## Combining Temporal Logic Systems (1994)

Venue: | Notre Dame Journal of Formal Logic |

Citations: | 29 - 2 self |

### BibTeX

@ARTICLE{Finger94combiningtemporal,

author = {Marcelo Finger and Dov Gabbay},

title = {Combining Temporal Logic Systems},

journal = {Notre Dame Journal of Formal Logic},

year = {1994},

volume = {37}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component one-dimensional temporal logics to the resulting two-dimensional temporal logic. Three basic logical properties are analysed, namely soundness, completeness and decidability. Each combination method is composed of three submethods that combine the languages, the inference systems and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying expressivity and varying degree of transference of logical properties. The temporalisation method and the independent combination method are shown to transfer all three basic logical properties. The method of full interlacing of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several...

### Citations

131 |
Basic Modal Logic
- Bull, Segerberg
- 1984
(Show Context)
Citation Context ... language, although it is known that no set of temporal operators can be expressively complete 2 over dyadic first-order language [Ven90]. Venema's [Ven90] two-dimensional temporal logic, Segerberg's =-=[Seg73]-=- two-dimensional modal logic and the temporalisation of a temporal logic are all broadly two-dimensional; so are the combined logics in Sections 6 and 7. In the study of one-dimensional temporal logic... |

130 |
Complexity of modal logics
- Spaan
- 1993
(Show Context)
Citation Context ...g73]. The complexity class of the decision problem for the combined logic is another interesting subject for study. For the independent combination of monomodal logics, such a study was done by Spaan =-=[Spa93]-=- and the conclusion was that the satisfiability problem of an independently combined logic is either reducible to that of one of the component logics, or it is PSPACE-hard or it is in NP. We believe a... |

126 | A propositional modal logic of time intervals
- Halpern, Shoham
- 1986
(Show Context)
Citation Context ...ition follows directly from Venema's proof that the valid formulae over the upper half two-dimensional plane are not enumerable for Z\Theta Z, N \Theta N and R\Theta R, which in its turn was based on =-=[HS86]-=-. Since there are sound, complete and decidable US-temporal logics over Z, N and R [Rey92], the general conclusion on full interlacing is the following. Theorem 6.2 (Full Interlacing) Completeness and... |

89 | Properties of independently axiomatizable bimodal logics
- Kracht, Wolter
- 1991
(Show Context)
Citation Context ... ( ~ T ; ! ) are in K, and ~ T is the (not necessarily disjoint) union of the sets of time points T and T that constitute each connected component; such a biordered flow of time has been discussed in =-=[KW91] for the-=- case of the independent combination of two mono-modal systems. A model structure for US \Phi �� U �� S over K \Phi K is a 4-tuple ( ~ T ; !; !; g), where ( ~ T ; !; ! ) 2 K \Phi K and g is an... |

86 | Basic tense logic - Burgess - 1984 |

77 | Adding a temporal dimension to a logic system
- Finger, Gabbay
- 1992
(Show Context)
Citation Context ...Imperial College, Department of Computing 180 Queen's Gate London SW7 2BZ, UK mf3@doc.ic.ac.uk dg@doc.ic.ac.uk Draft version 17 March 1994 Abstract This paper is a continuation of the work started in =-=[FG92]-=- on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component one-dimensional temporal logic... |

43 |
Expressive functional completeness in tense logic
- Gabbay
- 1981
(Show Context)
Citation Context ...c is expressively complete with respect to monadic first-order logic over N and over R, and that, with additional operators (the Stavi operators) , we can get expressive completeness over Q and K lin =-=[Gab81b]-=-. So expressive completeness is not transferred by full interlacing. It is easy to verify that the following formulae expressing the commutativity of future and past operators between the two dimensio... |

29 |
Expressiveness and completeness of an interval tense logic
- Venema
- 1990
(Show Context)
Citation Context ... two-dimensional plane T \Theta T in order to represent the database evolution. Another application of the two-dimensional plane (or its NW-semi-plane) is in the representation of intervals on a line =-=[Ven90]-=-. r q rr - \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma \Gamma 6 SE NW t 2 t 1 (t 1 ; t 2 ) Figure 8 Two-dimensional representation of inter... |

24 |
Fibred semantics and combinations of logics
- Gabbay
- 1992
(Show Context)
Citation Context ...l itself? The same question applies to other combination methods. Do they achieve transference of logical properties when one or both of the combined temporal of modal logics is not classical? Gabbay =-=[Gab92]-=- has recently posed that question in a very generic framework involving Labelled Deductive Systems (LDS) and found that in order to obtain the transference of completeness we do not need the full powe... |

13 |
An Axiomatisation for Until and Since over the reals without the IRR Rule
- Reynolds
(Show Context)
Citation Context ...o-dimensional plane are not enumerable for Z\Theta Z, N \Theta N and R\Theta R, which in its turn was based on [HS86]. Since there are sound, complete and decidable US-temporal logics over Z, N and R =-=[Rey92]-=-, the general conclusion on full interlacing is the following. Theorem 6.2 (Full Interlacing) Completeness and decidability do not transfer in general through full interlacing. It has to be noted that... |

12 | Axioms for tense logic I: "since" and "until - Burgess - 1982 |

11 | Handling Database Updates in Two-dimensional Temporal Logic
- Finger
- 1992
(Show Context)
Citation Context ...w of time of the underlying logic which, in this case, happens to be a temporal logic. The logic system we obtain by temporalising US-temporal logic is the two-dimensional temporal logic described in =-=[Fin92]-=-.sExample 4.3 N-dimensional temporal logic If we repeat the process started in the last two examples, we can construct an n-dimensional temporal logic T n (PL)=K (its alphabet including S n and U n ) ... |

10 |
G.: Transfer theorems for stratified multimodal logics
- Fine, Schurz
(Show Context)
Citation Context ...and Further Work With respect to combination of logics, the works in the literature that most closely approximate ours in spirit and aims, are those of Kracht and Wolter [KW91] and of Fine and Schurz =-=[FS91]-=-. Both works concentrated on monomodal logics, and investigated the transference of logical properties for only the method we called here independent combination. However, their work investigated seve... |

8 | A Conjectured Axiomatization of Two-dimensional Reichenbachian Tense Logic - Aqvist - 1979 |

7 | Changing the Past: Database Applications of Two-dimensional Temporal Logics
- Finger
- 1994
(Show Context)
Citation Context ...cur in A, then ` A. Theorem 6.1 (2D-completeness) There is a sound and complete axiomatisation over the class of full two-dimensional temporal models over K lin \Theta K lin . A proof can be found in =-=[Fin94]-=- showing that the axiomatisation above is sound and complete over K lin \Theta K lin . If K dis is the class of all linear and discrete flows, [Fin94] also shows completeness results for the classes K... |

7 |
Independent Propositional Modal Logics
- Thomason
- 1980
(Show Context)
Citation Context .... The independent combination generates a system that is twodimensional according to the first criterion but fails the second one, so it is not broadly two-dimensional. The following result is due to =-=[Tho80]-=- and is more general than the independent combination of two US-logics. Proposition 5.1 With respect to the validity of formulae, the independent combination of two modal logics is a conservative exte... |

6 |
An irreflexivity lemma
- Gabbay
- 1981
(Show Context)
Citation Context ...f ` hor(p)!A and p does not occur in A, then ` A IR2 if ` ver(p)!A and p does not occur in A, then ` A IR1 and IR2 are two-dimensional extensions of the irreflexivity inferrence rule (IRR) defined in =-=[Gab81a]-=- for the one-dimensional case: if ` pH:p!A and p does not occur in A, then ` A. Theorem 6.1 (2D-completeness) There is a sound and complete axiomatisation over the class of full two-dimensional tempor... |

4 | On some U; S-Tense Logics - Xu - 1988 |

2 |
The Decision Problem for Linear Logic
- Burgess, Gurevich
- 1985
(Show Context)
Citation Context ...L is to determine whether some A 2 L is a valid formula or not. We have the following results Theorem 2.1 ([Bur82, Xu88]) The Burgess-Xu axiomatisation is sound and complete over K lin . Theorem 2.2 (=-=[BG85]-=-) The logic US = hL US ; ` US ; j= US i is decidable over K lin . 3 Combining Logics As we have mentioned earlier, the combination of two one-dimensional temporal logics will generate a two-dimensiona... |