Structural and Computational Properties of Possibilistic Armstrong - PowerPoint PPT Presentation

Structural and Computational Properties of Possibilistic Armstrong Databases Seyeong Jeong, Haoming Ma, Ziheng Wei, Sebastian Link The University of Auckland, New Zealand ER 2020 Vienna, Austria

Context of the Work Apply possibility theory to schema design for uncertain data

Context of the Work Apply possibility theory to schema design for uncertain data Develop computational support for acquiring possibilistic functional dependencies that serve as meaningful input to schema design

Context of the Work Apply possibility theory to schema design for uncertain data Develop computational support for acquiring possibilistic functional dependencies that serve as meaningful input to schema design Mining constraints from data

Context of the Work Apply possibility theory to schema design for uncertain data Develop computational support for acquiring possibilistic functional dependencies that serve as meaningful input to schema design Mining constraints from data Iterative example-based acquisition (Armstrong relations)

Context of the Work Apply possibility theory to schema design for uncertain data Develop computational support for acquiring possibilistic functional dependencies that serve as meaningful input to schema design Mining constraints from data Iterative example-based acquisition (Armstrong relations) Validation of integrity requirements for uncertain data

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

The Big Picture

Possibilistic Relations and Functional Dependencies

Possibilistic Relations and Functional Dependencies

Possibilistic Relations and Functional Dependencies β 1 -cut Σ 1 MT → R α 4 -cut r 4 Tuples with α ≥ α 4

Possibilistic Relations and Functional Dependencies β 1 -cut Σ 1 MT → R α 4 -cut r 4 Tuples with α ≥ α 4 BCNF for Σ 1 R 1 = MTR with key MT R 2 = MTP with key MTP α 4 -lossless no α 4 -redundancy β 1 -preserving

Possibilistic Relations and Functional Dependencies β 2 -cut Σ 2 MT → R , RT → P α 3 -cut r 3 Tuples with α ≥ α 3

Possibilistic Relations and Functional Dependencies β 2 -cut Σ 2 MT → R , RT → P α 3 -cut r 3 Tuples with α ≥ α 3 BCNF for Σ 2 R 1 = RTP with key RT R 2 = RTM with key MT α 3 -lossless no α 3 -redundancy β 2 -preserving

Possibilistic Relations and Functional Dependencies β 3 -cut Σ 3 MT → R , RT → P , PT → M α 2 -cut r 2 Tuples with α ≥ α 2

Possibilistic Relations and Functional Dependencies β 3 -cut Σ 3 MT → R , RT → P , PT → M α 2 -cut r 2 Tuples with α ≥ α 2 BCNF for Σ 3 PTMR with keys MT , RT , PT α 2 -lossless no α 2 -redundancy β 3 -preserving

Possibilistic Relations and Functional Dependencies β 4 -cut Σ 4 MT → R , RT → P , P → M α 1 -cut r 1 Tuples with α = α 1

Possibilistic Relations and Functional Dependencies β 4 -cut Σ 4 MT → R , RT → P , P → M α 1 -cut r 1 Tuples with α = α 1 BCNF for Σ 4 R 1 = PM with key P R 2 = MTR with key MT R 3 = RTP with key RT α 1 -lossless no α 1 -redundancy β 4 -preserving

Computational Support for Finding Meaningful pFDs Definition (Possibilistic Armstrong Relation) A p-relation r over a p-schema ( R , α 1 > · · · > α k +1 ) is Armstrong for a given set Σ of pFDs over the p-schema if and only if for all pFDs ϕ over the p-schema, r satisfies ϕ if and only if Σ implies ϕ .

Computational Support for Finding Meaningful pFDs Definition (Possibilistic Armstrong Relation) A p-relation r over a p-schema ( R , α 1 > · · · > α k +1 ) is Armstrong for a given set Σ of pFDs over the p-schema if and only if for all pFDs ϕ over the p-schema, r satisfies ϕ if and only if Σ implies ϕ . R = { Project , Time , Manager , Room } β 1 > β 2 > β 3 > β 4 > β 5 Σ consists of: ( Manager , Time → Room , β 1 ) ( Room , Time → Project , β 2 ) ( Project , Time → Manager , β 3 ) ( Project → Manager , β 4 )

Computational Support for Finding Meaningful pFDs Definition (Possibilistic Armstrong Relation) A p-relation r over a p-schema ( R , α 1 > · · · > α k +1 ) is Armstrong for a given set Σ of pFDs over the p-schema if and only if for all pFDs ϕ over the p-schema, r satisfies ϕ if and only if Σ implies ϕ . Possibilistic Armstrong Relation for Σ R = { Project , Time , Manager , Room } Project Time Manager Room α i β 1 > β 2 > β 3 > β 4 > β 5 Eagle Mon, 9am Ann Aqua α 1 Σ consists of: Hippo Mon, 1pm Ann Aqua α 1 ( Manager , Time → Room , β 1 ) Kiwi Mon, 1pm Pete Buff α 1 Kiwi Tue, 2pm Pete Buff α 1 ( Room , Time → Project , β 2 ) Lion Tue, 4pm Gill Buff α 1 ( Project , Time → Manager , β 3 ) Lion Wed, 9am Gill Cyan α 1 ( Project → Manager , β 4 ) Lion Wed, 11am Bob Cyan α 2 Lion Wed, 11am Jack Cyan α 3 Lion Wed, 11am Pam Lava α 3 Tiger Wed, 11am Pam Lava α 4

Computational Support for Finding Meaningful pFDs Definition (Possibilistic Armstrong Relation) A p-relation r over a p-schema ( R , α 1 > · · · > α k +1 ) is Armstrong for a given set Σ of pFDs over the p-schema if and only if for all pFDs ϕ over the p-schema, r satisfies ϕ if and only if Σ implies ϕ . Possibilistic Armstrong Relation for Σ R = { Project , Time , Manager , Room } Project Time Manager Room α i β 1 > β 2 > β 3 > β 4 > β 5 Eagle Mon, 9am Ann Aqua α 1 Σ consists of: Hippo Mon, 1pm Ann Aqua α 1 ( Manager , Time → Room , β 1 ) Kiwi Mon, 1pm Pete Buff α 1 Kiwi Tue, 2pm Pete Buff α 1 ( Room , Time → Project , β 2 ) Lion Tue, 4pm Gill Buff α 1 ( Project , Time → Manager , β 3 ) Lion Wed, 9am Gill Cyan α 1 ( Project → Manager , β 4 ) Lion Wed, 11am Bob Cyan α 2 Lion Wed, 11am Jack Cyan α 3 Lion Wed, 11am Pam Lava α 3 Tiger Wed, 11am Pam Lava α 4

Computational Support for Finding Meaningful pFDs Definition (Possibilistic Armstrong Relation) A p-relation r over a p-schema ( R , α 1 > · · · > α k +1 ) is Armstrong for a given set Σ of pFDs over the p-schema if and only if for all pFDs ϕ over the p-schema, r satisfies ϕ if and only if Σ implies ϕ . Possibilistic Armstrong Relation for Σ R = { Project , Time , Manager , Room } Project Time Manager Room α i β 1 > β 2 > β 3 > β 4 > β 5 Eagle Mon, 9am Ann Aqua α 1 Σ consists of: Hippo Mon, 1pm Ann Aqua α 1 ( Manager , Time → Room , β 1 ) Kiwi Mon, 1pm Pete Buff α 1 Kiwi Tue, 2pm Pete Buff α 1 ( Room , Time → Project , β 2 ) Lion Tue, 4pm Gill Buff α 1 ( Project , Time → Manager , β 3 ) Lion Wed, 9am Gill Cyan α 1 ( Project → Manager , β 4 ) Lion Wed, 11am Bob Cyan α 2 Lion Wed, 11am Jack Cyan α 3 Lion Wed, 11am Pam Lava α 3 Perhaps, ( Project → Manager , β 3 ) should hold? Tiger Wed, 11am Pam Lava α 4

Computing Possibilistic Armstrong Relations maximal sets max Σ i ( R ) of R for FD set Σ i the subsets X ⊆ R such that for some attribute A ∈ R , X is maximal with the property that X → A is not implied by Σ i

Computing Possibilistic Armstrong Relations maximal sets max Σ i ( R ) of R for FD set Σ i the subsets X ⊆ R such that for some attribute A ∈ R , X is maximal with the property that X → A is not implied by Σ i As classically: introduce tuples that realize maximal sets

Computing Possibilistic Armstrong Relations maximal sets max Σ i ( R ) of R for FD set Σ i the subsets X ⊆ R such that for some attribute A ∈ R , X is maximal with the property that X → A is not implied by Σ i As classically: introduce tuples that realize maximal sets Strategy: For i = 1 , . . . , k , compute max i ( R ) for Σ i − Σ i − 1 with Σ 0 = ∅

Computing Possibilistic Armstrong Relations maximal sets max Σ i ( R ) of R for FD set Σ i the subsets X ⊆ R such that for some attribute A ∈ R , X is maximal with the property that X → A is not implied by Σ i As classically: introduce tuples that realize maximal sets Strategy: For i = 1 , . . . , k , compute max i ( R ) for Σ i − Σ i − 1 with Σ 0 = ∅ For i = k , . . . , 1, realize sets in max i ( R ) with tuples of degree α k +1 − i

Computing Possibilistic Armstrong Relations maximal sets max Σ i ( R ) of R for FD set Σ i the subsets X ⊆ R such that for some attribute A ∈ R , X is maximal with the property that X → A is not implied by Σ i As classically: introduce tuples that realize maximal sets Strategy: For i = 1 , . . . , k , compute max i ( R ) for Σ i − Σ i − 1 with Σ 0 = ∅ For i = k , . . . , 1, realize sets in max i ( R ) with tuples of degree α k +1 − i Algorithm returns a p-Armstrong relation for input pFD set Σ since every maximal set of Σ i is an agree set in r k +1 − i