Institut für Theoretische Informatik Lehrstuhl für Automatentheorie
DISUNIFICATION IN THE DESCRIPTION LOGIC EL:
Open Problem and Partial Solutions
Franz Baader, Stefan Borgwardt and Barbara Morawska
WARU 2015
Overview
Disunification (mod E)
Connection to Admissibility
Description Logic EL
Disunification in EL
Partial Results
Conclusions
WARU
Slide 2 of 19
Disunification (mod E)
Given: E a set of universally quantified equations between first order terms.
Disunification Problem
Input:
Γ := {s1 =? t1 ,
...,
|
{z
Γ+
sn =? tn ,
}
sn+1 6=? tn+1 ,
|
...,
{z
Γ−
sn+m 6=? tn+m }
}
Question: does a substitution σ exists, such that
E |= σ(s1 ) = σ(t1 ) ∧ . . . ∧ σ(sn ) = σ(tn )
and
E 6|= σ(sn+1 ) = σ(tn+1 ), . . . , E 6|= σ(sn+m ) = σ(tn+m )?
WARU
Slide 3 of 19
Connection to Admissibility
Given: a logic L
Admissibility
A rule Γ ⇒ ∆ is admissible iff for every subsitution σ,
L |= σ(Γ) implies L |= σ(∆0 ) for a non-empty set ∆0 , ∆0 ⊆ ∆.
or
A rule Γ ⇒ ∆ is not admissible iff there is a subsitution σ,
such that L |= σ(φ1 ) ∧ · · · ∧ σ(φn ) and L 6|= σ(ψ1 ), . . . L 6|= σ(ψm ) ,
where Γ := {φ1 , . . . , φn }, ∆ := {ψ1 , . . . , ψm }.
Disunification Problem
. . . φ1 ∧ · · · ∧ φm ≡?L > and ψ1 6≡? >, . . . ψm 6≡?L >.
WARU
Slide 4 of 19
Description Logic EL
Syntax of concept terms
Signature: concept names and role names. Constructors: >, u, ∃r
EL is a fragment of Km :
t
p
C ∧D
♦r .C
↔
↔
↔
↔
>
A
C uD
∃r.C
A is a concept name
r is a role name.
Semantics of EL: (∆I , I)
•
•
•
•
•
WARU
top: >I := ∆I ,
x
A I ⊆ ∆I ,
r I ⊆ ∆I × ∆I ,
r
CI
y
(C u D)I = C I ∩ D I ,
(∃r.C )I = {x ∈ ∆I | ∃y ∈ ∆I , (x, y) ∈ r I , y ∈ C I }.
Slide 5 of 19
Equivalence and subsumption of concepts
Subsumption & equivalence
C v D iff for every interpretation I, C I ⊆ D I .
C ≡ D iff C v D and D v C
Equational theory of EL
Associativity of u:
Commutativity of u:
Idempotence of u:
Identity for u:
Monotonicity of roles:
C1 u (C2 u C3 ) ≡ (C1 u C2 ) u C3
C1 u C2 ≡ C2 u C1
C1 u C1 ≡ C1
C1 u > ≡ C1
∃r.C1 u ∃r.C2 ≡ ∃r.C1 iff C1 u C2 ≡ C1
Deciding subsumption between concepts
• Polynomial time decision procedure for a subsumption (word problem) in EL.
• Unification in EL is NP-complete.
• Disunification in EL?
WARU
Slide 6 of 19
Disunification in EL: example
Two definitions of a Patient with severe head injury
• C1 ≡ Patient u ∃finding.(Head_injury u ∃severity.Severe),
• C2 ≡ Patient u ∃finding.(Severe_finding u Injury u ∃finding_site.Head)
Multiple unifiers
• σ1 := { Head_injury ≡ Injury u ∃finding_site.Head,
Severe_finding ≡ ∃severity.Severe}
• σ2 := { Head_injury ≡ Patient u Injury u ∃finding_site.Head,
Severe_finding ≡ Patient u ∃severity.Severe}
To filter out σ2 add a negative constraint:
Head_injury 6v? Patient
WARU
Slide 7 of 19
Disunification in EL: definition
Given disjoint sets of variables and constants and EL-concept terms C1 , . . . , Dn0
constructed over these sets,
Problem: Γ := {C1 v? D1 , . . . , Cm v? Dm } ∪ {C10 6v? D10 , . . . , Cn0 6v? Dn0 }
|
{z
Γ+
}
|
{z
Γ−
}
Solution: a substitution σ, such that:
σ(C1 ) v σ(D1 ), . . . , σ(Cm ) v σ(Dm ) and σ(C10 ) 6v σ(D10 ), . . . , σ(Cn0 ) 6v σ(Dn0 ).
Prefered is a ground solution: no variables in the range of σ.
WARU
Slide 8 of 19
Disunification in EL: Local solutions
Atoms
Atoms: A, ∃r.C , ∃r.>, . . . ,
where A is a variable or a constant, C is a concept, r is a role name.
Every concept term is a conjunction of atoms or is >.
Given a (dis)unification problem Γ, we define:
Local atoms
Local atoms are atoms of concept terms in Γ.
For example: C := Patient u ∃finding.(Head_injury u ∃severity.Severe).
Atoms of C :
Patient, ∃finding.(Head_injury u ∃severity.Severe), Head_injury, ∃severity.Severe
A substitution σ for variables in Γ is local
if atoms of concept terms in range(σ) are local.
WARU
Slide 9 of 19
An idea of a procedure for disunification in EL
Let Γ be a disunification problem in EL.
Let γ be a ground solution of Γ.
Theorem
There is a local solution σ of Γ+ such that for each variable in Γ,
γ(X) v σ(X)
Idea
1. Guess or compute σ.
2. Modify σ such that it solves Γ− .
Important property
C1 u · · · u Cm 6v D1 u · · · u Dn iff there is Dj , such that for each Ci , Ci 6v Dj
WARU
Slide 10 of 19
Disunification procedure: example
Example
Let A, B, C be constants.
Unification part
Γ+ := {X v? B,
Let σ = {X 7→ B,
A u B u C v? X,
∃r.X v? Y }
Y 7→ >}.
Disunification part
Γ− := {> 6v? Y ,
WARU
Y 6v? ∃r.B}
Slide 11 of 19
Disunification procedure: example
Unsolved
dissubsumption
> 6v? Y
• σ1 =
{X 7→ B, Y 7→ ∃r.U , U 7→ >}
• add > 6v? ∃r.U to Γ−
WARU
Extension of σ
Condition: The rule applies to
C 6v Y ∈ Γ− .
Action:
• adds ∃r.U to σ(Y ), for a
fresh variable U ,
• adds C 6v ∃r.U to Γ− .
• triggers Expansion of Γ
Slide 12 of 19
Disunification procedure: example
Initially solved
(dis)subsumptions
Expansion of Γ
∃r.X v? Y ∈ Γ+ ,
Y 6v? ∃r.B ∈ Γ−
Condition: Applies when an atom
∃r.U is added to σ(Y ).
• adds ∃r.X v? ∃r.U to
Γ+ .
It follows that X v? U is
added to Γ+ .
• adds ∃r.U 6v ∃r.B to
Γ− .
It follows that U 6v B is
added to Γ− .
Action:
• for C v? Y ∈ Γ+ , adds
C v? ∃r.U to Γ+ .
• for Y 6v D ∈ Γ− , adds
∃r.U 6v D to Γ− .
σ1 = {X 7→ B, Y 7→ ∃r.U , U 7→ >}
σ1 solves
Γ = {X v? B, A u B u C v? X, ∃r.X v? Y } ∪ {> 6v? Y ,
WARU
Y 6v? ∃r.B}.
Slide 13 of 19
Disunification procedure: example of non-termination
Let Σ = {r}.
Y 6v? X,
Γ0 = {Y
σ0 := {X 7→ >,
X 6v? Y }
Y 7→ >}
Y 6v? X triggers Extension of σ0 : σ1 (X) = ∃r.U1 .
Γ1 = Γ0 ∪ {Y 6v? ∃r.U1 }
σ1 := {X 7→ ∃r.U1 , Y 7→ >,
U1 7→ >}
X 6v? Y triggers Extension of σ1 : σ2 (Y ) = ∃r.U2
and this triggers Expansion of Γ1 :.
Γ2 = Γ0 ∪ {Y 6v? ∃r.U1 } ∪ {X 6v? ∃r.U2 } ∪ {∃r.U1 6v? ∃r.U2 , ∃r.U2 6v? ∃r.U1 }
U1 6v? U2 , U2 6v? U1 }
By Decomposition, Γ3 = Γ2 ∪ {U
σ2 := {X 7→ ∃r.U1 ,
WARU
Y 7→ ∃r.U2 ,
U1 7→ >,
U2 7→ >}
Slide 14 of 19
Local disunification
Restriction of solutions to local substitutions
σ is local iff for every variable X, σ(X) is a conjunction of local atoms,
i.e., atoms in Γ.
Restricted Extension rule
Local Extension:
Condition: The rule applies to C 6v? Y ∈ Γ− .
Action:
• adds D to σ(Y ), for a local atom D (not a variable),
• adds C 6v? D to Γ− .
• triggers Expansion of Γ
WARU
Slide 15 of 19
Dismatching: definition
Given disjoint sets of variables and constants and EL-concepts C1 , . . . , Dn0
constructed over these sets.
Problem: Γ := {C1 v? D1 , . . . , Cm v? Dm } ∪ {C10 6v? D10 , . . . , Cn0 6v? Dn0 }, such
|
{z
Γ+
}
|
{z
Γ−
}
that for each i ∈ {1, . . . , n}, Ci0 or Di0 is ground (contains no variables).
Solution: a substitution σ, such that:
σ(C1 ) v σ(D1 ), . . . , σ(Cm ) v σ(Dm ) and σ(C10 ) 6v σ(D10 ), . . . , σ(Cn0 ) 6v σ(Dn0 ).
Example
Head_injury 6v? Patient
WARU
Slide 16 of 19
Dismatching: reduction to local disunification
Getting rid of left-ground dissubsumptions
s = C 6v? X (C is a ground term)
• Guess a "witness" atom D for s: constant or ∃r.U (new variable)
• Add X v? D to Γ+
• Add C 6v? D to Γ− .
C 6v? ∃r.U
C 0 6v? U , where |C 0 | < |C |.
(Only polynomially many new variables needed.)
Output: Γ0 := {C1 v? D1 , . . . , Cm v? Dm } ∪ {X1 6v? D10 , . . . , Xn 6v? Dn0 }
WARU
Slide 17 of 19
Dismatching: reduction to local disunification
Theorem
A dismatching problem Γ has a solution iff a disunification problem Γ0 has a local
solution.
Idea of a proof
• If γ is a ground solution of Γ, then it is a solution of Γ0 .
• γ induces a local solution σ of the positive part of Γ0 , γ(X) v σ(X).
• Since
γ(Xi ) 6v Di0
and
γ(Xi ) v σ(Xi )
then (by transitivity of subsumption)
σ(Xi ) 6v Di0
Hence σ is a solution of Γ0 , and thus also of Γ.
WARU
Slide 18 of 19
Conclusions
We have seen:
• local disunification – as a way to reduce the number of local unifiers,
• in practice we use dismatching problems and these can be reduced to local
disunification,
• we have implemented local disunification in the unifier for EL, UEL.
UEL
http://uel.sourceforge.net
• Given two ontologies and a set of variables, computes a local unifier for two
concepts.
• Includes SAT reduction of local disunification.
• Rules-based local disunification still needs to be implemented.
Open problem
Disunification in EL
WARU
Slide 19 of 19