Computing Distributed Knowledge as the Greatest Lower Bound of Knowledge

Pinzón, Carlos; Quintero, Santiago; Ramírez, Sergio; Valencia, Frank

doi:10.1007/978-3-030-88701-8_25

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2022

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“…Note. This paper has been submitted to the LMCS special issue of RAMiCS 2021 as the extended version of [PQRV21] with full proofs of each result as well as with the new theoretical and experimental material mentioned in (2) in the above-listed mains results.…”

Section: Introductionmentioning

confidence: 99%

Computing Distributed Knowledge as the Greatest Lower Bound of Knowledge

Quintero¹,

Pinzón²,

Ramírez³

et al. 2022

Preprint

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Let L be a finite lattice and E(L) be the set of join endomorphisms of L. We consider the problem of given L and f, g ∈ E(L), finding the greatest lower bound f E(L) g in the lattice E(L). (1) We show that if L is distributive, the problem can be solved in time O(n) where n = |L|. The previous upper bound was O(n 2 ). ( 2) We provide new algorithms for arbitrary lattices and give experimental evidence that they are significantly faster than the existing algorithm. (3) We characterize the standard notion of distributed knowledge of a group as the greatest lower bound of the join-endomorphisms representing the knowledge of each member of the group. (4) We show that deciding whether an agent has the distributed knowledge of two other agents can be computed in time O(n 2 ) where n is the size of the underlying set of states. (5) For the special case of S5 knowledge, we show that it can be decided in time O(nαn) where αn is the inverse of the Ackermann function.

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Section: Introductionmentioning

confidence: 99%