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 Motivation 1 2 Merging Framework Prototype Implementation MELD 3 Application and Discussion 4 Conclusion 5 Redl C., Eiter T., Krennwallner T. (TU Vienna) Declarative Belief Set Merging using Merging Plans January 24, 2011 2 / 26

Motivation Outline Motivation 1 2 Merging Framework Prototype Implementation MELD 3 Application and Discussion 4 Conclusion 5 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 Motivation 1 2 Merging Framework Prototype Implementation MELD 3 Application and Discussion 4 Conclusion 5 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit Σ A belief set is a set B ⊆ Lit 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit Σ A belief set is a set B ⊆ Lit Σ Set of all belief sets A (Σ) over Σ : A (Σ) := 2 Lit 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit Σ A belief set is a set B ⊆ Lit Σ Set of all belief sets A (Σ) over Σ : A (Σ) := 2 Lit Σ 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit Σ A belief set is a set B ⊆ Lit Σ Set of all belief sets A (Σ) over Σ : A (Σ) := 2 Lit Σ 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 = (Σ c , Σ p ) ( Σ c ... constant symbols, Σ p ... predicate symbols) Set of all beliefs , i.e., all literals: Lit Σ A belief set is a set B ⊆ Lit Σ Set of all belief sets A (Σ) over Σ : A (Σ) := 2 Lit Σ 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 = KB 1 , . . . , KB n Associated collections of belief sets: BS ( KB 1 ) , . . . , BS ( KB n ) 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 = KB 1 , . . . , KB n Associated collections of belief sets: BS ( KB 1 ) , . . . , BS ( KB n ) 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 = KB 1 , . . . , KB n Associated collections of belief sets: BS ( KB 1 ) , . . . , BS ( KB n ) 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 = KB 1 , . . . , KB n Associated collections of belief sets: BS ( KB 1 ) , . . . , BS ( KB n ) 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: P 1 = { degree ( john , “ MSc ”) ←} vs. P 2 = { 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