#### DMCA

## Hybrid Metric Propositional Neighborhood Logics with Interval Length Binders

### Citations

2938 | Maintaining knowledge about temporal intervals
- Allen
- 1983
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Citation Context ...for the specification and verification of hardware [22] and of real-time systems [12]. A systematic analysis of the variety of relations between two intervals in a linear order was initiated by Allen =-=[1]-=-, who proposed the use of interval reasoning in systems for time management and planning. Allen identified the thirteen different binary relations between intervals on linear orders, hereafter referre... |

157 | A propositional modal logic of time intervals.
- Halpern, Shoham
- 1991
(Show Context)
Citation Context ...e Halpern-Shoham modal logic of Allen’s relations HS (whose satisfiability problem is undecidable under very general assumptions for the class of linear orders underlying the interval structures, see =-=[16]-=-), its hybrid metric version is, in fact, at least as expressive as the whole HS. To show that it suffices to define the operators 〈B〉, corresponding to the interval relation begins, and 〈E〉, correspo... |

88 |
Duration Calculus: A formal approach to real–time systems
- Chaochen, Hansen
- 2002
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Citation Context ... it. 1.2 A brief comparison with Duration Calculus The length variables and binders introduced in this paper bear natural resemblance with the interval length variables used in Duration Calculus (DC) =-=[11,18]-=- – an interval logic introduced by Zhou Chaochen, C.A.R. Hoare, and A.P. Ravn [12], extending Moszkowski’s ITL [22]. The original version of ITL involves only one, binary, modal operator C, called cho... |

62 |
Reasoning about digital circuits
- Moszkowski
- 1983
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Citation Context ...cessing, and constraint satisfaction problems. In particular, temporal logics with interval-based semantics have been proposed as a useful formalism for the specification and verification of hardware =-=[22]-=- and of real-time systems [12]. A systematic analysis of the variety of relations between two intervals in a linear order was initiated by Allen [1], who proposed the use of interval reasoning in syst... |

58 |
Reversibility and surjectivity problems of cellular automata,
- Kari
- 1994
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Citation Context ...,=)1, where the modal constant π is added, is undecidable, and then we show how the proof can be adapted to the fragment WHPNL(=)1. We will make use of the undecidability of the Finite Tiling Problem =-=[19]-=-. It is the problem of establishing whether, for a given set of tile types T = {t1, . . . , tk}, there exists a finite rectangle R = [0, X] × [0, Y ] = {(i, j) : i, j ∈ N ∧ 0 ≤ i ≤ X ∧ 0 ≤ j ≤ Y } for... |

25 | An adequate first order interval logic, in
- Chaochen, Hansen
- 1998
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Citation Context ...on of ITL that adds to the language state expressions, representing states of the system and assigned a duration – the length of the time period during which the system remains in the given state. In =-=[10]-=-, a version of DC based on Neighborhood Logic, denoted as DC/NL, has been introduced; it features the two interval neighborhood modalities r and l and subsumes the original DC. Both the interval log... |

24 |
Linear duration invariants
- Chaochen, Jingzhong, et al.
- 1994
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Citation Context ...ime logic, whose validity problem is decidable [20]. Subsequently, M. Fränzle and M. R. Hansen obtain a decidability result for a rich fragment of DC [14], extending the work of Zhou Chaochen et al. =-=[13]-=- on decidability of linear duration invariants to a much wider fragment of DC. In the Restricted Duration Calculus [17] denoted by RDC1, the length of the current intervals can be referred to by using... |

21 | Digitizing interval duration logic
- Chakravarty, Pandya
- 2003
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Citation Context ...in [23] that a specific fragment of IDL, denoted by LIDL−, is decidable, by using an automata-theoretic argument. The problem of deciding the validity of IDL formulae has been further investigated in =-=[9]-=-, where the authors propose a syntactic characterization of a subset of IDL-formulae that share a property called Strong Closure under Inverse Digitalization that allows one to effectively check the v... |

18 |
An optimal tableau-based decision algorithm for Propositional Neighborhood Logic
- Bresolin, Montanari, et al.
- 2007
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Citation Context ... c ≥ b such that M, [b, c] ψ; • M, [a, b] lψ iff there exists c ≤ a such that M, [c, a] ψ. The satisfiability problem of the various version of PNL has been shown to be decidable in [5,15]. In =-=[8]-=-, a tableau-based method has been presented for PNL. From now on, we only consider PNL and its extensions interpreted in the interval structure on N. We denote by δ : N×N → N the distance function on ... |

17 | G.: Decidable and Undecidable Fragments of Halpern and Shoham’s Interval Temporal Logic: Towards a Complete Classification
- Bresolin, Monica, et al.
- 2008
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Citation Context ... len>x). Thus, we have the following result. Theorem 4.1 The satisfiability problem for WHPNL, and hence for WHMPNL, interpreted over N is undecidable. 4.2 Undecidability of the fragments It is known =-=[3]-=- that any of the pairs of HS-modalities 〈B〉, 〈E〉 and 〈B〉, 〈E〉 is sufficient to establish undecidability. Thus, even very small fragments of WHD. Della Monica et al. / Electronic Notes in Theoretical C... |

16 | Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions
- Bresolin, Goranko, et al.
- 2009
(Show Context)
Citation Context ... by. Its satisfiability problem has been shown to be decidable (NEXPTIME-complete) when interpreted over various classes of linearly ordered sets, in particular, over domains based on natural numbers =-=[6]-=-; the results presented in the same paper and in [21] showed that all possible extensions of PNL with Allen’s modal operator make the logic undecidable, which means that PNL is maximal in terms of dec... |

16 | 2003b, ‘Propositional interval neighborhood temporal logics
- Goranko, Montanari, et al.
(Show Context)
Citation Context ...t is interpreted. One of the few known cases of decidable interval logics with truly interval semantics (not reducible to point-based semantics) is the Propositional Neighborhood Interval Logic (PNL) =-=[5,15]-=-. PNL is a fragment of HS with only two modal operators, corresponding to Allen’s relations meets and its inverse met by. Its satisfiability problem has been shown to be decidable (NEXPTIME-complete) ... |

14 |
Duration calculus: Logical foundations, Formal Aspects of Computing 9
- Hansen, Chaochen
- 1997
(Show Context)
Citation Context ... it. 1.2 A brief comparison with Duration Calculus The length variables and binders introduced in this paper bear natural resemblance with the interval length variables used in Duration Calculus (DC) =-=[11,18]-=- – an interval logic introduced by Zhou Chaochen, C.A.R. Hoare, and A.P. Ravn [12], extending Moszkowski’s ITL [22]. The original version of ITL involves only one, binary, modal operator C, called cho... |

14 | Maximal decidable fragments of Halpern and Shoham’s modal logic of intervals
- Montanari, Puppis, et al.
- 2010
(Show Context)
Citation Context ... decidable (NEXPTIME-complete) when interpreted over various classes of linearly ordered sets, in particular, over domains based on natural numbers [6]; the results presented in the same paper and in =-=[21]-=- showed that all possible extensions of PNL with Allen’s modal operator make the logic undecidable, which means that PNL is maximal in terms of decidability (as a matter of fact, there are extensions ... |

13 |
Deciding an interval logic with accumulated durations
- Fränzle, Hansen
- 2007
(Show Context)
Citation Context ...us (QDDT ), that is, the corresponding discrete-time logic, whose validity problem is decidable [20]. Subsequently, M. Fränzle and M. R. Hansen obtain a decidability result for a rich fragment of DC =-=[14]-=-, extending the work of Zhou Chaochen et al. [13] on decidability of linear duration invariants to a much wider fragment of DC. In the Restricted Duration Calculus [17] denoted by RDC1, the length of ... |

11 | Metric Propositional Neighborhood Logics: Expressiveness, Decidability, and Undecidability
- Bresolin, Monica, et al.
- 2010
(Show Context)
Citation Context ...of decidability (as a matter of fact, there are extensions of PNL that are non-elementarly decidable only if interpreted over finite prefixes of N and undecidable in most of the other cases [21]). In =-=[4,7]-=-, authors proposed a ‘metric’ extension of PNL, called Metric PNL (MPNL, for short), which involves special propositional letters expressing equality or inequality constraints on the length of the cur... |

10 | On decidability and expressiveness of propositional interval neighborhood logics
- Bresolin, Goranko, et al.
- 2007
(Show Context)
Citation Context ...t is interpreted. One of the few known cases of decidable interval logics with truly interval semantics (not reducible to point-based semantics) is the Propositional Neighborhood Interval Logic (PNL) =-=[5,15]-=-. PNL is a fragment of HS with only two modal operators, corresponding to Allen’s relations meets and its inverse met by. Its satisfiability problem has been shown to be decidable (NEXPTIME-complete) ... |

5 | Decidability of a hybrid duration calculus - Bolander, Hansen, et al. |

5 | A theory of duration calculus with application
- Hansen, Hung
(Show Context)
Citation Context ...esult for a rich fragment of DC [14], extending the work of Zhou Chaochen et al. [13] on decidability of linear duration invariants to a much wider fragment of DC. In the Restricted Duration Calculus =-=[17]-=- denoted by RDC1, the length of the current intervals can be referred to by using the equality constraint len=k (which can be expressed using C and len=1). This fragment turns out to be decidable over... |

4 | Right propositional neighborhood logic over natural numbers with integer constraints for interval lengths
- Bresolin, Montanari, et al.
- 2009
(Show Context)
Citation Context ...of decidability (as a matter of fact, there are extensions of PNL that are non-elementarly decidable only if interpreted over finite prefixes of N and undecidable in most of the other cases [21]). In =-=[4,7]-=-, authors proposed a ‘metric’ extension of PNL, called Metric PNL (MPNL, for short), which involves special propositional letters expressing equality or inequality constraints on the length of the cur... |

2 |
Ravn, A calculus of durations, Information Processing Letters 40
- Chaochen, Hoare, et al.
- 1991
(Show Context)
Citation Context ...action problems. In particular, temporal logics with interval-based semantics have been proposed as a useful formalism for the specification and verification of hardware [22] and of real-time systems =-=[12]-=-. A systematic analysis of the variety of relations between two intervals in a linear order was initiated by Allen [1], who proposed the use of interval reasoning in systems for time management and pl... |

1 |
Interval duration logic: Expressiveness and decidability, Electronic Notes in Theoretical Computer Science 65
- Pandya
- 2002
(Show Context)
Citation Context ...ecidable over almost all interesting classes of linear orders. Several versions of ITL and DC have been studied, aiming to obtain decidability for the satisfiability/validity problem. For example, in =-=[23]-=-, Pandya considers the interval-based version of DC, called Interval Duration Logic (IDL), that is generally undecidable; however, it has been proved in [23] that a specific fragment of IDL, denoted b... |