KBS Knowledge-Based Systems Group January 24, 2011 Redl C., Eiter - - PowerPoint PPT Presentation

Declarative Belief Set Merging using Merging Plans

Christoph Redl Thomas Eiter Thomas Krennwallner

{redl,eiter,tkren}@kr.tuwien.ac.at

KBS

Knowledge-Based Systems Group

January 24, 2011

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 1 / 26

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 2 / 26

Motivation

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 3 / 26

Motivation

Motivation

Usage of Multiple Knowledge Bases

No single point of truth Combining knowledge from different sources into a coherent view Possibly heterogeneous knowledge bases Contents may be contradicting

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 4 / 26

Motivation

Motivation

Usage of Multiple Knowledge Bases

No single point of truth Combining knowledge from different sources into a coherent view Possibly heterogeneous knowledge bases Contents may be contradicting

Examples

Judgment aggregation (discussed later) Merging of decision diagrams

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 4 / 26

Merging Framework

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 5 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ A belief set is a set B ⊆ LitΣ

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ A belief set is a set B ⊆ LitΣ Set of all belief sets A(Σ) over Σ: A(Σ) := 2LitΣ

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ A belief set is a set B ⊆ LitΣ Set of all belief sets A(Σ) over Σ: A(Σ) := 2LitΣ A collection of belief sets is a set B ⊆ A(Σ)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ A belief set is a set B ⊆ LitΣ Set of all belief sets A(Σ) over Σ: A(Σ) := 2LitΣ A collection of belief sets is a set B ⊆ A(Σ)

Definition (Knowledge Bases)

We abstract from a concrete language for knowledge bases KB

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Definition (Collections of Belief Sets)

Belief: atomic formula or a negated atomic formula Signature Σ = (Σc, Σp) (Σc ... constant symbols, Σp ... predicate symbols) Set of all beliefs, i.e., all literals: LitΣ A belief set is a set B ⊆ LitΣ Set of all belief sets A(Σ) over Σ: A(Σ) := 2LitΣ A collection of belief sets is a set B ⊆ A(Σ)

Definition (Knowledge Bases)

We abstract from a concrete language for knowledge bases KB Knowledge bases are identified with assigned collections of belief sets (their “semantics”): BS(KB) ⊆ A(Σ)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 6 / 26

Merging Framework

Belief Sets and Knowledge Bases

Example

KB = {dog(sue) ∨ cat(sue), female(sue)} Associated collections of belief sets depend on the semantics, e.g.,:

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 7 / 26

Merging Framework

Belief Sets and Knowledge Bases

Example

KB = {dog(sue) ∨ cat(sue), female(sue)} Associated collections of belief sets depend on the semantics, e.g.,: Minimal Herbrand models: BS(KB) = { {dog(sue), ¬cat(sue), female(sue)}, {¬dog(sue), cat(sue), female(sue)} }

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 7 / 26

Merging Framework

Belief Sets and Knowledge Bases

Example

KB = {dog(sue) ∨ cat(sue), female(sue)} Associated collections of belief sets depend on the semantics, e.g.,: Minimal Herbrand models: BS(KB) = { {dog(sue), ¬cat(sue), female(sue)}, {¬dog(sue), cat(sue), female(sue)} } Classically entailed literals: BS(KB) = { {female(sue)} }

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 7 / 26

Merging Framework

Merging Task

Collection of Knowledge Bases

Collection of knowledge bases: KB = KB1, . . . , KBn Associated collections of belief sets: BS(KB1), . . . , BS(KBn) Task: Integrate them into a single set of belief sets

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 8 / 26

Merging Framework

Merging Task

Collection of Knowledge Bases

Collection of knowledge bases: KB = KB1, . . . , KBn Associated collections of belief sets: BS(KB1), . . . , BS(KBn) Task: Integrate them into a single set of belief sets

Types of Mismatches

Naive union not always possible Mismatches:

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 8 / 26

Merging Framework

Merging Task

Collection of Knowledge Bases

Collection of knowledge bases: KB = KB1, . . . , KBn Associated collections of belief sets: BS(KB1), . . . , BS(KBn) Task: Integrate them into a single set of belief sets

Types of Mismatches

Naive union not always possible Mismatches:

language (syntactic) incompatibilities

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 8 / 26

Merging Framework

Merging Task

Collection of Knowledge Bases

Collection of knowledge bases: KB = KB1, . . . , KBn Associated collections of belief sets: BS(KB1), . . . , BS(KBn) Task: Integrate them into a single set of belief sets

Types of Mismatches

Naive union not always possible Mismatches:

language (syntactic) incompatibilities logical inconsistencies

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 8 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

The Problem

Different sources may use different vocabularies Syntactically equal beliefs may encode different information (homonyms) Syntactically different beliefs may encode the same information (synonyms)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 9 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

The Problem

Different sources may use different vocabularies Syntactically equal beliefs may encode different information (homonyms) Syntactically different beliefs may encode the same information (synonyms) Example: P1 = {degree(john, “MSc”) ←} vs. P2 = {deg(john, “Master of Science”) ←}

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 9 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

The Problem

Different sources may use different vocabularies Syntactically equal beliefs may encode different information (homonyms) Syntactically different beliefs may encode the same information (synonyms) Example: P1 = {degree(john, “MSc”) ←} vs. P2 = {deg(john, “Master of Science”) ←}

The Solution

Common signature: ΣC = (ΣC

c , ΣC p )

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 9 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

The Problem

Different sources may use different vocabularies Syntactically equal beliefs may encode different information (homonyms) Syntactically different beliefs may encode the same information (synonyms) Example: P1 = {degree(john, “MSc”) ←} vs. P2 = {deg(john, “Master of Science”) ←}

The Solution

Common signature: ΣC = (ΣC

c , ΣC p )

Convert the collection of belief sets Bi = BS(KBi) to a new collection over ΣC: B′

i = µi(Bi)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 9 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

The Problem

Different sources may use different vocabularies Syntactically equal beliefs may encode different information (homonyms) Syntactically different beliefs may encode the same information (synonyms) Example: P1 = {degree(john, “MSc”) ←} vs. P2 = {deg(john, “Master of Science”) ←}

The Solution

Common signature: ΣC = (ΣC

c , ΣC p )

Convert the collection of belief sets Bi = BS(KBi) to a new collection over ΣC: B′

i = µi(Bi)

Formally: A belief set conversion is a function µi : 2A(ΣKBi) → 2A(ΣC), 1 ≤ i ≤ n s.t. B′

i = B′ j iff they are considered to represent the same information

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 9 / 26

Merging Framework

Mismatch 1: Language Incompatibilities

Example (continued)

µ1(B) = B, µ2(B) = {{degree(X, “MSc”) | deg(X, “Master of Science”) ∈ B}∪ {degree(X, Y) | deg(X, Y) ∈ B, Y = “Master of Science”} | B ∈ B};

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 10 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Definition (Integrity Constraints)

Application-dependent integrity constraints are abstractly modeled as C ⊆ 2A(ΣC), s.t. B ⊆ A(ΣC) satisfies the constraints iff B ∈ C

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 11 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Definition (Integrity Constraints)

Application-dependent integrity constraints are abstractly modeled as C ⊆ 2A(ΣC), s.t. B ⊆ A(ΣC) satisfies the constraints iff B ∈ C

The Problem

We assume: Each source satisfies the constraints: Bi ∈ C for all 1 ≤ i ≤ n

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 11 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Definition (Integrity Constraints)

Application-dependent integrity constraints are abstractly modeled as C ⊆ 2A(ΣC), s.t. B ⊆ A(ΣC) satisfies the constraints iff B ∈ C

The Problem

We assume: Each source satisfies the constraints: Bi ∈ C for all 1 ≤ i ≤ n But the union may violate them:

n

• i=1

Bi ∈ C

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 11 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Definition (Integrity Constraints)

Application-dependent integrity constraints are abstractly modeled as C ⊆ 2A(ΣC), s.t. B ⊆ A(ΣC) satisfies the constraints iff B ∈ C

The Problem

We assume: Each source satisfies the constraints: Bi ∈ C for all 1 ≤ i ≤ n But the union may violate them:

n

• i=1

Bi ∈ C

The Solution

We introduce merging operators

• n,m :
• 2A(ΣC)n
• collections of belief sets

→ 2A(ΣC)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 11 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Definition (Integrity Constraints)

Application-dependent integrity constraints are abstractly modeled as C ⊆ 2A(ΣC), s.t. B ⊆ A(ΣC) satisfies the constraints iff B ∈ C

The Problem

We assume: Each source satisfies the constraints: Bi ∈ C for all 1 ≤ i ≤ n But the union may violate them:

n

• i=1

Bi ∈ C

The Solution

We introduce merging operators Maps n collections of belief sets to a new, integrated collection

• n,m :
• 2A(ΣC)n
• collections of belief sets

× D1 × . . . × Dm

• additional parameters

→ 2A(ΣC)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 11 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B}

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B} Example: {{a, b, c}, {a, d}} ∈ C

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B} Example: {{a, b, c}, {a, d}} ∈ C {{a, b, c}, {a, ¬d}} ∈ C

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B} Example: {{a, b, c}, {a, d}} ∈ C {{a, b, c}, {a, ¬d}} ∈ C {{a, b, ¬a}, {a, ¬d}} ∈ C

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B} Example: {{a, b, c}, {a, d}} ∈ C {{a, b, c}, {a, ¬d}} ∈ C {{a, b, ¬a}, {a, ¬d}} ∈ C Operator definition: (binary, no parameter, i.e., n = 2, m = 0)

• 2,0

∪ (B1, B2) = {B1 ∪ B2 | B1 ∈ B1, B2 ∈ B2, ∄A : {A, ¬A} ⊆ (B1 ∪ B2)} ,

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Mismatch 2: Logical Inconsistencies

Example: Operator definition

Idea: Union operator preserving consistency under classical semantics Integrity constraints: (formally) C = {B ⊆ A(Σ) | ∄B ∈ B : ∃A : {A, ¬A} ⊆ B} Example: {{a, b, c}, {a, d}} ∈ C {{a, b, c}, {a, ¬d}} ∈ C {{a, b, ¬a}, {a, ¬d}} ∈ C Operator definition: (binary, no parameter, i.e., n = 2, m = 0)

• 2,0

∪ (B1, B2) = {B1 ∪ B2 | B1 ∈ B1, B2 ∈ B2, ∄A : {A, ¬A} ⊆ (B1 ∪ B2)} ,

Application: B1 = {{a, b, c}, {¬a, c}} B2 = {{¬a, d}, {c, d}}

• 2,0

∪ (B1, B2) = { {a, b, ¬a, d} , {a, b, c, d}, {¬a, c, d}, {¬a, c, d}}

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 12 / 26

Merging Framework

Merging Plans

Hierarchical arrangement of operators:

Example

• 2

\

• 3

∪

• 1

¬

µ1(BS(KB1)) µ2(BS(KB2)) µ3(BS(KB3))

• 2

∪

µ4(BS(KB4)) µ5(BS(KB5))

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 13 / 26

Merging Framework

Merging Plans

Definition (Merging Plans)

The set MKB,Ω of merging plans over knowledge bases KB = KB1, . . . , KBn and a set Ω = {◦1, . . . , ◦n} of operators is the smallest set such that (i) each M ∈ KB, called atomic merging plan, is in MKB,Ω; (ii) if ◦n,m

i

∈ Ω, sj ∈ MKB,Ω and ak ∈ Di for 1 ≤ j ≤ n, 1 ≤ k ≤ m, then (◦n,m

i

, s1, . . . , sn, a1, . . . , am) ∈ MKB,Ω.

Example (continued)

M = (◦2

\, (◦3 ∪, (◦1 ¬, KB1), KB2, KB3), (◦2 ∪, KB4, KB5)).

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 14 / 26

Merging Framework

Merging Task

Definition (Merging Task)

A merging task is a quintuple T = KB, ΣC, µ, Ω, M

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 15 / 26

Merging Framework

Merging Task

Definition (Merging Task)

A merging task is a quintuple T = KB, ΣC, µ, Ω, M

Definition (Merging Task Result)

The result of a merging task T = KB, ΣC, µ, Ω, M, denoted as [ [T] ], is [ [T] ] = [µi(BS(M))]ΣC

p ,

if M ∈ KB, [◦n,m([ [T1] ], . . . , [ [Tn] ], a1, . . . , am)]ΣC

p ,

if M = (◦n,m, s1, . . . , sn, a1, . . . , am), where [B]ΣC

p = {{p(a1, . . . , an) ∈ BS | p = (¬)p′, p′ ∈ ΣC

p } | BS ∈ B} denotes the

projection of B to the atoms over ΣC

p , and Ti = KB, ΣC, µ, Ω, si, 1 ≤ i ≤ n.

Intuition

The result of a merging plan will be defined as the collection of belief sets delivered by the topmost operator

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 15 / 26

Merging Framework

Merging Plans

Example (continued)

M = (◦2

\, (◦3 ∪, (◦1 ¬, KB1), KB2, KB3), (◦2 ∪, KB4, KB5)).

Let KB1 = {a., b.}, KB2 = {x., y.}, KB3 = {¬a., c.}, KB4 = {a., x.}, KB5 = {c., x., y.} under answer-set semantics (x. is an abbreviation for x ← .)

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 16 / 26

Merging Framework

Merging Plans

Example (continued)

M = (◦2

\, (◦3 ∪, (◦1 ¬, KB1), KB2, KB3), (◦2 ∪, KB4, KB5)).

Let KB1 = {a., b.}, KB2 = {x., y.}, KB3 = {¬a., c.}, KB4 = {a., x.}, KB5 = {c., x., y.} under answer-set semantics (x. is an abbreviation for x ← .) Evaluation: [ [{KB1, . . . , KB5}, ΣC, µid, Ω, M] ] =

• 2

\

• [

[ [(◦3

∪, (◦1 ¬, KB1), KB2, KB3)]

] ], [ [ [(◦2

∪, KB4, KB5)]

] ]

• =
• 2

\

• 3

∪([

[ [(◦1

¬, KB1)]

] ], [ [ [KB2] ] ], [ [ [KB3] ] ]), [ [ [(◦2

∪, KB4, KB5)]

] ]

• =

· · · = ◦2

\ ({{¬a, ¬b, c, x, y}}, {{a, c, x, y}}) = {{¬a, ¬b}}.

([ [ [M] ] ] is an abbreviation for [ [{P1, . . . , P5}, ΣC, µid, Ω, M] ])

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 16 / 26

Prototype Implementation MELD

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 17 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex 2 Common signature:

A set of predicate symbols, constants are given implicitly

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex 2 Common signature:

A set of predicate symbols, constants are given implicitly

3 Belief Set Conversion functions:

rules under HEX-semantics; query the source (1) in the body; derive atoms

• ver common signature (2) in the head

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex 2 Common signature:

A set of predicate symbols, constants are given implicitly

3 Belief Set Conversion functions:

rules under HEX-semantics; query the source (1) in the body; derive atoms

• ver common signature (2) in the head

4 Merging operators: C++ functions in plugin libaries

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex 2 Common signature:

A set of predicate symbols, constants are given implicitly

3 Belief Set Conversion functions:

rules under HEX-semantics; query the source (1) in the body; derive atoms

• ver common signature (2) in the head

4 Merging operators: C++ functions in plugin libaries 5 Merging Plan: Plain text with hierarchical structure

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Towards Automated Evaluation

Goal

Define merging task formally Compute its result automatically ⇒ MELD System - MErging Library for DLVHEX

Realization of the Components

1 Knowledge bases: arbitrary source accessible from dlvhex 2 Common signature:

A set of predicate symbols, constants are given implicitly

3 Belief Set Conversion functions:

rules under HEX-semantics; query the source (1) in the body; derive atoms

• ver common signature (2) in the head

4 Merging operators: C++ functions in plugin libaries 5 Merging Plan: Plain text with hierarchical structure

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 18 / 26

Prototype Implementation MELD

Merging Task Language

Example: merging.mt

[common signature] predicate: a/0; predicate: b/0; predicate: c/0; predicate: p/1; predicate: q/3; [belief base] name:bb1; mapping: “some rule.”; % query external source here mapping: “q(X, Y, Z) :- &rdf[...](X, Y, Z).”; [belief base] name:bb2; source: “some program.hex”; % or within this program ...

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 19 / 26

Prototype Implementation MELD

Merging Task Language

Example: merging.mt (ctn’d)

[merging plan] {

• perator: setminus;

{

• perator: union;

{

• perator: neg;

{bb1}; }; {bb2}; {bb3}; }; {

• perator: union;

{bb4}; {bb5}; }; }

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 20 / 26

Prototype Implementation MELD

Advantages of the Framework

Support for Prototyping Applications

Reuse merging operators once

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 21 / 26

Prototype Implementation MELD

Advantages of the Framework

Support for Prototyping Applications

Reuse merging operators once Rapid prototyping of applications

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 21 / 26

Prototype Implementation MELD

Advantages of the Framework

Support for Prototyping Applications

Reuse merging operators once Rapid prototyping of applications Quick restructuring of merging plans, exchange of operators, parameter modification

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 21 / 26

Prototype Implementation MELD

Advantages of the Framework

Support for Prototyping Applications

Reuse merging operators once Rapid prototyping of applications Quick restructuring of merging plans, exchange of operators, parameter modification Automatic recomputation of result

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 21 / 26

Prototype Implementation MELD

Advantages of the Framework

Support for Prototyping Applications

Reuse merging operators once Rapid prototyping of applications Quick restructuring of merging plans, exchange of operators, parameter modification Automatic recomputation of result Experimenting with different merging plans

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 21 / 26

Application and Discussion

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 22 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

û ü

≥

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources

û ü

≥

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources Distance function: Compare collections of belief sets, e.g., Hamming distance

û ü

≥

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources Distance function: Compare collections of belief sets, e.g., Hamming distance Operator: Merged collection has minimal distance to sources

û ü

≥

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources Distance function: Compare collections of belief sets, e.g., Hamming distance Operator: Merged collection has minimal distance to sources

Fault diagnosis

x=1 y=1 cin=1 cout=1 s=0

expected: 1û

ü

& =1 =1 & ≥1

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources Distance function: Compare collections of belief sets, e.g., Hamming distance Operator: Merged collection has minimal distance to sources

Fault diagnosis

x=1 y=1 cin=1 cout=1 s=0

expected: 1û

ü

& =1 =1 & ≥1

Xor-gate 1 is defect Xor-gate 2 is defect Both are defect

Goal: Find a group decision

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Application and Discussion

Judgment Aggregation

Distance-based Merging Operators [Gabbay et al., 2009]

Idea: Integrated collection of belief sets should be similar to the sources Distance function: Compare collections of belief sets, e.g., Hamming distance Operator: Merged collection has minimal distance to sources

Fault diagnosis

x=1 y=1 cin=1 cout=1 s=0

expected: 1û

ü

& =1 =1 & ≥1

Xor-gate 1 is defect Xor-gate 2 is defect Both are defect

Goal: Find a group decision s.t. it is still be an explanations it is similar to individual opinions

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 23 / 26

Conclusion

Outline

1

Motivation

2

Merging Framework

3

Prototype Implementation MELD

4

Application and Discussion

5

Conclusion

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 24 / 26

Conclusion

Conclusion

Approach for merging of several collections of belief sets:

1

Belief set conversion functions

2

hierarchical merging plans with merging operators

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 25 / 26

Conclusion

Conclusion

Approach for merging of several collections of belief sets:

1

Belief set conversion functions

2

hierarchical merging plans with merging operators

Prototype implementation: MELD as a plugin for dlvhex

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 25 / 26

Conclusion

Conclusion

Approach for merging of several collections of belief sets:

1

Belief set conversion functions

2

hierarchical merging plans with merging operators

Prototype implementation: MELD as a plugin for dlvhex Applications: Judgment Aggregation, Merging of Decision Diagrams, ...

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 25 / 26

Conclusion

Conclusion

Approach for merging of several collections of belief sets:

1

Belief set conversion functions

2

hierarchical merging plans with merging operators

Prototype implementation: MELD as a plugin for dlvhex Applications: Judgment Aggregation, Merging of Decision Diagrams, ... URL: http://www.kr.tuwien.ac.at/research/dlvhex/meld.html

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 25 / 26

Conclusion

Conclusion

Approach for merging of several collections of belief sets:

1

Belief set conversion functions

2

hierarchical merging plans with merging operators

Prototype implementation: MELD as a plugin for dlvhex Applications: Judgment Aggregation, Merging of Decision Diagrams, ... URL: http://www.kr.tuwien.ac.at/research/dlvhex/meld.html

Advantages

Reusing of operators Evaluating different operators empirically Automatic recomputation of result Release user from routine tasks

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 25 / 26

Conclusion

References

Dov M. Gabbay, Odinaldo Rodrigues, Gabriella Pigozzi Connections between Belief Revision, Belief Merging and Social Choice In: Journal of Logic and Computation 19(3) (2009) Konieczny, S., P´ erez, R.P .: On the logic of merging. In: KR’98. (1998) 488–498 Redl, C.: Development of a belief merging framework for dlvhex. Master’s thesis, Vienna University of Technology (June 2010) http://www.ub.tuwien.ac.at/dipl/2010/AC07808210.pdf Eiter, T., Ianni, G., Schindlauer, R., Tompits, H.: dlvhex: A system for integrating multiple semantics in an answer-set programming framework. In: WLP’06. (2006) 206–210 Salzberg, S., Delcher, A.L., Fasman, K.H., Henderson, J.: A decision tree system for finding genes in DNA. Journal of Computational Biology 5(4) (1998) 667–680

Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 26 / 26