### Citations

634 |
Modal Logic,
- Blackburn, Rijke, et al.
- 2001
(Show Context)
Citation Context ... Sahlqvist formula. The proof of the ‘only if’ direction follows from lemmas 2 and 4 and the fact that all modally definable properties are preserved under disjoint unions and bounded morphisms (e.g. =-=[1]-=-). Together with the Sahlqvist completeness theorem it gives us that any modal logic axiomatizable by a single modal Horn formula is Kripke complete. The complexity of similar logics is studied in [5]... |

193 |
Manydimensional modal logics: theory and applications.
- Gabbay, Kurucz, et al.
- 2003
(Show Context)
Citation Context ...ion to a fragment with decidable implication, we are likely to obtain an algorithmic criterion for modal definability, as in this paper. Also this research is motivated by scrutinizing Theorem 5.9 of =-=[3]-=- saying that if L1 and L2 are Kripke complete and Horn axiomatizable unimodal logics, then L1 × L2 = [L1, L2] and studying whether Horn axiomatizability implies Kripke completeness. We give the positi... |

92 | Conjunctive query answering in the description logic EL using a relational database system.
- Lutz, Toman, et al.
- 2009
(Show Context)
Citation Context ...claim that there exists a directed tree T0 ∈ T such that for all T ∈ T there exists a surjective homomorphism from T0 to T . Let ∼ be the smallest equivalence relation on WD satisfying condition (cf. =-=[8]-=-) if there exists a, b, c, c′ such that aRDλ c, bR D λ c ′ and c ∼ c′, then a ∼ b. Define T0 = (W 0, (R0λ : λ ∈ Λ)) where W 0 = WD/ ∼, and for equivalence classes A,B ∈ W 0 AR0λB iff there exist a ∈ A... |

42 |
Tools and Techniques in Modal Logic,
- Kracht
- 1999
(Show Context)
Citation Context ...g the restricted universal quantifier (∀xi .λ xj)A ≡ ∀xi(xjRλxi → A), we can rewrite ED as ∀x0(∀x1 .λ(1) x0)(∀x2 .λ(2) xpr(2)) . . . (∀xn .λ(n) xpr(n))(αRλ0β). This is obviously a Kracht formula [6], =-=[7]-=-, so it is modally definable by a Sahlqvist formula. The proof of the ‘only if’ direction follows from lemmas 2 and 4 and the fact that all modally definable properties are preserved under disjoint un... |

38 |
Axiomatic classes in propositional modal logic, Algebra and logic (Fourteenth Summer Res.
- Goldblatt, Thomason
- 1974
(Show Context)
Citation Context ...Modal definability of first-order formulas has been intensively studied in modal logic, and even applied to automatic reasoning [9]. On the one hand, it has a nice Goldblatt-Thomason characterization =-=[4]-=-, on the other hand, the problem “decide whether a first-order formula is modally definable” is in general undecidable [2]. But the cause of this undecidability is in the undecidability of first-order... |

30 | How completeness and correspondence theory got married’,
- Kracht
- 1993
(Show Context)
Citation Context ... using the restricted universal quantifier (∀xi .λ xj)A ≡ ∀xi(xjRλxi → A), we can rewrite ED as ∀x0(∀x1 .λ(1) x0)(∀x2 .λ(2) xpr(2)) . . . (∀xn .λ(n) xpr(n))(αRλ0β). This is obviously a Kracht formula =-=[6]-=-, [7], so it is modally definable by a Sahlqvist formula. The proof of the ‘only if’ direction follows from lemmas 2 and 4 and the fact that all modally definable properties are preserved under disjoi... |

11 |
Chagrova L., “The Truth About Algorithmic Problems in Correspondence Theory
- Chagrov
- 2006
(Show Context)
Citation Context ...oning [9]. On the one hand, it has a nice Goldblatt-Thomason characterization [4], on the other hand, the problem “decide whether a first-order formula is modally definable” is in general undecidable =-=[2]-=-. But the cause of this undecidability is in the undecidability of first-order logic, so when we restrict attention to a fragment with decidable implication, we are likely to obtain an algorithmic cri... |

9 | On the complexity of elementary modal logics
- Hemaspaandra, Schnoor
- 2008
(Show Context)
Citation Context ...[1]). Together with the Sahlqvist completeness theorem it gives us that any modal logic axiomatizable by a single modal Horn formula is Kripke complete. The complexity of similar logics is studied in =-=[5]-=-. Consider two LfΛ-structures F 1 = (W 1, (R1λ : λ ∈ Λ)) and F 2 = (W 2, (R2λ : λ ∈ Λ)). A map g : W 1 → W 2 is called a homomorphism from F 1 to F 2 if, for any λ ∈ Λ and a, b ∈ W 1, aR1λb implies f(... |

6 | Query answering based on modal correspondence theory
- Zolin
- 2005
(Show Context)
Citation Context .... of the form K+ φ where φ is a modal Horn formula) is Kripke complete. Modal definability of first-order formulas has been intensively studied in modal logic, and even applied to automatic reasoning =-=[9]-=-. On the one hand, it has a nice Goldblatt-Thomason characterization [4], on the other hand, the problem “decide whether a first-order formula is modally definable” is in general undecidable [2]. But ... |