A comparison technique for ill-known time intervals

Reasoning and querying bounds on differences with layered preferences

“…Kamide and Koizumi [35] have recently proposed an inconsistency-tolerant probabilistic tree logic. Recently, Gammoudy et al [31] have proposed to model Allen's qualitative relations between fuzzy time intervals, while Billet et al [10] have considered "ill-known" time intervals.…”

“…As an example, we compute -1, 2, £⟨ [10,39], ⟩, ⟨ [11,37], 〉, ⟨ [12,36], ℎ ℎ⟩¤® ⊕ P]P. [10,11], ⊕ 3456 ℎ ℎ¤, £ [11,12], ⊕ 3456 ℎ ℎ¤, £ [12,24], ℎ ℎ ⊕ 3456 ℎ ℎ¤}}, for obtaining the AdI for the preference level , the ⊕ P]P. operator performs the union ⋃{ [10,11], [11,12], [12,24]}, and for obtaining the AdI for the preference level , it performs the union ⋃{ [11,12], [12,24] [10,15], [11,13], [12,12]}) = { [10,11], [11,12], [12,12], [12,13], [13,15] [10,15], [11,13], [12,12]⟩) = ( [10,39], { [10,35],…”

“…, e1 ], p^⟩¤⟩ with 1£k£r and 1£l£r), their extension is [10,15], 〉, ⟨ [11,13], 〉, ⟨ [12,12], ℎ ℎ〉¤{ = ⟨⟨ [10,11], ⟩, ⟨ [11,13], ⟩, ⟨ [12,12], ℎ ℎ⟩, ⟨ [12,13], ℎ ℎ⟩, ⟨ [13,35], ℎ ℎ⟩, ⟨ [35,36], ℎ ℎ⟩, ⟨ [36,37], ⟩, ⟨ [37,39], ⟩⟩, for obtaining the AdI for the preference level , the ⊙ P]P. operator performs the union ⋃{ [10,11], [11,12], [12,12], [12,13], [13,35], [35,36], [36,37], [37,39]}, for obtaining the AdI for the preference level , it performs the union ⋃{ [11,12], [12,12], [12,13], [13,35], [35,…”

“…], v ⟩⟩⟩ (with 1 ≤ ≤ and 1 ≤ ≤ ), the resume of C' and C'' is obtained as following (where Ç denotes the intersection operator between intervals): [10,39], 〉, ⟨ [11,37], 〉, ⟨ [12,36], ℎ ℎ〉⟩⟩ (see below), the result of the resume operation between them is ′ ⊕ 2 11 = ⟨ , 2, ⟨⟨ [10,24], 〉, ⟨ [11,24], 〉, ⟨ [12,24], ℎ ℎ〉⟩⟩. with that level, until it reaches the highest level for both constraints.…”

“…Let t1, t2 and t3 be time points, let granularity be minutes, and let KB be the following set of BoD constraints: KB={10£t2-t1£15, 20£t3-t2£30, 25£t3-t1£40}, i.e., t2 is between 10 and 15 minutes after t1, t3 is between 20 and 30 minutes after t2, and t3 is between 25 and 40 minutes after t1. As an example of extension and resume, the extension operation 10£t2-t1£15 ⊙ 20£t3-t2£30 gives as result the BoD 30£t3-t1£45 (intuitively, if the difference between t2 and t1 is in the range [10,15] and the difference between t3 and t2 is in the range [20,30], the implied difference between t3 and t1 is in the range [30,45]); the resume operation 30£t3-t1£45 ⊕ 25£t3-t1£40 gives as result the BoD 30£t3-t1£40 (intuitively, if the difference between t3 and t1 is both in the ranges [30,45] and [25,40], the implied difference is in the range [30,40] -see the definition of the operators in Section 3). KB is consistent, and {t1=0, t2=10, t3=30} is a scenario (solution) of KB.…”