The Semiring Framework for Database Provenance - PowerPoint Presentation

Slide1

The Semiring Framework for Database Provenance

5/21/2018

1

UW

Val Tannen

University of Pennsylvania

Slide2

5/21/20182UW

Collaborators

ORCHESTRA

TJ Green

RelationalAI

(was @

LogicBlox

)

Grigoris

Karvounarakis

RelationalAI

(was @

LogicBlox

)

Zack Ives

University of Pennsylvania

Other core papers

Nate Foster

Cornell University

Yael

Amsterdamer

Bar-

Ilan

University

Daniel

Deutch

Tel Aviv University

Tova

Milo

Tel Aviv University

Sudeepa

Roy

Duke University

Yuval

Moskovitch

Tel Aviv University

Slide3

5/21/2018For many, tracking data provenance means maintaining an “activity log”: record accessing of data items.

Many are satisfied, depending on what is recorded in each log entry!

A new,

both declarative (logic-based) and processing-based

, perspective arose in databases in the last 15+ years with the work of

Peter

Buneman

and others, including us.

This new perspective allows us to be more ambitious about applications of provenance

analysis. (As in algorithmic analysis.)

3

UW

Data provenance

Slide4

Binary trust5/21/2018UW4

mouse

gray

mouse

red

rat

gray

* Sue and Val are noted zoologists. ** Zack is a noted

computational

zoologist

cat

mouse

cat

rat

Sue’s notes *

Val’s notes *

cat

gray

cat

red

Zack **

computation

Yes

No

Yes

Yes

Yes

Yes

No

No

NoYes

food color

Slide5

Access control5/21/2018UW5

mouse

gray

mouse

red

rat

gray

Pub < Conf < Sec <

TSec

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

computation

TSec

TSec

Conf

Pub

Pub

Conf

TSec

Slide6

Confidence scores (non-binary trust)5/21/2018UW6

mouse

gray

mouse

red

rat

gray

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

computation

0.6

0.1

0.8

0.9

0.9

0.72

0.09

0.72 = max(0.9

× 0.8, 0.9 × 0.6)

0.09 = 0.9 × 0.1

Slide7

A simple model for data pricing5/21/2018UW7

mouse

gray

mouse

red

rat

gray

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

computation

$6

$1

$8

$10

$10

$16

$11

16 = min(10 +

8, 10 + 6)

11 = 10 + 1

Slide8

5/21/20188UWDo it once and use it repeatedly: provenance

Label (annotate) input items abstractly with

provenance tokens.

Provenance tracking

: propagate

expressions

(involving tokens)

(to annotate intermediate data and, finally, outputs)

Track

two distinct ways of using data items by computation primitives:

jointly

(this alone is basically like keeping a log)

alternatively

(doing both is essential; think trust)

Input-output compositional; Modular (in the primitives)

Later, we want to evaluate the provenance expressions to obtain binary trust, access control, confidence scores, data prices, etc.

Slide9

Database Provenance

V

R

View

V

=

Query

(

R

)

?

?

5/21/2018

UW

9

a

b

cd

b e

f g ea ca e

d

cd ef e A B C A C

Slide10

Provenance by propagating annotation of data items5/21/2018UW10

JOIN (on B)

…

a

b

c

…

p

The annotation

p

¢

r

means joint use of data annotated by p

and data annotated by r …a b c d

e

p

¢ r…R

R

⋈ S

S

…d b e …

r A B C

D B E A B C D E

Slide11

Another way to propagate annotations5/21/2018UW11

UNION

…

a

b

c

…

p

The annotation

p

+

r

means

alternative use

of data

…a b cp

+

r

…RR [ SS

…

a

b

c …r

A B C A B C A B C

Slide12

Another use of + 5/21/2018UW

12

…

a

b

c

1

p

…

a

b

c

2

r

…

a

b

c

3

…

s

…

a

bp + r + s

…

+

means alternative use of data PROJECT R

A B C ¼ABR A B

Slide13

Example: a positive querySPJU / UCQ / nonrecursive Datalog5/21/2018UW

13

a

b

c

p

d

b

e

r

f

g

e

s

a

c

(

p ¢ p + p

¢

p

) ¢ 0a e(p ¢ r ) ¢ 1

d

c

(r ¢ p ) ¢ 0d

e( r ¢ r +

r ¢ s + r ¢ r

) ¢ 1f e(

s

¢

s

+

s

¢

r

+

s

¢

s

)

¢

1

For selection we multiply

with two special annotations, 0 and 1

V

=



C=e

¼

AC

(

¼

AC

R

⋈

¼

BC

R

[

¼

AB

R

⋈

¼

BC

R )

R

A B C

A C

V

Slide14

Summary so farA space of annotations, K

K

-

relations

: every

tuple

annotated with some element from

K

.

Binary operations on K

: ¢

corresponds to joint use (

join

), and

+

corresponds to alternative use (union and projection).

We assume K contains special annotations 0 and 1. ‘‘Absent’’ tuples are annotated with

0!

1 is a ‘‘neutral’’ annotation (no restrictions).Algebra of annotations? What are the laws of (K, +, ¢, 0, 1) ?5/21/2018UW14

Slide15

Annotated relational algebraHence, for each commutative semiring K we have a K-

annotated relational algebra.

Proposition

. Above identities hold for queries on

K

-relations

iff

(

K

, +, ¢, 0, 1) is a commutative semiring15

5/21/2018

UW

DBMS query optimizers assume certain

equivalences

:

union is associative, commutative

join is associative, commutative, distributes over union

projections and selections commute with each other and with union and join (when applicable)Etc., but no R ⋈ R = R

[ R =

R (i.e., no idempotence, to allow for bag semantics)Equivalent queries should produce same annotations!

Slide16

What is a commutative semiring?An algebraic structure (K, +,

¢, 0, 1) where:

K is the domain

+ is associative, commutative, with

0

identity

¢

is associative, with

1

identity

semiring ¢ distributes over + a ¢ 0 = 0 ¢ a = 0

¢ is also

commutative

Unlike ring, no requirement for inverses to

+

16

5/21/2018

UW

Slide17

But are there useful commutative semirings?

(

B,

Ç,

Æ

,

?

,

>

)Set semantics(ℕ, +, ∙, 0, 1)Bag semantics

(BoolExp(

X),

Ç

,

Æ

, ?, >)Conditional tables (c-tables) [ImielinskiLipski 84](P(), [, Å, ;, )Probabilistic event tables [FuhrRölleke 97, Zimányi 97]175/21/2018UW

They capture several semantics:

Slide18

Using the commutative semiring axioms (I)5/21/2018UW18

a

b

c

p

d

b

e

r

f

g

e

s

a

c

(

p

¢ p + p

¢ p

)

¢ 0a ep ¢ r ¢ 1

d

c

r

¢ p ¢ 0d e

(r ¢ r + r ¢

s + r ¢ r ) ¢

1f e(s ¢ s

+

s

¢

r

+

s

¢

s

)

¢

1

R

A B C

A C

A C

A C

V

Slide19

Using the commutative semiring axioms (II)5/21/2018UW19

a

b

c

p

d

b

e

r

f

g

e

s

R

A B C

A Ca e

p ¢

r

d er ¢ r + r ¢ s

+

r

¢ r

f es ¢ s +

s ¢ r + s ¢ s

A C A C V

Slide20

Using … axioms (III): provenance polynomials5/21/2018UW20

a

b

c

p

d

b

e

r

f

g

e

s

R

A B C

A C A Ca e

pr

d

e2r2 + rsf e

rs

+ 2s2

A CMultivariate polynomials with - annotation tokens as indeterminates p, r, s - coefficients in N .Polynomials capture a remarkably general form of provenance.

V

Slide21

Provenance polynomials (I)5/21/2018UW21

a

b

c

p

d

b

e

r

f

g

e

s

R

A B C

A C A Ca e

pr

d

e2r2 + rsf e

rs

+ 2s2

A C VProvenance reading:Three ways to derive (d e): - two of them use r, twice, - the third uses r and s

, once each V = C=e¼AC( ¼ACR ⋈ ¼BCR [ ¼ABR ⋈

¼BCR )

Slide22

Provenance polynomials (II)5/21/2018UW22

(

N

[

X

], +,

¢

, 0, 1)

is the commutative

semiring

freely generated by X (universality property involving homomorphisms)Provenance polynomials are PTIME-computable (data complexity)

.

(query complexity depends on language and representation)

ORCHESTRA provenance (graph representation) about

30%

overhead

Monomials correspond to

logical derivations

(proof trees in non-rec. Datalog)

Slide23

Low-hanging fruit: deletion propagation5/21/2018UW23

a

b

c

p

d

b

e

r

f

g

e

s

R

A B C

A Ca epr

d

e2r2 + rsf ers + 2s2

Delete

(

d

b e) from R ?Set r = 0 ! a e

0d e0

f e2s2

f e2s2

A C

A C

We used this in

Orchestra

for update propagation

Q

Q

Q

Slide24

Wrong answers: diagnostic and repairs5/21/2018UW24

mouse

gray

mouse

red

rat

gray

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

Zack(x,z):-Sue(x,y),Val(y,z

)

r

s

t

p

q

p

∙

r+q

∙

tp∙s

Diagnostic: provenance of wrong answer: p∙r+q∙tFour minimal ways to make p∙r+q∙t=0p=q=0; r=q=0

p

=

t

=0;

r

=

t

=0

(maybe choose based on confidence or cost)

On planet

Erythro

, mice and rats are red!

Wrong answer!

Slide25

Specialize provenance for access control5/21/2018UW25

mouse

gray

mouse

red

rat

gray

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

Zack(x,z):-Sue(x,y),Val(y,z

)

r

s

t

p

q

pr+qt

ps

(

A

, min, max,

0,Pub) where A = Pub < Conf < Sec < TSec < 0X  A p, q Pub r,s  TSec

t



Conf

eval

:

N

[

X

]



A

eval(pr+qt

)=

Conf

eval(ps

)=

TSec

TSec

TSec

Conf

Conf

TSec

Pub

Pub

Slide26

Specialize provenance for confidence scores5/21/2018UW26

mouse

gray

mouse

red

rat

gray

cat

mouse

cat

rat

Sue’s notes

Val’s notes

cat

gray

cat

red

Zack

Zack(x,z):-Sue(x,y),Val(y,z

)

r

s

t

p

q

pr+qt

ps

V

= (

[0,1],

max, ∙, 0, 1) the Viterbi semiring0.60.1

0.8

0.72

0.09

0.9

0.9

Slide27

Some application semirings5/21/2018UW27

(

B

, Æ

,

Ç

,

>

,

?

) binary trust(N, +, ¢, 0, 1) multiplicity (number of derivations)(A, min, max, 0, Pub) access control

V

= ([0,1],

max,

∙

, 0, 1)

Viterbi

semiring (HMM) confidence scoresT = ([0, 1], min, +, 1, 0) tropical semiring (shortest paths) data pricingF = ([0,1], max, min, 0, 1) “fuzzy logic” semiring

Slide28

A menagerie of provenance semirings5/21/2018UW28

(

Which(X),

[, [*

,

;

,

;

*

) sets of contributing

tuples “Lineage” (1) [CWW00](Why(X), [, d, ;, {;}) sets of sets of … Witness why-provenance [BKT01](PosBool(X), Æ, Ç

, >, ?

) minimal sets of sets of… Minimal witness why-provenance [BKT01] also “Lineage” (2) used in probabilistic

dbs

[SORK11]

(

Trio(

X), +, ¢, 0, 1) bags of sets of … “Lineage” (3) [BDHT08,G09](B[X],+, ¢, 0, 1) sets of bags of … Boolean coeff. polynomials [G09](Sorp(X),+, ¢, 0, 1) minimal sets of bags of … absorptive polynomials [DMRT14](N[X], +, ¢, 0, 1) bags of bags of… universal provenance polynomials [GKT07]

Slide29

Two kinds of semirings in this framework5/21/2018UW29

Provenance

semirings

, e.g.,

(

N

[

X

], +,

¢

, 0, 1) provenance polynomials [GKT07](Why(X), [, d, ;, {;}) witness why-provenance [BKT01]Application semirings, e.g.,

(A, min, max, 0, Pub) access control

[FGT08]

V

= (

[0,1],

max,

∙, 0, 1) Viterbi semiring (HMM) [GKIT07]Provenance specialization relies on- Provenance semirings are freely generated by provenance tokens Query commutation with semiring homomorphisms

Slide30

Query commutation with homomorphisms query in QL homomorphism

h

:

K1



K

2

5/21/2018UW30K

1-Rel

K

1

-Rel

query

query

h

hK2-RelK2-Rel

QL

= RA+, Datalog [GKT07] and extensions [FGT08, GP10, ADT11a, T13, DMT15, GUKFC16, T17]

Slide31

A Hierarchy of Provenance Semirings [G09, DMRT14]N[X]

B

[X

]

Trio(

X

)

Why(

X

)

Which(X)PosBool(X)

most informative

least informative

Example:

2

x

2

y

+

xy + 5y2 +

xz + = 315/21/2018UWSorp(X)surjective semiring homomorphism, identity on X

absorption

absorption (

ab+a=a)  idemp. + idemp. x2y + xy + y2 + xz3xy + 5y + xzy + x

z

xy + y2+

xzxyz

 idemp.xy + y + xz  idemp. + idemp.A

T,VNB

Slide32

5/21/2018UW32

Slide33

From RA+ to DatalogImmediate consequence operator F

of a

Datalog program.

Incorporates the

edb

predicates, maps

idb

predicates to

idb

predicates.

It’s expressible in RA+. E.g., transitive closure F(T)

= E

[ ¼

1,3

(

E

⋈

T)Generalize to F: (K-Rel)n  (K-Rel)n (n=# of idb predicates)Solve certain (systems) of least fixed point equations over K-relations. T = F(T)

5/21/2018

UW33

Slide34

Find provenances of idb tuples

5/21/2018

UW

34

Slide35

Provenance of idb tuples

- introduce unknowns

Z

for the annotations of

idb

tuples

- solve system of fixed point equations over

K

; right-hand sides are polynomials in K[

Z].

Additional structure on

K

for these to have (unique) solutions?

5/21/2018

UW

35

Slide36

ω-continuous semirings

Semirings

K

such that the immediate consequence operator of any

Datalog

program has a least

fixpoint

on

K

-relations. Naturally ordered when x ≤

y

iff there exists

z

s.t

.

x+z = yis an order relation (all semirings seen here are naturally ordered)ω-complete also x0 ≤ x1 ≤ … ≤ xn ≤ … have l.u.b.’s (

sup’s)

ω-continuous moreover + and ¢ preserve those l.u.b.’s5/21/2018UW36

Slide37

Among our examplesMany of the semirings that interest us

B

,

T,

V

,

A

,

F

are already ω-continuous. (N, +,

¢

, 0, 1)

is not,

but its

“completion”

(N1= N [ {1} , +, ¢ , 0, 1) is.For provenance, the completion of N[X] is not

N1[X].

Instead of (finite) polynomials we need (possibly infinite) formal power series. They form an ω-continuous semiring N1[[X]]. Monomials still correspond to derivations trees. (Even transitive closure has infinitely many derivation trees if E has loops.)The completion of B[X] is B[[X]]. 5/21/2018UW

37

Slide38

Solution

5/21/2018

UW

38

Slide39

Absorptive polynomialsMost informative provenance semiring for Datalog

: (

N1

[[X]], +,

¢

, 0,1

)

(Infinite power series have finite representations as systems of polynomial equations.)

Absorption a + ab =

a

Absorptive polynomials Sorp(X

):

boolean

coefficients but only minimal degree monomials

x2y + xy + y2 + xz  xy + y2 + xzAbsorptive power series same as absorptive polynomials! Why? Order monomials by degree of each variable. In this infinite poset all antichains are finite! (Dickson’s Lemma)Sorp(X) is already ω

-continuous: provides provenance polynomials for Datalog.So is

PosBool(X), but Sorp(X) provenance also supports tropical and Viterbi semiring applications 5/21/2018UW39

Slide40

Provenance for aggregation5/21/2018

a 20+10

?

b

15+10+25

?

a 20

x

a

10

y

b

15

q

b

10

r

b

25

s

DesiderataCompatibility with set/bag semanticsFundamental property (commutation with homomorphisms)

Poly-size overhead! 1+2+4+…+2n-1 => 2n results

D S-

agg

D SSUM SGROUP BY D 40UW

Slide41

Solution inspired by (semi) linear algebra5/21/2018

a

x

20

+

y

10

?

b

q

15

+ r 10 +

s 25

?D S-agga 20x

a

10

y b 15q b 10

r

b

25

s D S41UW

(R, +, 0) is not a Prov(X)-semimodule, but…(K-Rel, [, ;) is a K-semimodule with the singletons as basis.Relations are the result of [-aggregation!What if (R, +, 0) were a Prov(X)-semimodule?

Slide42

Tensor product construction5/21/2018

a

x

⊗

20+

y

⊗

10

x

+

y

b

q

⊗15+r ⊗10+s ⊗2

5

q + r + sD S-aggEmbed a commutative monoid M (for sum, max or min) into a K-semimodule K ⊗

M (new values!)

Consistency: embedding should be faithful.42

UW

Slide43

Further aspects of the framework5/21/2018UW43

Extension to tree data (Nested Relational Calculus, structural recursion on trees, unordered

XQuery

) [FGT08]

Study of CQ/UCQ on provenance-annotated relations

[G09]

Extension to aggregates (poly-size overhead)

[ADT11a]

Poly-size provenance for

Datalog (circuits; PosBool(X

), Sorp(X)…)

[DMRT14]

Extension to data-dependent finite state processes

[DMT15]

Connections to

semiring

monad

[FGT08, T13] to semimodules [ADT11a] to tensor products [ADT11a, DMT15]

Slide44

Negative information; non-monotone operations (difference)5/21/2018UW44

Boolean expressions

[IL84].

Limited.

Add a binary operation corresponding to difference

m-semirings

(common gen. of set and bag difference)

[GP10]

spm-semirings (OPTIONAL in SPARQL) [GUKFC16] Encode difference by aggregation [ADT11a]

Different

equational theories, different algebraic optimizations

[ADT11b]

Still not clear how to track

negative information

. useful: non-answers (why not?), insertion propagation.

Logical model checking (“

provenance of … truth?”) negation as duality (NNFs), logical games ongoing work with Grädel [T16, T17]

Slide45

Current targets5/21/2018UW45

ANALYTICS COMPUTATIONS

“Fine-grained provenance for linear algebra operators”

Yan, T.,

Ives

TaPP

16

DISTRIBUTED SYSTEMS/NETWORK PROVENANCE

“Time-aware provenance for distributed systems”

, Zhou, Ding, Haeberlen, Ives, Loo TaPP 11“Diagnosing missing events in distributed systems with negative provenance”, Wu, Zhao, Haeberlen,

Zhou, Loo SIGCOMM 14

STATIC ANALYSIS OF SOFTWARE

“On abstraction refinement for program analyses in

Datalog

”

Zhang,

Mangal, Grigore, Naik PLDI 14

Slide46

Framework references (I) *5/21/2018UW46

[GKT07]

“Provenance semirings

” Green, Karvounarakis, Tannen PODS 07.

[GKIT07]

“Update exchange with mappings and provenance”

Green,

Karvounarakis

, Ives, Tannen VLDB 07.

[FGT08]

“Annotated XML: queries and provenance” Foster, Green, Tannen PODS 08.[G09]“Containment of conjunctive queries on annotated relations” Green ICDT 09.[GP10]“On database query languages for K-relations”, Geerts, Poggi J Appl. Logic 2010.* See also companion paper in PODS 2017 proceedings.

Slide47

Framework references (II)5/21/2018UW47

[ADT11a]

“Provenance for aggregate queries”, Amsterdamer

, Deutch, Tannen PODS 11.

[ADT11b]

“On the limitations of provenance for queries with difference”,

Amsterdamer

,

Deutch

, Tannen

TaPP 11[T13]“Provenance propagation in complex queries” Tannen Buneman Festschrift 2013[DMRT14]“Circuits for Datalog provenance”, Deutch, Milo, Roy, T. ICDT 14.[DMT15]“Provenance-based analysis of data-centric processes”Deutch, Moskovitch

, Tannen VLDB J. 2015

Slide48

Framework references (III)5/21/2018UW48

[GUKFC16]

“Algebraic structures for capturing the provenance of SPARQL queries”

Geerts, Unger, Karvounarakis

,

Fundulaki

,

Christophides

JACM 2016

[T16]

“About the provenance of truth” Tannen Simons Inst. Website 16https://simons.berkeley.edu/talks/val-tannen-2016-12-09[T17]“Provenance analysis for FOL model checking” Tannen SIGLOG News 2017

Slide49

Other references5/21/2018UW49

[IL84]

“Incomplete information in relational databases” Imieliński

, Lipski JACM 1984

[FR97]

“A probabilistic relational algebra”

Fuhr

,

Röllecke

TOIS 1997[Z97]“Query evaluation in probabilistic relational databases” Zimányi DDS 1997[CWW00]“Tracing the lineage of view data in a warehousing environment” Cui, Widom, Wiener TODS 2000[BKT01]“Why and where: a characterization of data provenance” Buneman, Khanna, Tan ICDT 2001[BDHTW08]

“Databases with uncertainty and lineage” Benjelloun, Das Sarma

, Halevy, Theobald, Widom

VLDB J. 2008

[SORK11]

“Probabilistic databases”

Suciu

,

Olteanu, Ré, Koch SLDM 2011 [SuciuOlteanuRéKoch 11]

Slide50

5/21/2018UW50

Thank you!