Non-Boolean MSO Queries

Overall idea

•

^{Query:Q(x1, . . . ,xn)with}free 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

Overall idea

•

^{Query:Q(x1, . . . ,xn)with}free 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

Overall idea

•

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

•

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

→ 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

•

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

Overall idea

•

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

•

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

→ 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

•

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

Overall idea

•

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

•

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

→ 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

•

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

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?”

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?”

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?”

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: {27→1, ∗ 7→0}

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

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: {2,77→1, ∗ 7→0}

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

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: {2,77→1, ∗ 7→0}

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

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?”

The tree automatonAaccepts

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: {27→1, ∗ 7→0}

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

The tree automatonArejects

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: {2,77→1, ∗ 7→0}

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

The tree automatonAaccepts

Set circuit

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^{Condition:Letν}be 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?

×

Set circuit

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^{Condition:Letν}be 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?

×

Set circuit

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^{Condition:Letν}be 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?

×

Set circuit

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^{Condition:Letν}be 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?

×

Set circuit

•

Tree automatonA, uncertain treeT, circuitC

•

Variable gatesofC: nodes ofT

•

^{Condition:Letν}be 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?

×

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?”

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?”

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?”

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?”

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?”

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?”

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?”

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?”

Circuits as factorized representations of query results

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

Example query:

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Semi-open question:what about memory usage?

Circuits as factorized representations of query results

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

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

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

Application to

Dans le document Enumerating MSO Query Results on Trees and Words (Page 28-65)