The Power of Tarski’s Relation Algebra on Treesjhellings.nl/files/foiks2018_1_slides.pdf1/35 The...

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The Power of Tarski’s Relation Algebra on Trees

Jelle Hellings1

Yuqing Wu2 Marc Gyssens1 Dirk Van Gucht3

1 Hasselt University2 Pomona College

3 Indiana University

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Relation algebra: graphs and queries

Alice

Bob

Carol

ParentOfParentOf

Dan

ParentOf

Faythe

ParentOf

Grace

ParentOf

PeggyFriendOf FriendOf

Victor

FriendOf

WorksWith

WendyFriendOf

FriendsOfGgp = π1[ParentOf ◦ ParentOf ◦ ParentOf ] ◦ FriendOf

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Relation algebra: other operators

ChildOf = ParentOf −1

AcquaintanceOf = FriendOf ∪WorksWith

WorkFriendOf = FriendOf ∩WorksWith

NonWorkFriend = FriendOf −WorksWith

Parent = π1[ParentOf ]

NonParent = π 1[ParentOf ]

GrandParentOf = ParentOf ◦ ParentOf

Also: constants ∅, id, di.

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Relation algebra and query languages

∅ id ∪ ◦ −1 π π ∩ − di

Tarski’s Relation Algebra, FO[3]

I On graphs: RPQs, 2RPQs, Nested RPQs, Graph XPath.I XPath of Benedikt et al.I Path+ of Wu et al.I Downward queries of Hellings et al.I Conditional and Navigational XPath.

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�estions about the relation algebra

�estions

I Are all these operators necessary?I What does each operator add?

What is the expressive power of fragments of the relation algebra?

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Expressive power of relation algebra

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Background: when are queries equivalent?

Two types of queries and equivalences

Path-queries. The exact query result is important:

FriendOf ∩WorksWith ≡path FriendOf − (FriendOf −WorksWith).

Boolean-queries. The existence of a query result is important:

FriendOf −1 ◦ ParentOf ≡bool π1[FriendOf ] ∩ π1[ParentOf ].

DefinitionLanguage L1 is z-subsumed by L2 if every query in L1 isz-equivalent to a query in L2 (denoted by L1 �z L2).

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Background: query fragments

Let F ⊆ {di, −1,π ,π ,∩,−}.I We write N(F): only allows ∅, id, `, ◦, ∪, and all operators in F.I We write N(F) to represent all operators expressible in N(F)

using the following basic rewrite rules:

π1[e] = π j[π 1[e]] = (e ◦ e−1) ∩ id = (e ◦ (id ∪ di)) ∩ id;π2[e] = π j[π 2[e]] = (e−1 ◦ e) ∩ id = ((id ∪ di) ◦ e) ∩ id;π i[e] = id − πi[e];

e1 ∩ e2 = e1 − (e1 − e2).

Examples

I N(π ) = N(π ,π ).I N(−1,−) = N(−1,π ,π ,∩,−).

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Results on graphs

Previous work by Fletcher et al.Relative expressive power of relation algebra fragments on graphs:I Path queries: each operator adds expressive power.I Boolean queries: −1 sometimes does not add expressive power.

Example: strong Boolean separationsConsider (E ◦ E) ∩ E:

G3,3: G4:

Without ∩ (or −): no query distinguishes G3,3 and G4.

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Our work: studying trees

GoalImprove understanding of the relation algebra by studying theexpressive power in restricted se�ings.

Trees

AdviserOf AdviserOf

AdviserOf AdviserOf

I Trees are hierarchical/nested structures.E.g. taxonomies, organizational charts, documents, file systems.

I Tree-based formal data models and query languagesE.g. XML data with XPath, JSON data, nested relational data.

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Initial classification

1-subtreereducible

2-subtreereducible

3-subtreereducible

downward non-downward

local

non-local

monotone

non-monotone

monotone

non-monotone

N(∩,−)

N(π ,∩)

N(π ,π ,∩,−) N(−1,π ,π ,∩)

N(−1,π ,∩)

N(di, −1,π )

N(di, −1,π ,π )

N(−1,π ,π ,∩,−)

N(di, −1,π ,∩)

N

I N(π ,π ,∩,−) studied by Hellings et al.I N(−1,π ) studied by Benedikt et al.I N(−1,π ,∩) studied by Wu et al.

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1-subtreereducible

2-subtreereducible

3-subtreereducible


local

non-local

monotone

non-monotone

monotone

non-monotone

N(∩,−)

N(π ,∩)

N(π ,π ,∩,−) N(−1,π ,π ,∩)

N(−1,π ,∩)

N(di, −1,π )

N(di, −1,π ,π )

N(−1,π ,π ,∩,−)

N(di, −1,π ,∩)

N

I N(π ,π ,∩,−) studied by Hellings et al.

I N(−1,π ) studied by Benedikt et al.I N(−1,π ,∩) studied by Wu et al.

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1-subtreereducible

2-subtreereducible

3-subtreereducible


local

non-local

monotone

non-monotone

monotone

non-monotone

N(∩,−)

N(π ,∩)

N(π ,π ,∩,−) N(−1,π ,π ,∩)

N(−1,π ,∩)

N(di, −1,π )

N(di, −1,π ,π )

N(−1,π ,π ,∩,−)

N(di, −1,π ,∩)

N

I N(π ,π ,∩,−) studied by Hellings et al.I N(−1,π ) studied by Benedikt et al.I N(−1,π ,∩) studied by Wu et al.

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Downward queries: Boolean and path queries

N()

N(π )

N(π )

N(∩,−)

N(π ,∩)

N(π ,π ,∩,−)

I N(π ,π ,∩,−) is downward.I −1 and di are not downward.

Theorem (Hellings et al.)Let F ⊆ {π ,π ,∩,−}.We have N(F) �path N(F − {∩,−}).

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Local queries

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Locality

DefinitionA query q is local if there exists k ≥ 0 such that, for every tree T , andfor all nodes m and n, (m, n) ∈ [[q]]T if and only if (m, n) ∈ [[q]]T′ , withT ′ the smallest subtree of T containing all nodes at distance at mostk from the nearest common ancestor of m and n.

I N(−1,π ,π ,∩,−) is local.I di is not local.

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Local queries: main resultsPath queries Boolean queries

N()

N(π )

N(π )

N(−1)

N(−1,π )

N(−1,π )

N(−1,−)

N(∩,−)

N(π ,∩)

N(π ,π ,∩,−)

N(−1,π ,∩)

N(−1,π ,π ,∩)

N(−1,π ,π ,∩,−)

N()

N(π ) = N(−1)

N(π )

N(−1,−)

N(∩,−)

N(π ,∩)N(−1,π ,∩)

N(π ,π ,∩,−)N(−1,π ,π ,∩)

N(−1,π ,π ,∩,−)

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Condition tree queries

DefinitionIf an expression always evaluates to a subset of id then it is called anode expression.

DefinitionA condition tree query Q is a tuple Q = (T ,C, s, t,γ ) withI T = (V, Σ,E) a labeled tree,I C is a set of node expressions,I s ∈ V is the source node,I t ∈ V is the target node,I γ ⊆ V × C is the node-condition relation.

γ (n) denotes the set {e1, . . . , ek} of conditions related to node n ∈ V .γ (n) also denotes the expression e1 ◦ . . . ◦ ek (≡path e1 ∩ . . . ∩ ek ).

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A condition tree query example

Consider the union-free expression

`−11 ◦ `

−12 ◦ π 2[`3] ◦ `3 ◦ `3 ◦ π1[`1] ◦ π1[`2].

This expression is path-equivalent to

`2 `3

`1 `3

`1 `2

s t

{π 2[`3]}

`3

`2 `3

`1 `3 `3

`1 `2

`1

`2 `3

`1

`4

`3

`1 `2

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Condition tree queries and local queries

Qtree(F) are the condition tree queries in which node conditions arerestricted to union-free expressions in N(F).

ClaimLet {−1,π } ⊆ F ⊆ {−1,π ,π }.Qtree(F) and union-free N(F) path-subsume each other.

PropositionLet F ⊆ {−1,π ,π } and Q be a condition tree query in Qtree(F).There exists a union-free expression e in N(F) such that e ≡path Q.

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Basic manipulations on condition tree queries

`

n

m

leaf-prune mγ (n) ◦ π1[` ◦ γ (m)]

`

n

m

root-prune mγ (n) ◦ π2[γ (m) ◦ `]

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Up-down queries

DefinitionA condition tree query Q = (T ,C, s, t,γ ) is an up-down query if alledges of T are on the unique path from s to t not taking into accountthe direction of the edges.

`3

`1

`2

`2

`1`2

s

tπ 2[`3]

pruning`1 `2

`1

s

t

π2[`3]

π1[`2]π1[`2 ◦ π 2[`3]]

LemmaLet {π } ⊆ F ⊆ {−1,π ,π }, and Q be a condition tree query in Qtree(F).There exists an up-down query Q ′ in Qtree(F) such that Q ≡path Q

′.

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Up-down queries and local queriesClaimLet {−1,π } ⊆ F ⊆ {−1,π ,π }.Qtree(F) and union-free N(F) path-subsume each other.

PropositionLet F ⊆ {−1,π ,π } and e be a union-free expression in N(F).There exists a condition tree query Q in Qtree(F) such that e ≡path Q.

Proof.

`1 `′1

r

n1

s

m1

t

direct rewrite γ (s) ◦ . . . ◦ γ (n1) ◦ `−11 ◦ γ (r)

◦ `′1 ◦ γ (m1) ◦ . . . ◦ γ (t)

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Path-queries: eliminate intersection

TheoremLet {−1,π } ⊆ F ⊆ {−1,π ,π ,∩}.We have N(F) �path N(F − {∩}).

Proof.

`1 `2

`3

r1

s1 n1

t1

Q1 ∩

` `

`1 `2

`3

r2

m1 m2

s2 m3

t2

Q2

merge ``1 `2

`3

r1

s1 n1

t1

Q1 ∩

`1 `2

`3

s2 m3

t2

Q ′2

π2[γ (r2) ◦ `]

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Path-queries: eliminate intersection

TheoremLet {−1,π } ⊆ F ⊆ {−1,π ,π ,∩}.We have N(F) �path N(F − {∩}).

Proof.

`1 `2

`3

r1

s1 n1

t1

Q1 ∩

`1 `2

`3

s2 m3

t2

Q ′2

π2[γ (r2) ◦ `]

merge nodes`1 `2

`3

γ (s1) ◦ γ (s2) γ (n1) ◦ γ (m3)

γ (t1) ◦ γ (t2)

γ (r1) ◦ π2[γ (r2) ◦ `]

Q1 ∩ Q2

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Boolean queries: eliminate projection or converse

PropositionN(π ) �bool N(

−1) and N(−1) �bool N(π ).

Proof.Wu et al.: union-free N(π ) is path-equivalence to Qtree().

`

s

tn

m

reset s, t

`

n

m

s, t

leaf-prune m s, t

π1[`]` ◦ `−1

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Bonus: condition tree queries on chains

DefinitionA condition tree query is a chain query if it is up-down and Tconsists of a single path.

` `

`′

r

s n

t

merge ``

`′

r

sγ (s) ◦ γ (n)

t

pruning `

`′sγ (s) ◦ γ (n) ◦ π2[γ (r) ◦ `]

t

LemmaLet {π } ⊆ F ⊆ {−1,π ,π }, and Q be a condition tree query in Qtree(F).There exists a chain query Q ′ in Qtree(F) such that Q ≡path Q

′.

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Bonus: eliminate di�erence on chains

TheoremLet {−1,π } ⊆ F ⊆ {−1,π ,π ,∩,−}.On chains, we have N(F) ≡path N(

−1,π ).

Proof.

`

`′

s1

t1

n

Q1 −

`

`′

s2

t2

m

Q2

di�erence `

`′

s

tγ (t1) ◦ γ (t2)

γ (n)

γ (s1)

`

`′

s

tγ (t1)

γ (n) ◦ γ (m)

γ (s1)

`

`′

s

tγ (t1)

γ (n)

γ (s1) ◦ γ (s2)

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Separation results

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Recognizing branches

DefinitionA k-subtree-reduction step on tree T consists of removing a subtreerooted at a node n with parent m, given that parent m has at least kother children isomorphic to n.

Theorem

1. N(di,∩) and N(−1,−) are closed under 3-subtree-reductions.

2. N(di, −1,π ,π ) is closed under 2-subtree-reductions.

3. N(−1,π ,π ,∩) is closed under 1-subtree-reductions.

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The subtree-reductions classification is strict

PropositionAlready on unlabeled trees, we have

1. N(di) �bool N(−1,π ,π ,∩),

2. N(−1,−) �bool N(−1,π ,π ,∩),

3. N(di,∩) �bool N(di, −1,π ,π ).

Proof.

1. e1 = E ◦ di ◦ di ◦ E.

2. e2 = (E−1 ◦ E) − id.

3. e3 = (((di ◦ E) ∩ di) ◦ ((di ◦ E) ∩ di)) ∩ di.

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Diversity and jumping in trees

PropositionAlready on unlabeled trees, N(di,∩) �bool N(

−1,π ,π ,∩,−).

Proof.e = P2,¬r ◦ di ◦ P2,¬r

P2,¬r = π2[E] ◦ P2

P2 = π1[S2]

S2 = (E ◦ di) ∩ E .

T1: T2:

r

m n

r′

m′ n′

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Diversity and jumping in chains

PropositionAlready on chains, we have

1. N(di, −1) �bool N(−1,π ,π ,∩,−);

2. N(di,π ) �bool N(−1,π ,π ,∩,−).

Proof.

1. e1 = (` ◦ `−1) ◦ di ◦ (` ◦ `−1).

2. e2 = π1[`] ◦ di ◦ π1[`].

C1: `` `′`′`′

nm

C2: ` `′`′

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Brute-force: path queries


1. N(−1) �path N(di,π ,π ),2. N(π ) �path N(

−1).

Proof.

1. e1 = E−1

2. e2 = π1[E] ◦ π2[E]

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Brute-force: Boolean queries


1. N(di, −1) �bool N(di) and N(di,π ) �bool N(di),2. N(di, −1,∩) �bool N(di,π ,π ,∩,−),3. N(di,π ) �bool N(di, −1).

Proof.

T1: T2: T3:

T ′1 : T ′2 : T ′3 :

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Conclusion and discussion

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Overview of resultsBoolean queries Path queries

di −1 π π ∩ − di −1 π π ∩ −

Localq

ueries

Dow

nwardqu

eries N() 7 7• 7• 7• 3• 3• 7• 7• 7• 7• 3• 3•

N(∩) 7 7• 7• 7• 3• 7• 7• 7• 7• 3•

N(∩,−) 7• 7• 7• 7• 7• 7• 7• 7•

N(π ) 7 3• 7• 3• 7• 7• 7• 7• 3• 7•

N(π ,∩) 7 3 7• 7• 7• 7• 7• 7•

N(π ,π ) 7 3• 3• 3• 7• 7• 3• 3•

N(π ,π ,∩) 7 3 3• 7• 7• 3•

N(π ,π ,∩,−) 7 7 7• 7•

N(−1) 7 3 7• 3 7• 7• 7• 7• 7 7•

N(−1,π ) 7 7• 3• 7• 7• 7• 3• 7•

N(−1,π ,∩) 7 7• 7• 7• 7• 7•

N(−1,π ,π ) 7 3 7 7• 3 7

N(−1,π ,π ,∩) 7 7 7• 7

N(−1,π ,π ,∩,−) 7 7•

N(di) 7 7 7• 7 7• 7 7 7• 7 7•

N(di,π ) 3• 7• 7 7• 7 7• 7 7•

N(di,π ,∩) 7 7• 7• 7 7• 7•

N(di,π ,π ) 3• 7 7 7 7 7

N(di,π ,π ,∩) 7 ? 7 ?N(di,π ,π ,∩,−) 7 7

N(di, −1) 7 7• 7 7• 7 7• 7 7•

N(di, −1,π ) 7• 7 7• 7• 7 7•

N(di, −1,π ,∩) 7• 7• 7• 7•

N(di, −1,π ,π ) 7 7 7 7

N(di, −1,π ,π ,∩) ? ?N(di, −1,π ,π ,∩,−)

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Conclusion

We fully established relationships between the local fragments.

Open problemLet {di,π ,∩} ⊆ F ⊆ {di, −1,π ,π ,∩} and let z ∈ {bool, path}.Do we have a collapse N(F ∪ {−}) �z N(F) or not?

LemmaOn unlabeled chains, we have N(di, −1,π ,π ,∩,−) �bool N(π ).

The Power of Tarski’s Relation Algebra on Treesjhellings.nl/files/foiks2018_1_slides.pdf1/35 The...

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