Nominal Schema DL 2011

Nominal Schemas for Integrating Rules and
Description Logics

Markus Krötzsch1 Frederick Maier2 Adila Krisnadhi2 Pascal Hitzler2
1 University of Oxford
2 Kno.e.sis Center – Wright State University

The 24th Description Logic Workshop – Barcelona, 2011

Nominal Schema as An Extension of Standard DLs

• Suﬃcient expressivity for standard DLs to incorporate (Datalog)
rule-based modeling.
• Seamless integration with standard DLs.
• Syntax is not hybrid, i.e., completely DL-based.
• Easy to incorporate in current OWL standard.

• Semantics is not complicated.
• Decidable.
• A large tractable fragment is identiﬁed.

Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 2

What is Nominal Schema?

• Nominal-like DL syntax.
• {v} where v is a variable.

• Semantically behaves like DL-safe variable (without hybrid combination
with rule syntax)
• Treated as a “macro” representing named individuals in the knowledge
base.

• Results in the paper:
• N2ExpTime-completeness of SROIQV = SROIQ + nominal schema,
i.e., no harder than SROIQ. (V is to indicate nominal schema extension).
• Tractable fragment of DLs SROELV n (n ≥ 0).
• Expressing DL-safe datalog with nominal schema


Example

• Consider the (hybrid) knowledge base K :

hasParent(x, y) ∧ hasParent(x, z) ∧ married(y, z) → C (x)
hasParent(mary, john)
(∃hasParent.∃married.{john})(mary)


Example



• FOL semantics of SWRL: K |= C (mary)
• Undecidable in general


Example




• DL-safe rules semantics: K |= C (mary),
• John’s spouse is not named by any constant.
• K |= C (mary) if z is DL-safe while x and y can be arbitrary values


Example




• DL-safe rules semantics: K |= C (mary),
• John’s spouse is not named by any constant.
• K |= C (mary) if z is DL-safe while x and y can be arbitrary values

• With nominal schemas, the rule can be expressed as:
∃hasParent.{z} ∃hasParent.∃married.{z} C


SROIQV = SROIQ + Nominal Schema (Syntax)

• Signature of SROIQV: NI , NC , Nr , NV .

• Concept expression:

C ::= | ⊥ | NC | {NI } | {NV } | ¬C | C C|C C|
∃R.C | ∀R.C | ∃S.Self | k S.C | k S.C

k ∈ N, R (S) is a (simple) SROIQ role, incl. the universal role U .

• Knowledge base axioms are as usual as in SROIQ with regular RBoxes


SROIQV = SROIQ + Nominal Schema (Semantics)

• Standard semantics for concept and roles with variables interpreted as
placeholder of named individuals:
• embed a restricted form of variable assignment to the interpretation;
• equivalent to replacing nominal schemas with finitely many nominals they
can represent and applying the standard SROIQ semantics to the result.

• ground(α): the set of all axioms obtained by uniformly replacing nominal
schemas in α with nominals in NI
ground(KB) := α∈KB ground(α)

• I |= α iff I |= ground(α) and I |= KB iff I |= ground(KB).

• Satisfaction and entailment are defined as usual.

• Grounding does not affect restrictions on simplicity and regularity.


Complexity of reasoning in SROIQV

• Grounding provides direct (and naive) approach for reasoning in
SROIQV:
• yields exponentially larger SROIQ KB after grounding;
• (not tight) complexity upper bound is exponentially larger than that of
SROIQ, i.e., N3ExpTime.

• N2ExpTime-hardness follows from SROIQ.
• Retaining N2ExpTime upper bound by extending the N2ExpTime
upper bound proof for SROIQ from [Kazakov, KR08]: exponential
reduction to satisﬁability of theories of C 2 (two-variable fragment of FOL
with counting quantiﬁers) that is in NExpTime
• transform axioms (except complex RIAs) into normal form
• eliminate RIAs using automata construction
• express remaining axioms as C 2 theories


N2ExpTime upper bound

• Given a SROIQV KB, depth of ground(KB) = depth of KB.

• ground(KB) is in SROIQ, hence the exponential reduction applies.

• The resulting C 2 theories T is equisatisfiable to ground(KB) that is
equisatisfiable with KB

• The size of T is bounded by a function that is linear in the size of
ground(KB) and exponential in the depth of KB.

• The size of ground(KB) is exponential in the size of KB.

• Thus, the size of T is bounded exponentially in the size and depth of KB

• Satisfiability of T is in NExpTime, hence satisfiability of KB is in
N2ExpTime


A tractable fragment: consideration

• Consider adding nominal schemas to the tractable SROEL
• Normalization destroys complex dependencies between nominal schemas.
• Exponential blow up due to grounding nominal schemas is unavoidable.

• Limiting the occurrences of all nominal schemas (with global constant
from the language) trivially yields to tractability.
• Non-trivial tractability result: only limit (globally) the occurrences of
problematic nominal schemas
• no limit on occurrences of safe nominal schemas.
• inspired from tree-shapedness notion from the rule language ELP
[Krötzsch,et.al., ISWC08]


Safe Environment of Nominal Schema

x, y, z ∈ NV
∃reviews.({x} ∃hasAuthor.{y} ∃atVenue.{z})
∃submits.(∃hasAuthor.{y} ∃atVenue.{z})
∃hasConflictingAssignedPaper.{x}

v

x u

y

z


p ∈ NI y, z ∈ NV
∃reviews.({p} ∃hasAuthor.{y} ∃atVenue.{z})
∃hasConflictingAssignedPaper.{p}

v

p p u

y

z


p ∈ NI y, z ∈ NV
∃reviews.({p} ∃hasAuthor.{y} {p} ∃atVenue.{z})

v

p p u

y

z

Safe Environment
• An occurrence of a nominal schema {x} in a concept C is safe if
• C has a sub-concept of the form {a} ∃R.D for some a ∈ NI
• D contains the occurrence of {x} but no other occurrence of any nominal
schema.

• {a} ∃R.D is a safe environment S(a, x) for this occurrence of {x}.
• A nominal schema {x} is safe for a SROIQV TBox axiom C D if
• {x} does not occur in D, and
• at most one occurrence of {x} in C is not safe.


Safe Environment
• An occurrence of a nominal schema {x} in a concept C is safe if
• C has a sub-concept of the form {a} ∃R.D for some a ∈ NI
• D contains the occurrence of {x} but no other occurrence of any nominal
schema.

• {a} ∃R.D is a safe environment S(a, x) for this occurrence of {x}.
• A nominal schema {x} is safe for a SROIQV TBox axiom C D if
• {x} does not occur in D, and
• at most one occurrence of {x} in C is not safe.
• The DL SROELV n :
• concept/roles: SROIQV concepts/roles using only , ⊥, , ∃, Self, U ,
{a}, nominal schemas
• no , ¬, ∀, number restrictions, inverse roles
• TBox axioms: SROIQV TBox axioms with SROELV n concept/roles,
and at most n nominal schemas are not safe for each axiom.
• KB axioms: uses only SROELV n axioms


Transformation Example (1)
• Assume KB has only p, a, b as named individuals. Consider an axiom α:
∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z}))



• For each nominal schema {x} safe for α with safe environments Si (ai , x),
i = 1, . . . , , introduce a fresh concept name Ox,α
• Oy,α , Oz,α



• For each individual c in KB, add polynomially many ground axioms ( for
empty conjunction):
∃U .Si (ai , c) ∃U .({c} Ox,α )
i=1



• For each individual c in KB, add polynomially many ground axioms ( for
empty conjunction):
∃U .Si (ai , c) ∃U .({c} Ox,α )
i=1

∃U .({p} ∃hasAuthor.{p}) ∃U .({p} Oy,α )
∃U .({p} ∃hasAuthor.{a}) ∃U .({a} Oy,α )
∃U .({p} ∃hasAuthor.{b}) ∃U .({b} Oy,α )
∃U .({p} ∃atVenue.{p}) ∃U .({p} Oz,α )
∃U .({p} ∃atVenue.{a}) ∃U .({a} Oz,α )
∃U .({p} ∃atVenue.{b}) ∃U .({b} Oz,α )




• Let C be the LHS of α. For each nominal schema {x} safe for α:
• replace all safe occurrences S(a, x) in C by {a}



∃reviews.{p}
∃submits.(∃hasAuthor.Oy,α ∃atVenue.Oz,α )

• replace the non-safe occurrence (if any) of {x} in C by Ox,α



∃reviews.{p}
∃U .Oy,α ∃U .Oz,α

• add a new conjunct ∃U .Ox,α to C .



∃reviews.{p}
∃U .Oy,α ∃U .Oz,α

• add a new conjunct ∃U .Ox,α to C .

• After the above steps, C has only nominal schemas (if any) that are not safe for
α. We thus ground α.


Tractability Result

• Given a SROELV n knowledge base KB, the size of KB after unfolding
is exponential in n and polynomial in the size of KB before unfolding.

• KB before transformation is equisatisfiable with KB after transformation.

• A knowledge base is unsatisfiable iff it entails {a} ⊥ for arbitrary
a ∈ NI .

• KB after transformation is in SROEL(×).

• Instance retrieval for SROEL(×) is polynomial [Krötzsch, IJCAI11].

• If n is constant, satisfiability of KB is P-complete.


Expressing DL-safe Rules with Nominal Schemas

• dl(C (t)) := ∃U .({t} C)

• dl(R(s, t)) := ∃U .({s} ∃R.{t})

• dl(p(t1 , . . . , tk )) := ∃U .(∃p1 .{t1 } ··· ∃pk .{tk })

• dl(B → H ) := F∈B dl(F ) dl(H )

• For a set of DL-safe rules RB, dl(RB) := B→H ∈RB dl(B → H ).

• RB is semantically equivalent dl(RB)

• If RB is a set of n-variable rules with n constant, then satisﬁability of
RB is P-complete.


Summary

• Conclusion:
• Nominal schemas provides suﬃcient expressivity to incorporate rule-based
modeling.
• DL-safe datalog with arbitrary arity is covered.
• SROIQV is no harder than SROIQ
• Tractable reasoning is possible with SROELV n .

• Outlook:
• Concrete serialization formats (currently proposed).
• Deferred grounding for inference algorithms (ongoing work).


Recall: N2ExpTime upper bound proof from SROIQ

1 SROIQ axioms can be transformed in linear time into normal form
where R(i) , S1 , S2 ∈ Nr , S1 , S2 simple:

A ∀R.B Ai Bj S1 S2
A n S.B A ≡ {a} R1 R−
A n S.B A ≡ ∃S.Self R1 ◦ · · · ◦ Rn R

2 For each non-simple role R, construct an NFA AR that accepts all role
chains (viewed as strings) that imply R
• The number of states of AR is bounded exponentially in the depth of KB:
maximum cardinality of any strict linear order of role names that witnesses
regularity of KB.


Recall: N2ExpTime upper bound proof from SROIQ (cont.)

3 Drop complex RIAs and replace each A ∀R.B with axioms (with NFA
AR = (QR , NR , ∆R , q0R , FR )):
S
{A Xq0R } ∪ {Xq B | q ∈ FR } ∪ {Xq ∀S.Xq | q → q ∈ ∆R }

For each axiom A ∀R.B:
• The number of new axioms is linearly bounded by |QR | + |∆R |
• |∆R | is linearly bounded by |QR | and |Nr |
• |QR | is exponentially bounded by the depth of KB

The overall size of KB after adding new axioms is now bounded by a
function that is linear in the size of original KB and exponential in the
depth of original KB

4 Rewrite KB into C 2 (not increase the size of KB)


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