Database Tuples Play Cooperative Games

Ester Livshits

Joint work with:

Leopoldo Bertossi, Benny Kimelfeld, Alon Reshef, Moshe Sebag

Ester Livshits Oxford Data and Knowledge Seminar 2

AUTHOR

Name Affiliation

Alice UCLA

Bob NYU

Cathy MIT

David UCSD

Ellen NYU

INSTITUTE

Name STATE

UCLA CA

UCSD CA

NYU NY

MIT MA

PUBLICAION

Author Paper

Alice A

Alice B

Cathy C

Cathy D

David C

CITATIONS

PAPER CITS

𝑞 𝑧, 𝑤 :−AUTHOR 𝑥, 𝑦 , INSTITUE 𝑦, ′CA′ , PUBLICATION 𝑥, 𝑧 , CITATIONS 𝑧, 𝑤

PAPER CITS

Why we obtained a

particular answer?

Why we did not obtain

some other answer?

AUTHOR

Name Affiliation

Alice UCLA

Bob NYU

Cathy MIT

David UCSD

Ellen NYU

INSTITUTE

Name STATE

UCLA CA

UCSD CA

NYU NY

MIT MA

PUBLICAION

Author Paper

Alice A

Alice B

Cathy C

Cathy D

David C

CITATIONS

PAPER CITS

Why we obtained a

particular answer?

Why we did not obtain

some other answer?

Which tuples in the database

explain the query result?

Measuring Contribution

➢ Causal responsibility [Meliou et al. 2010]

❖ 𝑡 is a counterfactual cause for 𝑞 if 𝐷 ⊨ 𝑞 and 𝐷 ∖ {𝑡} ⊨ 𝑞

❖ 𝑡 is an actual cause for 𝑞 if 𝐷 ∖ Γ ⊨ 𝑞 and 𝐷 ∖ {Γ ∪ {𝑡}} ⊨ 𝑞for some Γ ⊆ 𝐷 ∖ {𝑡}

❖ The responsibility of 𝑡 is 1

1+|Γmin|

➢ Not extendable to aggregate queries

➢ May be counterintuitive

Is there a path from a to b?

Contingency

Measuring Contribution

➢ Causal effect [Salimi et al. 2016]

❖ See the database as a probabilistic database

❖ CE 𝑡 = E 𝑞 𝑡 ∈ 𝐷) − E 𝑞 𝑡 ∉ 𝐷)

What makes the choice of a contribution score a good one?

Shapley Value

➢ A widely known profit-sharing formula in cooperative game theory

➢ Introduced by Lloyd Shapley in 1953

➢ Applied in various areas beyond cooperative game theory:

❖ Pollution responsibility in environmental management

❖ Influence measurement in social network analysis

❖ Identifying candidate autism genes

❖ Bargaining foundations in economics

❖ Takeover corporate rights in law

❖ Explanations in machine learning

Shapley Value

Ester Livshits 8

Set 𝐴 of players: Wealth function 𝑣:𝒫 𝐴 → ℝ:

How to distribute the total

wealth among the players?

Machine learning

Query answering

Inconsistency

Features Prediction

Tuples Answer

Tuples Measure

[Lundberg, Lee 2017]

[L, Kimelfeld 2021]

[L et al. 2020]

Oxford Data and Knowledge Seminar

Shapley Value

Shapley 𝐴, 𝑣, 𝑎 =

𝐵⊆𝐴∖{𝑎}

𝐵 ! 𝐴 − 𝐵 − 1 !

𝐴 !𝑣 𝐵 ∪ 𝑎 − 𝑣 𝐵

The Shapley value is the expected delta

due to the addition in a random permutation

Shapley Value for Database Queries

➢ Which tuples in the database explain the query result?

AUTHOR

Name Affiliation

Alice UCLA

Bob NYU

Cathy MIT

David UCSD

Ellen NYU

INSTITUTE

Name STATE

UCLA CA

UCSD CA

NYU NY

MIT MA

PUBLICAION

Author Paper

Alice A

Alice B

Cathy C

Cathy D

David C

CITATIONS

PAPER CITS

𝑞 𝑧, 𝑤 :−AUTHOR 𝑥, 𝑦 , PUBLICATION 𝑥, 𝑧 , CITATIONS(𝑧, 𝑤)

SUM𝑤⟨𝑞 𝑧, 𝑤 ⟩

Players

Wealth function

𝑆𝑉 𝐴𝑙𝑖𝑐𝑒 = 20𝑆𝑉 𝐶𝑎𝑡ℎ𝑦 = 14.67𝑆𝑉 𝐵𝑜𝑏 = 2.67𝑆𝑉 𝐷𝑎𝑣𝑖𝑑 = 2.67𝑆𝑉 𝐸𝑙𝑙𝑒𝑛 = 0

AUTHOR

Name Affiliation

Alice UCLA

Bob NYU

Cathy MIT

David UCSD

Ellen NYU

INSTITUTE

Name STATE

UCLA CA

UCSD CA

NYU NY

MIT MA

PUBLICAION

Author Paper

Alice A

Alice B

Cathy C

Cathy D

David C

CITATIONS

PAPER CITS

AUTHOR

Name Affiliation

Alice UCLA

Bob NYU

Cathy MIT

David UCSD

Ellen NYU

INSTITUTE

Name STATE

UCLA CA

UCSD CA

NYU NY

MIT MA

PUBLICAION

Author Paper

Alice A

Alice B

Cathy C

Cathy D

David C

CITATIONS

PAPER CITS

𝑞():−AUTHOR 𝑥, 𝑦 , INSTITUE 𝑦, ′CA′ , PUBLICATION 𝑥, ′A′ , CITATIONS ′A′, 18

➢ Explaining Query Answers

➢ Computational Complexity

➢ Responsibility to Inconsistency

Outline

Computational Complexity

➢ A CQ 𝑞 is hierarchical if for every two existential variables 𝑥 and 𝑦:

❖ 𝐴𝑡𝑜𝑚𝑠 𝑥 ⊆ 𝐴𝑡𝑜𝑚𝑠 𝑦 or

❖ 𝐴𝑡𝑜𝑚𝑠 𝑦 ⊆ 𝐴𝑡𝑜𝑚𝑠 𝑥 or

❖ 𝐴𝑡𝑜𝑚𝑠 𝑥 ∩ 𝐴𝑡𝑜𝑚𝑠 𝑦 = ∅

𝑞1():−𝑅 𝑥, 𝑦 , 𝑆(𝑥, 𝑧)

Query Hierarchical Non-hierarchical

SJFCQ PTIME FP#P-complete

SJFCQ with

negationsPTIME FP#P-complete

sum \ count PTIME FP#P-complete

[L et al.

ICDT 2020]

[Reshef et al.

PODS 2020]

𝑞():−AUTHOR 𝑥, 𝑦 , INSTITUTE 𝑦, ′CA′ , PUBLICATION 𝑥, 𝑧

𝑦 𝑧

SJFCQ with

𝑞2():−𝑅 𝑥 , 𝑆 𝑥, 𝑦 , 𝑇(𝑦)

[L et al.

ICDT 2020]

[Reshef et al.

PODS 2020]

𝑞():−AUTHOR 𝑥, 𝑦 , INSTITUTE 𝑦, ′CA′ , PUBLICATION 𝑥, 𝑧

Conjunctive Queries

➢ To prove hardness, we consider the simplest non-hierarchical query

𝑞𝑅𝑆𝑇(): −𝑅 𝑥 , 𝑆 𝑥, 𝑦 , 𝑇(𝑦)

➢ Reduction from counting independent sets in a bipartite graph

Conjunctive Queries

➢ Each instance provides us with an equation over |IS(𝑔, 𝑘)|

➢ |IS(𝑔, 𝑘)| - number of independent sets of size 𝑘 in 𝑔

SJFCQ with

[L et al.

ICDT 2020]

[Reshef et al.

PODS 2020]

𝑞():−AUTHOR 𝑥, 𝑦 , ¬INSTITUTE 𝑦, ′CA′ , PUBLICATION 𝑥, 𝑧

SJFCQ with

[L et al.

ICDT 2020]

[Reshef et al.

PODS 2020]

𝑞 𝑧, 𝑤 :−AUTHOR 𝑥, 𝑦 , PUBLICATION 𝑥, 𝑧 , CITATIONS(𝑧, 𝑤)SUM𝑤⟨𝑞 𝑧, 𝑤 ⟩

SJFCQ with

[L et al.

ICDT 2020]

[Reshef et al.

PODS 2020]

𝑞 𝑧, 𝑤 :−AUTHOR 𝑥, 𝑦 , PUBLICATION 𝑥, 𝑧 , CITATIONS(𝑧, 𝑤)MAX𝑤⟨𝑞 𝑧, 𝑤 ⟩, MIN𝑤⟨𝑞 𝑧, 𝑤 ⟩, AVERAGE𝑤⟨𝑞 𝑧, 𝑤 ⟩

Hardness can be extended to

general numerical queries

➢ Computing the Shapley value is often hard

➢ The picture is more positive when allowing approximation

➢ Generalizes to unions of CQs

Approximation Complexity

Pr𝑓(𝑥)

1 + 𝜖≤ 𝐴 𝑥, 𝜖, 𝛿 ≤ (1 + 𝜖)𝑓(𝑥) ≥ 1 − 𝛿

SJFCQ PTIME FPRAS

sum \ count PTIME FPRAS

➢ Additive approximation via Monte Carlo sampling

➢ Also a multiplicative approximation due to the “gap property”

➢ Does not hold when allowing negation

➢ Negation fundamentally changes the complexity picture!

Pr 𝑓 𝑥 − 𝜖 ≤ 𝐴 𝑥, 𝜖, 𝛿 ≤ 𝑓 𝑥 + 𝜖 ≥ 1 − 𝛿

For every tuple 𝑡 in the database 𝐷:

Shapley(𝑡)=0 or Shapley(𝑡)≥1

𝑝(|𝐷|)

➢ With negation, the contribution can be negative

Register

Student Course

Alice OS

Alice AI

Bob OS

Cathy DB

Cathy IC

Student

𝑞(): −Student 𝑥 , ¬TA 𝑥 , Register(𝑥, 𝑦)

In some cases, deciding whether Shapley(𝑡)≠0 is hard

➢ Causal effect [Salimi et al. 2016]

❖ See the database as a probabilistic database

❖ CE 𝑡 = E 𝑞 𝑡 ∈ 𝐷) − E 𝑞 𝑡 ∉ 𝐷)

➢ Coincides with the Banzhaf Power Index [Banzhaf 1965]

➢ Our complexity results extend to this measure

Ester Livshits Oxford Data and Knowledge Seminar

Banzhaf Power Index

➢ Explaining Query Answers

➢ Computational Complexity

➢ Responsibility to Inconsistency

Outline

Inconsistent Databases➢ A database is inconsistent if it violates integrity constraints

Cullen Douglas

dbo:birthPlace

▪ dbr:California

▪ dbr:Florida

Marion Jones

dbo:height

▪ 1.524

▪ 1.778

Irene Tedrow

dbo:deathPlace

▪ dbr:California

▪ dbr:Hollywood,_Los_Angeles

▪ dbr:New_York_City

Inconsistent Databases

➢ Imprecise data sources

❖ Crowd, Web pages, social encyclopedias, sensors, …

➢ Imprecise data generation

❖ natural-language processing, sensor/signal processing, image recognition, …

➢ Conflicts in data integration

❖ Crowd + enterprise data + KB + Web + ...

➢ Data staleness

❖ Entities change address, status, ...

➢ And so on…

Quantify the extent to which

integrity constraints are violated

Reliability estimationHow reliable is a new data source?

Progress indicationProgress bar for data repairing

Action prioritizationWhich tuples are mostly

responsible for inconsistency?

How can we quantify the

responsibility of individual tuples

to inconsistency?

Inconsistency measure

Responsibility sharing

mechanism

to inconsistency?

mechanism

How to Measure Inconsistency?

➢ Several measures proposed by the KR and DB communities

❖ The drastic measure – 1 if inconsistent, 0 otherwise [Thimm 2017]

❖ #minimal inconsistent subsets [Hunter and Konieczny 2008]

❖ #problematic tuples [Grant and Hunter 2011]

❖ Minimal #tuples to remove to satisfy the constraints [Grant and Hunter 2013], [Bertossi 2018]

❖ #maximal consistent subsets [Grant and Hunter 2011]

➢ What makes a measure a good one? [L et al. SIGMOD 2021]

to inconsistency?

mechanism

Shapley Value

Measure lhs chainNo lhs chain,

tractable c-repairother

drastic PTIME FP#P-complete

inconsistentPTIME

#problematic

tuplesPTIME

cardinality

repairPTIME Open NP-hard

#repairs PTIME FP#P-complete

FD: birthCity → birthState

Tractable Measures

➢ 𝐼𝑀𝐼 - Number of minimal inconsistent subsets

Train Departs Arrives Time Duration

𝑓1 16 NYP BBY 1030 315

𝑓2 16 NYP PVD 1030 250

𝑓3 16 PHL WIL 1030 20

𝑓4 16 PHL BAL 1030 70

𝑓5 16 PHL WAS 1030 120

𝑓6 16 BBY PHL 1030 260

𝑓7 16 BBY NYP 1030 260

𝑓8 16 BBY WAS 1030 420

𝑓9 16 WAS PVD 1030 390

Train Time → Departs

Train Time Duration → Arrives

𝑓7 𝑓1 𝑓3 𝑓9 𝑓2 𝑓5 𝑓8 𝑓6

Tractable Measures

➢ 𝐼𝑀𝐼 - Number of minimal inconsistent subsets

𝑓1 16 NYP BBY 1030 315

𝑓2 16 NYP PVD 1030 250

𝑓3 16 PHL WIL 1030 20

𝑓4 16 PHL BAL 1030 70

𝑓5 16 PHL WAS 1030 120

𝑓6 16 BBY PHL 1030 260

𝑓7 16 BBY NYP 1030 260

𝑓8 16 BBY WAS 1030 420

𝑓9 16 WAS PVD 1030 390

𝑓7 𝑓1 𝑓3 𝑓9 𝑓2 𝑓5 𝑓8 𝑓6

𝑓 increases the value of 𝐼𝑀𝐼 by 𝑘 if

𝑘 of the previous tuples conflict with it

Tractable Measures

➢ 𝐼𝑃 - Number of problematic tuples

𝑓1 16 NYP BBY 1030 315

𝑓2 16 NYP PVD 1030 250

𝑓3 16 PHL WIL 1030 20

𝑓4 16 PHL BAL 1030 70

𝑓5 16 PHL WAS 1030 120

𝑓6 16 BBY PHL 1030 260

𝑓7 16 BBY NYP 1030 260

𝑓8 16 BBY WAS 1030 420

𝑓9 16 WAS PVD 1030 390

𝑓7 𝑓1 𝑓3 𝑓9 𝑓2 𝑓5 𝑓8 𝑓6

𝑓 increases the value of 𝐼𝑝 by 𝑘 if

(𝑘 − 1) of the previous tuples:

(1) conflict with 𝑓,

(2) do not conflict with other

tuples that occur before 𝑓.

inconsistentPTIME

#problematic

tuplesPTIME

cardinality

{𝑩 → 𝐴,𝑩𝑪 → 𝐷,𝑩𝑪𝑮 → 𝐸,𝑩𝑪𝑭 → 𝐻}

𝐵 ⊆ 𝐵, 𝐶 ⊆ {𝐵, 𝐶, 𝐹} 𝐵, 𝐶, 𝐺 ⊈ 𝐵, 𝐶, 𝐹 , {𝐵, 𝐶, 𝐹}⊈ 𝐵, 𝐶, 𝐺

{𝑩 → 𝐴,𝑩𝑪 → 𝐷,𝑩𝑪𝑭 → 𝐸}

Left-Hand Side Chain

𝑓1 16 NYP BBY 1030 315

𝑓2 16 NYP PVD 1030 250

𝑓3 16 PHL WIL 1030 20

𝑓4 16 PHL BAL 1030 70

𝑓5 16 PHL WAS 1030 120

𝑓6 16 BBY PHL 1030 260

𝑓7 16 BBY NYP 1030 260

𝑓8 16 BBY WAS 1030 420

𝑓9 16 WAS PVD 1030 390

Train, Time

Departs

Duration

NYP PHL BBY WAS

16, 1030

315 250 20 70 120 260 420 390

BBY PVD WIL BAL WAS PHL NYP WAS

Arrives

Independent

branchesConflicting

branches

inconsistentPTIME

#problematic

tuplesPTIME

cardinality

{𝑩 → 𝐴,𝑩𝑪 → 𝐷,𝑩𝑪𝑮 → 𝐸,𝑩𝑪𝑭 → 𝐻}

𝐵 ⊆ 𝐵𝐶 ⊆ {𝐵𝐶𝐹} 𝐵𝐶𝐺 ⊈ 𝐵𝐶𝐹 , {𝐵𝐶𝐹}⊈ 𝐵𝐶𝐺

{𝑩 → 𝐴,𝑩𝑪 → 𝐷,𝑩𝑪𝑭 → 𝐸}

Efficiency: σ𝑎∈𝐴 Shapley 𝐴, 𝑣, 𝑎 = 𝑣(𝐴)

drastic PTIME FPRAS

inconsistentPTIME

#problematic

tuplesPTIME

cardinality

repairPTIME FPRAS No FPRAS

#repairs PTIME Open

Would imply an FPRAS for #MIS in a bipartite

graph – long standing open problem

➢ Two situations where we wish to quantify the responsibility of tuples:

❖ Query answering

❖ Database inconsistency

➢ We treat the contribution from the viewpoint of game theory

➢ We investigated the computational complexity

Ester Livshits Oxford Data and Knowledge Seminar

Concluding Remarks

Thank you for listening!

Questions?

Database Tuples Play Cooperative Games

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