Existential Rules Combined approach for EL Micha el Thomazo - - PowerPoint PPT Presentation

Existential Rules

Existential Rules

Combined approach for EL Micha¨ el Thomazo November 22nd, 2013 Dresden

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Existential Rules

Flow of the course

What has been seen until now:

◮ forward chaining approach ◮ backward chaining approach

Today’s topic: a case where none of the already presented approaches are applicable.

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Reference for today’s course

Conjunctive Query Answering in the Description Logic EL Using a Relational Database System, Lutz, Toman, Wolter, IJCAI 09

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ELHdr

NC and NR disjoint sets of (atomic) concept and role names.

EL concept

An EL concept is built as follows:

◮ an atomic concept C ∈ NC ◮ top concept ⊤ ◮ the intersection of two concepts C1 ⊓ C2 ◮ ∃r.C, where r is a role name and C an EL concept

An ELHdr TBox is a set of concept inclusions C1 ⊑ C2 and of role inclusions r1 ⊑ r2, as well as a set of domain and range restrictions ( dom(r) ⊑ C and ran(r) ⊑ C ′).

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EL and existential rules

Any EL ontology can be expressed thanks to a set of existential rules. → consider the standard translation into first-order logic

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EL and rule applications

Let us consider the following EL concept inclusion: C ⊑ ∃r.C, and the following ABox: C(a).

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EL and query rewriting

Let us consider the following EL concept inclusion: ∃r.C ⊑ C and the following query: ∃xC(x) ∧ D(x).

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The combined approach (1)

In the combined approach, both the query and the data are

• modified. We look for F ′ and q′ such that:

F, R | = q ⇔ F ′ | = q′.

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The combined approach (2)

It has been applied for

◮ ELHdr ⊥ (today’s course) ◮ DL-Lite ◮ generalizations of guarded rules

Advantages:

◮ allow to deal with more expressive ontologies ◮ avoid some blow up inherent to query rewriting approaches

Main limitation:

◮ the data is modified

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Existential Rules

Some technical assumptions

• 1. queries contain only individual names that occur in the KB
• 2. TBoxes do not contain domain restrictions
• 3. TBoxes contain exactly one range restriction per role name
• 4. if K |

= r ⊑ s and ran(r) ⊑ C, ran(s) ⊑ D are in T , then C ⊑T D

• 5. there are no r, s ∈ NR with r = s such that K |

= r ⊑ s and K | = s ⊑ r All these assumptions can be made without loss of generality.

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Some notations

◮

sub(T ): subconcepts appearing in T

◮

rol(R): roles appearing in T

◮

Ind(A): individuals appearing in A

◮

ranT (r): unique concept C such that ran(r) ∈ T

◮

ran(T ) = { ranT (r) | r ∈ rol(T )}

◮

NIaux = {xC,D | C ∈ ran(T ) and D ∈ sub(T )}

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Enriching the data

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First step: canonical model for instance queries

◮ ∆IK := Ind(A) ∪ NIaux ◮ aIK := a for all a ∈ Ind(A) ◮ AIK := {a ∈ Ind(A) | K |

= A(a)} ∪ {xC,D ∈ NIaux | C ⊓ D ⊑ A}

◮ r IK = r IK

1

∪ r IK

2

∪ r IK

3

◮ r IK

1

:= {(a, b) ∈ Ind(A) × Ind(A) | s(a, b) ∈ A and K | = s ⊑ r}

◮ r IK

2

:= {(a, xC,D) ∈ Ind(A) × NIaux | K | = ∃s.D(a), ran(s) = C, K | = s ⊑ r}

◮ r IK

3

:= {(xC,D, xC′,D′) ∈ NIaux × NIaux | K | = C ⊓ D ⊑ ∃r.D′, ran(s) = C ′, K | = s ⊑ r}

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Existential Rules

First problem

Consider K1 = (T1, A1) with T1 = {A ⊑ A}, and A1 = {B(a)}. Let q1 = ∃u(B(v) ∧ A(u)). x⊤,A ∈ AIK1 and IK1 | = q1[a] but a is not a certain answer. Solved by considering only the “connected” part → Ik

R.

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Existential Rules

Second problem

Consider K2 = (T2, A2) with T2 = {A ⊑ ∃r.B ⊓ ∃s.B} and A2 = {A(a)}. Let consider q2 = ∃u(r(v, u) ∧ s(v, u). IK2 | = q2[a], but a is not a certain answer. Question: why does IK2 | = q2[a] hold?

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Solving them by unraveling (1)

We define Ind(A)I = {aI | a ∈ Ind(A)}. A path in I is a finite sequence d0r1d1 . . . rndn such that:

◮ d0 ∈ Ind(A)I; ◮ (di, di+1) ∈ rI i+1 for all i < n.

Denoted by pathsA(I).

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Solving them by unraveling (2)

Let R be the set of role inclusion in K. The (A, R)-unraveling J

• f I is defined as follows:

◮ ∆J = pathsA(I) ◮ aJ = aI for all a ∈ Ind(A) ◮ AJ = {p | tail(p) ∈ AI} ◮ rJ = rJ 1 ∪ rJ 2 ◮ rJ 1 = {(d, e) | d, e ∈ Ind(A) ∧ (d, e) ∈ rI} ◮ rJ 2 = {(p, p · se) | p, p · se ∈ ∆J and R |

= s ⊑ r} The unraveling of Ir

K is denoted by UK.

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Motivation for the second step

We want to represent the canonical model thanks to a relational database. We thus need a finite canonical model. The unraveling is infinite in general. We rewrite the query in order to regain soundness.

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Rewriting the query

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Recognizing errors

The aim is to exclude some of the answer provided with the canonical model for instance queries. We make use of the following equivalence relation:

The equivalence relation ∼q

∼q is the smallest relation on terms(q) that include identity, is transitive and satisfies the following closure relation: If r1(s, t), r2(s′, t′) ∈ q and t ∼q t′, then s ∼q s′

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Existential Rules

pre and in

For an equivalence class ζ of ∼q, we define the following two sets:

◮

pre(ζ) = {t | r(t, t′) ∈ q for some r ∈ NR and t′ ∈ ζ}

◮

in(ζ) = {r | r(t, t′) ∈ a for some t ∈ term(q) and t′ ∈ ζ}

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Implicant and prime implicant

For R ⊆ NR and r ∈ NR, r is an implicant of NR if for any s ∈ R, R | = r ⊑ s. It is called a prime implicant if for all implicant r′ of R such that r′ = r, R | = r ⊑ r′,

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Some relevant situations

We define:

◮

Fork= us the set if pairs ( pre(ζ), ζ), where pre(ζ) is of cardinality at least two;

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Existential Rules

Some relevant situations

We define:

◮

Fork= us the set if pairs ( pre(ζ), ζ), where pre(ζ) is of cardinality at least two;

◮

Fork= is the set of variables v ∈ qvar(q) such that there is no implicant of in([v]);

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Existential Rules

Some relevant situations

We define:

◮

Fork= us the set if pairs ( pre(ζ), ζ), where pre(ζ) is of cardinality at least two;

◮

Fork= is the set of variables v ∈ qvar(q) such that there is no implicant of in([v]);

◮

ForkH is the set of pairs (I, ζ) such that pre(ζ) = ∅, there is a prime implicant of in(ζ) that is not contained in in(ζ) and I is the set of all prime implicants of in(ζ)

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Existential Rules

Some relevant situations

We define:

◮

Fork= us the set if pairs ( pre(ζ), ζ), where pre(ζ) is of cardinality at least two;

◮

Fork= is the set of variables v ∈ qvar(q) such that there is no implicant of in([v]);

◮

ForkH is the set of pairs (I, ζ) such that pre(ζ) = ∅, there is a prime implicant of in(ζ) that is not contained in in(ζ) and I is the set of all prime implicants of in(ζ)

◮

Cyc is the set of variables v ∈ qvar(q) such that there are r0(t0, t′

0), . . . , rm(tm, t′ m), . . . , rn(tn, t′n) ∈ q, n, m ≥ 0 with

v ∼q ti for some i ≤ n, T ′

i ∼q ti+1 for all i < n and t′ n ∼q tm.

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The rewriting

We rewrite q = ∃u(ψ) into ∃u(ψ ∧ ϕ1 ∧ ϕ2 ∧ ϕ3), where:

◮ ϕ1 = v∈ avar(q)∪ Fork=∪ Cyc ¬ Aux(v) ◮ ϕ2 = ({t1,...,tk},ζ)∈ Fork=( Aux(tζ) → 1≤i<k ti = ti+1) ◮ ϕ3 = (I,ζ)∈ ForkH( Aux(tζ) → r∈I r(tpre ζ

, tζ)) where tζ is a representative of ζ and t

pre

ζ

is an arbitrary element of pre(tζ)

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Formalization of the goal

We prove the following equivalence: Ir

K |

= q∗

R[a1, . . . , ak] ⇔ UK |

= a[a1, . . . , ak]

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Two properties of models

Let A be an ABox. Let I be a model where AuxI = ∆I \ Ind(A)I.

◮ I is A-connected if every d ∈ ∆I equals tail(p) for some

p ∈ pathsA(I)

◮ I is split if d ∈ AuxI and (d, d′) ∈ rI imply d′ ∈ AuxI, for

all r ∈ NR and d, d′ ∈ ∆I

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The theorem

Let I be split and A-connected and let I′ be the (A, R)-unraveling of I. Let q be a k-ary conjunctive query. Then the following holds for all a1, . . . , ak ∈ Ind(A): I | = q∗

R[a1, . . . , ak] ⇔ I′ |

= q[a1, . . . , ak]

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