### Citations

2936 | Maintaining knowledge about temporal intervals
- Allen
- 1983
(Show Context)
Citation Context ...t here, but we refer the reader to the recent state-of-the-art references on it [35, 50]. An important early work in the formal study of purely interval-based temporal ontology and reasoning in AI is =-=[2]-=-, where Allen considers the family of binary relations arising between two intervals in a given linear order, subsequently called Allen’s relations. Besides these, the natural and important operation ... |

622 |
Modal Logic
- Blackburn, Rijke, et al.
- 2000
(Show Context)
Citation Context ...iven modal operator in a given fragment, we use a standard technique in modal logic, based on the notion of bisimulation and the invariance of modal formulae with respect to bisimulations (see, e.g., =-=[5]-=-). Let F be an HS fragment. An F -bisimulation between two interval models M = 〈I(D),V〉 and M′ = 〈I(D′),V ′〉 over AP is a relation Z ⊆ I(D) × I(D′) satisfying the following properties: • local conditi... |

351 |
The Classical Decision Problem
- Borger, Graedel, et al.
(Show Context)
Citation Context ...lve the problem, one must find a function f : O → T such that right( f (n,m)) = le f t( f (n + 1,m)) and up( f (n,m)) = down( f (n,m + 1)). The undecidability of the tiling problem for O is proved in =-=[6]-=- from that of the tiling problem for Z × Z (known to be co-r.e. complete by a reduction from the halting problem of a Turing machine), through the tiling problem for N × N, by application of König’s L... |

47 |
The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse.
- Benthem
- 1991
(Show Context)
Citation Context ...d the philosophical roots of interval-based temporal reasoning can be dated back to Zeno and Aristotle [46]. Already Zeno noted that in an interval-based setting, several of his paradoxes ’disappear’ =-=[4]-=-, like the flying arrow paradox (“if at each instant the flying arrow stands still, how is movement possible?”) and the dividing instant dilemma (“if the light is on and it is turned off, what is its ... |

17 | G.: Decidable and Undecidable Fragments of Halpern and Shoham’s Interval Temporal Logic: Towards a Complete Classification
- Bresolin, Monica, et al.
- 2008
(Show Context)
Citation Context ...ively axiomatizable. Subsequently, the techniques proving such undecidability results were sharpened to apply to a multitude of – sometimes surprisingly simple and inexpressive – fragments of HS, see =-=[8, 28, 37, 38]-=-. The underlying technical reason for these undecidability results can be found in the very nature of purely interval-based temporal reasoning, where all atomic propositions, and therefore all formula... |

11 | Metric Propositional Neighborhood Logics: Expressiveness, Decidability, and Undecidability
- Bresolin, Monica, et al.
- 2010
(Show Context)
Citation Context ... Right PNL (RPNL for short), with a special attention to decidability and expressive completeness issues. Such a work has been subsequently extended to the family of metric extensions of the full PNL =-=[7, 10]-=-. The most expressive language in that family, called Metric PNL (MPNL, for short) features a set of special atomic propositions representing integer constraints (equalities and inequalities) on the l... |

8 | G.: Undecidability of Interval Temporal Logics with the Overlap Modality
- Bresolin, Monica, et al.
- 2009
(Show Context)
Citation Context ...arry out Halpern and Shoham’s idea of encoding Turing machine configurations and consequently, to yield undecidability [37]. More recently, a number of other HS fragments have been proved undecidable =-=[8, 9, 11, 12, 28, 38, 39]-=- by means of suitable reductions from known undecidable problems. The most widely applied such reductions have been constructed from several variants of the tiling problem: theN×N tiling problem [8], ... |

5 | G.: Undecidability of the logic of Overlap relation over discrete linear orderings
- Bresolin, Monica, et al.
(Show Context)
Citation Context ...arry out Halpern and Shoham’s idea of encoding Turing machine configurations and consequently, to yield undecidability [37]. More recently, a number of other HS fragments have been proved undecidable =-=[8, 9, 11, 12, 28, 38, 39]-=- by means of suitable reductions from known undecidable problems. The most widely applied such reductions have been constructed from several variants of the tiling problem: theN×N tiling problem [8], ... |

4 |
Metric propositional neighborhood interval logics on natural numbers. Software and Systems Modeling (SoSyM), accepted for publication, January 2011 (doi: 10.1007/s10270-011-0195-y, online since
- Bresolin, Monica, et al.
- 2011
(Show Context)
Citation Context ... Right PNL (RPNL for short), with a special attention to decidability and expressive completeness issues. Such a work has been subsequently extended to the family of metric extensions of the full PNL =-=[7, 10]-=-. The most expressive language in that family, called Metric PNL (MPNL, for short) features a set of special atomic propositions representing integer constraints (equalities and inequalities) on the l... |

2 |
Two Sorted Point-Interval Temporal Logic. Accepted for publication on
- Balbiani, Goranko, et al.
- 2011
(Show Context)
Citation Context ...The reader is referred to [4] for a detailed philosophical-logical comparative discussion of both approaches, while a recent study and technical exploration of the two-sorted approach can be found in =-=[3]-=-. One of the first applications of interval-based logical formalisms – to the specification and verification of hardware components – is Propositional Interval Temporal Logic (PITL), introduced by Mos... |