Regular and Sufficient Bounds of Finite Domain Constraints for Skeptical C-Inference

Beierle, Christoph; Kutsch, Steven

doi:10.1007/978-3-319-60042-0_52

Cited by 5 publications

(8 citation statements)

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“…While it has been conjectured that in general the number of conditionals u = |R| in the given knowledge base is sufficient [8], in the following sections we will show that this is not the case.…”

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 81%

“…When we are not interested in capturing all c-representations as done by a regular CR u (R), but aim at capturing c-inference instead, we can specify a maximal impact value from this inference perspective. Definition 9 (sufficient, minimally sufficient [4,8]) Let R be a knowledge base and let u ∈ N. Then CR u (R) is called sufficient iff for all formulas A, B we have…”

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

“…Proposition 4 (regular vs. sufficient [8]) Let R be consistent and u ∈ N. If CR u (R) is regular, then it is also sufficient; the reverse is not true in general.…”

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

“…If CR u (R) is regular, it is also sufficient according to Proposition 3. For showing the other direction, the knowledge base [8]. The verification and falsification behavior as well as all solution vectors and their induced ranking models for the CSP CR 2 (R 2 ) are given in Table 3.…”

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

“…In [4], c-inference under a maximal impact factor u is introduced as skeptical inference taking not all c-representations, but only those crepresentations which are solutions of the restricted constraint satisfaction problem CR u (R) into account. Besides introducing system W, the second major objective of this article is to investigate upper and lower bounds u ∈ N that are sufficient for a knowledge base R in the sense that c-inference under the maximal impact u fully realizes skeptical c-inference [8].…”

Section: Introductionmentioning

confidence: 99%

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Nonmonotonic reasoning from conditional knowledge bases with system W

Komo

Beierle

2021

Ann Math Artif Intell

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For nonmonotonic reasoning in the context of a knowledge base $\mathcal {R}$ R containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of $\mathcal {R}$ R into account. System Z uses only the unique minimal ranking model of $\mathcal {R}$ R , and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of $\mathcal {R}$ R into account. C-representations constitute the subset of all ranking models of $\mathcal {R}$ R that are obtained by assigning non-negative integer impacts to each conditional in $\mathcal {R}$ R and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of $\mathcal {R}$ R . We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to $\mathcal {R}$ R having n conditionals if there is at least one world verifying all conditionals in $\mathcal {R}$ R . In contrast to the previous conjecture that the number of conditionals in $\mathcal {R}$ R is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in $\mathcal {R}$ R for fully capturing c-inference.

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Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 81%

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

“…Proposition 4 (regular vs. sufficient [8]) Let R be consistent and u ∈ N. If CR u (R) is regular, then it is also sufficient; the reverse is not true in general.…”

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

Section: Definition 6 (C-inference Under Maximal Impact Value |∼ Cumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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