### Citations

2796 | Computational Complexity
- Papadimitriou
- 1994
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Citation Context ...e and therefore a logic is exactly a (decision) problem in the usual sense in complexity theory. That is, as a language viewed as a set of strings built upon a given alphabet. By definition (see e.g. =-=[Pap94]-=-), for 3 Also called GL (for Gödel and Löb), KW, K4W, PrL. 2 any fragment of classical logic that is C-hard with respect to C′ many-one reductions4, there is a mapping f inC′ such that any modal for... |

304 |
Proof Methods for Modal and Intuitionistic Logics.
- Fitting
- 1983
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Citation Context ...φ), 1) ∈ T . Theorem 3 is a mere consequence of Lemma 4 and Lemma 5. Its proof uses the sequent calculi GS4 and GT whereas in [CCM97] the proofs manipulate Fitting’s non prefixed calculi for S4 and T =-=[Fit83]-=-. Observe the map h is a variant of a map defined in [Fit88]. Let us write h′(φ) to denote the formula h(φ, (mwn(φ) + 1).mwp(φ), 1). By close examination of the definition of h′(φ), 1. computing h′(φ)... |

266 | Modal languages and bounded fragments of predicate logic
- Andreka, Benthem, et al.
- 1998
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Citation Context ...ned by translating T into FO2, which is known to be decidable (see e.g. [Mor75]). Furthermore, the formula obtained by reduction belongs to the decidable guarded fragment of classical logic (see e.g. =-=[ANB98]-=-) for which a resolution decision procedure has been defined in [Niv98]. In [Boo93, Chapter 12], a (non polynomial-time) transformation from Grz into G is defined. By using renamings of subformulae, i... |

245 | Modal logic - Chagrov, Zakharyaschev - 1997 |

237 | The computational complexity of provability in systems of modal propositional logic. - Ladner - 1977 |

151 | The Logic of Provability - Boolos - 1993 |

151 | Tableau methods for modal and temporal logics - Goré - 1999 |

134 |
Display logic,
- Belnap
- 1982
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Citation Context ...lise our method to handle other “second order” propositional modal logics like S4.3.1 using the calculi from [Gor94] (see also [DG99] for a generalisation and extension in the Display Logic framework =-=[Bel82]-=-). This paper is a completed version of [DG98]. 2 Basic Notions In the present paper, we assume that the modal formulae are built from a countably infinite set For0 def= {pi,j : i, j ∈ ω} of atomic pr... |

129 | Why is modal logic so robustly decidable - Vardi - 1997 |

89 | A Resolution Calculus for Modal Logics - Ohlbach - 1988 |

72 |
Provability interpretations of modal logic
- Solovay
- 1976
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Citation Context ...able. For instance, the modal logic K augmented with the McKinsey axiom is captured by the framework presented in [Ohl93]. Similarly, the provability logic G3 that admits arithmetical interpretations =-=[Sol76]-=- is treated within the set-theoretical framework defined in [dMP95]. Both techniques in [Ohl93,dMP95] use a version of classical logic augmented with a theory. Alternatively, G can also be translated ... |

57 |
First-order modal tableaux.
- Fitting
- 1988
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Citation Context ...ctive logics. Both reductions proceed via an analysis of the proofs in cut-free sequent calculi from the literature. The second reduction is a slight variant of the one presented in [CCM97] (see also =-=[Fit88]-=-). The reduction announced in the title can be obtained by translating T into FO2, which is known to be decidable (see e.g. [Mor75]). Furthermore, the formula obtained by reduction belongs to the deci... |

57 | Modal tableau calculi and interpolation - Rautenberg - 1983 |

56 |
Complexity results for classes of quantificational formulas.
- Lewis
- 1980
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Citation Context ...inC′ such that any modal formula φ ∈ L iff f(φ) is valid in such a first-order fragment. From the facts that G is in PSPACE (see e.g. [BH94,Lad77]), validity in FO2 isNEXPTIME-hard [Für81] (see also =-=[Lew80]-=-) and PSPACE ⊆ NEXPTIME, it is easy to conclude that there exists a polynomial-time transformation from G into validity in FO2. As is well-known, this illustrates the difference between the fact that ... |

45 |
Modal logic and classical logic. Bibliopolis
- BENTHEM
- 1983
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Citation Context ...nted with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined in [BH94] and then a translation from K4 into classical logic (see e.g. =-=[Ben83]-=-). The fact that G can be translated into a decidable fragment of classical logic follows from a purely complexity theory viewpoint, as shown next. Take a modal logic L that is in the complexity class... |

43 |
Modal Theorem Proving
- Abadi, Manna
- 1986
(Show Context)
Citation Context ...∧( ∧ ψ∈Γ ′′ 2nh(ψ, n−1, 0)))⇒ (h(2φ, n−1, 1)∨ ∨ ψ∈∆′ h(ψ, n, 1)). For ψ ∈ Γ ′′, 2nh(ψ, n−1, 0) occurs negatively in ϕ and h(2φ, n−1, 1) occurs positively in ϕ. By the Monotonicity of Entailment Lemma =-=[AM86]-=-, (( ∧ ψ∈Γ ′ h(ψ, n, 0))∧( ∧ ψ∈Γ ′′ 2nh(ψ, n, 0)))⇒ (h(2φ, n, 1)∨ ∨ ψ∈∆′ h(ψ, n, 1)) ∈ T By completeness of GT , we get that h(Γ ′, n, 0), h(2Γ ′′, n, 0) ` h(2φ, n, 1), h(∆′, n, 1) has a cut-free proo... |

40 |
On languages with two variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
- Mortimer
- 1975
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Citation Context ...uction is a slight variant of the one presented in [CCM97] (see also [Fit88]). The reduction announced in the title can be obtained by translating T into FO2, which is known to be decidable (see e.g. =-=[Mor75]-=-). Furthermore, the formula obtained by reduction belongs to the decidable guarded fragment of classical logic (see e.g. [ANB98]) for which a resolution decision procedure has been defined in [Niv98].... |

23 |
A resolution decision procedure for the guarded fragment.
- Nivelle
- 1998
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Citation Context ... [Mor75]). Furthermore, the formula obtained by reduction belongs to the decidable guarded fragment of classical logic (see e.g. [ANB98]) for which a resolution decision procedure has been defined in =-=[Niv98]-=-. In [Boo93, Chapter 12], a (non polynomial-time) transformation from Grz into G is defined. By using renamings of subformulae, it is easy to extract from that transformation, an O(n.log n)-time trans... |

23 | Gentzen Method in Modal Calculi I, - Ohnishi, Matsumoto - 1957 |

22 | Cut-free sequent and tableau systems for propositional Diodorean modal logics
- Goré
- 1994
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Citation Context ...des logics like G and Grz which have been shown to have “arithmetical” interpretations as well as logics like S4.3.1 which have interpretations as logics of linear time (without a next-time operator) =-=[Gor94]-=-. Somewhat surprisingly, faithful translations into classical logic (usually augmented with theories) have been found for some propositional modal logics even when these logics are characterized by cl... |

20 | A set-theoretical translation method for polymodal logics
- d’Agostino, Montanari, et al.
- 1995
(Show Context)
Citation Context ...xiom is captured by the framework presented in [Ohl93]. Similarly, the provability logic G3 that admits arithmetical interpretations [Sol76] is treated within the set-theoretical framework defined in =-=[dMP95]-=-. Both techniques in [Ohl93,dMP95] use a version of classical logic augmented with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined... |

20 | Raisonnement Automatique en Logique Modale et Algorithmes d'Unification - Herzig - 1989 |

19 | A resolution-based decision procedure for extensions of K4.
- Ganzinger, Hustadt, et al.
- 1999
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Citation Context ...nt delineated by the translation is decidable. The only known first-order decision procedure for that particular fragment except the one that mimicks the rules for K4 is the one recently published in =-=[GHMS98]-=-. Therefore, blind translation is not useful if this means giving up decidability. Moreover, it is well-known that many decidable propositional modal logics are characterised by classes of Kripke fram... |

17 | The computational complexity of the unconstrained limited domino problem (with implications for logical decision problems). In Logic and machines: decision problems and complexity, - Furer - 1984 |

15 | Combining Hilbert style and semantic reasoning in a resolution framework,” in - Ohlbach - 1998 |

13 | Methods for Automated Theorem Proving in Nonclassical Logics - Morgan |

12 | On modal systems having arithmetical interpretations, The - Avron - 1984 |

12 | Optimized Modal Translation and Resolution,
- Schmidt
- 1997
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Citation Context ...in the guarded fragment of classical logic (see e.g. [ANB98]), for which a proof procedure based on resolution is proposed in [Niv98]. Alternatively, after translating Grz into T, the techniques from =-=[Sch97]-=- could also be used to translate T into classical logic. These are possibilities to obtain a decision procedure for Grz using theorem provers for classical logic. 15 We are currently investigating whe... |

9 |
A translation from the modal logic of provability into K4
- Balbiani, Herzig
- 1994
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Citation Context ...th techniques in [Ohl93,dMP95] use a version of classical logic augmented with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined in =-=[BH94]-=- and then a translation from K4 into classical logic (see e.g. [Ben83]). The fact that G can be translated into a decidable fragment of classical logic follows from a purely complexity theory viewpoin... |

9 | Resolution-based calculi for modal and temporal logics - Nonnengart - 1996 |

9 |
Optimized translation of multi modal logic into predicate logic
- Ohlbach
- 1993
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Citation Context ...hen these logics are characterized by classes of frames that are not first-order definable. For instance, the modal logic K augmented with the McKinsey axiom is captured by the framework presented in =-=[Ohl93]-=-. Similarly, the provability logic G3 that admits arithmetical interpretations [Sol76] is treated within the set-theoretical framework defined in [dMP95]. Both techniques in [Ohl93,dMP95] use a versio... |

8 | A polynomial translation of S4 into T and contraction-free tableaux for S4
- Cerrito, Mayer
- 1997
(Show Context)
Citation Context ...li for these respective logics. Both reductions proceed via an analysis of the proofs in cut-free sequent calculi from the literature. The second reduction is a slight variant of the one presented in =-=[CCM97]-=- (see also [Fit88]). The reduction announced in the title can be obtained by translating T into FO2, which is known to be decidable (see e.g. [Mor75]). Furthermore, the formula obtained by reduction b... |

8 |
Improved decision procedures for the modal logics K, T and S4
- Hudelmaier
- 1996
(Show Context)
Citation Context ...((n.log n)3). As a side-effect, we obtain an O(n.log n)-time transformation from Grz into S4 and an O((n.log n)3)-time transformation from Grz into T. Using the space upper bound for S4-validity from =-=[Hud96]-=-, we obtain that Grz requires only space in O(n2.(log n)3). We are not aware of any tighter bound for Grz in the literature. Furthermore, our purely proof-theoretical analyses of the cut-free sequent-... |

7 | Theoremhood-preserving maps characterizing cut elimination for modal provability logics
- Demri, Goré
- 2002
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Citation Context .... As we intend to report in a longer paper, it is also possible to generalise our method to handle other “second order” propositional modal logics like S4.3.1 using the calculi from [Gor94] (see also =-=[DG99]-=- for a generalisation and extension in the Display Logic framework [Bel82]). This paper is a completed version of [DG98]. 2 Basic Notions In the present paper, we assume that the modal formulae are bu... |

7 | Relations between propositional normal modal logics: an overview
- Goré, Heinle, et al.
- 1997
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Citation Context ... K4 def= K + 2p⇒ 22p, S4 def= K4 + 2p⇒ p and Grz def= S4 + 2(2(p⇒ 2p)⇒ p)⇒ 2p. Numerous variants of the system Grz 4 (having the same set of theorems) can be found in the literature (see for instance =-=[GHH97]-=-). We call GT , GS4 and GGrz the cut-free versions of the Gentzen-style calculi defined in [OM57,Avr84] where the sequents are built from finite sets of formulae. Moreover, the weakening rule is absor... |

6 |
Gentzen-type and resolution rules part I: propositional logic
- Mints
(Show Context)
Citation Context ...The main contribution of this paper is the definition of an O(n.log n)-time transformation from Grz into S4, using cut-free sequent-style calculi for these respective logics. Renaming techniques from =-=[Min88]-=- are used in order to get the O(n.log n)-time bound. Then, we present a cubic-time transformation from S4 into T, again using the cutfree sequent-style calculi for these respective logics. Both reduct... |

2 | The Range of Modal Logic - An Essay in Memory of George Gargov - Benthem - 1999 |

1 |
An O((n.log n)3)-time transformation from Grz into decidable fragments of classical first-order logic
- Demri, Goré
- 1998
(Show Context)
Citation Context ...propositional modal logics like S4.3.1 using the calculi from [Gor94] (see also [DG99] for a generalisation and extension in the Display Logic framework [Bel82]). This paper is a completed version of =-=[DG98]-=-. 2 Basic Notions In the present paper, we assume that the modal formulae are built from a countably infinite set For0 def= {pi,j : i, j ∈ ω} of atomic propositions using the usual connectives 2, ¬, ⇒... |