs that manipulate dynamicallyallocated objects via pointers and destructive pointer updates. The main feature of this logic is the ability to limit the neighborhood of a node that is reachable via a regular expression from a designated node. The logic is closed under boolean operations (entailment, negation) and has a finite model property. The key technical result is the proof of decidability. We show how to express preconditions, postconditions, and loop invariants for some interesting programs. It is also possible to express properties such as disjointness of datastructures, and lowlevel heap mutations. Moreover, our logic can express properties of arbitrary datastructures and of an arbitrary number of pointer fields. The latter provides a way to naturally specify postconditions that relate the fields on the entry of a procedure to the field on the exit of a procedure. Therefore, it is possible to use the logic to automatically prove partial correctness of programs performing lowlevel heap mutations.
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 22 (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).
On the complexity of hybrid logics with binders
 Proc. of the 19th CSL, 2005, LNCS 3634 (2005
, 2005
"... Abstract. Hybrid logic refers to a group of logics lying between modal and firstorder logic in which one can refer to individual states of the Kripke structure. In particular, the hybrid logic HL(@, ↓) is an appealing extension of modal logic that allows one to refer to a state by means of the give ..."
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Cited by 21 (0 self)
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Abstract. Hybrid logic refers to a group of logics lying between modal and firstorder logic in which one can refer to individual states of the Kripke structure. In particular, the hybrid logic HL(@, ↓) is an appealing extension of modal logic that allows one to refer to a state by means of the given names and to dynamically create new names for a state. Unfortunately, as for the richer firstorder logic, satisfiability for the hybrid logic
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 21 (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.
Hybridizing a logical framework
 In International Workshop on Hybrid Logic 2006 (HyLo 2006), Electronic Notes in Computer Science
, 2006
"... The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good r ..."
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Cited by 20 (1 self)
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The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good representations of state change. We describe and argue for the usefulness of an extension of LF by features inspired by hybrid logic, which has several benefits. For one, it shows how linear logic features can be decomposed into primitive operations manipulating abstract resource labels. More importantly, it makes it possible to realize a metalogical framework capable of reasoning about stateful deductive systems encoded in the style familiar from prior work with LLF, taking advantage of familiar methodologies used for metatheoretic reasoning in LF.Acknowledgments From the very first computer science course I took at CMU, Frank Pfenning has been an exceptional teacher and mentor. For his patience, breadth of knowledge, and mathematical good taste I am extremely thankful. No less do I owe to the other two major contributors to my programming languages
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 16 (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
Tableaux for Quantified Hybrid Logic
 METHODS FOR MODALITIES 2, WORKSHOP PROCEEDINGS, NOVEMBER 2930, 2001. ILLC
, 2002
"... We present a (sound and complete) tableau calculus for Quantified Hybrid Logic (QHL). QHL is an extension of orthodox quantified modal logic: as well as the usual # and # modalities it contains names for (and variables over) states, operators @s for asserting that a formula holds at a named state ..."
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Cited by 16 (4 self)
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We present a (sound and complete) tableau calculus for Quantified Hybrid Logic (QHL). QHL is an extension of orthodox quantified modal logic: as well as the usual # and # modalities it contains names for (and variables over) states, operators @s for asserting that a formula holds at a named state, and a binder that binds a variable to the current state. The firstorder component contains equality and rigid and non rigid designators. As far as we are aware, ours is the first tableau system for QHL. Completeness
Hybrid logics on linear structures: Expressivity and complexity
 In Proceedings TIME 2003
, 2003
"... We investigate expressivity and complexity of hybrid logics on linear structures. Hybrid logics are an enrichment of modal logics with certain firstorder features which are algorithmically well behaved. Therefore, they are well suited for the specification of certain properties of computational sys ..."
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Cited by 14 (4 self)
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We investigate expressivity and complexity of hybrid logics on linear structures. Hybrid logics are an enrichment of modal logics with certain firstorder features which are algorithmically well behaved. Therefore, they are well suited for the specification of certain properties of computational systems. We show that hybrid logics are more expressive than usual modal and temporal logics on linear structures, and exhibit a hierarchy of hybrid languages. We determine the complexities of the satisfiability problem for these languages and define an existential fragment of hybrid logic for which satisfiability is still NPcomplete. Finally, we examine the linear time model checking problem for hybrid logics and its complexity. 1