Semantic Diff as the Basis for Knowledge Base Versioning

Semantic Diff as the Basis forKnowledge Base Versioning

Enrico Franconi1 Thomas Meyer2 Ivan Varzinczak2

1Free University of Bozen/BolzanoBolzano, Italy

2Meraka Institute, CSIRPretoria, South Africa

Franconi, Meyer, Varzinczak (FUB/Meraka) Semantic Diff for KB Versioning 1 / 24

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Regulations or norms

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Motivation

Knowledge Base

Ontology (DL, RDF)

Agents’ beliefs

Need for a versioning system

Motivation

Issues

Maintaining different versionsI Parsimonious representation

Reasoning with versionsI In which of the KBs does α hold,

but not β?

Difference between versionsI How they differ in meaning

Motivation

Issues

but not β?

Motivation

Issues

but not β?

Motivation

Issues

but not β?

Outline

1 Logical Preliminaries

2 Knowledge Base VersioningSemantic DiffA General FrameworkCompiled Representation

3 ConclusionContributionsFuture Work

Outline

Logical Preliminaries

Knowledge bases

A knowledge base K is a (possibly infinite) set of formulas

Cn(K) = {α | K |= α}

Cn(.) is called Tarskian iff it satisfies

I Inclusion: X ⊆ Cn(X )

I Idempotence: Cn(Cn(X )) ⊆ Cn(X )

I Monotonicity: X ⊆ Y implies Cn(X ) ⊆ Cn(Y )

[α] = {β | α ≡ β}

Knowledge bases

Cn(K) = {α | K |= α}

[α] = {β | α ≡ β}

Knowledge bases

Cn(K) = {α | K |= α}

[α] = {β | α ≡ β}

Knowledge bases

Cn(K) = {α | K |= α}

[α] = {β | α ≡ β}

Outline

Semantic Diff

Difference in meaning between knowledge bases K and K′

Analogy with the Unix diff commandI diff distinguishes between syntactically different files

Semantic diff highlights the difference in (logical) meaning

Assume a logic with a Tarskian consequence relation

Example

Let the (propositional) knowledge bases:

K1 = {p, q} and K2 = {p, p → q}

K1 and K2 differ in syntax

But K1 and K2 convey the same meaning (K1 ≡ K2)

Semantic Diff

Example

K1 = {p, q} and K2 = {p, p → q}

Semantic Diff

Example

K1 = {p, q} and K2 = {p, p → q}

Semantic Diff

Example

K1 = {p, q} and K2 = {p, p → q}

Characterizing Semantic Diff

KBs closed under logical consequence

(P1) K = Cn(K) and K′ = Cn(K′)

Semantic diff of K and K′: pair 〈A,R〉A is the add-set of (K,K′)

R as the remove-set of (K,K′)

(P2) K′ = (K ∪ A) \ R

Minimal change and no redundancy

(P3) A ⊆ K′

(P4) R ⊆ K

Duality of semantic diff

(P5) K = (K′ ∪ R) \ A

‘Undo’ operation when moving between versions

(P3) A ⊆ K′

(P4) R ⊆ K

(P5) K = (K′ ∪ R) \ A

(P3) A ⊆ K′

(P4) R ⊆ K

(P5) K = (K′ ∪ R) \ A

Definition

K and K′ knowledge bases, A and R sets of sentences

〈A,R〉 is semantic diff compliant w.r.t. (K,K′) iff (K,K′) and 〈A,R〉satisfy Postulates (P1)–(P5)

(P2) K′ = (K ∪ A) \ R

(P3) A ⊆ K′

(P4) R ⊆ K

(P5) K = (K′ ∪ R) \ A

Definition

K and K′ knowledge bases, A and R sets of sentences

〈A,R〉 is semantic diff compliant w.r.t. (K,K′) iff (K,K′) and 〈A,R〉satisfy Postulates (P1)–(P5)

(P2) K′ = (K ∪ A) \ R

(P3) A ⊆ K′

(P4) R ⊆ K

(P5) K = (K′ ∪ R) \ A

Characterizing Semantic DiffSpecific construction for the semantic diff operator:

Definition

The ideal semantic diff of (K,K′) is the pair 〈A,R〉, where

A = K′ \ K and R = K \ K′

Neither A nor R are logically closed:

Example

Let K = Cn(p ∧ q) and K′ = Cn(¬q)

A = {[¬q], [¬p ∨ ¬q]}

R = {[p ∧ q], [p], [q], [p ↔ q], [p ∨ q], [¬p ∨ q]}

p ∨ ¬q ∈ Cn(A), p ∨ ¬q ∈ Cn(R), but p ∨ ¬q /∈ A and p ∨ ¬q /∈ R

In fact, for any ideal semantic diff 〈A,R〉, > /∈ A and > /∈ R

Definition

A = K′ \ K and R = K \ K′

Example

A = {[¬q], [¬p ∨ ¬q]}

R = {[p ∧ q], [p], [q], [p ↔ q], [p ∨ q], [¬p ∨ q]}

Definition

A = K′ \ K and R = K \ K′

Example

A = {[¬q], [¬p ∨ ¬q]}

R = {[p ∧ q], [p], [q], [p ↔ q], [p ∨ q], [¬p ∨ q]}

Definition

A = K′ \ K and R = K \ K′

Example

A = {[¬q], [¬p ∨ ¬q]}

R = {[p ∧ q], [p], [q], [p ↔ q], [p ∨ q], [¬p ∨ q]}

Definition

A = K′ \ K and R = K \ K′

Example

A = {[¬q], [¬p ∨ ¬q]}

R = {[p ∧ q], [p], [q], [p ↔ q], [p ∨ q], [¬p ∨ q]}

There is a unique ideal semantic diff associated with any two KBs

Theorem

Let 〈A,R〉 be the ideal semantic diff of K and K′. Then

〈A,R〉 is semantic diff compliant with respect to (K,K′)

〈A,R〉 is unique w.r.t. (K,K′)

Corollary

For the ideal semantic diff 〈A,R〉 of (K,K′), A ∩ R = ∅

Ideal semantic diff and symmetric difference: (K′ \ K) ∪ (K \ K′)

Corollary

For the ideal semantic diff 〈A,R〉 of (K,K′), 〈A,R〉 = 〈∅, ∅〉 iff K = K′

Theorem

Corollary

Theorem

Corollary

Theorem

Corollary

Outline

A Framework for Knowledge Base Versioning

Scenario:

n versions, K1, . . . ,Kn, of a KB that need to be stored

A core knowledge base Kc

For 1 ≤ i , j ≤ n:

Ideal semantic diff of (Ki ,Kj): 〈Dij ,Dji 〉Ideal semantic diff of (Kc ,Ki ): 〈Dci ,Dic〉

From Properties

(P2) K′ = (K ∪ A) \ R

(P5) K = (K′ ∪ R) \ A

The add-set Dij of (Ki ,Kj) is also the remove-set of (Kj ,Ki )

The remove-set Dji of (Ki ,Kj) is also the add-set of (Kj ,Ki )

Scenario:

For 1 ≤ i , j ≤ n:

From Properties

(P2) K′ = (K ∪ A) \ R

(P5) K = (K′ ∪ R) \ A

Scenario:

For 1 ≤ i , j ≤ n:

From Properties

(P2) K′ = (K ∪ A) \ R

(P5) K = (K′ ∪ R) \ A

In order to access any version, it is sufficient:

To store Kc , and

To store Dic and Dci for all Ki s.t. 1 ≤ i ≤ n

By Theorem 1, Ki = (Kc ∪ Dci ) \ Dic for every i s.t. 1 ≤ i ≤ n

• 〈Dc1,D1c〉

• 〈Dc2,D2c〉

• 〈Dc3,D3c〉•〈Dc4,D4c〉

•〈Dc5,D5c〉•

〈Dc6,D6c〉

To store Kc , and

• 〈Dc1,D1c〉

• 〈Dc2,D2c〉

• 〈Dc3,D3c〉•〈Dc4,D4c〉

•〈Dc5,D5c〉•

〈Dc6,D6c〉

To store Kc , and

• 〈Dc1,D1c〉

• 〈Dc2,D2c〉

• 〈Dc3,D3c〉•〈Dc4,D4c〉

•〈Dc5,D5c〉•

〈Dc6,D6c〉

A Framework for Knowledge Base VersioningWe can generate the ideal semantic diff of Ki and Kj

Proposition

Dij = (Dcj \ Dci ) ∪ (Dic \ Djc) and Dji = (Dci \ Dcj) ∪ (Djc \ Dic)

〈Dc1,D1c〉

〈Dci ,Dic〉

〈Dcj ,Djc〉

〈Dcn,Dnc〉

〈Dn1,D1n〉

〈Dnj ,Djn〉

〈D1i ,Di1〉

〈Dij ,Dji 〉

A Framework for Knowledge Base VersioningWe can generate the ideal semantic diff of Ki and Kj

Proposition

Dij = (Dcj \ Dci ) ∪ (Dic \ Djc) and Dji = (Dci \ Dcj) ∪ (Djc \ Dic)

〈Dc1,D1c〉

〈Dci ,Dic〉

〈Dcj ,Djc〉

〈Dcn,Dnc〉

〈Dn1,D1n〉

〈Dnj ,Djn〉

〈D1i ,Di1〉

〈Dij ,Dji 〉

Outline

Compiled Representation

Our characterization of Semantic Diff is in the knowledge level

Need for a compiled representation of KBs and the diffs

Computationally, a compiled format is required: F (K)

Given any representation of Ki and Kj , look for an intermediaterepresentation of the ideal semantic diff 〈I (Dij), I (Dji )〉

From Ki together with this intermediate representation of the idealsemantic diff, generate Kj

From this intermediate representation generate the ideal semantic diff〈Dij ,Dji 〉

Given any representation of Ki and Kj , look for an intermediaterepresentation of the ideal semantic diff 〈I (Dij), I (Dji )〉 such that:

With the intermediate representation

We can also generate one KB from another

Theorem

F (Ki ) = (F (Kj) \ I (Dji )) ∪ I (Dij)

= (F (Kj) ∪ I (Dij)) \ I (Dji )

We can generate the ideal diff (details in the NMR’10 paper)

We can get I (Dij) and I (Dji )

Theorem

For 1 ≤ i , j ≤ n, I (Dij) = (I (Dcj) \ I (Dci )) ∪ (I (Dic) \ I (Djc))

Theorem

F (Ki ) = (F (Kj) \ I (Dji )) ∪ I (Dij)

= (F (Kj) ∪ I (Dij)) \ I (Dji )

Theorem

F (Ki ) = (F (Kj) \ I (Dji )) ∪ I (Dij)

= (F (Kj) ∪ I (Dij)) \ I (Dji )

Theorem

Outline

Contributions

Groundwork for a semantic-driven notion of versioning

I Intuitive, simple and general

Notion of semantic diff applicable to a large class of KR languages

I Our results hold for any KB in a Tarskian logic

Parsimonious representation

I Core KB: sufficient to reconstruct any of the versions

I Diff between KBs: no direct access to any of the versions

I This holds for any syntactic representation (see the NMR’10 paper)

Contributions

Outline

Ongoing and Future Work

How to choose the core knowledge base Kc

Which normal forms are more appropriate

Experiments with realistic data for evaluation of the approach

Ontology versioning in Description Logics

Reference

E. Franconi, T. Meyer, I. Varzinczak. Semantic Diff as the Basis forKnowledge Base Versioning. Workshop on Nonmonotonic Reasoning(NMR), 2010.

Thank you!

Reference

E. Franconi, T. Meyer, I. Varzinczak. Semantic Diff as the Basis forKnowledge Base Versioning. Workshop on Nonmonotonic Reasoning(NMR), 2010.

Thank you!

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