Description Logics and Logics

Enrico Franconi franconi@cs.man.ac.uk + +

Summary

• Description Logics

• K_{(m)}, multi-modal Normal Modal Logic K • Propositional Dynamic Modal Logics

• Propositional μ-calculus

• Propositional Temporal Modal Logics

• L̈^{n} FOL fragments

• Guarded Fragment of FOL

+ +

What is a Description Logic

A logical system, based on objects (individuals), classes (concepts), and relationships (roles), constituted by:

• a description language, which specifies how to construct concept and relationship expressions,

• a knowledge base language, which specifies properties of objects, concepts, and relationships,

• a set of (decidable) reasoning services over a knowledge base, with sound and complete procedures.

+ +

An example

Σ:

TBox

∃TEACHES. Course ⊑ ¬Undergrad ⊔ Prof

ABox

TEACHES(mary, cs415), Course(cs415),
Undergrad(mary)

Σ ⊧ Prof(mary)

+ +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ + 5-a +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ + 5-b +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ + 5-c +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ + 5-d +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ + 5-e +

Description Logics: syntax

C, D → A |  A  (primitive concept)
¬C |  (not C)  (complement)
C ⊓ D |  (and C D …)  (conjunction)
C ⊔ D |  (or C D …)  (disjunction)
∀R. C |  (all R C)  (universal quant.)
∃R. C |  (some R C)  (existential quant.)

≥n R. C |

            (atleast n R C)

(minimum cardin.)

≤n R. C |

            (atmost n R C)

(maximum cardin.)

f : C | (in f C) (selection)

f ↑ |

(undefined f) (undefinedness)

{ a, b … }

(oneof a b …)

(enumeration)

R, S → P |

P

(primitive role)

R ⊔ S |

(or R S)

(disjunction)

R ∘ S | (compose R S) (role chain)

R^{∗} |

            (trans R)

(transitive closure)

id(C) |

            (self C)

(role identity)

R^{−1} |

            (inverse R)

(inverse role)

f f (feature)

f, g → p |

p

(primitive feature)

f ∘ g

            (compose f g)

(feature chain)

TBox: (C ˙⊑ D), (R ˙⊑ S)

ABox: C(a), R(a, b) ALCIQFOreg

+ +

Description Logics: semantics

A^{I} ⊆ Δ^{I}  A(γ)
(¬C)^{I} = Δ^{I} \ C^{I}  ¬F_{C}(γ)
(C ⊓ D)^{I} = C^{I} ∩ D^{I}  F_{C}(γ) ∧ F_{D}(γ)
(C ⊔ D)^{I} = C^{I} ∪ D^{I}  F_{C}(γ) ∨ F_{D}(γ)
(∀R. C)^{I} = \{{i ∈ Δ^{I} | \atop ∀j. R^{I}(i, j) ⇒ C^{I}(j)\}}  ∀x. F_{R}(γ, x) ⇒ F_{C}(x)
(∃R. C)^{I} = \{{i ∈ Δ^{I} | \atop ∃j. R^{I}(i, j) ∧ C^{I}(j)\}}  ∃x. F_{R}(γ, x) ∧ F_{C}(x)
(≥n R. C)^{I} = \{{i ∈ Δ^{I} | \atop ♯\{j | R^{I}(i, j) ∧ C^{I}(j)\} ≥ n\}} ∃^{n}x. F_{R}(γ, x) ∧ F_{C}(x)
(≤n R. C)^{I} = \{{i ∈ Δ^{I} | \atop ♯\{j | R^{I}(i, j) ∧ C^{I}(j)\} ≤ n\}} ¬∃^{n+1}x. F_{R}(γ, x) ∧ F_{C}(x)
(f : C)^{I} = {i ∈ dom f^{I} | C^{I}(f^{I}(i))}  ∃x. F_{f} (γ, x) ∧ F_{C}(x)
(f ↑)^{I} = Δ^{I} \ dom f^{I}  ¬∃x. F_{f} (γ, x)
{ a, b … }^{I} = {a^{I}, b^{I} …}  γ = a ∨ γ = b ∨ …
P^{I} ⊆ Δ^{I} × Δ^{I}  P (α, β)
(R ⊔ S)^{I} = R^{I} ∪ S^{I}  F_{R}(α, β) ∨ F_{S}(α, β)
(R ∘ S)^{I} = R^{I} ∘ S^{I}  ∃x. F_{R}(α, x) ∧ F_{S}(x, β)
(R^{∗})^{I} = (R^{I})^{∗}  …
(id(C))^{I} = {(i, i) ∈ Δ^{I} × Δ^{I} | C^{I}(i)}  F_{C}(α) ∧ α = β
(R^{−1})^{I} = \{{(i, j) ∈ Δ^{I} × Δ^{I} | \atop R^{I}(j, i)\}}  F_{R}(β, α)
f^{I} = \{{(i, j) ∈ Δ^{I} × Δ^{I} | \atop f^{I}(i) = j\}}  F_{f} (α, β)
p^{I} =

Δ^{I} ⟼^{partial} Δ^{I}

p(α) = β
(f ∘ g)^{I} = f^{I} ∘ g^{I} ∃x. F_{f} (α) = x ∧ F_{g}(x) = β

+ + 6-a +

Description Logics: semantics

A^{I} ⊆ Δ^{I}  A(γ)
(¬C)^{I} = Δ^{I} \ C^{I}  ¬F_{C}(γ)
(C ⊓ D)^{I} = C^{I} ∩ D^{I}  F_{C}(γ) ∧ F_{D}(γ)
(C ⊔ D)^{I} = C^{I} ∪ D^{I}  F_{C}(γ) ∨ F_{D}(γ)
(∀R. C)^{I} = \{{i ∈ Δ^{I} | \atop ∀j. R^{I}(i, j) ⇒ C^{I}(j)\}}  ∀x. F_{R}(γ, x) ⇒ F_{C}(x)
(∃R. C)^{I} = \{{i ∈ Δ^{I} | \atop ∃j. R^{I}(i, j) ∧ C^{I}(j)\}}  ∃x. F_{R}(γ, x) ∧ F_{C}(x)
(≥n R. C)^{I} = \{{i ∈ Δ^{I} | \atop ♯\{j | R^{I}(i, j) ∧ C^{I}(j)\} ≥ n\}} ∃^{n}x. F_{R}(γ, x) ∧ F_{C}(x)
(≤n R. C)^{I} = \{{i ∈ Δ^{I} | \atop ♯\{j | R^{I}(i, j) ∧ C^{I}(j)\} ≤ n\}} ¬∃^{n+1}x. F_{R}(γ, x) ∧ F_{C}(x)
(f : C)^{I} = {i ∈ dom f^{I} | C^{I}(f^{I}(i))}  ∃x. F_{f} (γ, x) ∧ F_{C}(x)
(f ↑)^{I} = Δ^{I} \ dom f^{I}  ¬∃x. F_{f} (γ, x)
{ a, b … }^{I} = {a^{I}, b^{I} …}  γ = a ∨ γ = b ∨ …
P^{I} ⊆ Δ^{I} × Δ^{I}  P (α, β)
(R ⊔ S)^{I} = R^{I} ∪ S^{I}  F_{R}(α, β) ∨ F_{S}(α, β)
(R ∘ S)^{I} = R^{I} ∘ S^{I}  ∃x. F_{R}(α, x) ∧ F_{S}(x, β)
(R^{∗})^{I} = (R^{I})^{∗}  …
(id(C))^{I} = {(i, i) ∈ Δ^{I} × Δ^{I} | C^{I}(i)}  F_{C}(α) ∧ α = β
(R^{−1})^{I} = \{{(i, j) ∈ Δ^{I} × Δ^{I} | \atop R^{I}(j, i)\}}  F_{R}(β, α)
f^{I} = \{{(i, j) ∈ Δ^{I} × Δ^{I} | \atop f^{I}(i) = j\}}  F_{f} (α, β)
p^{I} =

Δ^{I} ⟼^{partial} Δ^{I}

p(α) = β
(f ∘ g)^{I} = f^{I} ∘ g^{I} ∃x. F_{f} (α) = x ∧ F_{g}(x) = β

+ +

Another Example

 ∃TEACHES. Course  ˙⊑ (Student ⊓ ∃DEGREE. Bs) ⊔ Prof
            Prof  ˙⊑ ∃DEGREE. Ms
      ∃DEGREE. Ms  ˙⊑ ∃DEGREE. Bs
          Ms ⊓ Bs  ˙⊑ ⊥

TEACHES(paul,cs415), Course(cs415),
(∃FRIEND. ({ paul } ⊓ ≤1DEGREE))(john)
Σ ⊧ Student(paul)

+ +

ALC and K_{(m)}

ALC K_{(m)}
C^{I} is a set of individuals α_{C}^{I} is a set of worlds
R^{I} is a set of pairs of individuals R is an accessibility relation
A P_{A}
C ⊓ D α_{C} ∧ α_{D}
C ⊔ D α_{C} ∨ α_{D}
¬C ¬α_{C}
∀R. C □_{R} α_{C}
∃R. C ◇_{R} α_{C}
o ∈ C^{I} I, o ⊧ α_{C}

∃T. C _⊑ ¬U ⊔ P ◇_{T}C → ¬U ∨ P

U(m), T(m, c), C(c)

Σ ⊧ P(m)

+ +

Some Results on satisfiability

ALC  ⟺ K_{(m)}

∩

ALC^{+}  ⟺ K_{(m)} ∪ S4

∩

⋯

ALC  ⟺ K_{(m)}

∩

ALCO  ⟺ K_{(m)} with nominals
ALC  ⟺ K_{(m)}

∩

ALC with axioms ⟺ K_{(m)} theories

• ALC: PSPACE-complete

• ALC^{+}: PSPACE-complete

• ALCO: PSPACE-complete

• ALC with axioms: EXPTIME-complete

+ +

ALCreg and PDL

• The domain of the interpretation is to be read as a set of program states.

• Concepts are to be interpreted as the set of states in which they hold.

• Roles are to be interpreted as nondeterministic programs.

+ +

ALCreg and PDL

• ∀R. C as [R]C: “whenever program R terminates, proposition C holds on termination”.

• R_{1} ∘ R_{2} as “run R_{1} and R_{2} consecutively”.

• R_{1} ⊔ R_{2} as “nondeterministically do R_{1} or R_{2}”.

• R^{⋆} as “repeat program R a nondeterministically chosen number of times ≥ 0”.

• id(C) as “proceed without changing the program state iff proposition C holds”.

• R^{−1} as “run R in reverse” (ALCIreg and CPDL).

+ + 11-a +

ALCreg and PDL

• ∀R. C as [R]C: “whenever program R terminates, proposition C holds on termination”.

• R_{1} ∘ R_{2} as “run R_{1} and R_{2} consecutively”.

• R_{1} ⊔ R_{2} as “nondeterministically do R_{1} or R_{2}”.

• R^{⋆} as “repeat program R a nondeterministically chosen number of times ≥ 0”.

• id(C) as “proceed without changing the program state iff proposition C holds”.

• R^{−1} as “run R in reverse” (ALCIreg and CPDL).

+ +

Internalization of axioms

\begin{matrix}ψ ⊧ φ & ⇝ & ⊧ [ν] ψ ⇒ φ \\ C ˙⊑ D ⊧ φ & ⇝ & ⊧ [ν] (α_{C} ⇒ α_{D}) ⇒ α_{φ} \end{matrix}

ν ≐ (R_{1} ∨ R_{1}^{−1} ∨ … ∨ R_{n} ∨ R_{n}^{−1})^{⋆} • Reasoning with theories is reduced to

satisfiability of single formulas

• The complexity does not change

+ +

Results: satisfiability

ALCreg  ⟺ PDL

⇑

ALCIreg ⟺ CPDL

⇑

ALCFIreg^{−} ⟺ DCPDL

⇑

ALCFIreg

⇑

ALCQIreg^{−} ⟺ CPDL + graded modalities

⇑

ALCQIreg

• Satisfiability is EXPTIME-complete

(subsumption and logical implication wrt a free TBox can be reduced to satisfiability).

• CPDL has the finite model property, DCPDL not.

+ +

Results: satisfiability with individuals

ALCreg  ⟺ PDL

⇑

ALCQOreg  ⟺ PDL + graded modalities and nominals
ALCIreg  ⟺ CPDL

⇑

ALCIOreg  ⟺ CPDL + nominals
ALCIreg  ⟺ CPDL

⇑ ALCQIreg + ABox

Satisfiability is EXPTIME-complete.

+ +

So what?

• DL extend modal logic in interesting ways:

– Reasoning in DL is always reasoning with theories.

– Nominals.

– Graded modalities.

• Results:

– ALC theories are EXPTIME-complete.

– ALCQIreg theory reduces to ALCreg theory (DCPDL to PDL).

– ALCIOreg theory reduces to ALCreg theory (CPDL + nominals to PDL).

– ALCQIreg + Abox theory reduces to ALCreg theory.

+ +

Inductive Definitions

  • An Empty-List is a List.

• A Node, that has exactly one SUCCESSOR that is a List, is a List.

• Nothing else is a LIST.

Node≐ ¬Empty-List
List≐ Empty-List ⊔ (Node ⊓ ≤1SUCCESSOR ⊓ ∃SUCCESSOR. List)

Δ = {a, b, nil}

Node^{I} = {a, b}
Empty-List^{I} = {nil}
SUCCESSOR^{I} = {⟨a, nil⟩, ⟨b, b⟩}

With descriptive semantics:

List^{I} = {a, nil}; List^{I} = {a, b, nil}

With least fixpoint semantics:

List^{I} = {a, nil}

+ +

Propositional μ-calculus

Node≐ ¬Empty−List
List≐ μX. (Empty−List ⊔ (Node ⊓ ≤1SUCCESSOR ⊓ ∃SUCCESSOR. X))

C → … | μX. C | νX. C | X

μALC ⟺ Propositional μ-calculus

⇑ μALCQI

• Satisfiability is EXPTIME-complete.

• Can express well-foundedness of relations, useful to describe finite structures.

• Open problems: individuals.

+ +

Tense Logic (point ontology)

• Tense logic is a propositional modal logic, interpreted over temporal structure T = (P, <), where P is a set of time points and < is a strict partial order on P.

• Mortal ⊑ LivingBeing ⊓ ∀LIVES-IN. Place ⊓ (LivingBeing U (□^{+}¬LivingBeing))

• Satisfiability in ALCT – the combination of tense logic with K_{(m)} – over a linear, unbounded, and discrete temporal structure has the same complexity as its base (PSPACE-complete).

• Satisfiability in ALCQIT reg with ABox – the combination of tense logic with ALCQIreg with ABox – over a linear, unbounded, and discrete temporal structure has the same complexity as its base (EXPTIME-complete).

+ +

HS: Interval Temporal Propositional Modal Logic

• HS is a propositional modal logic interpreted over an interval set T_{<}^{∗}, defined as the the set of all closed intervals [u, v] ≐ {x ∈ P | u ≤ x ≤ v, u ≠ v} in some temporal structure T .

• HS extends propositional logic with modal formulæ ⟨R⟩ϕ and [R]ϕ – where R is a basic Allen’s algebra temporal relation:

i  j before (i, j) meets (i, j) overlaps (i, j) starts (i, j) during (i, j) finishes (i, j)

• Mortal≐ LivingBeing ∧ ⟨after⟩. ¬LivingBeing

• Satisfiability HS is undecidable for the most interesting classes of temporal structures.

• Therefore, HS ∪ K_{(m)} is undecidable.

+ +

Decidable Interval Temporal Description Logics

The combination of ALCF and HS^{∗} :

• HS^{∗}:

– No universal quantification, or restricted to homogeneous properties: □(=, starts, during, finishes). ψ

– Allows for temporal variables:

◇ x^{→} TN(x^{→}). ψ

ψ@x

• Global roles – denoting temporal independent properties.

• Logical implication in the combined language is decidable (PSPACE-hard); satisfiability is PSPACE-complete.

• Logical implication in TL-F is NP-complete.

• Useful for event representation and plan recognition.

+ +

The Block World Domain

1  2  1  2 Initial State  Approach
1  2  2 1 Grasp  Final State

  [figure: Holding-Block(OBJ1); Clear-Block(OBJ2); ON(OBJ1, OBJ2); Clear-Block(OBJ1); Clear-Block(OBJ1); Stack(OBJ1, OBJ2); w; x; y; v; z; ♯]
Stack  ≐

◇(x y z v w) \begin{matrix}(♯ finishes x)(♯ meets y)(♯ meets z) \\ (v overlaps ♯)(w finishes ♯)(v meets w). \\ ( & (⋆OBJECT2 : Clear-Block)@x ⊓ (⋆OBJECT1∘ON =^{↓} ⋆OBJECT2)@y ⊓ (⋆OBJECT1 : Clear-Block)@v ⊓ (⋆OBJECT1 : Holding-Block)@w ⊓ (⋆OBJECT1 : Clear-Block)@z) \end{matrix}

+ +

L̈^{n} FOL fragments

• L̈^{n} is the set of function-free FOL formulas with equality and constants, with only unary and binary predicates, and which can be expressed using at most n variable symbols.

• Satisfiability of L̈^{3} formulas is undecidable.

• Satisfiability of L̈^{2} formulas is NEXPTIMEcomplete.

+ +

The DL description logic

ALCQIreg

- the transitive closure operator,

- number restriction operators, + propositional calculus on roles, + the concept (R ⊆ S).

• The DL description logic and L̈^{3} are equally expressive.

• The DL^{−} description logic (i.e., DL without the composition operator) and L̈^{2} are equally expressive.

• Open problem: relation between DL including cardinalities and C̈^{n}– adding counting quantifiers to L̈^{n}.

+ +

Guarded Fragments of FOL

The guarded fragment GF of FOL is defined as:

1. Every relational atomic formula is in GF

2. GF is propositionally closed

3. If x, y are tuples of variables, α(x, y) is atomic, and ψ(x, y) is a formula in GF, such that free(ψ) ⊆ free(α) = {x, y}, then the following formulae are in GF: ∃y. α(x, y) ∧ ψ(x, y) ∀y. α(x, y) → ψ(x, y)

The guarded fragment contains the modal fragment of FOL (and Description Logics); a weaker definition (LGF) is needed to include temporal logics.

+ +

Properties of GF

• GF has the finite model property

• GF and LGF have the tree model property

• Many important model theoretic properties which hold for FOL and the modal fragment, do hold also for GF and LGF

• Satisfiability is decidable for GF and LGF (deterministic double exponential time complete)

• Bounded-variable or bounded-arity fragments of GF and LGF (which include Description Logics) are in EXPTIME.

• GF with fix-points is decidable.

+ +

That’s not all, folks…

• Defaults and non monotonic logic • Other modal extensions

• Autoepistemic logic

• Formal ontology

• Decision procedures

• Database theory

• …

+ +

References

• Donini, F., Lenzerini, M., Nardi, D., Schaerf, A., ‘Reasoning in Description Logics’, Principles of Knowledge Representation and Reasoning, editor, G. Brewka, Studies in Logic, Language and Information, CLSI Publications, pp 193-238, 1996.

• Schild, K., ‘A correspondence theory for terminological logics: preliminary report’, IJCAI-91, pp 466-471, 1991.

• De Giacomo, G., Decidability of Class-Based Knowledge Representation Formalisms, Ph.D. thesis, Dipartimento di Informatica e Sistemistica, Universita degli Studi di Roma ”La Sapienza”, 1995.

• De Giacomo, G., Lenzerini, M., ‘TBox and ABox Reasoning in Expressive Description Logics’, KR-96, pp 316-327, 1996.

• De Giacomo, G., Lenzerini, M., ‘A Uniform

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Framework for Concept Definitions in Description Logics’, Journal of Artificial Intelligence Research, Vol 6, pp 87-110, 1997.

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