Orders on intervals over partially ordered sets: extending Allen’s algebra and interval graph results

Zapata, Francisco; Kreinovich, Vladik; Joslyn, Cliff; Hogan, Emilie

doi:10.1007/s00500-013-1010-1

Cited by 5 publications

(3 citation statements)

References 11 publications

(6 reference statements)

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“…Combined with Theorem 69, this gives us a full classification of the CSP for first-order expansions of the basic relations of the po-time analogue of the interval algebra. This generalisation of the interval algebra has been studied by Zapata et al [ZKJH13]. Kompatscher and Van Pham describe the tractable fragments of the random partial order with the aid of polymorphisms.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Complexity Classification Transfer for CSPs via Algebraic Products

Bodirsky¹,

Jönsson²,

Martin³

et al. 2022

Preprint

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We study the complexity of infinite-domain constraint satisfaction problems: our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure A can be transferred to a classification of the CSPs of first-order expansions of another structure B. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSP for first-order expansions of the n-fold algebraic power of (Q; <). This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of (Q; <) and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen's Interval Algebra, the n-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyse with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the AI literature.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

Complexity Classification Transfer for CSPs via Algebraic Products

Bodirsky¹,

Jönsson²,

Martin³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In closing this section, we note that while these ordering relations ≤ S , ⊆, and ≤ W are defined here for real intervals in the chain R, ≤ , in fact they are also available for general intervals in arbitrary posets P, ≤ , something which we have started to explore elsewhere [30]. Table 1: Relations among quantities of interval difference depending on order relation, assume in all cases that no two of x * , x * , y * , and y * are equal.…”

Section: Numeric Intervals: Operations and Ordersmentioning

confidence: 99%

Interval-Valued Rank in Finite Ordered Sets

Joslyn

Pogel

Purvine

2016

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We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the bottom and from the top. Our rank functions are thus interval-valued, and always exist, even for non-graded posets, providing order homomorphisms to an interval order on the interval-valued ranks. The concept of rank width arises naturally, allowing us to identify the poset region with point-valued width as its longest graded portion (which we call the "spindle"). A standard interval rank function is naturally motivated both in terms of its extremality and on pragmatic grounds. Its properties are examined, including the relationship to traditional grading and rank functions, and methods to assess comparisons of standard interval-valued ranks.

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“…Gonzalez-Pachon et al [42] used the interval goal programming model to aggregate preference ranking information to obtain the ranking results of the schemes. Zapata et al [43] extended the ordering of Allen's algebra to intervals in an arbitrary partially ordered set. Pouzet and Zaguia [44] described ordered groups such that the ordering is a semiorder, and they introduced threshold groups generalizing totally ordered groups.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making

Ruan,

Gong,

Chen

2023

Axioms

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Probabilistic interval ordering, as a helpful tool for expressing positive and negative information, can effectively address multi-attribute decision-making (MADM) problems in reality. However, when dealing with a significant number of decision-makers and decision attributes, the priority relationships between different attributes and their relative importance are often neglected, resulting in deviations in decision outcomes. Therefore, this paper combines probability interval ordering, the prioritized aggregation (PA) operator, and the Gauss–Legendre algorithm to address the MADM problem with prioritized attributes. First, considering the significance of interval priority ordering and the distribution characteristics of attribute priority, the paper introduces probability interval ordering elements that incorporate attribute priority, and it proposes the probabilistic interval ordering prioritized averaging (PIOPA) operator. Then, the probabilistic interval ordering Gauss–Legendre prioritized averaging operator (PIOGPA) is defined based on the Gauss–Legendre algorithm, and various excellent properties of this operator are explored. This operator considers the priority relationships between attributes and their importance level, making it more capable of handling uncertainty. Finally, a new MADM method is constructed based on the PIOGPA operator using probability intervals and employs the arithmetic–geometric mean (AGM) algorithm to compute the weight of each attribute. The feasibility and soundness of the proposed method are confirmed through a numerical example and comparative analysis. The MADM method introduced in this paper assigns higher weights to higher-priority attributes to establish fixed attribute weights, and it reduces the impact of other attributes on decision-making results. It also utilizes the Gauss AGM algorithm to streamline the computational complexity and enhance the decision-making effectiveness.

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Orders on intervals over partially ordered sets: extending Allen’s algebra and interval graph results

Cited by 5 publications

References 11 publications

Complexity Classification Transfer for CSPs via Algebraic Products

Complexity Classification Transfer for CSPs via Algebraic Products

Interval-Valued Rank in Finite Ordered Sets

Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making

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