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Preliminary Results on The Identity Problem in Description Logic Ontologies Adrian Nuradiansyah Franz Baader, Daniel Borchmann Technische Universitt Dresden July 21, 2017 Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 1

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Preliminary Results on The Identity Problem in Description Logic Ontologies

Adrian Nuradiansyah Franz Baader, Daniel Borchmann Technische Universität Dresden July 21, 2017

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 1 / 1

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Identification Problem

combined ← − − − − − →

Anonymized Survey Data (anonymous individuals) Employee Database (known individuals) Background Knowledge

Ontology

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Identification Problem

combined ← − − − − − →

Anonymized Survey Data (anonymous individuals) Employee Database (known individuals) Background Knowledge

Ontology known a

identity of

← − − − − − − anonymous x

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Identification Problem

An attacker still can access some axioms in the ontology s.t. he knows: x {Female} logic privacy

A:

+ linda {Female} pattie {Female} john {Male} jim {Male}

expert expert

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Identification Problem

An attacker still can access some axioms in the ontology s.t. he knows: x {Female} logic privacy

A:

+ linda {Female} pattie {Female} john {Male} jim {Male}

expert expert

∃expert.{logic} ⊑ VerTeam

T :

∃expert.{privacy} ⊑ SecTeam Female ⊑ ¬Male SecTeam ≡ {linda, john, jim} VerTeam ≡ {linda, john, pattie}

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 3 / 1

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Identification Problem

An attacker still can access some axioms in the ontology s.t. he knows: x {Female} logic privacy

A:

+ linda {Female} pattie {Female} john {Male} jim {Male}

expert expert

∃expert.{logic} ⊑ VerTeam

T :

∃expert.{privacy} ⊑ SecTeam Female ⊑ ¬Male SecTeam ≡ {linda, john, jim} VerTeam ≡ {linda, john, pattie} consequence: x ˙ = linda w.r.t. O

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 3 / 1

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The Identity Problem

Identity Problem

Given a, b ∈ NI and an ontology O. Check whether aI = bI for all models I of O. It is denoted by (O | = a ˙ =b).

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The Identity Problem

Identity Problem

Given a, b ∈ NI and an ontology O. Check whether aI = bI for all models I of O. It is denoted by (O | = a ˙ =b). Not all DLs are able to derive equalities between two individuals :(

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DLs without Equality Power

Definition

L is a DL without equality power if there are no ontologies O formulated in L and two distinct individuals a, b, ∈ NI s.t. O | = a ˙ =b.

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DLs without Equality Power

Definition

L is a DL without equality power if there are no ontologies O formulated in L and two distinct individuals a, b, ∈ NI s.t. O | = a ˙ =b.

Theorem

Every DL translated to a first-order logic without equality predicate is a DL without equality power

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DLs without Equality Power

Definition

L is a DL without equality power if there are no ontologies O formulated in L and two distinct individuals a, b, ∈ NI s.t. O | = a ˙ =b.

Theorem

Every DL translated to a first-order logic without equality predicate is a DL without equality power

They are: ALC and its fragments: EL, FL0, FLE, . . . SRI: extending ALC with inverse roles, role axioms, role compositions, and transitive roles.

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DLs with Equality Power

ALCO: lifting up an individual into a concept Example: Case of Employee.

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DLs with Equality Power

ALCO: lifting up an individual into a concept Example: Case of Employee. ALCQ: restricting the number of successors of a domain element Example: O = ({PhDstudent ⊑ ≤ 1supervised.⊤}, {supervised(adrian, y), supervised(adrian, franz)})

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 6 / 1

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DLs with Equality Power

ALCO: lifting up an individual into a concept Example: Case of Employee. ALCQ: restricting the number of successors of a domain element Example: O = ({PhDstudent ⊑ ≤ 1supervised.⊤}, {supervised(adrian, y), supervised(adrian, franz)}) CFDnc: featuring functional dependencies Functional Dependencies: if two individuals agree on some attributes, then they are unique. Example: O = ({A ⊑ A : f → id}, {A(a), A(x),f (a) = b, f (x) = b})

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 6 / 1

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How to solve the identity problem?

Problem Reduction 1 (Upper Bound)

Identity reduced − − − − − → Instance for all DLs with equality power. O1 | = a ˙ =b iff (O1 ∪ A(a)) | = A(b), where A ∈ NC is new

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How to solve the identity problem?

Problem Reduction 1 (Upper Bound)

Identity reduced − − − − − → Instance for all DLs with equality power. O1 | = a ˙ =b iff (O1 ∪ A(a)) | = A(b), where A ∈ NC is new

Problem Reduction 2 (Lower Bound)

Instance reduced − − − − − → Identity in ALCO and ALCQ HornSAT reduced − − − − − → Identity in CFDnc

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 7 / 1

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How to solve the identity problem?

Problem Reduction 1 (Upper Bound)

Identity reduced − − − − − → Instance for all DLs with equality power. O1 | = a ˙ =b iff (O1 ∪ A(a)) | = A(b), where A ∈ NC is new

Problem Reduction 2 (Lower Bound)

Instance reduced − − − − − → Identity in ALCO and ALCQ HornSAT reduced − − − − − → Identity in CFDnc

Complexity Results

ExpTime-complete in ALCO and ALCQ NExpTime-complete in ALCOIQ PTime-complete in CFDnc Complexities of identity and instance problem are not the same in ALC= allowing {a ˙ =b | a, b ∈ NI} ⊆ A → PTime vs ExpTime-hard

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 7 / 1

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The View-based Identity Problem

A rôle-based access control scenario:

A partially visible ontology OI

combined ← − − − − − →

Anonymized Survey Data (anonymous individuals) Employee Databases (known individuals) Background knowledge

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 8 / 1

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The View-based Identity Problem

A rôle-based access control scenario:

A partially visible ontology OI

combined ← − − − − − →

Anonymized Survey Data (anonymous individuals) Employee Databases (known individuals) Background knowledge At rôle ˆ r1

  • queries through Oˆ

r1 ⊆ OI

  • obtains View Vˆ

r1

switch

− − → . . . switch − − →

At rôle ˆ rk

  • queries through Oˆ

rk ⊆ OI

  • obtains View Vˆ

rk

At rôle ˆ rk+1, is the identity of an anonymous x hidden w.r.t. Vˆ

r1, . . . , Vˆ rk ? Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 8 / 1

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Query Answering and View

Let NI = NKI ∪ NAI, where NKI and NAI are the sets of known and anonymous individuals, respectively.

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Query Answering and View

Let NI = NKI ∪ NAI, where NKI and NAI are the sets of known and anonymous individuals, respectively. Let x ∈ NAI. The identity of x w.r.t. an ontology OI is idn(x, OI) = {a ∈ NKI | OI | = x ˙ =a}

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 9 / 1

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Query Answering and View

Let NI = NKI ∪ NAI, where NKI and NAI are the sets of known and anonymous individuals, respectively. Let x ∈ NAI. The identity of x w.r.t. an ontology OI is idn(x, OI) = {a ∈ NKI | OI | = x ˙ =a} Given OI, Oˆ

r ⊆ OI accessed by a user with a rôle ˆ

r, and a (subsumption or retrieval) query q, the answer to q w.r.t. ˆ r is: ans(q, ˆ r) := {true}, if q = C ⊑ D and Oˆ

r |

= C ⊑ D, ans(q, ˆ r) := ∅, if q = C ⊑ D and Oˆ

r |

= C ⊑ D, ans(q, ˆ r) := {a ∈ NI | Oˆ

r |

= C(a)}, if q = C, ans(q, ˆ r) := {(a, b) ∈ NI × NI | Oˆ

r |

= r(a, b)}, if q = r ∈ NR.

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 9 / 1

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Query Answering and View

Let NI = NKI ∪ NAI, where NKI and NAI are the sets of known and anonymous individuals, respectively. Let x ∈ NAI. The identity of x w.r.t. an ontology OI is idn(x, OI) = {a ∈ NKI | OI | = x ˙ =a} Given OI, Oˆ

r ⊆ OI accessed by a user with a rôle ˆ

r, and a (subsumption or retrieval) query q, the answer to q w.r.t. ˆ r is: ans(q, ˆ r) := {true}, if q = C ⊑ D and Oˆ

r |

= C ⊑ D, ans(q, ˆ r) := ∅, if q = C ⊑ D and Oˆ

r |

= C ⊑ D, ans(q, ˆ r) := {a ∈ NI | Oˆ

r |

= C(a)}, if q = C, ans(q, ˆ r) := {(a, b) ∈ NI × NI | Oˆ

r |

= r(a, b)}, if q = r ∈ NR. Given a rôle ˆ r, a view is a total function Vˆ

r : dom(Vˆ r) → 2NI ∪ 2NI ×NI ∪ {{true}}, where

View definition dom(Vˆ

r) is a finite set of queries.

Vˆ

r(q) is a finite set of answers for all q ∈ dom(Vˆ r). Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 9 / 1

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How to solve the View-based Identity Problem?

Canonical Ontology

The canonical ontology C(Vˆ

r1, . . . , Vˆ rk) of Vˆ r1, . . . , Vˆ rkis defined as

C(Vˆ

r1, . . . , Vˆ rk) := (T , A) where

T := {C ⊑ D | Vˆ

ri(C ⊑ D) = {true} for some i, 1 ≤ i ≤ k}

A := {C(a) | a ∈ Vˆ

ri(C) for some i, 1 ≤ i ≤ k} ∪

{r(a, b) | (a, b) ∈ Vˆ

ri(r) for some i, 1 ≤ i ≤ k}.

Hidden Identity

The identity of x ∈ NAI is hidden w.r.t. Vˆ

r1, . . . , Vˆ rk iff

idn(x, C(Vˆ

r1, . . . , Vˆ rk)) = ∅.

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Future Work

Probabilistic-based Reasoning Two individuals are equal with certain probability. Subjective probabilistic in DLs with equality power is more suitable

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 11 / 1

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Future Work

Probabilistic-based Reasoning Two individuals are equal with certain probability. Subjective probabilistic in DLs with equality power is more suitable

Adrian Nuradiansyah Description Logic Workshop 2017 July 21, 2017 11 / 1

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Future Work

Probabilistic-based Reasoning Two individuals are equal with certain probability. Subjective probabilistic in DLs with equality power is more suitable Anonymizing Description Logic Ontologies Generalizing concepts/nominals on the right hand side of GCIs as specific as possible.

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Thank You

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