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## A Principle for Incorporating Axioms into the First-Order Translation of Modal Formulae (2003)

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Venue: | Automated Deduction—CADE-19, volume 2741 of Lecture Notes in Artificial Intelligence |

Citations: | 20 - 7 self |

### Citations

54 | Resolution-based methods for modal logics.
- Nivelle, Schmidt, et al.
- 2000
(Show Context)
Citation Context ...ith remarks on implementation and a summary. It is assumed that the reader has a basic familiarity with modal logic, firstorder logic and resolution. The notation used in this paper is the same as in =-=[2]-=-. Due to lack of space all proofs of theorems are omitted but can be found in [15]. 2 The idea of the principle The idea of the principle presented in this paper can be situated in the general framewo... |

42 |
Tools and Techniques in Modal Logic,
- Kracht
- 1999
(Show Context)
Citation Context ...dality (which can be further reduced to local satisfiability in K (m) ). On a modal level the axiomatic translation is closely related to the reduction functions introduced independently by Kracht in =-=[10]-=- and developed further in [11]. Kracht's reduction functions are defined di#erently, the most important di#erence being that they do not use new symbols for subformulae. Renaming would lead to an impr... |

39 | The two-variable guarded fragment with transitive relations. To appear in
- Ganzinger, Meyer, et al.
- 1999
(Show Context)
Citation Context ...#p #x (#y (R(x, y) # holds(p, y)) (5) # #y (R(x, y) # (#z (R(y, z) # holds(p, z))))) #p #x holds(#p # p, x) # #p #x (holds(#p, x) # holds(p, x)) (6) # #p #x (#y (R(x, y) # holds(p, y)) # holds(p, x)) =-=(7)-=- If we replace the holds(p, x) literals in (5) and (7) by literals of the form P (x) and the #p quantifier by #P then what we have is the standard translations of 4 and T . The standard translations o... |

36 | Complexity of terminological reasoning revisited
- Lutz
- 1999
(Show Context)
Citation Context ...cut clauses are not necessary for completeness. For the logics KT , KB, KD, Kalt 1 one can observe that the axiomatic reduction to K is a set of non-logical axioms which are acyclic. Thus a result of =-=[13]-=- for the corresponding description logic ALC with acyclic TBoxes implies: Theorem 15. The computational complexity of the satisfiability problem of each of KT , KB, KD, Kalt 1 and their fusions is PSP... |

34 | Deciding Regular Grammar Logics with Converse Through First-Order Logic.
- Demri, Nivelle
- 2005
(Show Context)
Citation Context ...at is the smallest set of instances necessary. Such questions are of general significance to the e#ciency of proof methods exploiting the axiomatic translation. There is a (non-obvious) connection to =-=[4]-=-, worth exploring further. Demri and De Nivelle show that modal regular grammar logics with converse are decidable by reduction to GF 2 . For a given modal logic it is not immediate whether it is in f... |

26 |
The complexity of regularity in grammar logics and related modal logics,
- Demri
- 2001
(Show Context)
Citation Context ...lar grammar modal logic with converse for a larger class of modal logics. Such a connection is useful because regular grammar modal logics have been shown to have a number of nice properties by Demri =-=[3]-=-. 8 Conclusions The attractiveness of the translation principle described in this paper is its amenability for automation. The translation is not di#cult to implement. As a prover any existing first-o... |

23 | On the Relation of Resolution and Tableaux Proof Systems for Description Logics.
- Hustadt, Schmidt
- 1999
(Show Context)
Citation Context ...his immediately gives practical decision procedures for testing satisfiability of the modal logics considered in the previous section and their fusions. With the blocking inference rule introduced in =-=[9]-=- hyperresolution can also be turned into a decision procedure for the axiomatic translation of the considered logics. 7 Other consequences and related work The combination of the axiomatic translation... |

15 | Combining Hilbert style and semantic reasoning in a resolution framework,” in
- Ohlbach
- 1998
(Show Context)
Citation Context ...of space all proofs of theorems are omitted but can be found in [15]. 2 The idea of the principle The idea of the principle presented in this paper can be situated in the general framework of Ohlbach =-=[14]-=- for the combination of Hilbert-style deduction and semantic reasoning. In this framework, if a first-order semantics is known for some of the logical connectives, the semantic definitions are used as... |

14 | Eliminating \converse" from converse PDL
- Giacomo
- 1996
(Show Context)
Citation Context ...n of a formula, or eliminating a relation with special properties by internalisation, is of course not new. Often, reductions between logics can be seen to use this principle. For example, De Giacomo =-=[1]-=- embedded CPDL into PDL by using an axiomatic encoding. Demri and Goré [5] embed the provability logics G and Grz into GF 2 . Both aressecond-order modal logics. This shows that with the axiomatic tra... |

13 | Judgment and consequence relations
- Kracht
(Show Context)
Citation Context ...educed to local satisfiability in K (m) ). On a modal level the axiomatic translation is closely related to the reduction functions introduced independently by Kracht in [10] and developed further in =-=[11]-=-. Kracht's reduction functions are defined di#erently, the most important di#erence being that they do not use new symbols for subformulae. Renaming would lead to an improvement of complexity of the r... |

10 | Normal forms and proofs in combined modal and temporal logics,” in
- Hustadt, Dixon, et al.
- 2000
(Show Context)
Citation Context ...n be used to combine di#erent styles of reasoning systems. For example, the axiomatic translation was used in a decision procedure for the join of discrete linear temporal logic with subsystems of S5 =-=[8]-=-. The system combines temporal resolution for the temporal formulae and first-order resolution on the modal formulae. The idea of incorporating a theory into the translation of a formula, or eliminati... |

7 |
Eliminating "converse" from Converse PDL
- Giacomo
- 1996
(Show Context)
Citation Context ...n of a formula, or eliminating a relation with special properties by internalisation, is of course not new. Often, reductions between logics can be seen to use this principle. For example, De Giacomo =-=[1]-=- embedded CPDL into PDL by using an axiomatic encoding. Demri and Gore [5] embed the provability logics G and Grz into GF 2 . Both are second-order modal logics. This shows that with the axiomatic tra... |

6 |
Decidability results in non-classical logics I
- Gabbay
- 1975
(Show Context)
Citation Context ...logics considered in the previous section, and their fusions, are decidable and have the finite model property. Decidability of extensions of K with the axioms T , B, D, 4, 5, alt 1 are wellknown. In =-=[6]-=- the decidability of the logics K4 # , K5 # and Kalt #1 ,#2 1 was shown by using a reduction to S#S. It is also well-known that the operation of fusing normal propositional modal logics preserves deci... |

4 | Tractable transformations from modal provability logics into rst-order logic
- Demri, Gore
- 1999
(Show Context)
Citation Context ...nalisation, is of course not new. Often, reductions between logics can be seen to use this principle. For example, De Giacomo [1] embedded CPDL into PDL by using an axiomatic encoding. Demri and Gore =-=[5]-=- embed the provability logics G and Grz into GF 2 . Both are second-order modal logics. This shows that with the axiomatic translation it is possible to get a first-order translation of second-order m... |

3 | Notes on the space requirements for checking satisfiability in modal logics
- Kracht
(Show Context)
Citation Context ...lexity of the satisfiability problem of each of KT , KB, KD, Kalt 1 and their fusions is PSPACE-complete. More and better space bounds can be given by using the axiomatic translation and the ideas of =-=[12]-=-. The axiomatic translation can be used to combine di#erent styles of reasoning systems. For example, the axiomatic translation was used in a decision procedure for the join of discrete linear tempora... |