When Can We Answer Queries Using Result-Bounded Data Interfaces?

(1)

When Can We Answer Queries

Using Result-Bounded Data Interfaces?

Antoine Amarilli¹, Michael Benedikt² December 10th, 2018

1Télécom ParisTech

2Oxford University

(2)

Problem: Answering Queries Using Web Services

Directory service DBLP service

Query

Findresearchers from adepartment

Findpapers from aresearcher

Find allpapers written byresearchers from mydepartment?

•

We have severalWeb servicesthat expose data

•

We want to answer aqueryusing the Web services

→ How can werephrasethe query against the Web services?

2/16

(3)

Directory service DBLP service Query

?

•

(4)

Directory service DBLP service Query

?

•

2/16

(5)

Formalizing the Problem

Service schema:

DBLP(author,title,year)

Input author? Michael Benedikt OK

author title year

Michael Benedikt Goal-Driven Query Answering ... 2018 Michael Benedikt Form Filling Based on ... 2018 Michael Benedikt How Can Reasoners Simplify ... 2018 Michael Benedikt When Can We Answer Queries ... 2018

. . . . . . . . .

•

Model each service as arelation

•

Some attributes areinputs

•

Access the service by giving abindingfor the input attributes

•

The service returnsall tuples that match thebinding

Query:conjunctive query over the relations

?

Find all papers written by researchers from my department? Q(t) :∃a yDirectory(MyDept,a)∧DBLP(a,t,y)

Constraints:express logical relationships between the services

!

Every researcher from the directory is in DBLP Σ :∀d aDirectory(d,a)→ ∃t y DBLP(a,t,y)

(6)

Service schema:

. . . . . . . . .

•

?

!

3/16

(7)

Service schema:

Input author? OK

. . . . . . . . .

•

?

!

(8)

Service schema:

. . . . . . . . .

•

?

!

3/16

(9)

Service schema:

. . . . . . . . .

•

?

!

(10)

Service schema:

. . . . . . . . .

•

?

Find all papers written by researchers from my department?

Q(t) :∃a yDirectory(MyDept,a)∧DBLP(a,t,y)

!

3/16

(11)

Service schema:

. . . . . . . . .

•

?

Find all papers written by researchers from my department?

Q(t) :∃a yDirectory(MyDept,a)∧DBLP(a,t,y)

!

Every researcher from the directory is in DBLP

Σ :∀d aDirectory(d,a)→ ∃t y DBLP(a,t,y) ^3/16

(12)

Existing Solutions

Given theservice schemaS, thequeryQand theconstraintsΣ we want to find amonotone planforQusing the services ofS

Example:Find all papers written by researchers from my department? withDirectory(department,person) andDBLP(author,title,year) What is amonotone plan?

•

^{Access the}^services^{by giving}^bindings

•

Evaluate monotonerelational algebra

•

The plan iscorrectif it returnsQ(D) on any databaseDthat satisfiesΣ

Example:

T₁⇐Directory⇐MyDept; T₂⇐DBLP⇐π_person(T₁); T₃⇐π_title(T₂);

ReturnT₃

Extensive literatureabout how to reformulate queries to monotone plans and about thecomplexity

depending on the constraint language

4/16

(13)

Example:Find all papers written by researchers from my department?

withDirectory(department,person) andDBLP(author,title,year)

What is amonotone plan?

•

Example:

T₁⇐Directory⇐MyDept; T₂⇐DBLP⇐π_person(T₁); T₃⇐π_title(T₂);

ReturnT₃

(14)

withDirectory(department,person) andDBLP(author,title,year) What is amonotone plan?

•

Example:

T₁⇐Directory⇐MyDept;

T₂⇐DBLP⇐π_person(T₁); T₃⇐π_title(T₂);

ReturnT₃

4/16

(15)

withDirectory(department,person) andDBLP(author,title,year) What is amonotone plan?

•

Example:

T₁⇐Directory⇐MyDept;

T₂⇐DBLP⇐π_person(T₁); T₃⇐π_title(T₂);

ReturnT₃

(16)

New Challenge: Result Bounds

•

Real Web services do not returnall matching tuplesfor accesses!

→ Plan results arenondeterministicand may not becorrect! Formalization:DBLP(author,title,year) has aresult boundof 1000

•

If an access matches≤1000tuples then they areallreturned

•

If it matches>1000tuples then we get1000of them (random)

→How to reformulate queries using result-bounded services?

5/16

(17)

•

(18)

•

→ Plan results arenondeterministicand may not becorrect!

Formalization:DBLP(author,title,year) has aresult boundof 1000

•

5/16

(19)

•

(20)

Formal Problem Statement Input:

•

Service schemaSof relation names and attributes withinput attributesand optionally aresult bound

→ Directory(department,person)

→ DBLP(author,title,year) with bound 1000

•

Conjunctive queryQ

→ Q(t) :∃a yDirectory(MyDept,a)∧DBLP(a,t,y)

•

ConstraintsΣ

→ ∀d aDirectory(d,a)→ ∃t yDBLP(a,t,y)

Output:

•

Is there a monotone plan forQonSunderΣ? We study:

→ What is thecomplexityof deciding plan existence, depending on the constraint language?

→ Inwhich waysare result-bounded services useful?

6/16

(21)

•

Conjunctive queryQ

•

ConstraintsΣ

→ ∀d aDirectory(d,a)→ ∃t yDBLP(a,t,y) Output:

•

Is there a monotone plan forQonSunderΣ?

We study:

→ Inwhich waysare result-bounded services useful?

(22)

•

Conjunctive queryQ

•

ConstraintsΣ

→ ∀d aDirectory(d,a)→ ∃t yDBLP(a,t,y) Output:

•

Is there a monotone plan forQonSunderΣ? We study:

→ Inwhich waysare result-bounded services useful? ^6/16

(23)

Summary of Results

→ We showschema simplificationresults that describe when result bounds on services can beremoved

→ We showcomplexity resultsfor many constraint languages on deciding the existence of monotone plans

Fragment Simplification Complexity

Inclusion dependencies(IDs) Existence-check EXPTIME-complete Bounded-width IDs Existence-check NP-complete Functional dependencies(FDs) FD NP-complete

FDs and UIDs Choice NP-hard, inEXPTIME

Equality-free FO Choice Undecidable

Frontier-guarded TGDs Choice 2EXPTIME-complete

→Let’s see theschema simplification resultsand proof techniques

(24)

7/16

(25)

(26)

→Let’s see theschema simplification resultsand proof techniques_7/16

(27)

Existence-Check Simplification

Idea:use result-bounded services tocheck the existenceof tuples

•

Schema: DBLP(author,title,year) with bound 1000

•

^Query^Q^:Has Michael Benedikt published something?

•

^Plan:^access^DBLPwith “Michael Benedikt” and check if empty

A schemaSwith constraintsΣisexistence-check simplifiableif any query that has a plan onSunderΣstill has a plan

on theexistence-check approximation:

•

For each relationDBLP(author,title,year) with a result bound, create a new relationDBLP_check(author)

•

Add two IDs inΣto relateDBLP_check andDBLP:

∀aDBLP_check(a)↔ ∃t yDBLP(a,t,y)

•

Forbid direct accesses toDBLP(so the result bound is irrelevant)

(28)

•

^Plan:^access^DBLPwith “Michael Benedikt” and check if empty A schemaSwith constraintsΣisexistence-check simplifiableif any query that has a plan onSunderΣstill has a plan

•

8/16

(29)

•

(30)

•

8/16

(31)

•

(32)

Existence-Check Simplification Results

Theorem

Any schemaSwith constraintsΣinInclusion dependencies(IDs) isexistence-check simplifiable

→Under IDs, result-bounded services only serve asexistence checks

As the existence-check approximation hasno result bounds, we can reduce to the classical setting and deduce:

Corollary

For any schemaSwith result bounds,IDconstraintsΣ, and queryQ, deciding the existence of a monotone plan isEXPTIME-complete

9/16

(33)

Existence-Check Simplification Results

Theorem

Any schemaSwith constraintsΣinInclusion dependencies(IDs) isexistence-check simplifiable

→Under IDs, result-bounded services only serve asexistence checks As the existence-check approximation hasno result bounds,

we can reduce to the classical setting and deduce:

Corollary

For any schemaSwith result bounds,IDconstraintsΣ, and queryQ, deciding the existence of a monotone plan isEXPTIME-complete

(34)

FD Simplification

Result-bounded services cando morethan existence checks:

•

^Schema^S: directory that returnsaddressesandphone numbers

→ Dir2(name,address,phone) with bound 1000

•

Constraints:aFunctional dependency(FD)φ:name→address

→ Each person has at most one address

•

^Query:Find the address of “Michael Benedikt”

•

^Plan: ^access^Dir2, return theaddressof any obtained tuple We callSandΣFD-simplifiableif any query that has a plan onSandΣstill has one on theFD approximation:

•

For each relationDir2(name,address,phone) with result bound, create a new relationDir2_FD(name,address) that outputs

the attributesdeterminedinΣby the input attributes

•

Forbid accesses onDir2and add IDs withDir2_FDlike before

∀n aDir2_FD(n,a)↔ ∃pDir2(n,a,p)

10/16

(35)

•

(36)

•

10/16

(37)

•

^Plan:^access^Dir2, return theaddressof any obtained tuple

We callSandΣFD-simplifiableif any query that has a plan onSandΣstill has one on theFD approximation:

•

(38)

•

^Plan:^access^Dir2, return theaddressof any obtained tuple We callSandΣFD-simplifiableif any query that has a plan onSandΣstill has one on theFD approximation:

•

10/16

(39)

•

(40)

•

10/16

(41)

FD Simplification Results

Theorem

Any schemaSwith constraintsΣinFunctional dependencies(FDs) isFD simplifiable

→ Under FDs, result-bounded services are only useful to access outputs that arefunctionally determined (i.e., we are guaranteed to have only one result)

Again, there areno result boundsleft in the FD approximation, so we can use this result to show:

Corollary

For any schemaSwith result bounds,FDconstraintsΣ, and queryQ, deciding the existence of a monotone plan isNP-complete

(42)

FD Simplification Results

Theorem

Any schemaSwith constraintsΣinFunctional dependencies(FDs) isFD simplifiable

→ Under FDs, result-bounded services are only useful to access outputs that arefunctionally determined (i.e., we are guaranteed to have only one result)

Again, there areno result boundsleft in the FD approximation, so we can use this result to show:

Corollary

For any schemaSwith result bounds,FDconstraintsΣ, and queryQ, deciding the existence of a monotone plan isNP-complete

11/16

(43)

Choice Simplification

With expressive constraints, theFD approximationisnot enough:

Lemma

There is a service schemaS, queryQ, andTGDsΣsuch that Qisnot FD-simplifiable(hence, notexistence-check-simplifiable)

A less drastic simplification is thechoice simplification:

•

For every service with a result bound, change the bound to be1

→Intuition:It’s important to getsome tupleif one exists

(44)

Choice Simplification

With expressive constraints, theFD approximationisnot enough:

Lemma

There is a service schemaS, queryQ, andTGDsΣsuch that Qisnot FD-simplifiable(hence, notexistence-check-simplifiable) A less drastic simplification is thechoice simplification:

•

For every service with a result bound, change the bound to be1

→Intuition:It’s important to getsome tupleif one exists

12/16

(45)

Choice Simplification Results

Theorem

Any schemaSwith constraintsΣinequality-free first-order logic (e.g., TGDs) ischoice simplifiable

Thus, plan existence is decidable fordecidable FO fragments, e.g., frontier-guarded TGDs(FGTGDs):

Corollary

For any schemaSwith result bounds, queryQ, andFGTGDsΣ deciding the existence of a monotone plan is2EXPTIME-complete We can also show choice approximability for another fragment:

Theorem

Any schemaSwith constraintsΣthat areFDsandunary IDs(UIDs) ischoice simplifiable

→This implies that plan existence isdecidablefor FDs and UIDs

(46)

Theorem

Corollary

For any schemaSwith result bounds, queryQ, andFGTGDsΣ deciding the existence of a monotone plan is2EXPTIME-complete

We can also show choice approximability for another fragment: Theorem

13/16

(47)

Theorem

Corollary

For any schemaSwith result bounds, queryQ, andFGTGDsΣ deciding the existence of a monotone plan is2EXPTIME-complete We can also show choice approximability for another fragment:

Theorem

(48)

Overview of Proof Techniques

•

Show that result bounds can be axiomatized insimpler ways:

• Ensure that doing thesame accesstwice returns thesame result

• Only write thelower bound:“ifiresults exist theniare returned”

•

Show that plan existence can be rephrased asanswerability:

→ If a databaseIsatisfiesQandI⁰ hasmore accessible datathanI thenI⁰should satisfyQas well

•

Show simplification results using ablowup technique:

• Start with acounterexample to answerabilityon the simplification

• Blow it upto a counterexample on the original schema

•

^{Reduce to}query containment under constraints

→ Study the result of the translation to show complexity bounds

14/16

(49)

•

→ If a databaseIsatisfiesQandI⁰ hasmore accessible datathanI thenI⁰ should satisfyQas well

•

(50)

•

14/16

(51)

•

(52)

Other Results in the Paper

•

Better complexity bounds using alinearization technique for query containment underIDs + side information

Theorem

Plan existence underbounded-width IDsisNP-complete

Theorem

Plan existence underUIDs and FDsisNP-hardandinEXPTIME

•

^{Result for}arity-2 constraints Theorem

Plan existence isdecidableforguarded two-variable FO + counting

•

Results when the database is assumed to befinite

•

Results fornon-monotone plans(= with relational difference)

•

Example of FO constraints that arenot choice simplifiable

15/16

(53)

•

Better complexity bounds using alinearization technique for query containment underIDs + side information Theorem

Theorem

•

(54)

•

Theorem

•

15/16

(55)

•

Theorem

•

(56)

•

Theorem

•

15/16

(57)

•

Theorem

•

(58)

•

Theorem

•

Example of FO constraints that arenot choice simplifiable _15/16

(59)

Summary and future work

•

^Problem:Given a schema of services withresult bounds, logical constraints, and a query, is there a plan to answer it?

•

Simplification resultsto remove bounds, andcomplexity results

→ What are other possible uses of thelinearization technique?

→ Do the bounds matter inpractice? (approximate plans?) Thanks for your attention!

(60)

•

Simplification resultsto remove bounds, andcomplexity results Fragment Simplification Complexity

→ Do the bounds matter inpractice? (approximate plans?) Thanks for your attention!

16/16

(61)

•

→ Do the bounds matter inpractice? (approximate plans?)

Thanks for your attention!

(62)

•

→ Do the bounds matter inpractice? (approximate plans?)

Thanks for your attention! ^16/16

(63)

References

Benedikt, M., Leblay, J., Cate, B. t., and Tsamoura, E. (2016).

Generating plans from proofs: the interpolation-based approach to query reformulation.

Morgan & Claypool.