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Deciding regular grammar logics with converse through firstorder logic
 JOURNAL OF LOGIC, LANGUAGE AND INFORMATION
, 2005
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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...
Tractable Transformations from Modal Provability Logics into FirstOrder Logic
 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 1999
"... We define a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of firstorder definable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that are not firstorder definable. The class LF inclu ..."
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Cited by 4 (1 self)
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We define a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of firstorder definable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that are not firstorder definable. The class LF includes provability logics with deep arithmetical interpretations. Using Belnap's prooftheoretical framework Display Logic we characterise the "pseudodisplayable" subclass of LF and show how to define polynomialtime transformations from each such LF into the corresponding L, and hence into firstorder classical logic. Theorem provers for classical firstorder logic can then be used to mechanise deduction in these "psuedodisplayable second order" modal logics.
Abstract
"... We define display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use the natural translation of MNTL into the minimal tense logic of inequality (L�=) which is known to be properly displayable by application of Kracht’s ..."
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We define display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use the natural translation of MNTL into the minimal tense logic of inequality (L�=) which is known to be properly displayable by application of Kracht’s results. The rules of the display calculus δMNTL for MNTL mimic those of the display calculus δL� = for L�=. We show that every MNTLvalid formula admits a cutfree derivation in δMNTL. We also show that a restricted display calculus δ − MNTL, is not only complete for MNTL, but it enjoys cutelimination for arbitrary sequents. Finally, we give a weak Sahlqvisttype theorem for two semantically defined extensions of MNTL. Using Kracht’s techniques we obtain sound and complete display calculi for these two extensions based upon δMNTL and δ − MNTL respectively. The display calculi based upon δMNTL enjoy cutelimination for valid formulae only, but those based upon δ − MNTL enjoy cutelimination for arbitrary sequents. 1
An O((n log n)³)time transformation from Grz into decidable fragments of classical firstorder logic
, 1998
"... The provability logic Grz is characterized by a class of modal frames that is not firstorder definable. We present a simple embedding of Grz into decidable fragments of classical firstorder logic such as FO 2 and the guarded fragment. The embedding is an O((n.log n)³)time transformation that ne ..."
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The provability logic Grz is characterized by a class of modal frames that is not firstorder definable. We present a simple embedding of Grz into decidable fragments of classical firstorder logic such as FO 2 and the guarded fragment. The embedding is an O((n.log n)³)time transformation that neither involves first principles about Turing machines (and therefore is easy to implement), nor the semantical characterization of Grz (and therefore does not use any secondorder machinery). Instead, we use the syntactic relationships between cutfree sequentstyle calculi for Grz, S4 and T. We first translate Grz into T, and then we use the relational translation from T into FO 2 .
APower and Limits of Structural Display Rules
"... What can (and what cannot) be expressed by structural display rules? Given a display calculus, we present a systematic procedure for transforming axioms into structural rules. The conditions for the procedure are given in terms of (purely syntactic) abstract properties of the base calculus and thus ..."
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What can (and what cannot) be expressed by structural display rules? Given a display calculus, we present a systematic procedure for transforming axioms into structural rules. The conditions for the procedure are given in terms of (purely syntactic) abstract properties of the base calculus and thus the method applies to large classes of calculi and logics. If the calculus satisfies certain additional properties we prove the converse direction thus characterising the class of axioms that can be captured by structural display rules. Determining if an axiom belongs to this class or not is shown to be decidable. Applied to the display calculus for tense logic, we obtain a new proof of Kracht’s Display Theorem I.
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"... An O((n.log n)3)time transformation from Grz into decidable fragments of classical firstorder logic ..."
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An O((n.log n)3)time transformation from Grz into decidable fragments of classical firstorder logic