Overview of the Course

Part D

Approximate reasoning on tableaux

Holger Wache

Overview of the Course

A. Semantic Web in general and OWL syntax

B. OWL Semantics (DLs) and tableaux reasoning

C. Approximate reasoning in general

D. Approximate reasoning on tableaux

E. Approximate resolution for OWL

ReasoningDL- Reasoning Ontology

Semantic Web Systems in General

Input Output

Description Logics

„ C

u

„ C

t

„

¬

„

∃ R.C

„

…

C:=D

„ Human := MantWoman

„ Man := ¬Woman

„ Father := Manu

∃ hasChild.Human CvD

„ Father v Human

„ Father v Woman

DL- Reasoning

Problem

Input Output

t

Toy Examples

Real World

„ Performance

… DL-Reasoning in EXPTIME (OWL DL

in NEXPTIME) Ontology

Answers Query

Reasoning Method Knowledge

Base

DL- Reasoning

Ontology

Problem

„ Robustness

… For example during query answering Possible

Answers

Input Output

DL- Reasoning

Ontology

Input DL- Output

Reasoning Ontology

Input Output

Approximation Approaches

‰

Language Weakening

‰

Language Weakening

‰

Approximate Deduction

‰

Approximate Deduction

‰

Theory

Approximation

‰

Theory

Approximation

+ +

‰

Approximate Input

‰

Approximate Input

DL- Reasoning

Ontology

Input DL- Output

Reasoning Ontology

Input Output

Approximation Approaches

‰

Language Weakening

‰

Language Weakening

‰

Approximate Deduction

‰

Approximate Deduction

‰

Theory

Approximation

‰

Theory

Approximation

+ +

‰

Approximate Input

‰

Approximate

Input

Approximation through Language Weakening

Input Reasoning Output

Method Knowledge

Base

DL- Reasoning

Ontology

OWL-FULL

OWL-DLP

A-Box

Role- free DataBase

Queries

T-Box

DLP- Reasoning

Approximation on A-Boxes

Normal A-Box

Role-Free A-Box

Querying in Role-Free A-Boxes

Query

Instances contained in the query

Instances to be tested if contained

Instance retrieval

„ For role-free A-Boxes a:Q↔ I_av Q; Iaconcept description of instance a

„ Try to replace DL reasoning as much as possible with

(scalable) DB retrieval

… Classify Q [DL REASONING]

… For all C v Q: a:C→a:Q [DATABASE RETRIEVAL]

… If needed check border instances

[DL REASONING]

Q

DL- Reasoning

Ontology

Input DL- Output

Reasoning Ontology

Input Output

Approximation Approaches

‰

Language Weakening

‰

Language Weakening

‰

Approximate Deduction

‰

Approximate Deduction

‰

Theory Approximation

‰

Theory Approximation

+ +

‰

Approximate Input

‰

Approximate Input

Approximate Entailment by Cadoli- Schaerf

„

S-1-entailment:

interpret everything outside of S as false

„

S-3-entailment:

interpret everything outside of S as true (or normal)

S

L

x ¬x

S1 S3

0/0 1/0 1/1

0/1

Cadoli-Schaerf for DL

„ Description Logic is subset of First Order Logic

„ Cadoli Schaerf can directly be applied

„ Needs to implement a specialized reasoner ….

² ^S

Approximate Reasoning

Description Logic (DL) = Tableaux Reasoning

{((C u ¬E) t (D u F)) u E}

{(C u ¬E) t (D u F), E}

{D u F, E}

{C u ¬E, E}

{C, ¬E, E} {D, F, E}

Approximating Inference Rule

x

{((C u ¬E) t(D u F)) u E}

{(C u ¬E) t(D u F), E}

{D u F, E}

{C u ¬E, E}

{C, ¬E, E} {D, F, E}

Approximating Query

x

{((C u ¬E) t (D u F)) u E}

OR

Same effect when query is modified

Approximate Reasoning through modification of the query

Cadoli-Schaerf for DL

„

Instead of using a different entailment

„

Modify the query

„

Existing reasoners can be used

(C u ¬E) t (D u F)

Cadoli-Schaerf for DL

Cadoli-Schaerf

Modify Query (C u ¬E) t (D u F)

=

² ^S

DL- Reasoning

Ontology

Input DL- Output

Reasoning Ontology

Input Output

Approximation Approaches

‰

Language Weakening

‰

Language Weakening

‰

Approximate Deduction

‰

Approximate Deduction

‰

Theory

Approximation

‰

Theory

Approximation

+ +

‰

Approximate Input

‰

Approximate Input

Query

Approximate Deduction through Simplification

„ Simplify query

„ Simple query ⇒ fast query answering

„ Simple query ⇒

approximated answers

„ Continuously complete query

„ Anytime behavior

t Simplified Query

Reasoner

Quality

How to simplify?

Query = … u Φ u … u ( …t Ψ t …)

First Idea:

Omit some parts (e.g. Φ, Ψ) First Idea:

Omit some parts (e.g. Φ, Ψ)

Q

⊆ Q

Q

⊆ Q

Q

Q

Query = … u Φ u … u ( …t Ψ t …)

How to simplify? (II)

Second Idea:

Rewrite some parts (e.g. Φ, Ψ) Second Idea:

Rewrite some parts (e.g. Φ, Ψ)

Q

⊆ Q

>

Q

⊆ Q

>

Q

⊆ Q

Cadoli-Schaerf Approximation for DLs

„

Well-founded theory with (theoretical) nice results

„

BUT:How to increase the complexity?

„ Levels i = nested quantifiers

„ C_i= Ommit all exist qantifiers greater or equal level i

S3-Approximation for Description Logics (ALE)

C₀

C₁

C₂ L₁ L₀

Application

Application: Classification

„

Central process Classify Query Q

„

Contained in

… Generating the subsumption hierarchy

… Instance Retrieval

„

Generates a sequence of subsumption tests

9

9 9

8 9

8 8

Q

Approximating the Classification

„

„

Cadoli-Schaerf ensures:

Level := 0 Compute

Level := Level+1 Unsatifyable?

Max Level?

t

f

f t

TRUE

FALSE

?

Implementation

RACER

DIG Interface

FACT FACT

subsumes/satisfy Taxonomy Classify

Approximate Query

Mixed Results:

Classifying in TAMBIS

„

Application: Classification of Concepts

⇒ sequence of subsumption test: C v D

8 181

157 32

≈24 ≈0

≈0 ≈279

157 32

8 149

Further Results

Query = … u Φ u … u ( …t Ψ t …) Problem: Term Collapsing

„ Term D

… From the subsumption hierarchy

… Very often also conjunction of subterms

„ Term C

… to be classified;

… very often conjunction of subterms

… e.g. conjunctive queries

Queries have very often this structure

Query = … u Φ u … u ( …t Ψ t …) Problem: Term Collapsing

„ Term D

… From the subsumption hierarchy

… Very often also conjunction of subterms

> > >

„ Term C

… to be classified;

… very often conjunction of subterms

… e.g. conjunctive queries

Query = … u Φ u … u ( …t Ψ t …) Problem: Term Collapsing

„ Term D

… From the subsumption hierarchy

… Very often also conjunction of subterms

> >

„ Term C

… to be classified;

… very often conjunction of subterms

… e.g. conjunctive queries

>

Classifying in TAMBIS (IV)

65 = 35,9%

157 = 100% 190 = 62,1%

Term Collapsing:

Approximation and Proof Reuse

Proof for Approximation

Proof for Normal Case

Proof for Approximation + Extension

≤ ≤

No Time may be saved

No Time may be saved

Proofs can only be extended at the leaves Proofs can only be extended at the leaves

Lessons learned

„

Avoid Term Collapsing

…Replace ψ with something else than > or ⊥

„

Find better places to rewrite

…Ontology-adapted φ?

…Heuristic to select the rewriting places?

Special Case: Instance Retrieval

Focused Case: Instance Retrieval

„

Find all instances a which belongs to a query Q:

a:Q

„

Tool InstanceStore:

…Try to replace DL reasoning as much as possible with (scalable) DB retrieval

…Only applyable to role-free A-Boxes

a:Q↔I_av Q; I_aconcept description of instance a

„

Boolean Conjunctive Queries

…q₁∧...∧q_n, where q₁,...,q_nare of the form x:C or hx,yi:R

…Restrict to those which can be converted to a concept expression C

New Approximation Method:

Heuristic Ordering of Conjuncts

„ Compute a ranking value for each conjunct

„ Order the conjuncts q₁,Λ,q_n according to their value

„ Complete approximated query with more and more expensive conjuncts

Reasoner

Φ(q₁)

q_n q₁ q_i Φ(q_i) Φ(q_n) Φ(q₁) Φ(q_i) Φ(q_n) < <

Query q_n q_i q₁∧..∧ ∧..∧

∧..∧ ^∧

How to order conjuncts?

„

Recall

…Node expansion

…Depth-First and Breadth-First Search

„

Furthermore

…According to the needed computation time for each conjunction

„ Estimate the computation time a priori

…According to the possible search space reduction

„ Prefer conjuncts which eliminate a lot of instances

father-of Female degree father-of

Female

father-of Female

works-at degree

father-of Female

works-at degree

awarded-by {vu}

How to estimate the computation

costs

Effects of Cadoli-Schaerf for Subsumption

C v D

Semantics

(C v D) ^⊥

C v D ↔ ² C u ¬D

Effects of Cadoli-Schaerf for Subsumption

C v D

Semantics

(C v D) ^⊥

(C v D) ^⊥ ↔ ² ( C u ¬D ) ^⊥

C v D

(C v D) ^⊥

Effects of CS for Subsumption:

Term Collapsing

Semantics

Term collapsing

Effects of new Approximation

Semantics

not

changed;

C v D

(C (I _a v v Q) D) ^{Δ Δ} (I _a v Q)

only

Evaluation Scenario

DB Instance

Store Gene Ontology

DL Rasoner:

FACT++

Aproxx Server Q1, …, Q8

Q9, …, Q18

Results: Subsumption tests

Results: Time

Summary

„

Achieving scalability through approximation and modularization/distribution

„

S1&S3 approximation seems not to work in practical situations

„

Heuristic approximation seems to be better

Performance improvements (only) in restricted cases?!

„

What’s about Robustness?

Answers Query

Reasoning Method Knowledge

Base

DL- Reasoning

Ontology

Problems tackled in KWEB

„ Robustness

… For example during query answering Possible

Answers

WP1.1 Use-Case 1 “Human Resources” and WP2.1

„

Collaborative Answering for supporting users

„

Approximation as a basic technology

„

Feature: Distributed reasoning possible Feature: Matching of different ontologies

Job request:

Semantic Web professional with 10 years business experiences, solid background in Trust and Security, finished studying with a excellent grade, not married, and not older than 21 years

Job request:

Semantic Web professional with 10 years business experiences, solid background in Trust and Security, finished studying with a excellent grade, not married, and not older than 21 years

L3S Use-Case “E-learning”

(KWEB portal: REASE)

Ressource request (somehow completed):

I’m looking for a lecture with subject

“approximation for the Semantic Web”

hold in Slovak. I have experiences in propositional logics and I followed a course about Timbuktu.

Ressource request (somehow completed):

I’m looking for a lecture with subject

“approximation for the Semantic Web”

hold in Slovak. I have experiences in propositional logicsand I followed a course about Timbuktu.

Ressource:

Lecture with “approximation for the Semantic Web” in the description (only a part of that talks) hold in Danish. It needs experiences in description logics and OWL.

Ressource:

Lecture with “approximation for the Semantic Web” in the description (only a part of that talks) hold in Danish. It needs experiences in description logicsand OWL.

Collaborative Query Answering

„

Assisting users in formulating their query

„

Technically: Query reformulation

„

How to transform?

„

Technologically similar to

approximation

Query

Transformation

Answers Answers

Logic Approximation

Idea: Rewriting rules

Rewrite

Rewrite

Rewrite

Rewrite Rewrite Rewrite

Rewrite Re

write

User- Centric Domain-

Centric

Rewrite Rule

„

Three parts

„

Invocation:

1. If “Pattern” matches (part-of) query, …

2. … replace matched pattern with

“replaced-by” …

3. … if “Condition” is satisfied

Pattern

Replaced-By

Condition

Rewriting rules: Example

PATTERN

{Resource} subject {Subject}

WHERE

Subject Like Value^^xsd:string,

WITH

NEWTitle = concat("*",concat(Value,"*")) REPLACE-BY

{Resource} subject {Subject}, {Resource} title {TMPTitle}

WHERE

TMPTitle Like NEWTitle

Rewrite Rule Example Applied

SELECT * FROM

{Resource} subject {Subject}, {Resource} title {TMPTitle},

{Resource} title {Title},

… WHERE

SELECT * FROM

{Resource} subject {Subject}, {Resource} title {Title},

… WHERE

Subject Like "inference engines",

…

DL- Reasoning

Ontology

Input DL- Output

Reasoning Ontology

Input Output

Approximation Approaches

‰

Language Weakening

‰

Language Weakening

‰

Approximate Deduction

‰

Approximate Deduction

‰

Theory

Approximation

‰

Theory

Approximation

+ +

‰

Approximate Input

‰

Approximate Input

Ontology

Output Reasoning

Method

Approximation through Knowledge Compilation

Input

Ontology (optimized)

Reasoning (optimized)

Thank you for your

attention!