## Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization? (1999)

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Citations: | 15 - 4 self |

### BibTeX

@MISC{Demri99sequentcalculi,

author = {Stéphane Demri},

title = {Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?},

year = {1999}

}

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### Abstract

. We define sequent-style calculi for nominal tense logics characterized by classes of modal frames that are first-order definable by certain \Pi 0 1 -formulae and \Pi 0 2 -formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cut-formulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...

### Citations

272 |
Proof methods for modal and intuitionistic logics
- Fitting
- 1983
(Show Context)
Citation Context ...o represent sets of positive literals in first-order logic. It partly explains why numerous calculi can be viewed as a “clever translation” 4 into classical logic (see e.g. [Gen92]). For instance, in =-=[Fit83]-=-, a prefix is defined as a (non-empty) sequence of natural numbers. A sequence i1 . . . in ∈ ω ∗ (n ≥ 1) can be understood (for example for the modal logic S4) as the set 5 {R(ai1...im , ai1...i m ′ )... |

231 | Introduction to Logic - Rescher - 1964 |

173 | Labeled Deductive Systems - Gabbay - 1996 |

145 | A semantical analysis of modal logic. i: Normal modal propositional calculi - Kripke - 1963 |

125 | Tableau methods for modal and temporal logics
- Goré
- 1999
(Show Context)
Citation Context ...sequent calculi defined in the present paper are based on a completely different approach: we rather use the nominals as “implicit prefixes”. In that sense, our calculi are explicit systemssfollowing =-=[Gor99]-=- but without introducing any extra proof-theoretical device that does not belong to the object modal language. Furthermore, the calculi defined in this paper does not differ very much in spirit with t... |

94 | Time and modality - Prior - 1957 |

88 | Automated deduction in nonclassical logic - Wallen - 1990 |

74 | Internalizing labeled deduction - Blackburn - 1998 |

72 | Modal logic with names - Gargov, Goranko - 1993 |

52 | The taming of the cut: classical refutations with analytic cut
- D’Agostino, Mondadori
- 1994
(Show Context)
Citation Context ...n. For any nominal tense logic L from the class C Π 0 2 definedsin this paper, we define a sequent-style calculus, say GL, that is based on the sequent-style counterpart of the calculus KE defined in =-=[dM94]-=-. Our calculi admit a cut rule satisfying the following nice computational properties. When reading the proof upwards, at any stage of the construction of the proof, (CR1) the number of potential cut-... |

48 | Strongly analytic tableaux for normal modal logics - Massacci - 1994 |

44 |
Nominal tense logic. Notre Dame
- Blackburn
- 1993
(Show Context)
Citation Context ...nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57,RU71]) by adding nominals (also called names) to the language (see e.g. =-=[Bla93]-=-). Nominals are understood as atomic propositions that hold true in a unique world of the Kripke-style models. The nominal tense logics are quite expressive since not only do they extend the standard ... |

42 | Free Variable Tableaux for Propositional Modal Logics
- Beckert, Goré
- 1997
(Show Context)
Citation Context ...φ has a proof Π with the “concise representation” of the positive literals, then φ has a proof Π ′ with the representation of literals “in extension” where |Π ′ | is in O(|Π| 3 ). The length of the 4 =-=[BG97]-=- is one of the rare papers where such a relationship is explicitly recognized. 5 Since ω ∗ and ω have the same cardinality, without any loss of generality, we can assume that the individual constants ... |

34 | Labelled Propositional Modal Logics: Theory and Practice - Basin, Matthews, et al. - 1997 |

33 | Modal Logic and Classical Logic, Bibliopolis - Benthem - 1983 |

29 | Tableau calculi for hybrid logics - Tzakova - 1999 |

22 | Modal tableaux with propagation rules and structural rules - Castilho, Cerro, et al. - 1997 |

21 |
A logic for reasoning about relative similarity
- Konikowska
- 1997
(Show Context)
Citation Context ...presence of nominals in the modal language to use “implicit prefixes” in the proof systems. As far as we know, the idea of using such implicit prefixes when nominals are involved is due to Konikowska =-=[Kon97]-=-. In [Kon97], Rasiowa-Sikorski-style calculi for relative similarity logics are defined. Herein, we generalize the use of implicit prefixes to a class of nominal tense logics and we introduce various ... |

20 | Investigations into the complexity of some propositional calculi - D’Agostino - 1990 |

20 |
and weakness of the modal display calculus
- Power
- 1996
(Show Context)
Citation Context ...]. Indeed, we associate syntactically rules to formulas defining relational theories. However, we are able to capture all the conditions on frames for the properly displayable modal logics defined in =-=[Kra96]-=-. We wish also to thank one of the referees for pointing us to [Bla98,Tza99] where tableau-style calculi having technical similarities with ours have been defined. 2 Nominal tense logics Given a count... |

18 | Don’t Eliminate Cut - Boolos - 1984 |

17 | A resolution-based decision procedure for extensions of K4 - Ganzinger, Hustadt, et al. - 2001 |

17 | Labelled tableaux for multi-modal logics - Governatori - 1995 |

16 | Display calculi for nominal tense logics
- Demri, Goré
- 2002
(Show Context)
Citation Context ...for (non nominal) tense logics can be for instance found in [RU71,Kra96,Heu98,BG98] but these calculi do not treat the nominal case and they do not consider so large a class of logics as C Π 0 2 . In =-=[DG99]-=-, display calculi for nominal tense logics have been defined and cut is not only eliminable but also a strong normalization theorem is established. For all the calculi designed in the present paper, c... |

13 | A labelled sequent system for tense logic kt - Bonnette, Goré - 1998 |

12 | Sequent Calculi for Proof Search in Some Modal Logics, Theory - Implementation - Applications - Heuerding - 1998 |

11 | Natural deduction for non-classical logics
- Basin, Matthews, et al.
- 1998
(Show Context)
Citation Context ...irst-order definable classes of frames considered in [Rus96,CFdCGH97] are C Π 0 2 - definable and C Π 0 2 contains all the modal logics (in their nominal tense version) defined with Horn clauses from =-=[BMV98]-=-. Furthermore, for any nominal tense logic L = 〈NTL(G, H), C〉 such that C is first-order definable by a finite set Φ of restricted Π 0 2 -formulae, it is known that the L-validity problem can be trans... |

11 | Modal Logics as Labelled Deductive Systems - Russo - 1996 |

6 |
Normal Multimodal Logics: Automated Deduction and Logic Programming
- Baldoni
- 1998
(Show Context)
Citation Context ...l in the application of the (ψ)-rule. (ψ)sCondition 3.(b) states that the nominals corresponding to the y i ’s are new on the branch. The (ψ)-rule can be viewed as a generalization of the “ρ-rule” in =-=[Bal98]-=- and of the “Horn relational rule” in [BMV97,BMV98]. More generally, the (ψ)-rules merely encodes the logical consequence relation of the first-order relational theory of L (as also done in [Gen92]). ... |

3 | A tableau–like proof procedure for normal modal logics - Ognjanović - 1994 |

2 |
Analytic proof systems for classical and modal logics of restricted quantification
- Gent
- 1992
(Show Context)
Citation Context ...prefixes as a compact way to represent sets of positive literals in first-order logic. It partly explains why numerous calculi can be viewed as a “clever translation” 4 into classical logic (see e.g. =-=[Gen92]-=-). For instance, in [Fit83], a prefix is defined as a (non-empty) sequence of natural numbers. A sequence i1 . . . in ∈ ω ∗ (n ≥ 1) can be understood (for example for the modal logic S4) as the set 5 ... |

1 |
Improved decision procedures for the modal logics K, KT and S4
- Hudelmaier
- 1996
(Show Context)
Citation Context ...ant (even in the worst-case) when dealing with NP-hard problems (and a fortiori with PSPACE-hard problems). Of course, this is highly significant to establish tight complexity upper bounds as done in =-=[Hud96]-=-. In [Kri63,CFdCGH97] and [Heu98, Chapter 4], some of the graphical representations of the sets of (positive) first-order literals enjoy some conciseness property comparable to the one for prefixes. 3... |

1 | A labelled sequent systems for tense logic K t - Bonnette, Gor'e - 1998 |