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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....
Back and Forth Between Modal Logic and Classical Logic
, 1994
"... Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. ..."
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Cited by 30 (3 self)
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Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, kvariable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the Avalues for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...
Dynamic Bits And Pieces
, 1997
"... Arrow Logic remains PSPACEcomplete. Further arrow axioms can easily lead to 36 undecidability. (In private correspondence, Marx has also announced EXPTIME complexity for the original Guarded Fragment, via a reduction to CRS over 'locally cube' models.) Marx and Venema 1996 is a systematic stateo ..."
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Cited by 26 (3 self)
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Arrow Logic remains PSPACEcomplete. Further arrow axioms can easily lead to 36 undecidability. (In private correspondence, Marx has also announced EXPTIME complexity for the original Guarded Fragment, via a reduction to CRS over 'locally cube' models.) Marx and Venema 1996 is a systematic stateoftheart presentation of manydimensional modal logic, including bridges with algebraic logic, as well as many key techniques for dynamic logic, broadly conceived. Ter Meulen 1995 proposes a concise framework for temporal representation in natural language that may be viewed as an alternative dynamification of temporal logic, using an extra, intermediate level of representation. Successive formulas algorithmically generate successive 'dynamic aspect trees', for which there is a notion of 'succesful embedding' into standard temporal models. Valid inference can then be defined as verification of the conclusion by any succesful embedding for the DAT of the premise sequence. This alternative d...
Modal Logic In Two Gestalts
, 1998
"... We develop a translationbased view dual of modal logic as the study of intensional languages that are at the same time interesting expressive and decidable parts of standard logical systems. This tandem approach improves our understanding of modal logic  while at the same time, it extends the ran ..."
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Cited by 7 (3 self)
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We develop a translationbased view dual of modal logic as the study of intensional languages that are at the same time interesting expressive and decidable parts of standard logical systems. This tandem approach improves our understanding of modal logic  while at the same time, it extends the range of modal notions and techniques into broader areas of standard logic. 1 Translation as a Way of Life 1.1 Basic modal logic and the modal fragment of FOL Modal languages as used today can be considered a species of their own, inhabiting the realm of Intensional Logic. But they can also be translated into fragments of standard logical languages, mostly firstorder, sometimes higherorder or infinitary. These translations reflect the truth conditions for modal operators in possible worlds models. The urexample is the basic modal language of possibility and necessity, whose standard translation ST inspired Correspondence Theory (van Benthem 1976, 1985): an existential modality <>p goes to a...
Equational axioms of test algebra
 Computer Science Logic, 11th International Workshop, CSL ’97, volume 1414 of LNCS
, 1997
"... We presentacomplete axiomatization of test algebra ([24, 18, 29]), the twosorted algebraic variant of Propositional Dynamic Logic (PDL, [21, 7]). The axiomatization consists of adding a nite number of equations to any axiomatization of Kleene algebra ([15, 26, 17, 4]) and algebraic translations of ..."
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Cited by 4 (0 self)
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We presentacomplete axiomatization of test algebra ([24, 18, 29]), the twosorted algebraic variant of Propositional Dynamic Logic (PDL, [21, 7]). The axiomatization consists of adding a nite number of equations to any axiomatization of Kleene algebra ([15, 26, 17, 4]) and algebraic translations of the Segerberg ([27]) axioms for PDL. Kleene algebras are not nitely axiomatizable ([25, 6]), so our result does not give us a nite axiomatization of test algebra: in fact, no nite equational axiomatization exists. We alsopresent a singlesorted version of test algebra, using the notion of dynamic negation ([9, 2, 11]), to which the previous results carry over. 1
Quantifiers and Operations on Modalities and Contexts
 KR’98: Principles of Knowledge Representation and Reasoning
, 1997
"... We can reason about theories, as well as in them. Many natural phenomena can be captured by theories, often by introducing a modality, true just of the theory. Rather than treat these phenomena as the sentences true in this modality, we can treat the entire theory or modality as an object. This p ..."
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We can reason about theories, as well as in them. Many natural phenomena can be captured by theories, often by introducing a modality, true just of the theory. Rather than treat these phenomena as the sentences true in this modality, we can treat the entire theory or modality as an object. This point of view was proposed by McCarthy, who proposed calling these reified modalities contexts. Contexts, viewed as theories or modalities, have structural properties, and associated natural structural operations, such as union and intersection. These structural properties are most naturally captured by an algebra, reminiscent of relation or dynamic algebra , a set of operations that can be applied to contexts to form new contexts. We introduce an algebra, reminiscent of relation or dynamic algebra, but which acts on modalities, not relations. This algebra allows us to construct new modalities from simpler modalities. We use the natural correspondence between a modality and the underl...
Safety for Bisimulation in Monadic SecondOrder Logic
 Dept. of Philosophy, Utrecht University
, 1996
"... We characterize those formulas of MSO (monadic secondorder logic) that are safe for bisimulation: formulas defining binary relations such that any bisimulation is also a bisimulation with respect to these defined relations. Every such formula is equivalent to one constructed from ¯ calculus tests ..."
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We characterize those formulas of MSO (monadic secondorder logic) that are safe for bisimulation: formulas defining binary relations such that any bisimulation is also a bisimulation with respect to these defined relations. Every such formula is equivalent to one constructed from ¯ calculus tests, atomic actions and the regular operations. The proof uses a characterization of completely additive ¯calculus formulas: formulas OE(p) that distribute over arbitrary unions. It turns out that complete additivity is equivalent to distributivity over countable unions. For FOL (firstorder logic) a similar theorem is shown (giving an alternative proof to the original of [4]). Here though distributivity over finite unions is sufficient. This enables us to show that the characterization of safe FOLformulas carries over to the setting of finite models. 1 Introduction The identification of processes with transition systems has been a fruitful one in computer science. In this perspective, stat...
Bisimulation Respecting FirstOrder Operations.
"... We identify two features of common process algebra operations: their firstorder flavour and the fact that they respect bisimulation in a uniform manner. For this purpose two notions are introduced: first, a notion of firstorder definable operations on process graphs and second, respect for sequenc ..."
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We identify two features of common process algebra operations: their firstorder flavour and the fact that they respect bisimulation in a uniform manner. For this purpose two notions are introduced: first, a notion of firstorder definable operations on process graphs and second, respect for sequence extension. In the first part of the paper those firstorder definable operations are characterised whose defining formulas respect sequence extension. The second part uses the resulting format to calculate modal preconditions: it gives an algorithm that reduces modal truth in the output of certain process graph operations to modal truth in the inputs of this operation. 1 Introduction This paper fits into the recent tradition ([5, 6, 8]) of attempts to apply modal logical techniques to process algebra ([2]). These attempts are motivated by two observations. First, rooted transition systems (or process graphs) play a role in both process algebra and modal logic. In process algebra, process ...