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Reasoning challenges in description logic Roman Kontchakov and - - PowerPoint PPT Presentation

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Reasoning challenges in description logic Roman Kontchakov and Michael Zakharyaschev Department of Computer Science , Birkbeck College London http://www.dcs.bbk.ac.uk/~{roman,michael} Description Logic DL is a (large) family of knowledge

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Reasoning challenges in description logic

Roman Kontchakov and Michael Zakharyaschev

Department of Computer Science, Birkbeck College London http://www.dcs.bbk.ac.uk/~{roman,michael}

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Description Logic

DL is a (large) family of knowledge representation & reasoning formalisms

  • more expressive than propositional logic
  • less expressive than first-order logic

(≈ decidable modal, hybrid logics)

  • developed by the KR community for applications in AI

Application-driven equilibrium: expressiveness vs. computational costs Recent applications:

  • Ontologies (or terminologies) in medicine, bioinformatics, ...
  • Semantic Web
  • Ontology-based data access and integration
  • Ontology maintenance, module extraction, versioning, updates, ...

Web Ontology Language (OWL) W3C standards OWL 1 (2004), OWL 2 (2009) OWL = DL + XML

  • r

DL = OWL − XML

Moscow, 23 August 2010 1

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DL architecture

Knowledge Base (KB)

TBox (terminological box, schema)

Man ≡ Human ⊓ Male Appendicitis ⊑ Disease ⊓ ∃morphology.Inflam ...

ABox (assertion box, data)

Man(john) hasChild(john, mary) ...

Inference System Interface

Moscow, 23 August 2010 2

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Description logic constructs

  • Alphabet:

– concept names A0, A1, ...

(e.g., Person, Female, ...)

– role names R0, R1, ...

(e.g., hasChild, loves, ...)

– individual names a0, a1, ...

(e.g., john, mary, ...)

– concept constructs: ⊤, ⊓, ¬, ∃, ∀, ≥ q, ...

(e.g., Person ⊓ Female)

– role constructs: R−, R ◦ S, ...

(e.g., isChildOf)

– axiom construct: ⊑

(e.g., Man ⊑ Person)

  • Concepts:

– concept names –

⊤, ⊥, ¬C, C ⊓ D, ∀R.C, ∃R.C, ≥ qR.C,

where C, D are concepts and R a role Examples: Person ⊓ Female, Person ⊓ ¬Female, Person ⊓ ∃hasChild.⊤, Person ⊓ ∀hasChild.Male

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Description logic semantics

  • (standard Tarski-style) interpretation is a structure I = (∆I

, ·I)

– ∆I is the domain of I (a non-empty set) – ·I is an interpretation function that maps: ∗ concept name Ai → subset AI

i

  • f ∆I

(AI

i ⊆ ∆I)

∗ role name Ri → binary relation RI

i

  • ver ∆I

(RI

i ⊆ ∆I × ∆I)

∗ individual name ai → element aI

i

  • f ∆I

(aI

i ∈ ∆I)

  • interpretation of complex concepts in I:

– (⊤)I = ∆I and (⊥)I = ∅ – (¬C)I = ∆I \ CI – (C ⊓ D)I = CI ∩ DI – (∀R.C)I = {x ∈ ∆I | ∀y ∈ ∆I (x, y) ∈ RI → y ∈ CI } – (∃R.C)I = {x ∈ ∆I | ∃y ∈ CI (x, y) ∈ RI} – (≥ qR.C)I =

  • x ∈ ∆I | ♯{y ∈ CI | (x, y) ∈ RI} ≥ q
  • Moscow, 23 August 2010

4

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TBoxes

statements about how concepts and roles are related to each other A TBox T is a finite set of terminological axioms:

  • C ⊑ D

C is subsumed by D (concept inclusion)

  • R ⊑ S

R is a subrole of S (role inclusion) an interpretation I satisfies an axiom – I | = C ⊑ D iff CI ⊆ DI – I | = R ⊑ S iff RI ⊆ SI An interpretation I is a model of T if I satisfies every axiom of T

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ABoxes

assert knowledge about individuals An ABox A is a finite set of assertional axioms

  • C(a)

concept assertion for an individual

  • R(a, b)

role assertion for a pair of individuals an interpretation I satisfies an assertion – I | = C(a) iff aI ∈ CI – I | = R(a, b) iff (aI, bI) ∈ RI An interpretation I is a model of a knowledge base K = (T , A) if I satisfies every axiom of T and A

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OWL ontology example

  • Prot´

eg´ e 4.0 a free, open source ontology editor http://protege.stanford.edu/ where you can also find a library of ontologies (tutorials explaining how to use Prot´ eg´ e are at http://www.co-ode.org/resources/tutorials/ )

  • built-in ontology reasoners

FaCT++, Pellet

  • r

HermiT http://owl.man.ac.uk/factplusplus/ http://pellet.owldl.com/ http://hermit-reasoner.com/

Moscow, 23 August 2010 7

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Reasoning problems

Concept satisfiability: given K = (T , A) and C, decide whether there is I | = K with CI = ∅ Subsumption: given T and concepts C, D, decide whether T | = C ⊑ D

i.e., ∀I (I | = T → I | = C ⊑ D)

Instance checking: given K = (T , A), C and an individual a from A, decide whether K | = C(a) Exercise: show that these three problems are reducible to each other Conjunctive query answering: given a KB K = (T , A), a CQ q( x) and a tuple

  • a of individual names from A, decide whether

K | = q( a) Query answering is typically a harder problem than the other three

Moscow, 23 August 2010 8

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First-order translation

A ❀ A(x) ¬C ❀ ¬C(x) C ⊓ D ❀ C(x) ∧ D(x) ∀R.C ❀ ∀y

  • R(x, y) → C(y)
  • ∃R.C

❀ ∃y

  • R(x, y) ∧ C(y)
  • ≥ qR.C

❀ ∃≥qy (R(x, y) ∧ C(y)) C ⊑ D ❀ ∀x

  • C(x) → D(x)
  • DL is embeddable into the 2-variable fragment of first-order logic with counting

(full FOL is undecidable; this fragment is in NExpTime)

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Unique name assumption (UNA)

An interpretation I is a model of a KB K = (T , A) under the UNA if I | = K and aI

i = aI j , for any distinct object names ai and aj occurring in A

OWL: a more flexible approach

  • UNA is dropped (so no restrictions on interpretations of object names)
  • User is provided with the constructs = (sameAs) and = (differentFrom)

to explicitly impose constraints on individual names

  • UNA is expressible: add ai = aj to A, for all distinct ai and aj in A

Price of = Have to check whether a = b in A under given equality constraints Equivalent to reachability in undirected graphs, which is LOGSPACE-complete

(Reingold 2008) . . . just peanuts for most DLs, but not for DL-Lite & OWL 2 QL. . .

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The history of description logic so far

. . . – mid 1990s: efficient reasoning cannot afford full Booleans sub-Boolean DLs with ⊓ and ∀ are enough FL, AL, . . . combined complexity ≤ NP mid 1990s – 2005 ‘efficient’ reasoning possible for ExpTime DLs (FaCT,...) full Booleans and other constructs SHIQ, SHOIN (≈ OWL 1), SROIQ (≈ OWL 2) ≥ EXPTIME mid 2005 – . . . new challenges: answering queries & HUGE ontologies Horn DLs with ⊓ and ∃ DL-Lite and EL families ≤ P

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Ontology-based data access

Aim: to achieve logical transparency in accessing data – hide from the user where and how data is stored – present only a conceptual view of the data – query the data sources through the conceptual model using RDBMSs

AcademicStaff

Lecturer Module teaches

subclass

r a n g e domain

  • ntology

data sources

Moscow, 23 August 2010 12

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Designing DL for conceptual data modelling

empCode: Integer salary: Integer

Employee Manager AreaManager TopManager 1..1 1..* boss

projectName: String

Project 3..* 1..1 1..1 worksOn manages 1..*

{disjoint, complete}

Translating into DL:

TopManager ⊑ Manager AreaManager ⊑ ¬TopManager Manager ⊑ AreaManager ⊔ TopManager Employee ⊑ ∃salary.⊤ ⊤ ⊤ ∃salary−.⊤ ⊤ ⊤ ⊑ Integer ≥ 2 salary.⊤ ⊤ ⊤ ⊑ ⊥ Project ⊑ ≥ 3 worksOn−.⊤ ⊤ ⊤ manages ⊑ worksOn CEO ⊓ (≥ 5 worksOn.⊤ ⊤ ⊤) ⊓ ∃manages.⊤ ⊤ ⊤ ⊑ ⊥ (integrity constraint)

DL-Lite

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Basic DL-Lite logics

  • 1. DL-LiteN

bool

R ::= P | P − B ::= ⊥ | A | ≥ qR C ::= B | ¬C | C1 ⊓ C2 TBox axioms C1 ⊑ C2

  • 2. DL-LiteN

horn

TBox axioms B1 ⊓ · · · ⊓ Bn ⊑ B

  • 3. DL-LiteN

krom

TBox axioms B1 ⊑ B2 B1 ⊑ ¬B2 ¬B1 ⊑ B2

  • 4. DL-LiteN

core = DL-LiteN horn ∩ DL-LiteN krom

under UNA combined complexity sat.: NP data comp. instance: in AC0 data comp. query: coNP combined complexity: P data comp. instance: in AC0 data comp. query: in AC0

  • comb. comp.: NLOGSPACE

d.c. instance: in AC0 d.c. query: coNP

  • comb. comp.: NLOGSPACE

d.c. instance: in AC0 d.c. query: in AC0

DL-Litebool, DL-Litehorn, DL-Litekrom, DL-Litecore:

  • nly

∃R available

Moscow, 23 August 2010 14

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Observations and examples

DL-Lite can only speak about the domains and ranges of binary relations, and how many successors and predecessors a point can have but not about the types of these successors/predecessors; types are defined uniformly by domain/range constraints Examples. Describe the models of the following KBs:

  • 1. T = {⊤ ⊑ ∃R,

≥ 2R ⊑ ⊥}, A = ∅

  • 2. T = {A ⊑ ¬∃R−,

A ⊑ ∃R, ∃R− ⊑ ∃R, ≥ 2−R ⊑ ⊥}, A = {A(a)}

  • Infinite models are required; no finite model property
  • Tree model property
  • Can be simulated by first-order formulas with one variable

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Bisimulations for DL-LiteN

bool

Let I and J be two interpretations. A relation ̺ ⊆ ∆I × ∆J is called a lite-bisimulation between I and J if (concept) for every concept name A, if x̺y then x ∈ AI iff y ∈ AJ (role) for every role R, if x̺y then x ∈ (= qR)I iff y ∈ (= qR)J

where q ∈ N ∪ {∞}

(I, x) ∼ (J , y) if there is a lite-bisimulation ̺ between I and J with x̺y DL-LiteN

bool concepts are invariant under lite-bisimulations, that is,

if (I, x) ∼ (J , y) then x ∈ CI iff y ∈ CJ , for every concept C A first-order formula ϕ(x) is equivalent to a DL-LiteN

bool concept

iff ϕ(x) is invariant under lite-bisimulations

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Global lite-bisimulations for DL-LiteN

bool

A lite-bisimulation relation ̺ between I and J is global if – for every x ∈ ∆I there is y ∈ ∆J with x̺y, and – for every y ∈ ∆J there is x ∈ ∆I with x̺y I is lite-bisimilar to J , I ∼ J , if there is a global lite-bisimulation between I and J DL-LiteN

bool TBoxes are invariant under global lite-bisimulations, that is,

if I ∼ J then I | = T iff J | = T , for every DL-LiteN

bool TBox T

Given I and x ∈ ∆I, let tI(x) = {C | x ∈ CI} — the type of x in I TI = {tI(x) | x ∈ ∆I} — set of all types in I I ∼ J iff TI = TJ

models are determined by their types ❀ 1-ary predicates

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Tree model propety

Every model of a DL-LiteN

bool TBox is globally lite-bisimilar to a tree-shaped model

Examples. Construct a tree-shaped model which is globally lite-bisimilar to t1 t2 t3

R R R

where t1, t2, t3 are distinct types Tree models of DL-LiteN

bool KBs?

ABox A B

Why is the tree-model property so important?

Moscow, 23 August 2010 18

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Embedding DL-Lite into 1-variable FO logic

Satisfiability of DL-LiteN

bool KBs is NP-complete (for combined complexity)

Proof DL-LiteN

bool

K ❀ K† (a universal 1-variable FO formula)

T = {A ⊑ ∃P − , ∃P − ⊑ A, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P − , ∃P ⊑ A}, A = {A(a), P (a, a′)}

∀x

  • (A(x)→E1P −(x))∧(E1P −(x)→A(x))∧(A(x)→E2P (x))∧¬E2P −(x)∧(E1P (x) → A(x))

∧ (E2P (x)→E1P (x)) ∧ (E2P −(x)→E1P −(x)) ∧ (E1P (x)→E1P −(dp−)) ∧ (E1P −(x)→E1P (dp))

  • ∧ A(a) ∧ E1P (a) ∧ E1P −(a′)

(∃P )I = ∅ iff (∃P −)I = ∅ ∃x E1P (x) ↔ ∃x E1P −(x)

. .

dp− dp a a′ | = K†

. .

dp− dp a a′ | = K†

. .

dp− dp a a′ | = K† | = K

K is satisfiable iff K† is. K† computed in LogSpace. K† says that – ∃ appropriate dr – ∀ point is of proper type

Moscow, 23 August 2010 19

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DL-Lite Horn, Krom and core

(under UNA)

For DL-LiteN

horn KBs K, the translation K† is a conjunction of formulas of the form

(horn) ∀x

  • A1(x) ∧ · · · ∧ An(x) → A(x)
  • Satisfiability of Horn formulas is P-complete (combined complexity)

For DL-LiteN

krom KBs K, the translation K† is a conjunction of formulas of the form

(krom) ∀x

  • A1(x) → A2(x)
  • ,

∀x

  • A1(x) → ¬A2(x)
  • ,

∀x

  • ¬A1(x) → A2(x)
  • Satisfiability of Krom formulas is NLogSpace-complete (combined complexity)

For DL-LiteN

core KBs K, the translation K† is a conjunction of formulas of the form

(core) ∀x

  • A1(x) → A2(x)
  • ,

∀x

  • A1(x) → ¬A2(x)
  • Satisfiability of core formulas is NLogSpace-complete (combined complexity)

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Canonical models for DL-LiteN

horn and DL-LiteN core

For a consistent DL-LiteN

horn KB K = (T , A), the canonical model IK

is constructed as follows:

  • 1. take the ABox and add ≥ qR to t(a) if q-many R-arrows start from a in A
  • 2. ‘saturate’ the existing types by applying the rules in T
  • 3. for every x, if (≥ qR) ∈ t(x) but there are < q R-arrows starting from x,

draw the missing R-arrows to fresh points and add ∃R− to their types

  • 4. go to Step 2
  • If I |

= K then there is a map h: ∆IK → ∆I such that, for all x, y ∈ ∆IK, basic concepts B and roles R, – if x ∈ BIK then h(x) ∈ BI; – if (x, y) ∈ RIK then (h(x), h(y)) ∈ RI

  • K |

= q( a) iff IK | = q( a)

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DL-Lite with role hierarchies

DL-LiteF

core (only functionality)

is NLogSpace-complete for combined complexity and in AC0 for data complexity DL-LiteHF

core

(DL-LiteF

core + R1 ⊑ R2)

is ExpTime-complete for combined complexity and P-complete for data complexity Example: A1 ⊓ A2 ⊑ C can be simulated by the axioms: A1 ⊑ ∃R1 A2 ⊑ ∃R2 R1 ⊑ R12 R2 ⊑ R12 ≥ 2 R12 ⊑ ⊥ ∃R−

1

⊑ ∃R−

3

∃R3 ⊑ C R3 ⊑ R23 R2 ⊑ R23 ≥ 2 R−

23 ⊑ ⊥ Moscow, 23 August 2010 22

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DL-Lite(HN )

α

: pushing the limits of DL-Lite

  • role inclusions + number restrictions

if R has a proper sub-role in T then T contains no negative occurrences of ≥ q R or ≥ q R− with q ≥ 2

  • positive occurrences of qualified number restrictions ≥ q R.C

≥ q R.C ≥ q R.C if ≥ q R.C occurs in T then T contains no negative occurrences of ≥ q′ R or ≥ q′ inv(R) with q′ ≥ 2

no TBox can contain both a functionality constraint ≥ 2 R ⊑ ⊥ and ≥ q R.C, for any q ≥ 1

  • role disjointness, symmetry, asymmetry, reflexivity and irreflexivity constraints

all these extensions do not change the complexity

in particular, same complexity of DL-Lite(RN )

α

and DL-LiteN

α

  • NB. transitive roles do not change the combined complexity

(NLogSpace-hard for data complexity)

Moscow, 23 August 2010 23

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DL-Lite without UNA

Without UNA, satisfiability of DL-LiteN

α KBs is NP-complete w.r.t. both

combined and data complexity, for any α ∈ {core, krom, horn, bool}

source of non-determinism: different ways of identifying ABox individuals

Lower bound: by reduction of monotone 1-in-3 3SAT n

k=1(ak,1 ∨ ak,2 ∨ ak,3)

a1,1 a1,2 a1,3 an,1 an,2 an,3

q q q

c1 cn

a1, a2, . . . , am

  • ✉

✉ ✉ ✉ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ❅ ■ ❅ ❅ ❅ ❅ ■ ✻ ✻

  • ✒
  • ✒

A = {ak,i = ak,j | i = j} ∪ {P (ck, ak,j) | k ≤ n, j ≤ 3} T = {≥ 4P ⊑ ⊥} Answer is yes iff there is a (true) variable ai in the given CNF such that Kai = (T , A ∪ {P (ck, ai) | k ≤ n}) is satisfiable without UNA NB: One can get rid of = in A

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The DL-Lite family: complexity-scape

. .

UNA no role inclusions horn core krom bool F N AC0 P

CONP

instance checking data complexity EXPTIME NP P NLOGSPACE satisfiability combined complexity

Legend

query answering

CONP

query answering = instance checking

. .

no UNA no role inclusions F N F N AC0 P

CONP

instance checking data complexity EXPTIME NP P NLOGSPACE satisfiability combined complexity

Legend

query answering

CONP

query answering = instance checking

. .

with/without UNA role inclusions F N F N F N AC0 P

CONP

instance checking data complexity EXPTIME NP P NLOGSPACE satisfiability combined complexity

Legend

query answering

CONP

query answering = instance checking

Moscow, 23 August 2010 25

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OWL 2 QL

‘An OWL 2 profile is a trimmed down version of OWL 2 that trades some expressive power for the efficiency of reasoning’ ‘OWL 2 QL is aimed at applications that use very large volumes of instance data, and where query answering is the most important reasoning task. In OWL 2 QL, conjunctive query answering can be implemented using conventional relational database systems.’ OWL 2 QL = DL-LiteH

core

with/without UNA with = (but no =) with (a)symmetric, (ir)reflexive and disjoint roles (but no transitive roles) Why not DL-LiteH

horn? Moscow, 23 August 2010 26

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OWL 2 EL

‘The OWL 2 EL profile is designed as a subset of OWL 2 that

  • is particularly suitable for applications employing ontologies that define

very large numbers of classes and/or properties,

  • captures the expressive power used by many such ontologies, and
  • for which ontology consistency, class expression subsumption, and

instance checking can be decided in polynomial time.’ For example, OWL 2 EL provides class constructors that are sufficient to express the very large biomedical ontology SNOMED CT (≈ 400.000 axioms) Pericardium ⊑ Tissue ⊓ ∃cont in.Heart Pericarditis ⊑ Inflammation ⊓ ∃has loc.Pericardium Inflammation ⊑ Disease ⊓ ∃acts on.Tissue Disease ⊓ ∃has loc.∃cont in.Heart ⊑ Heartdisease ⊓ NeedsTreatment

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Basic EL

EL concepts: C ::= ⊤ | ⊥ | A | ∃R.C | C1 ⊓ C2 EL TBoxes: finite sets of CIs C1 ⊑ C2 EL ABoxes: finite sets of assertions C(a), R(a, b) Concept satisfiability: given T , C, decide whether there is I | = T with CI = ∅ Subsumption: given T and concepts C, D, decide whether T | = C ⊑ D Instance checking: given a KB K = (T , A), C and an individual a from A, decide whether K | = C(a) Reducible to each other! Conjunctive query answering: given a KB K = (T , A), a CQ q( x) and a tuple

  • a of individual names from A, decide whether

K | = q( a)

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Observations and examples

EL can specify some positive information about types of points, viz:

that a point belongs to certain concepts

(but not that it does not belong to a concept);

that there is an outgoing R-arrow which ends in a certain concept

(but not that all outgoing R-arrows end in the concept);

that some concepts are disjoint

Example. Describe the models of the following KBs: T = {A ⊑ B1, B1 ⊑ ∃R.B1, ∃R.B1 ⊑ B2, B1 ⊓ B2 ⊑ ∃S.B2}, A = {A(a)}

  • Finite models are enough (finite model property)
  • Tree model property (but infinite!)
  • Not ‘local’ as DL-Lite; one-variable first-order formulas are not enough

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Simulations for EL

Let I and J be two interpretations. A relation ̺ ⊆ ∆I × ∆J is called a simulation of I in J if (concept) for every concept name A, if x̺y then x ∈ AI ⇒ y ∈ AJ (role) for every role name R, if x̺y then (x, x′) ∈ RI ⇒ ∃y′ (y, y′) ∈ RJ and x′̺y′ (I, x) (J , y) if there is a simulation ̺ of I in J with x̺y EL concepts are preserved under simulations, that is, if (I, x) (J , y) then x ∈ CI ⇒ y ∈ CJ , for every concept C EL concepts cannot distinguish between (I, x) and (J , y) if (I, x) (J , y) and (J , y) (I, x)

What are the differences between DL-Lite and EL?

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SLIDE 32

Tree canonical models for EL

(basically the same construction as for DL-LiteN

horn)

For a consistent EL KB K = (T , A), the canonical model IK is constructed as follows

  • 1. ‘saturate’ the existing types (starting with A) by applying the rules in T
  • 2. for every x, if ∃R.C ∈ t(x) but no R-arrow from x leads to C,

draw an R-arrow to a fresh point and add C to its type

  • 3. go to Step 1
  • If I |

= K then there is a map h: ∆IK → ∆I such that, for all x, y ∈ ∆IK, concept and role names A and R, – if x ∈ AIK then h(x) ∈ AI; – if (x, y) ∈ RIK then (h(x), h(y)) ∈ RI

  • K |

= q( a) iff IK | = q( a) IK can be infinite

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SLIDE 33

Compact canonical models for EL

ABox A a C TBox T ⊤ ⊑ ∃R.A, ⊤ ⊑ ∃R.B Canonical model IK aC

  • A
  • B
  • A
  • B
  • A
  • B

R R R R R R

Compact canonical model CK aC wA A wB B

R R R R R R IK is obtained by unravelling CK; (CK, a) (IK, a)

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Constructing CK Compact canonical interpretation CK:

Con(K) = the set of all concepts in K ∆CK = Ind(A) ∪ {wC | C ∈ Con(K)} wC is a witness for C ACK = {a | K | = A(a)} ∪ {wC | T | = C ⊑ A}

(A a concept name)

RCK = {(a, b) | R(a, b) ∈ A} ∪

(R a role name)

{(a, wC) | K | = ∃R.C(a)} ∪ {(wC, wD) | T | = C ⊑ ∃R.D}

  • Can be constructed in polynomial time in the size of K
  • Inconsistency can be detected during construction

❀ Satisfiability of EL KBs is PTime-complete

Moscow, 23 August 2010 33

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SLIDE 35

EL++ and OWL 2 EL

EL can be extended, without losing tractability , with

role implications R1 ◦ · · · ◦ Rn ⊑ R

(e.g., R ◦ R ⊑ R means transitivity)

range restrictions ⊤ ⊑ ∀R.C

(the range of R is in C)

domain restrictions ⊤ ⊑ ∀R−.C

(the domain of R is in C)

nominals {a}, a an individual name

≈ OWL 2 EL Extensions with any of the constructs C ⊔ D, ∀R.C, ≥ qR, R−, symmetric roles result in ExpTime-hard reasoning Exercise: construct an ELI (EL + inverse roles) KB K with CK of exponential size

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SLIDE 36

Ontology-based data access: the story so far

  • Next generation of information systems:

instance data + ontologies Reasoning problem: answering queries over knowledge & data

  • Instance queries

q = C(x)

  • ver a TBox T and an Abox A

an ABox individual a is an answer iff T , A | = C(a)

Example T = {Boss ⊑ Employee}, A = {Boss(bob)}, q = Emploee(x)

‘list all employees’

Answer: x = bob

(not an answer over A alone)

T , A | = C(a) iff there is no I | = T ∪ A such that I | = ¬C(a) iff T ∪ A ∪ {¬C(a)} is not satisfiable Instance checking is as complex as satisfiability checking

Moscow, 23 August 2010 35

slide-37
SLIDE 37

The story so far: more complex queries

  • Conjunctive queries

q = ∃ y ϕ( x, y),

where ϕ( x, y) is a conjunction of atoms A(z), R(z, z′) with z, z′ ∈ x ∪ y

  • x are the answer variables,
  • y the quantified variables

a tuple a of ABox individuals is an answer iff I | = ∃ y ϕ( a, y) for every I | = T ∪ A usually more complex than satisfiability

complexity of answering CQs without quantified variables?

  • Positive existential queries

q = ∃ y ϕ( x, y), ϕ may contain both ∧ and ∨

(but no ¬)

  • General FO queries

may contain ∧, ∨, ¬, ∀, ∃

no good: validity of FO formulas is undecidable

? description logics for which ontology-based query answering is

(1) as efficient as database query answering and (2) based on relational database management systems

Moscow, 23 August 2010 36

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SLIDE 38

Answering CQs in DL-LiteN

bool: exercise

T :

Research ⊑ ∃worksIn, ∃worksIn− ⊑ Project, Project ⊑ ∃manages−, ∃manages ⊑ Academic ⊔ Visiting, ∃teaches ⊑ Academic ⊔ Research, Academic ⊑ ∃teaches ⊓ ≤ 1 teaches, Research ⊓ Visiting ⊑ ⊥, ∃writes ⊑ Academic ⊔ Research,

A = {teaches(a, b), teaches(a, c)} q = ∃y

  • (∃teaches)(y) ∧ (≤ 1 teaches)(y)
  • is there anybody who teaches precisely one module?

T ′ = T ∪ {Visiting ⊑ ≥ 2 writes} Disjunction is (NP-) hard even for data complexity Only Horn logics can be suitable for ontology-based data access

Moscow, 23 August 2010 37

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SLIDE 39

Approach 1: query rewriting

Given a CQ q( x) over T , rewrite q( x) into an FO query q′( x) such that for all A and a, T , A | = q[ a] iff A | = q′[ a] conjunctive query q TBox T + union of conjunctive queries q′ ABox A ABox A ‘Maximal’ DLs for which query answering is in FO (=AC0) for data complexity: DL-Lite(HN )

horn

under UNA

and DL-LiteH

horn

without UNA

Moscow, 23 August 2010 38

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SLIDE 40

Query rewriting (cont.)

Want: all tuples a of individuals in A such that IK | = q( a)

where IK is the canonical model of K = (T , A)

Can: query the ABox A (using an RDBMS) To construct the canonical model IK:

  • 1. take the ABox

A B

  • 2. apply TBox axioms to ABox

∃R ∃S

  • 3. satisfy the existential quantifiers

by introducing ‘fresh’ witnesses q′ should incorporate T

Moscow, 23 August 2010 39

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SLIDE 41

Query rewriting: exercise

Compute the rewriting q′ for the following CQ and TBox: q(x) = x A y1 y2 y3 y4

R S S R R S

T = {B ⊑ ∃R, B ⊑ ∃S, ∃R ⊑ A}

  • r

q(x) = ∃y1, y2, y3, y4

  • A(x) ∧ R(x, y1) ∧ S(y1, x) ∧

R(y1, y2) ∧ S(y1, y3) ∧ R(y4, y2) ∧ S(y4, y3)

  • Hint: Consider all possible locations for y1, y2, y3, y4 in the canonical model

(in ABox or the tree part)

Moscow, 23 August 2010 40

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SLIDE 42

Exercise (cont.)

Suppose y1 is in the ABox, while y2, y3, y4 are in the tree part qabox x A y1 y1 y2 y3 y4

R S S R R S

qtree T = {B ⊑ ∃R, B ⊑ ∃S, ∃R ⊑ A}

  • Which concepts at y1 can ensure that there is a match for qtree in

the canonical model?

  • Which concepts at x can ensure A?

rewritten query for this partition: A ∨ B ∨ ∃R x y1 B

R S take disjunction of such queries for all partitions

Moscow, 23 August 2010 41

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SLIDE 43

Query rewriting: summary

Off-the-shelf RDBMSs can be used for CQ answering in DL-Lite working systems available (Quonto, Requiem, Presto) Experimental results: not scalable for large DL-Litecore ontologies complexity paradox? Reason: q over (T , A) ❀T q′ over A with |q′| = O(|T | · |q|)|q| is it optimal? Is data complexity a proper measure? (in RDBMSs, typical queries are relatively small...)

Take the structure of A, T , q into account? Bounded treewidth? ...

The rewriting approach is not applicable to other tractable DLs, e.g., EL

why?

Moscow, 23 August 2010 42

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SLIDE 44

Approach 2: data completion

conjunctive query q TBox T + FO query q′ + ABox A ABox A′

  • Extend ABox to the canonical model of (T , A)
  • Encode it as a finite structure A′
  • Rewrite q into q′ to ensure that the answers to q′ over A′ are correct

❀ combined approach

Moscow, 23 August 2010 43

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SLIDE 45

Compact canonical models (example)

ABox A a A b

S

TBox T A ⊑ ∃R, ∃S− ⊑ B, ∃R− ⊑ ∃S, ∃S− ⊑ ∃S Canonical model IK a A bB

S

  • R

S

  • S

S

B B B ‘Compact’ canonical model CK a A bB

S

cR

R

cS

S S

B

S

IK is obtained by ‘unravelling’ CK

Does CK give correct answers to queries?

Moscow, 23 August 2010 44

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SLIDE 46

Querying CK

IK a A bB

S

  • R

S

  • S

S

B B B CK a A bB

S

cR

R

cS

S S

B

S

q y0 y4 y1

R

y3 y2

S S S

What is the answer to q over CK? What is the answer to q over IK?

Find an FO expressible condition for such situations

a cS− cR cR cS

Moscow, 23 August 2010 45

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SLIDE 47

Tree witnesses

Given K = (T , A), q and R(x, y) ∈ q,

  • ne can compute (in polynomial time) a partial function

fR(x,y) : terms(q) → {cS | S used in K} ∪ {ε} such that

  • if fR(x,y) does not exist then y cannot be mapped to cR
  • if y is mapped to cR in CK and fR(x,y)(z) is defined then

– if fR(x,y)(z) = ε then we must have x = z – otherwise z must be mapped to fR(x,y)(z)

In the previous example, fR(y1,y2)(y3) = ε fR(y,y) does not exists

Moscow, 23 August 2010 46

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SLIDE 48

Query rewriting for DL-LiteN

horn (1)

rewrite a given CQ q = ∃ u ϕ into an FO query q† such that

  • answers to q over IK

= answers to q† over CK

  • |q†| = O(|q| · |T |)

q† = ∃ u (ϕ∧ϕ1∧ϕ2∧ϕ3)

ϕ1 =

  • v /

∈ u

  • R a role in T

(v = cR)

‘all answer variables must get ABox values’

  • NB. if ϕ1 is replaced with

ϕ′

1 =

  • v /

∈ u

¬aux(v), where aux is a new relation containing all cR, then |q†| = O(|q|)

Moscow, 23 August 2010 47

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SLIDE 49

Query rewriting for DL-LiteN

horn (2)

ϕ2 =

  • R(x,y)∈q

fR(x,y) does not exist

(y = cR)

if no tree witness exists then y cannot be mapped to a non-ABox element

ϕ3 =

  • R(x,y)∈q

fR(x,y) exists

  • (y = cR)

→

  • fR(x,y)(z)=ε

(z = x)

  • Moscow, 23 August 2010

48

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SLIDE 50

Exercises

Exercise 1: compute q′ for the exercise on page 45

ϕ1 = ϕ2 = ⊤ ϕ3 = (y2 = cS) → (y1 = y3)

Exercise 2: Use the rewriting and combined approaches for the following KB and query: T : A: {Student(a), Professor(b), teaches(b, a)} Student ⊑ ∃hasTutor ∃teaches− ⊑ Student Professor ⊑ ∃teaches ∃hasTutor− ⊑ Professor q(x) = teaches(x, y), hasTutor(y, z), hasTutor(u, z)

Moscow, 23 August 2010 49

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SLIDE 51

Query answering in DL-Lite(HN )

horn

what can we do with role inclusions? Reduce positive existential queries over DL-Lite(HN )

horn

KBs to unions of (exponentially many) CQs over DL-LiteN

horn KBs

Step 1. DL-Lite(HN )

horn

KB K = (T , A) ❀ DL-LiteN

horn KB K = (Th, A)

by replacing all R ⊑∗ S with ∃R ⊑ ∃S

(⊑∗ is the transitive closure of ⊑)

Step 2. Positive existential q over K ❀ union of CQs qh over CKh: – replace each R(t, t′) in q with

  • S⊑∗R

S(t, t′) – convert result into disjunctive normal form (exponential blowup) ≤ r|q| conjuncts, where r is the depth of ⊑∗ K | = q( a) iff CKh | = qh

is there a polynomial rewriting?

Moscow, 23 August 2010 50

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SLIDE 52

Other applications

  • CK can be constructed by first-order queries

❀ pure polynomial rewriting for DL-LiteN

core

  • without the UNA, the technique is applicable to query answering in DL-Lite(HF)

horn

(which is P-complete for data complexity)

  • experiments show that the approach is competitive

with executing the original query over the data

(the formulas ϕ1–ϕ3 introduce additional selection conditions on top of the original query)

Open questions

– is the exponential blowup unavoidable for role inclusions? – is the exponential blowup unavoidable for positive existential queries? – for which DLs pure rewriting can be polynomial?

Moscow, 23 August 2010 51

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SLIDE 53

Query rewriting in EL

The query rewriting approach cannot work for EL because already instance checking in EL is PTime-complete w.r.t. data complexity Lower bound: by reduction of PTime-complete entailment for Horn CNF E.g., ϕ = (a1 ∧ a2 → a3) ∧ (a2 → a1) ∧ a2 is encoded by the ABox Aϕ a1 a2 a3 c1 c2

P R S S P R

T and the (ϕ-independent) TBox T : T = {∃S.(∃P.T ⊓ ∃R.T ) ⊑ T } ϕ | = ai iff (T , Aϕ) | = T (ai)

Moscow, 23 August 2010 52

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SLIDE 54

Compact canonical models for EL

ABox A a C TBox T ⊤ ⊑ ∃R.A, ⊤ ⊑ ∃R.B Canonical model IK aC

  • A
  • B
  • A
  • B
  • A
  • B

R R R R R R

Compact canonical model CK aC wA A wB B

R R R R R R IK is obtained by unravelling CK

Difference from DL-Lite: multiple R-successors of non-ABox points

Moscow, 23 August 2010 53

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SLIDE 55

Query rewriting for EL

rewrite a given CQ q = ∃ u ϕ into an FO query q† such that

  • answers to q over IK

= answers to q∗ over CK

  • |q∗| = O(|q| · |T |)

q† = ∃ u (ϕ∧ϕ1∧ϕ2∧ϕ3)

ϕ1: answer variables and variables in cycles in q must be mapped to ABox ϕ2: if x1 x2 x3

R R

in q and x2 is mapped outside the ABox then x1 = x3 ϕ3: if x1 x2 x3

R S

in q and R = S then x2 must be mapped to ABox

Moscow, 23 August 2010 54

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SLIDE 56

Query rewriting for EL: example 1

ABox A a A bA

R

TBox T A ⊑ ∃R.A CK a A bA wA A

R R R R

q(x) = ∃y

  • R(x, y) ∧ R(y, y)
  • answers x = a,

x = b q∗(x) = ∃y

  • R(x, y) ∧ R(y, y) ∧ ABox(x) ∧ ABox(y)
  • no answer

Moscow, 23 August 2010 55

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SLIDE 57

Query rewriting for EL: example 2

ABox A a A bA

R

TBox T A ⊑ ∃R.A CK a A bA wA A

R R R R

q(x, x′) = ∃y

  • R(x, y) ∧ R(x′, y) ∧ R(x, x′)
  • answers x = a,

x′ = b q∗(x) = ∃y

  • R(x, y) ∧ R(x′, y) ∧ R(x, x′) ∧

ABox(x) ∧ ABox(x′) ∧

  • ¬ABox(y) → x = x′

no answer

Moscow, 23 August 2010 56

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SLIDE 58

Developing and maintaining ontologies

  • versions:

comparing logical consequences over some common vocabulary Σ not the syntactic form of the axioms (as in diff)

  • refinement:

adding new axioms but preserving the relationships between terms of a certain part Σ of the vocabulary

  • reuse:

importing an ontology and using its vocabulary Σ as originally defined

(relationships between terms of Σ should not change)

  • module extraction:

computing a subset M (ideally as small as possible) of an ontology T that ‘says’ the same about Σ as T

new types of reasoning problems

Moscow, 23 August 2010 57

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SLIDE 59

DL-Lite: Description Logic for Databases

A fragment of a conceptual schema:

disj cov Staff Research Visiting Academic ProjectManager △ △ ▽ Project

❄

worksOn 1..* 1..*

✻

manages 1..2

Translating into DL:

∃ manages.⊤ ⊤ ⊤ ⊑ ProjectManager ∃ manages−.⊤ ⊤ ⊤ ⊑ Project Project ⊑ ∃ manages−.⊤ ⊤ ⊤ ≥ 3 manages−.⊤ ⊤ ⊤ ⊑ ⊥ Research ⊓ Visiting ⊑ ⊥ Academic ⊑ ProjectManager ProjectManager ⊑ Academic ⊔ Visiting . . .

R ::= P | P − B ::= ⊥ | Ai | ∃R | ≥ q R C ::= B | ¬C | C1 ⊓ C2 | C1 ⊔ C2 DL-Litebool TBox axioms: C1 ⊑ C2 ABox assertions: C(a), R(b, c) Essentially positive existential queries: ∃ yϕ( x, y), built from C(t), R(t, t′), ∧, ∨

Moscow, 23 August 2010 58

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SLIDE 60

Σ-entailment and Σ-inseparability

Let T1 and T2 be TBoxes and Σ a signature (concept and role names) When do T1 and T2 ‘say’ the same about Σ?

  • T1 Σ-concept entails T2 if, for all Σ-concept inclusions C ⊑ D,

T1 c

Σ T2

T1 | = C ⊑ D implies T2 | = C ⊑ D

  • T1 Σ-query entails T2 if, for all Σ-queries q(

x) and ABoxes A, T1 q

Σ T2

(T1, A) | = q( a) implies (T2, A) | = q( a),

for all

a

  • . . .
  • T1 Σ-model entails T2 if,

for all Σ-interpretations I, T1 m

Σ T2

∃ I1 ⊇ I I1 | = T1 implies ∃ I2 ⊇ I I2 | = T2

  • T1 and T2 are SΣ (concept/query/model) inseparable if

T1 ≡S

Σ T2

T1 S

Σ T2 and T2 S Σ T1 Moscow, 23 August 2010 59

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SLIDE 61

Σ-inseparability: Examples

Example 1.

Σ = {Lecturer, Course} T1 = ∅, T2 = {Lecturer ⊑ ∃teaches, ∃teaches− ⊑ Course}

  • Is

T1 ≡c

Σ T2 ?

  • Is

T1 ≡q

Σ T2 ?

Take A = {Lecturer(a)}, q = ∃y Course(y). Then (T1, A) | = q but (T2, A) | = q

Example 2.

Σ = {Lecturer} T1 = ∅, T2 = {Lecturer ⊑ ∃teaches, Lecturer ⊓ ∃teaches− ⊑ ⊥}

  • Is

T1 ≡c

Σ T2 ?

  • Is

T1 ≡q

Σ T2 ?

Take A = {Lecturer(a)}, q = ∃y ¬Lecturer(y). Then (T1, A) | = q and (T2, A) | = q

Moscow, 23 August 2010 60

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SLIDE 62

Σ-inseparability: Examples (cont.)

Example 3. Let T1 contain the axioms

Research ⊑ ∃worksOn, ∃worksOn− ⊑ Project, Project ⊑ ∃manages−, ∃manages ⊑ Academic ⊔ Visiting, ∃teaches ⊑ Academic ⊔ Research, Academic ⊑ ∃teaches ⊓ ≤ 1 teaches, Research ⊓ Visiting ⊑ ⊥, ∃writes ⊑ Academic ⊔ Research, T2 = T1 ∪ {Visiting ⊑ ≥ 2 writes} and Σ = {teaches}

  • T1 ≡c

Σ T2

T2 | = Visiting ⊑ Academic, but nothing new in the signature Σ

  • T1 ≡q

Σ T2:

A = {teaches(a, b), teaches(a, c)} q = ∃x

  • (∃teaches)(x) ∧ (≤ 1 teaches)(x)
  • ‘is there anybody who teaches precisely one module?’

. .

b b b b

Research Visiting Project

a b c (T1, A) | = q (I | = (T1, A) but I | = q) (T2, A) | = q

. .

b b b b

Research Visiting ⊔ Academic Project

a b c

. .

b b b b b

Research Academic Project

a b c

Moscow, 23 August 2010 61

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SLIDE 63

Σ-entailment: semantic criteria

Let Q be a set of numerical parameters and Σ a signature ΣQ-concepts B: Ai ∈ Σ and (≥ q R) with q ∈ Q and R ∈ Σ ΣQ-type t t t is a set of ΣQ-concepts containing B or ¬B (but not both), for all B For a TBox T , a ΣQ-type t t t is T -realisable if t t t is satisfied in a model of T

(i.e., there is a I of T and a point w in it such that w ∈ BI iff B ∈ t t t)

a set Ξ of ΣQ-types is precisely T -realisable if there is a model of T realising precisely the types from Ξ

  • Theorem. Let Q denote the set of parameters occurring in T1 ∪ T2

T1 Σ-concept entails T2 iff every T1-realisable ΣQ-type is T2-realisable T1 Σ-query entails T2 iff every precisely T1-realisable set Ξ of ΣQ-types is precisely T2-realisable

Moscow, 23 August 2010 62

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SLIDE 64

Σ-inseparability: complexity

Theorem.

  • Deciding Σ-concept and Σ-query inseparability is

Πp

2-complete

  • Deciding Σ-model inseparability is

elementary and CONEXPTIME-hard

  • Can be simpler for various fragments of DL-LiteN

bool

E.g. deciding Σ-concept and Σ-query inseparability for DL-LiteN

horn is

CONP-complete

  • NB. Πp

2-completeness means that the problem can be encoded as

satisfiability of ∀∃ quantified Boolean formulas Various QBF solvers can be used to check Σ-concept and Σ-query inseparability

  • NB. Inseparability is much harder for ALC and other non-‘lite’ DLs

(2EXPTIME-complete for ALC, undecidable for ALCQIO)

Moscow, 23 August 2010 63

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SLIDE 65

Encoding Σ-concept entailment in QBF

Let T be a TBox, Q a set of numerical parameters and t t t a sig(T )Q-type ‘t t t0 is T -realisable with t t t1, . . . ,t t tn being witnesses’

propositional formula

= ΦT (b0, b1, . . . , bn)

bj is the vector of all propositional variables B∗ of the type t t tj

Then the condition ‘every T1-realisable ΣQ-type t t t is T2-realisable’ is described by the following QBF ∀bΣQ

  • ∃bT1\ΣQ

∃bT1

1 . . . ∃bT1 n1 ΦT1(bΣQ

· bT1\ΣQ , bT1

1 , . . . , bT1 n1)

→ ∃bT2\ΣQ ∃bT2

1 · · · ∃bT2 n2 ΦT2(bΣQ

·bT2\ΣQ , bT2

1 , . . . , bT2 n2)

  • (bΣQ

is the ΣQ-part of b0 and bTi\ΣQ contains the rest of the variables)

Moscow, 23 August 2010 64

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SLIDE 66

Experiments

TBox instances

(standard Department Ontology + ICNARC)

  • no. of

axioms basic concepts series description instances

T1 T2 T1 T2 Σ NN

T1 does not Σ-concept entail T2

840 59–308 74–396 47–250 49–300 5–103 YN

T1 Σ-concept but not Σ-query entails T2

504 56–302 77–382 44–246 58–298 6–89 YY

T1 Σ-query entails T2

624 43–178 43–222 40–158 40–188 5–64

QBF solvers

  • sKizzo 0.8.2

(http://skizzo.info/)

  • 2clsQ

(http://www.cs.toronto.edu/~fbacchus/)

  • yQuaffle

(http://www.princeton.edu/~chaff/quaffle.html)

  • QuBE 6.4/7.0

(http://www.star.dist.unige.it/)

  • AQME

(http://www.mind-lab.it/aqme/)

  • depQBF

(http://fmv.jku.at/depqbf/) Σ-concept entailment QBF Σ-query entailment QBF

series

variables clauses variables clauses NN 1,469–48,631 2,391–74,621 1,715–60,499 5,763–1,217,151 YN 1,460–46,873 2,352–71,177 1,755–59,397 7,006–1,122,361 YY 1,006–16,033 1,420–23,363 1,202–20,513 2,963–204,889

number of clauses is

linear quadratic

(in the number of roles)

Moscow, 23 August 2010 65

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SLIDE 67

Experimental results: percentage of solved instances

Σ-concept entailment Σ-query entailment

YY

%

10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s

YN

%

10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s

NN

%

10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s 10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s 10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s 10 20 30 40 50 60 70 80 90 100 1 2 4 8 16 32 64 128 256 512 s

Moscow, 23 August 2010 66

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SLIDE 68

Forgetting

studied under different names: forgetting, uniform interpolation, variable elimination. . .

A DL L admits forgetting (has uniform interpolation) if, for every T in L and every Σ, there exists TΣ in L with sig(TΣ) ⊆ Σ such that T and TΣ are Σ-concept inseparable in L Theorem Both DL-Litebool and DL-Litehorn have uniform interpolation and the uniform interpolant can be constructed in exponential time DL-Liteu

bool:

C ::= . . . | ∃C | . . .

(universal modality)

e.g., (≥ 2 teaches) ⊑ ∃ ∃ ∃(∃teaches ⊓ ≤ 1 teaches)

TΣ with sig(TΣ) ⊆ Σ is a uniform interpolant of T w.r.t. Σ in DL-Liteu

bool if

T | = C ⊑ D iff TΣ | = C ⊑ D, for every C ⊑ D in DL-Liteu

bool with sig(C ⊑ D) ⊑ Σ

T ′ Σ-query entails T iff T ′ | = C ⊑ D, for each C ⊑ D ∈ TΣ Theorem For every T in DL-Litebool and every Σ one can construct a uniform interpolant TΣ of T w.r.t. Σ in DL-Liteu

bool in time exponential in T Moscow, 23 August 2010 67

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SLIDE 69

What is a module?

Let S be an inseparability relation, T a TBox and Σ a signature. M ⊆ T is

  • an SΣ-module of T

if M ≡S

Σ T

  • a self-contained SΣ-module of T

if M ≡S

Σ∪sig(M) T

  • a depleting SΣ-module of T

if ∅ ≡S

Σ∪sig(M) T \ M

M is a minimal module of T if it can’t be made smaller Facts:

  • depleting ≡q

Σ-module

⇒ ⇒ ⇒ self-contained ≡q

Σ-module

⇒ ⇒ ⇒ ≡q

Σ-module

  • self-contained ≡c

Σ-module

⇒ ⇒ ⇒ ≡c

Σ-module

  • There is precisely one minimal depleting ≡q

Σ-module

  • There may be (exponentially) many minimal modules of other types

Moscow, 23 August 2010 68

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SLIDE 70

Modules for Σ = {Publisher}

(1) Publisher ⊑ ∃pubHasDistrib (11) BookUser ⊑ User (2) ∃pubHasDistrib− ⊑ Distributor (12) User ⊑ ∃hasRole (3) Publisher ⊑ ¬Distributor (13) ∃hasRole− ⊑ Role (4) ∃pubHasDistrib ⊑ Publisher (14) Role ⊑ ¬Publisher (5) Publisher ⊑ ≤ 1 pubHasDistrib (15) User ⊑ ¬Publisher (6) Role ⊑ ¬Distributor (16) Role ⊑ ¬User (7) User ⊑ ¬Distributor (17) User ⊑ ∃userAdmedBy (8) Publisher ⊑ ∃pubAdmedBy (18) ∃userAdmedBy− ⊑ AdmUser (9) ∃pubAdmedBy− ⊑ AdmUser ⊔ BookUser (19) ∃userAdmedBy ⊑ User (10) AdmUser ⊑ User (20) ∃pubAdmedBy ⊑ Publisher

the minimal Sc

Σ-module is ∅

minimal Sq

Σ-modules of T :

MD , MR and MU the minimal depleting Sq

Σ-module is T Moscow, 23 August 2010 69

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SLIDE 71

Module extraction algorithms

  • minimal SΣ-module

input T , Σ M := T for each α ∈ M do if M \ {α} ≡S

Σ M then M := M \ {α}

end for

  • utput M

NB: depends

  • n the order
  • f axioms in T
  • minimal depleting SΣ-module

input T , Σ T ′ := T ; Γ := Σ; W := ∅ while T ′ \ W = ∅ do choose α ∈ T ′ \ W W := W ∪ {α} if W ≡S

Γ ∅ then

T ′ := T ′ \ {α}; W := ∅; Γ := Γ ∪ sig(α) endif end while

  • utput T \ T ′

Moscow, 23 August 2010 70

slide-72
SLIDE 72

Practical minimal module extraction

34

MCM

54

MQM

294

MDQM

465

⊤ ⊥M

206

SRS

598

Prompt

393

E-mod

Core (1283)

39

MCM

66

MQM

319

MDQM

351

⊤ ⊥M

117

SRS

509

Prompt

191

E-mod

Umbrella (1247)

200 400 600

Module sizes and standard deviation for |Σ| = 10

Moscow, 23 August 2010 71

slide-73
SLIDE 73

Complexity of Σ-entailment

  • Σ-concept and Σ-query entailment are EXPTIME-complete for EL;
  • for EL TBoxes consisting of (possibly cyclic) concept definitions only,

Σ-concept entailment is tractable;

  • in ALC and ALCQI, Σ-concept entailment is 2EXPTIME-complete;
  • in ALCQIO, Σ-concept entailment is undecidable;
  • complexity of Σ-query entailment is currently unknown

for expressive DLs such as ALC;

  • Σ-model entailment is undecidable for EL (and all its extensions).

Moscow, 23 August 2010 72