Hybrid Knowledge Bases
Under Paraconsistent Well-Founded Semantics
Tobias Kaminski
CENTRIA - Universidade Nova de Lisboa
EMCL Student Workshop, Vienna
February 19, 2014
Overview
Motivation
Hybrid MKNF Knowledge Bases
Well-Founded MKNF Semantics
Immediate Consequence Operator
Dealing with Inconsistencies
Logic FOUR
Paraconsistent Reasoning
Conclusion
Hybrid Knowledge Bases
Bringing two worlds together...
Example: Customs Risk Assessment
ToxicChemical
v
∃Contains.ToxicSubstance
∃Contains.ToxicSubstance
v
ProvenRisk
ProvenRisk t PotentialRisk
v
Risk
ToxicChemical(Pesticide)
monitor(x) ← product(x), Risk(x).
PotentialRisk(x) ← product(x), not labelled(x).
product(food) ←
product(Pesticide) ←
labelled(Pesticide) ←
Problem with Inconsistencies
Even if the DL and the LP are consistent alone,
contradictions can easily occur due to:
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complex interactions between the two components,
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integration of different data sources and many users.
An issue arises due to the Principle of Explosion:
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A formula ψ is a logical consequence of a formula φ, iff every
model for φ is also a model for ψ.
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A contradictory formula has no models in FOL, so every
formula can be derived from it.
⇒ Under classical semantics, knowledge bases containing a
contradiction are useless.
How to deal with inconsistencies?
There are two strategies for handling inconsistencies:
I Revise the KB to make it consistent again.
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How to do this is a difficult problem by itself.
Some knowledge is lost in the process.
Reject the principle of explosion by defining a
paraconsistent semantics.
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Inconsistencies can be modeled (often by introducing an
additional truth-value >).
The explosive behavior can be “encapsulated”.
The user can be informed about which facts can only be
derived from a contradiction.
Desired properties of Hybrid Knowledge Bases
1 The coupling should be tight, flexible and faithful wrt. the
semantics of each component.
⇒ Hybrid MKNF Knowledge Bases fulfill all of these
requirements [Motik and Rosati 2010].
2 The computation of models should be of low complexity.
⇒ Well-founded version of the MKNF semantics with low
complexity has been developed at UNL [Knorr et al. 2011].
3 It should deal with inconsistencies in a sensible way.
⇒ Paraconsistent version of Hybrid MKNF under the stable
model semantics exists [Huang et al. 2011].
Developing a paraconsistent well-founded semantics for Hybrid
MKNF is the topic of my Master’s thesis.
Hybrid MKNF Knowledge Bases are build over
a common logic for Rules and Ontologies
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Hybrid MKNF is based on the logic of
Minimal Knowledge and Negation as Failure [Lifschitz 1991].
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It extends first-order logic with two modal operators:
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Hybrid MKNF KBs are parametric on the DL.
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K: checks whether a formula is known to be true.
not: checks whether a formula is not known to be true.
It must be translatable into a function-free FOL formula with
equality.
Two restrictions:
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The underlying DL has to be decidable.
The rules must be DL-safe.
Syntax of a Hybrid MKNF Knowledge Base
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A hybrid MKNF rule r has the form:
H ← A1 , . . . , An , not B1 , . . . , not Bm
where H, Ai , and Bi are function free first-order atoms.
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A program P is a finite set of MKNF rules.
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A hybrid MKNF knowledge base K is a pair (O, P).
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The ground instantiation of K is the KB KG = (O, PG ),
where PG is the grounding of P.
Hybrid MKNF under Well-Founded Semantics
A Well-Founded Semantics (WFS) has several advantages
for large-scale Semantic Web applications:
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Data complexity is polynomial if DL component is tractable.
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A unique model always exists.
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It allows for the top-down processing of queries.
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Only the relevant part of the model has to be computed.
It has been implemented in the NoHR-Plugin for Protégé 1
and the DL EL+
⊥.
However, the Stable Model Semantics allows for a more
expressive language and more consequences can be derived.
http://centria.di.fct.unl.pt/nohr/
How to derive consequences
from the Rules and the Ontology together?
Given
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a positive, ground hybrid MKNF KB KG = (O, PG ) and
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a subset S of atoms in PG ,
the immediate consequence operator TKG (S) derives
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all atoms in the head of a rule with true body (given S) and
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all atoms from PG that can be derived from O ∪ S.
TKG (S) has a least fixpoint.
Example: Least Fixpoint of TKG
ProvenRisk t PotentialRisk
v
Risk
monitor(food) ← Risk(food).
PotentialRisk(food).
TKG ↑ 0 = ∅
TKG ↑ 1 = {PotentialRisk(food)}
TKG ↑ 2 = {PotentialRisk(food), Risk(food)}
TKG ↑ 3 = {PotentialRisk(food), Risk(food), monitor (food)}
TKG ↑ ω = {PotentialRisk(food), Risk(food), monitor (food)}
How to handle default negation?
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Assume a subset S of program atoms to be true.
Interpret default negation wrt. to S.
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Rules with not Bi in the body and Bi ∈ S can be deleted.
All negative literals can be deleted from the remaining rules.
⇒ We obtain the positive knowledge base KG /S and
can apply TKG .
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We define the operator ΓKG (S) = TKG /S ↑ ω.
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In even iterations of ΓKG , the set of true atoms is increased.
In odd iterations of ΓKG , the set of “non-false” atoms is
decreased.
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Here, Coherence has to be assured ⇒ Γ0KG .
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
P1 = ΓKG Γ0KG (P0 )
=
{Risk(Pesticide)}
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
P1 = ΓKG Γ0KG (P0 )
=
{Risk(Pesticide)}
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
P1 = ΓKG Γ0KG (P0 )
=
{Risk(Pesticide)}
Γ0KG (P1 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
Example: Computation of the Well-Founded Partition
ProvenRisk t PotentialRisk
v
Risk
PotentialRisk(food)
←
not labelled(food).
labelled(food)
←
not Risk(food).
monitor(food)
←
ProvenRisk(food).
Risk(Pesticide)
P0
=
∅
Γ0KG (P0 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
P1 = ΓKG Γ0KG (P0 )
=
{Risk(Pesticide)}
Γ0KG (P1 )
ΓKG Γ0KG (P1 )
=
{Risk(Pesticide), labelled(food), PotentialRisk(food), Risk(food)}
=
{Risk(Pesticide)} = P1
P2 =
The well-founded partition is:
({Risk(Pesticide)}, {not monitor (food), not ProvenRisk(food)})
Example: Inconsistent Hybrid Knowledge Base
ResolvedRisk
v
¬Monitor
Monitor(food)
←
PotentialRisk(food)
←
Monitor(food).
labelled(food)
←
toxicChemical(food).
ResolvedRisk(food)
←
The well-founded partition is:
({Monitor (food), PotRisk(food), ResRisk(food), toxicChemical(food), labelled(food)}, ∅})
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Principle of Explosion:
⇒ All atoms that can be derived to be true, are also false at
the same time!
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Under a paraconsistent semantics, the explosive behavior of
the ontology could be prevented.
Belnap’s bilattice FOUR [Belnap 1977]
Example: Paraconsistent MKNF Reasoning
ResolvedRisk
v
¬Monitor
Monitor(food)
←
PotentialRisk(food)
←
Monitor(food).
labelled(food)
←
toxicChemical(food).
ResolvedRisk(food)
←
The well-founded partition is:
({ResolvedRisk(food), Monitor (food), PotentialRisk(food)},
{not Monitor (food), not PotentialRisk(food), not toxicChemical(food), not labelled(food)})
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The explosive behavior can be prevented.
Facts not depending on an inconsistency can be just true.
Facts only derivable from an inconsistency are true and false.
So far, inconsistencies are only propagated in the program
component.
Conclusion
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Hybrid MKNF Knowledge Bases allow for a tight integration
of rules and ontologies.
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Models can be computed efficiently by means of the WFS for
Hybrid MKNF.
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By altering the consequence operator we can already perform
paraconsistent derivations.
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This behavior has to be investigated more closely.
For this, a paraconsistent MKNF model theory corresponding
to paraconsistent WFS has to be developed.
Finally, paraconsistent reasoning should be implemented in the
NoHR-Plugin.
Thank you for your attention!
Questions/Comments/Suggestions?
References
Alferes, J. J., Knorr, M., & Swift, T. (2013). Query-driven
procedures for hybrid mknf knowledge bases. ACM
Transactions on Computational Logic (TOCL).
Belnap Jr, N. D. (1977). A useful four-valued logic. In Modern
uses of multiple-valued logic.
Huang, S., Li, Q., & Hitzler, P. (2011). Paraconsistent semantics
for hybrid MKNF knowledge bases. In Web reasoning and
rule systems.
Knorr, M., Alferes, J. J., & Hitzler, P. (2011). Local closed world
reasoning with description logics under the well-founded
semantics. Artificial Intelligence.
Lifschitz, V. (1991). Nonmonotonic databases and epistemic
queries. In IJCAI.
Maier, F., Ma, Y., & Hitzler, P. (2012). Paraconsistent OWL and
related logics. Semantic Web.
Motik, B., & Rosati, R. (2010). Reconciling description logics and
rules. Journal of the ACM (JACM).
Paraconsistent Interpretation of the Ontology
adapted from [Huang et al. 2011]
Interpretation of atoms:

>



t
I (P(t1 , ..., tn )) =
u



f
iff
iff
iff
iff
P(t1 , ..., tn ) ∈ I
P(t1 , ..., tn ) ∈ I
P(t1 , ..., tn ) 6∈ I
P(t1 , ..., tn ) 6∈ I
and
and
and
and
¬P(t1 , ..., tn ) ∈ I
¬P(t1 , ..., tn ) 6∈ I
¬P(t1 , ..., tn ) 6∈ I
¬P(t1 , ..., tn ) ∈ I
Interpretation of formulas:
I
I
I
I
I
I
|=p
|=p
|=p
|=p
|=p
P(t1 , . . . , tn )
¬ϕ
ϕ1 ∧ ϕ2
∃x : ϕ
ϕ1 ⊃ ϕ2
iff
iff
iff
iff
iff
I (P(t1 , . . . , tn )) ∈ {t, >}
I (ϕ) ∈ {f, >}
I |=p ϕ1 and I |=p ϕ2
I |=p ϕ[α/x] for some α ∈ ∆
I 6|=p ϕ1 or I |=p ϕ2
We adopt |=p in the immediate consequence operator.
⇒ This already results in a paraconsistent behavior of Hybrid MKNF.