Enumerating MSO Query Results on Trees and Words

(1)

A Circuit-Based Approach to Efficient Enumeration:

Enumerating MSO Query Results on Trees and Words

Antoine Amarilli¹, Pierre Bourhis², Louis Jachiet², Stefan Mengel³, Matthias Niewerth⁴

May 20, 2019

1Télécom ParisTech

2CNRS CRIStAL

(2)

Roadmap of this talk

•

Stefan has presented enumeration ford-DNNF set circuits

•

^{This talk:}introduce MSO query evaluation on trees and build aset circuitthat represents the answers to enumerate

•

Using the previous talkwe can reprove the result: Theorem [Bagan, 2006, Kazana and Segoufin, 2013]

We can enumerate the answers of MSO queries on trees with linear-time preprocessing and constant delay.

•

^Also:new results oncombined complexity: Theorem [Amarilli et al., 2019a, Amarilli et al., 2019b]

We can enumerate the results of anautomatonon a tree with:

• Preprocessinglinearin the tree andpolynomialin the automaton

• Delayconstantin the tree andpolynomialin the automaton

•

^{Next talk:}how to supportupdates(and provenew results)

(3)

•

Using the previous talkwe can reprove the result: Theorem [Bagan, 2006, Kazana and Segoufin, 2013]

•

(4)

•

Using the previous talkwe can reprove the result:

Theorem [Bagan, 2006, Kazana and Segoufin, 2013]

•

(5)

•

(6)

•

(7)

Boolean MSO on trees

(8)

Boolean query evaluation on trees

Data: a treeTwhere nodes have a color from an alphabet

?

^Query^Q^{: a}^sentence^{in monadic}

second-order logic (MSO)

•P (x)means “xis blue”

•x→ymeans “xis the parent ofy”

“Is there both a pink and a blue node?”

∃x y P (x)∧P (y)

i

Result: TRUE/FALSE indicating if the treeTsatisfies the queryQ

Computational complexityas a function ofT (the queryQisfixed)

(9)

?

∃x y P (x)∧P (y)

i

(10)

?

∃x y P (x)∧P (y)

i

(11)

?

∃x y P (x)∧P (y)

i

Computational complexityas a function ofT

(12)

Monadic second-order logic (MSO)

•

^P ^(x)^{means “x}is blue”; alsoP (x),P (x)

•

^x^→^y^{means “}^xis the parent ofy”

•

Propositional logic:formulas withAND∧,OR∨,NOT¬

• P (x)∧P (y)means“Nodexis pink and nodeyis blue”

•

First-order logic:addsexistential quantifier∃and universal quantifier∀

• ∃x y P (x)∧P (y)means“There is both a pink and a blue node”

•

Monadic second-order logic (MSO):addsquantifiers over sets

• ∃S∀x S(x)means“there is a setScontaining every elementx”

• Can expresstransitive closurex→^∗y, i.e.,“xis an ancestor ofy”

• ∀x P (x)⇒ ∃y P (y)∧x→^∗y

means“There is a blue node below every pink node”

(13)

•

• ∀x P (x)⇒ ∃y P (y)∧x→^∗y

(14)

•

• ∀x P (x)⇒ ∃y P (y)∧x→^∗y

(15)

•

(16)

Tree automata

Tree alphabet:

•

Bottom-up deterministictree automaton

•

“Is there both a pink and a blue node?”

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

Initial function: ⊥ P B

•

Transitions(examples): P

P ⊥

> B P

⊥

(17)

Tree automata

Tree alphabet:

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P ⊥

> B P

⊥

(18)

Tree automata

Tree alphabet:

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P ⊥

> B P

⊥

(19)

Tree automata

Tree alphabet:

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P ⊥

> B P

⊥

(20)

Tree automata

Tree alphabet:

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P ⊥

> B P

⊥

(21)

Tree automata

Tree alphabet:

B

⊥ P ⊥

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P ⊥

> B P

⊥

(22)

Tree automata

Tree alphabet:

B

⊥ P ⊥

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

Transitions(examples):

P P ⊥

>

B P

⊥

(23)

Tree automata

Tree alphabet:

B

⊥ P

P ⊥

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P P ⊥

>

B P

⊥

(24)

Tree automata

Tree alphabet:

>

B

⊥ P

P ⊥

•

^States:^{⊥,^B,^P,^>}

•

Final states: {>}

•

P P ⊥

>

B P

⊥

(25)

Boolean MSO query evaluation via automata

Theorem [Thatcher and Wright, 1968]

MSOandtree automatahave the sameexpressive poweron trees

→ Given a Boolean MSO query, we can compute a tree automaton that accepts precisely the trees on which the query holds

→ Complexity (in the query) is generally nonelementary Corollary

Evaluating a Boolean MSO query on a tree is inlinear timein the tree

(26)

→ Complexity (in the query) is generally nonelementary

Corollary

(27)

→ Complexity (in the query) is generally nonelementary Corollary

(28)

Set Circuits for

Non-Boolean MSO Queries

(29)

Overall idea

•

^Query:^Q(x¹^{, . . . ,}^xⁿ⁾^withfree variablesx₁, . . . ,x_n

•

^Goal:find all tuplesa₁, . . . ,a_nsuch thatQ(a₁, . . . ,a_n)holds

→ Addspecial nodes: for each nodenand variablex_i, add a noden_iwhich is coloredrediffx_i is the noden

n n⁰ n⁰⁰

⇒

n

•

n⁰ n⁰⁰ n₁ n₂

•

Rewrite the query to aBoolean querywhich uses the new nodesn_i to read the valuation ofx_i

•

This can be done inlinear timein the input tree

•

Remark: same construction for freesecond-order variables

(30)

Overall idea

•

→ Addspecial nodes: for each nodenand variablex_i, add a noden_iwhich is coloredrediffx_i is the noden

n n⁰ n⁰⁰

⇒

n

•

n⁰ n⁰⁰ n₁ n₂

•

(31)

Overall idea

•

→ Addspecial nodes: for each noden and variablex_i, add a noden_iwhich is coloredrediffx_i is the noden

n n⁰ n⁰⁰

⇒

n

•

n⁰ n⁰⁰ n₁ n₂

•

(32)

Overall idea

•

n n⁰ n⁰⁰

⇒

n

•

n⁰ n⁰⁰ n₁ n₂

•

(33)

Overall idea

•

n n⁰ n⁰⁰

⇒

n

•

n⁰ n⁰⁰ n₁ n₂

•

(34)

Uncertain trees

Now: Boolean query on a tree where the color of nodes isuncertain 1

5

6 7 2

4 3

Avaluationof a tree decides whether to keep(1) ordiscard(0) node labels Valuation: {2,3,77→1, ∗ 7→0}

A:“Is there both a pink and a blue node?”

(35)

Uncertain trees

5

6 7 2

4 3

Avaluationof a tree decides whether to keep(1) ordiscard(0) node labels

Valuation: {2,3,77→1, ∗ 7→0}

(36)

Uncertain trees

5

6 7 2

4 3

(37)

Uncertain trees

5

6 7 2

4 3

Avaluationof a tree decides whether to keep(1) ordiscard(0) node labels Valuation: {27→1, ∗ 7→0}

(38)

Uncertain trees

5

6 7 2

4 3

Avaluationof a tree decides whether to keep(1) ordiscard(0) node labels Valuation: {2,77→1, ∗ 7→0}

(39)

Uncertain trees

5

6 7 2

4 3

(40)

Uncertain trees

5

6 7 2

4 3

The tree automatonAaccepts

(41)

Uncertain trees

5

6 7 2

4 3

Avaluationof a tree decides whether to keep(1) ordiscard(0) node labels Valuation: {27→1, ∗ 7→0}

The tree automatonArejects

(42)

Uncertain trees

5

6 7 2

4 3

The tree automatonAaccepts

(43)

Set circuit 1

5 6 7 2

4 3

Set circuit:

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^Condition:^Let^νbe a valuation ofT, thenA acceptsν(T)iff the setS(g0)of the output gateg₀ contains{g∈C|ν(g) =1}.

Query:Is there both a pink and a blue node?

×

∪ 7

2 3

×

× ×

∪ ∪ ∪ ∪

> 1 > 4 > 5 > 6

(44)

Set circuit 1

5 6 7 2

4 3

Set circuit:

•

×

∪ 7

2 3

×

× ×

∪ ∪ ∪ ∪

> 1 > 4 > 5 > 6

(45)

Set circuit 1

5 6 7 2

4 3

Set circuit:

•

×

∪ 7

2 3

×

× ×

∪ ∪ ∪ ∪

> 1 > 4 > 5 > 6

(46)

Set circuit 1

5 6 7 2

4 3

Set circuit:

•

×

∪ 7

2 3

×

× ×

∪ ∪ ∪ ∪

> 1 > 4 > 5 > 6

(47)

Set circuit 1

5 6 7 2

4 3

Set circuit:

•

×

∪ 7

2 3

×

× ×

∪ ∪ ∪ ∪

(48)

Building provenance circuits on trees

Theorem

For any bottom-uptree automatonAand inputtreeT, we can build ad-DNNF set circuitofAonTinO(|A| × |T|)

•

^Alphabet:

•

^Automaton:“Is there both a pink and a blue node?”

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

> P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(49)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(50)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(51)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(52)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(53)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(54)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

∪ ∪ ∪ ∪

⊥ B P >

×

(55)

Theorem

•

^Alphabet:

•

^States:

{⊥,B,P,>}

•

^Final: ^{>}

•

Transitions:

>

P ⊥

P P ⊥

n

∪ ∪ ∪ ∪

⊥ B P >

×

(56)

Circuits as factorized representations of query results

→ Theset circuitofQis now afactorized representation which describes all the tuples that makeQtrue

Example query:

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results: X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

Semi-open question:what about memory usage?

(57)

→ Theset circuitofQis now afactorized representation which describes all the tuples that makeQtrue Example query:

Q(X₁,X₂) :P (x)∧P (y)

Data: 1

2 3

Results: X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(58)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results: X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(59)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(60)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(61)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(62)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(63)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(64)

Q(X₁,X₂) :P (x)∧P (y) Data:

1

2 3

Results:

X1 X2

1 2

1 3

Provenance circuit:

× X₁(1) ∪

X₂(2) X₂(3)

{X₂(2),X₂(3)}

{(X₁(1),X₂(2)),(X₁(1),X₂(3))}

(65)

Application to

Pattern Matching in Texts

(66)

Problem statement: Pattern matching in texts Data:atextT

Antoine Amarilli Description Name Antoine Amarilli. Handle: a3nm. Identity Born 1990-02-07.

French national. Appearance as of 2017. Auth OpenPGP. OpenId. Bitcoin. Contact Email and XMPP a3nm@a3nm.net Affiliation Associate professor of computer science (office C201-4) in the DIG team of Télécom ParisTech, 46 rue Barrault, F-75634 Paris Cedex 13, France. Studies PhD in computer science awarded by Télécom ParisTech on March 14, 2016. Former student of the École normale supérieure.

test@example.com More Résumé Location Other sites Blogging: a3nm.net/blog Git: a3nm.net/git ...

?

^Query:^a^pattern^Pgiven as a regular expression P:= ␣ [a-z0-9.]^∗ @ [a-z0-9.]^∗ ␣

i

^Output:the list ofsubstringsofT that matchP: [186,200i, [483,500i, . . .

Goal:

•

^bevery efficientinT (constant-delay)

•

^bereasonably efficientinP(polynomial-time)

(67)

?

i

Goal:

•

(68)

?

i

Goal:

•

(69)

?

i

^Output:the list ofsubstringsofT that matchP: [186,200i, [483,500i, . . . Goal:

•

(70)

Reducing to MSO

•

^A^text^{is just a}^treewith a simpler shape

•

^Aregular expression patterncan be expressed in MSO

→ More generally: regular expressions withvariables

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

Translate to aword automaton(withcapture variables)

1 2 3 4

• α

a β

b γ

•

→ The MSO result implies: Theorem [Florenzano et al., 2018]

We can enumerate all matches of a regular expression pattern on a tree with linear preprocessing and constant delay

→ The resulting set circuit is abinary decision diagram, i.e., each×-gate has only one input which is not a variable

(71)

Reducing to MSO

•

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

1 2 3 4

• α

a β

b γ

•

(72)

Reducing to MSO

•

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

1 2 3 4

• α

a β

b γ

•

(73)

Reducing to MSO

•

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

1 2 3 4

• α

a β

b γ

•

(74)

Reducing to MSO

•

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

1 2 3 4

• α

a β

b γ

•

→ The MSO result implies:

Theorem [Florenzano et al., 2018]

(75)

Reducing to MSO

•

→ Example:P:=•^∗ α a^∗ β b^∗ γ •^∗

•

1 2 3 4

• α

a β

b γ

•

→ The MSO result implies:

Theorem [Florenzano et al., 2018]

(76)

Efficiency in the query

We have shown linear preprocessing and constant delay in thedata; but what about thequery?

•

For general MSO queries:nonelementarycomplexity

•

For regular expressions:exponential(determinization)

•

However: our methods adapt tonondeterministic automata

• Constant-delay enumeration for set circuits without assuming determinismbut bounding some notion ofwidth

Theorem [Amarilli et al., 2019a, Amarilli et al., 2019b] We can enumerate all matches of a nondeterministic tree automaton on a tree with

• Delayconstantin the tree andpolynomialin the automaton Corollary: enumeration for regular expression patterns on text

(77)

•

(78)

•

(79)

•

Theorem [Amarilli et al., 2019a, Amarilli et al., 2019b]

We can enumerate all matches of a nondeterministic tree automaton on a tree with

(80)

Implementation (ongoing internship by Rémi Dupré)

•

Prototype to find matches of aregular expressionin atext

•

https://github.com/remi-dupre/enum-spanner-rs

•

Work-in-progress

•

Open questions / projects:

• What aboutmemory usage? (we cannot keep the whole index)

• Output matchesin streaming? (problem: duplicates)

• Can we enumerateother notions of matches?

→ factors ofmaximal/minimal size

→ distinct matchingstrings

→ etc.

• Whichapplication domainsneed this?

• Are there goodbenchmarks?

(81)

Implementation (ongoing internship by Rémi Dupré)

•

Prototype to find matches of aregular expressionin atext

•

https://github.com/remi-dupre/enum-spanner-rs

•

Work-in-progress

•

Open questions / projects:

• What aboutmemory usage? (we cannot keep the whole index)

• Output matchesin streaming? (problem: duplicates)

• Can we enumerateother notions of matches?

→ factors ofmaximal/minimal size

→ distinct matchingstrings

→ etc.

• Whichapplication domainsneed this?

• Are there goodbenchmarks?