Recall: Semantics of DL
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Semantics defined by interpretations. We consider ALCN once more.
An interpretation I = (4, ·I), where
– Δ is the domain (a non-empty set)
– ·I is an interpretation function such that (for ALCN ):
Reasoning in Description Logic
• AI ⊆ 4
(where A is a class name)
• RI ⊆ 4 × 4 (where R is a Role (property) name)
• iI ∈ 4 (where i is an Individual name)
Jacques Fleuriot
• ⊥I = ∅ and >I = 4
• (¬C)I = 46 CI
• (C u D)I = CI ∩ DI and (C t D)I = CI ∪ DI
• (∀R. C)I = {a ∈ 4 | ∀b. (a, b) ∈ RI → b ∈ CI}
• (∃R. C)I = {a ∈ 4 | ∀b. (a, b) ∈ RI ∧ b ∈ CI}
• (· nR. C)I = {a ∈ 4 | k { b | (a, b) ∈ RI ∧ b ∈ CI } k · n}
• (≥ nR. C)I = {a ∈ 4 | k { b | (a, b) ∈ RI ∧ b ∈ CI } k ≥ n}
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Example Interpretation for ALCN
Recall: ABox and TBox
(there may be others)
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4 = {leo, jerome, ralph, lulu, bart, tania, robert}
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GiraffesI = {jerome, lulu, robert}
LionsI = {leo}
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eatsI = {(leo, jerome), (leo, lulu), (leo, robert), (tania, ralph), (tania, jerome)}
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A set of DL sentences (making up the KB) about a domain D is divided
into:
– TBox: general knowledge with axioms of the form:
• C v D, C ≡ D, R v S, R ≡ S and R+ v R
Then we have (for instance) that:
where C, D are concepts, R, S are roles, and R+ is a set of transitive
roles.
(¬Giraffes)I = 46 GiraffesI
= {leo, jerome, ralph, lulu, bart, tania, robert}6 { jerome, lulu, robert}
Example: Sibling ≡ Brother t Sister
= {leo, ralph, bart, tania}
– ABox: assertions (specific knowledge):
(∀eats. Giraffes)I = {a ∈ 4 | ∀b. (a, b) ∈ eatsI → b ∈ GiraffesI} = {leo}
• jerome: Giraffe
(Giraffes u Lions)I = GiraffesI ∩ LionsI = ∅ = ⊥I
• (leo, jerome) : eats
(· 2 eats. Giraffes)I = ??
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Rasoning about TBoxes
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Satisfiability: A concept C is satisfiable wrt TBox T iff
there is an interpretation I of T such that CI ≠ ∅. Such an
interpretation is known as a model for C.
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Subsumption: A concept C is subsumed by a concept D
wrt TBox T iff CI⊆ DI for all interpretations I of T. This
can be denoted by T ² C v D.
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Equivalence: A concept C is equivalent to a concept D wrt
TBox T iff CI = DI for all interpretations I of T. This can
be denoted by T ² C ≡ D.
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Reduction to Subsumption
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Subsumption (traditionally) acts as the basic reasoning
mechanism in DL systems.
– This is sufficient in order to implement other inferences since
they can be reduced to subsumption checking (is C v D?) by
the following theorem (provided the DL system contains u and
⊥) :
For concepts C and D, we have
C is unsatisfiable wrt TBox T iff T ² C v ⊥
Disjointness: Two concepts C and D are disjoints wrt TBox
T iff CI ∩ DI = ∅ for all interpretations I of T.
T ² C ≡ D iff T ² C v D and T ² D v C
C and D are disjoints wrt TBox iff T ² C u D v ⊥
Note: In the special case where the TBox is empty, we can write ² C v D and
² C ≡ D, where appropriate.
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Reduction to Unsatisfiability
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Reducing Unsatisfiability
If the DL system allows complement (negation), subsumption,
equivalence, and disjointness can be reduced to a satisfiability
problem:
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For concepts C and D, we have
Let C be a concept. Then the following are equivalent
with respect to a TBox T :
C is subsumed by D wrt TBox T iff C u ¬D is unsatisfiable
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Unsatisfiability is a special case of each of the following:
subsumption, equivalence, and disjointness:
T ² C ≡ D iff both C u ¬D and ¬C u D are unsatisfiable
1.
C is unsatisfiable;
C and D are disjoints wrt TBox T iff C u D is unsatisfiable
2.
T ²Cv⊥;
3.
T ²C≡⊥;
4.
C and > are disjoint.
So, to obtain decision procedures for the above, we just need
algoritms that decide the satisfiability of concepts
– DL needs to support u and negation (¬) for arbitrary concepts
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Reasoning Tasks for ABoxes
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Structural Subsumption Algorithm(s)
ABox contains two types of assertions: concept assertions of
the form a: C and role assertions of the form (a,b): R.
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Algorithms usually have two phases:
– normalize the concepts to be tested for subsumption
– compare the syntactic structure of the normal forms
An ABox A is consistent wrt a TBox T iff there is an
interpretation that is a model for both A and T.
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Normal Forms:
– An atomic concept A is in normal form
Instance checking: An assertion α is entailed by A, written A ²
α, iff every interpretation that satisfies, that is, every model of
A, also satisfies α.
– The negation of an atomic ¬A concept is in normal form
– ⊥ is in normal form
– A concept C, in a DL with only u, ∀, ⊥, ¬, is in normal form if it has
the following form:
– If is α of the form a: C then, we have A ² a: C iff A ∪ {¬ a: C} is
inconsistent.
A1 u … u Am u ∀R1. C1 u … u ∀Rn. Cn
– A concept is C satisfiable iff {a: C} is consistent wrt to ABox A for
an arbitrary a.
where A1, … , Am are distinct atomic concept names different
from ⊥ and R1, … , Rn are distinct role names and C1, … , Cn
are concepts in normal form.
Note: a: C and (a,b): R are often written as C(a) and R(a,b) respectively.
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Subsumption Check Algoritm:
Comparison
Converting to Normal Form
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The second phase of the algorithm for ALC :
The conversion to normal form for previous DL relies on a number of
identities:
– ⊥ is subsumed by every concept;
– C is subsumed by D (C v D) if:
– AuA≡A
– AuB≡BuA
– A u (B u C) ≡ (A u B) u C
– ∀R.(A u B) ≡ (∀R. A) u (∀R. B)
•
B1 u … u Bk u ∀S1. D1 u … u ∀Sr. Dr is the normal form of D
1. for all i, 1 · i · k there exists j, 1 · j · m such that Bi = Aj
– Au⊥≡⊥
2. for all i, 1 · i · r there exists j, 1 · j · n such that Si = Rj and
Cj v D i
To deal with number restrictions, we need additional rules such as:
– m > n → (· nR. C) u (≥ mR. C) ≡ ⊥
– n ≥ 1 → (≥ nR. C) u (∀R. ⊥) ≡ ⊥
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A1 u … u Am u ∀R1. C1 u … u ∀Rn. Cn is the normal form of C
and
– A u ¬A ≡ ⊥
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•
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Exercise: Work out the normal form of: (∀R.¬A) u A u (∀R. A u (∀R. B))
To deal with number restrictions (for ALCN), we need additional
rules such as:
(≥ nR. C) v (≥ mR. C) iff n ≥ m
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Examples and Remark
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CvC
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CuDvC
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∀R. (C u D) v ∀R.C
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A u B u ∀R.(C u D u ∀S. E) u ∀T. ⊥ v B u ∀T. F u ∀ R.(C u ∀ S. E)
On Tableau Algorithms
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These algorithms do not do subsumption testing directly
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Negation is used to reduce subsumption to (un)satisfiability of
concepts using the fact that:
C v D iff C u ¬D is unsatisfiable
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They act as decision procedures for unsatisfiability
– if a formula is decidable, they can exhibit a model
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For DLs with full negation (i.e. not just applied to concept names) and
full existential restriction ∃R. C, structural subsumption cannot be used.
Instead, one can use a tableau-based algorithm.
Idea: Incrementally build model by decomposing negation
normal form (nnf) formula in a top-down fashion. All
possibilities are looked by applying a number of
‘transformation rules’ exhaustively until no more rules apply.
If no obvious contradiction or clash is found, then the formula
is satisfiable, otherwise, it is unsatisfiable.
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Tableau Algorithm: Informal View
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List all assertions in ABox A.
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Apply specific completion rules and add resulting assertions to A
yielding A’.
Negation Normal Form
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• ¬(C u D) ≡ ¬C t ¬D
– Completion rules can be:
De Morgan’s laws
• ¬(C t D) ≡ ¬C u ¬D
• deterministic yielding a uniquely determined set of assertions
• ¬∀R.C ≡ ∃R. ¬C
• non-deterministic yielding several alternative sets of assertions,
each set corresponding to a branch of the tableau
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An ALC formula can be transformed into negation normal
form (nnf) using the following identities:
• ¬∃R. C ≡ ∀R. ¬C
Apply the completion rules until either a contradiction or clash
occurs in every branch, or there is a completed branch where no
more rule is applicable.
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In nnf, the negation is pushed as far inward as possible
until it is applied to a concept name.
– A completed branch of the tableau gives a model for the completed
system.
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Transformation Rules for
Satisfiability Algorithm
Clash
branching
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A clash corresponds to a logical contradiction.
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It occurs when a set of assertions contains both
C(a) and ¬C(a).
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A clash is an unsatisfiable set of assertions. The
tableau algorithm closes a branch as soon as a
clash is encountered.
Recall:
C(a) is a: C
R(a,b) is (a,b): R
In algorithm:
(C1 u C2)(x) stands for
x : (C1 u C2)
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Example
Example
Does ∀R. (C u D) v ∀R. C u ∀R. D ?
Does ∃R. C u ∃R. D v ∃R.(C u D)?
Check whether ∀R. (C u D) u ¬(∀R. C u ∀R. D) is unsatisfiable using the tablean
algorithm:
Tableau:
1.
( ∀R. (C u D) u ¬(∀R. C u ∀R. D))(x)
2.
(∀R. (C u D) u (¬∀R. C t ¬∀R. D))(x)
by De Morgan’s law
3.
(∀R. (C u D) u (∃R. ¬C t ∃R. ¬D))(x)
by nnf
4.
(∀R. (C u D))(x)
5.
(∃R. ¬C t ∃R. ¬D)(x)
(∃R. ¬C)(x)
6.
by
→∀ –rule to 4:
by
by
→∃ –rule
by
→∃ -rule
(C u D)(z)
by
→∀ –rule to 4
9b’.
C(z)
by
→u -rule
9b.
D(z)
R(x, y)
7b.
R(x, z)
7a’.
¬C(y)
7b’.
¬C(z)
(C u D)(y)
8.
9a.
C(y)
9b.
D(y)
Tableau:
(∃R. ¬D)(x)
7a.
8.
→u -rule
by
→u –rule
X
X
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Expressive Roles
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Reading
The DL(s) considered only allow for atomic roles. The expressivity of the
languages can be extended by allowing role constructors that build
complex roles from atomic ones:
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Inverse role: (R−)I = {(b, a) ∈ 4 × 4 | (a, b) ∈ RI}
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Role composition:
(R ◦ S )I = {(a, c) ∈ 4 × 4 | ∃b. (a, b) ∈ RI ∧ (b, c) ∈ SI }
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Transitive closure: (R+)I =
∪n≥1 (RI)n
(RI)1 = RI
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F. Baader and W. Nutt. Basic Description Logics. In Baader,
Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The
Description Logics Handbook, chapter 2, pages 43-95. Cambridge
University Press, 2003. Available from MASWS web-page.
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F. Baader and U. Sattler. Tableau Algorithms for Description
Logics. Proceedings of the International Conference on
Automated Reasoning with Tableaux and Related Methods
(Tableaux 2000), Vol. 1847, Springer-Verlag. Available at
http://citeseer.ist.psu.edu/baader00tableau.html
(RI)n+1 = (RI)n ◦ RI
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etc.
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In general, there’s a price to pay in terms of reasoning/deciding a formula.
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