On the proof theory for Description Logics

On the Proof Theory for Description Logics

Alexandre Rademaker

March 30, 2010

Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 1 / 58

Description Logics

Description Logics

Notations and Formalism for KR
FOL −→ Semantic-Network −→ Conceptual-Graphs −→ DLs


Description Logics

Description Logics

Decidable Fragments of FOL
Binary (Roles) and unary (Concepts) predicate symbols, R(x, y )
and C(y ).
Prenex Guarded formulas (∀y (R(x, y ) → C(y )),
∃y (R(x, y ) ∧ C(y ))).
Non-trivial extensions (transitive Closure R ∗ ).
Essentially propositional (Tboxes), but may involve reasoning on
individuals (Aboxes).


Description Logics

ALC semantics

Given by: I = (∆I , I )
I = ∆I
⊥I = ∅
(¬C)I = ∆I C I
(C D)I = C I ∩ DI
(C D)I = C I ∪ DI
(∃R.C)I = {a ∈ ∆I | ∃b.(a, b) ∈ R I ∧ b ∈ C I }
(∀R.C)I = {a ∈ ∆I | ∀b.(a, b) ∈ R I → b ∈ C I }


Description Logics

A T-Box on Family Relationships

Woman ≡ Person Female
Man ≡ Person ¬Woman
Mother ≡ Woman ∃hasChild.Person
Father ≡ Man ∃hasChild.Person
Parent ≡ Father Mother
Grandmother ≡ Mother ∃hasChild.Parent
MotherWithoutDaughter ≡ Mother ∀hasChild.¬Woman
MotherInTrouble ≡ Mother ≥ 10hasChild


Description Logics

Some definitions

Definition
The concept description D subsumes the concept description C,
written C D, if and only if C I ⊆ D I for all interpretations I.

Definition
C is satisfiable if and only if there exists an interpretation I such that
C I = ∅.


Description Logics

Some definitions

Definition
C is valid or a tautology if and only if for all interpretation I, C I ≡ ∆I .

Definition
C and D are equivalent, written C ≡ D, if and only if C D and D C.


The Sequent Calculus SCALC

An ALC Sequent Calculus: main motivation

DL have implemented reasoners and editors. However, they do
not have good, if any, support for explanations.
Simple Tableaux (without analytical cuts) cannot produce short
proofs (polynomially lengthy proofs). Sequent Calculus (SC) (with
the cut rule) has short proofs.
We have an industrial application problem with explanations
requirement.



Labeled Formulas

LB → ∀R | ∃R
L → LB, L | ∅
φlc → L φc

Each labeled ALC concept has an straightforward ALC concept
equivalent. For example:

∃R2 .∀Q2 .∃R1 .∀Q1 .α ≡ ∃R2 ,∀Q2 ,∃R1 ,∀Q1 α



SCALC

Structural rules

α⇒ α ⊥⇒ α
∆⇒ Γ ∆⇒ Γ
weak-l weak-r
∆, δ ⇒ Γ ∆ ⇒ Γ, γ
∆, δ, δ ⇒ Γ ∆ ⇒ Γ, γ, γ
contraction-l contraction-r
∆, δ ⇒ Γ ∆ ⇒ Γ, γ
∆1 , δ1 , δ2 , ∆2 ⇒ Γ ∆ ⇒ Γ1 , γ1 , γ2 , Γ2
perm-l perm-r
∆1 , δ2 , δ1 , ∆2 ⇒ Γ ∆ ⇒ Γ1 , γ2 , γ1 , Γ2
∆1 ⇒ Γ1 , L α L α, ∆ ⇒ Γ
2 2
cut
∆1 , ∆2 ⇒ Γ1 , Γ2



SCALC

Rules on role quantiﬁcation

∆, L,∀R α ⇒ Γ ∆ ⇒ Γ, L,∀R α
∀-l ∀-r
∆, L (∀R.α)L2 ⇒ Γ ∆ ⇒ Γ, L (∀R.α)

∆, L,∃R α ⇒ Γ ∆ ⇒ Γ, L,∃R α
∃-l ∃-r
∆, L (∃R.α) ⇒ Γ ∆ ⇒ Γ, L (∃R.α)



SCALC

Boolean Rules

∆, ∀L α, ∀L β ⇒ Γ ∆ ⇒ Γ, ∀L α ∆ ⇒ Γ, ∀L β
-l -r
∆, ∀L (α β) ⇒ Γ ∆ ⇒ Γ, ∀L (α β)

∆, ∃L α ⇒ Γ ∆, ∃L β ⇒ Γ ∆ ⇒ Γ, ∃L α, ∃L β
-l -r
∆, L (α β) ⇒ Γ ∆ ⇒ Γ, ∃L (α β)

∆ ⇒ Γ, ¬L α ∆, ¬L α ⇒ Γ
¬-l ¬-r
∆, L ¬α ⇒ Γ ∆ ⇒ Γ, L ¬α



SCALC

Generalization rules

δ⇒ Γ ∆⇒ γ
+∃R
prom-∃ prom-∀
δ ⇒ +∃R Γ +∀R
∆⇒ +∀R γ



SCALC example

Doctor ⇒ Doctor
prom-∀
∀child
Doctor ⇒ ∀child Doctor
weak-l
, ∀child Doctor ⇒ ∀child Doctor
¬-r
⇒ ∃child ¬Doctor ,∀child Doctor
∃-r
⇒ ∃child.¬Doctor , ∀child Doctor
prom-∃
∃child
⇒ ∃child (∃child.¬Doctor ), ∃child,∀child Doctor
¬-l
∃child
, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child,∀child Doctor
∀-l
∃child
, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child ∀child.Doctor
∃-r
∃child
, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
∃child
∀-l
, ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
∃-l
∃child. , ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
-l
∃child. ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor



SCALC soundness

Theorem (SCALC is sound)
Considering Ω a set of sequents, a theory or a TBox, let an Ω-proof be
any SCALC proof in which sequents from Ω are permitted as initial
sequents (in addition to the logical axioms). The soundness of SCALC
states that if a sequent ∆ ⇒ Γ has an Ω-proof, then ∆ ⇒ Γ is satisﬁed
by every interpretation which satisﬁes Ω. That is,

if Ω SALC ∆ ⇒ Γ then Ω |= T (δ) T (γ)
δ∈∆ γ∈Γ

for all interpretation I.



SCALC completeness

ALC sequent calculus deduction rules without labels behave
exactly as sequent calculus rules for classical propositional logic.
The derivation of the rule of necessitation
⇒α prom-∀
⇒ ∀R α ∀-r
⇒ ∀R.α
The axiom
∀R.(α β) ≡ ∀R.α ∀R.β



SCALC Cut elimination

We follow Gentzen’s original proof for cut elimination. Let δ be a
labeled formula. An inference of the following form is called mix with
respect to δ:
∆1 ⇒ Γ1 ∆2 ⇒ Γ2
(δ)
∆1 , ∆∗ ⇒ Γ∗ , Γ2
2 1




Quasi-mixes

Lα ⇒ Γ1 ∆2 ⇒ Γ2 Lα
∆1 ⇒ Γ1 ⇒ Γ2
+∃R ∗
(L α, ∃R,L α) (∃R,L α,L α)
∃R,L α, ∆∗
2 ⇒ Γ1 , Γ2 ∆1 ⇒ Γ∗ , +∃R Γ2
1

∆1 ⇒ L α ∆2 ⇒ Γ2 ∆1 ⇒ Γ1 ∆2 ⇒ L α
(L α, ∀R,L α) (∀R,L α, L α)
+∀R
∆1 , ∆∗ ⇒ Γ2
2 ∆1 , +∀R ∆∗ ⇒ Γ∗ , ∀R,L α
2 1




∗
Deﬁnition (The SCALC system)
∗
We call SCALC the new system obtained from SCALC by replacing the
cut rule by the quasi-mix rules.

Lemma
∗
The systems SCALC and SCALC are equivalent, that is, a sequent is
∗
SCALC -provable if and only if that sequent is also SCALC -provable.




T
Definition (SCALC system)
SCALC was defined with initial sequents of the form α ⇒ α with α a
ALC concept definition (logical axiom). However, it is often convenient
to allow for other initial sequents. So if T is a set of sequents of the
form ∆ ⇒ Γ, where ∆ and Γ are sequences of ALC concept
T
descriptions (non-logical axioms), we define SCALC to be the proof
system defined like SCALC but allowing initial sequents to be from T
too.




Deﬁnition (Free-quasi-mix free proof)
∗T
Let P be an SCALC -proof. A formula occurring in P is anchored (by an
T -sequent) if it is a direct descendent of a formula occurring in an
initial sequent in T . A quasi-mix inference in P is anchored if either: (i)
the mix formulas are not atomic and at least one of the occurrences of
the mix formulas in the upper sequents is anchored, or (ii) the mix
formulas are atomic and both of the occurrences of the mix formulas in
the upper sequents are anchored. A quasi-mix inference which is not
anchored is said to be free. A proof P is free-quasi-mix free if it
contains no free quasi-mixes.




Theorem (Free-mix Elimination)
∗T
Let T be a set of sequents. If SCALC ∆ ⇒ Γ then there is a
∗T
free-quasi-mix free SCALC -proof of ∆ ⇒ Γ.

Lemma
∗T
If P is a proof of S (in SCALC ) which contains only one free-quasi-mix,
occurring as the last inference, then S is provable without any free-mix.

The Theorem is obtained from the Lemma by simple induction over the
number of quasi-free-mix occurring in a proof P.




We prove the last lemma by lexicographically induction on the ordered
triple (grade,ldegree,rank ) of the proof P. We divide the proof into two
main cases, namely rank = 2 and rank > 2 (regardless of grade and
ldegree).


Comparing SCALC with other deduction systems

The Structural Subsumption algorithm

SSA deals with normalized concepts in FL0 ( , ∀R.C), and can
be extended to ALN (⊥, ¬A, ≤ R, ≥ R).
any concept description in FL0 can be transformed into an
equivalent in normal form.
Given C and D, C D iff the following two conditions holds:

C ≡ A1 ... Am ∀R1 .C1 ... ∀Rn .Cn
D ≡ B1 ... Bk ∀S1 .D1 ... ∀Sl .Dl

for 1 ≤ i ≤ k , ∃j, 1 ≤ j ≤ m such that Bi = Aj
for 1 ≤ i ≤ l, ∃j, 1 ≤ j ≤ n such that Si = Rj and Cj Di



The Tableaux algorithm

Strongly based on the use of individuals and FOL Tableaux.
C D iff C0 ≡ C ¬D is unsatisﬁable. C0 in negation normal
form.
I
Try to construct a ﬁnite interpretation I such that C0 = ∅.
The individuals must satisfy the constraints (C0 clauses).

(+) It provides counter-model
(−) super-polynomial lengthy proofs



Comparing with SSA

Each step taken by a bottom-up construction of a SCALC proof
corresponds to a step towards this matching by means of the SSA.

⇒ S1 D1 R1 C
1
A1 ⇒ B1 ⇒ ∀S1 .D1R1 C
1
∀R1 .C1 , A1 ⇒ B1 ∀R1 .C1 ⇒ ∀S1 .D1
A1 , ∀R1 .C1 ⇒ B1 A1 , ∀R1 .C1 ⇒ ∀S1 .D1
A1 , ∀R1 .C1 ⇒ B1 ∀S1 .D1
A1 ∀R1 .C1 ⇒ B1 ∀S1 .D1



Reﬁning SCALC towards Automatic Theorem Proving

[]
SCALC was extended to SALC in order to be able to construct a
counter-model from unsuccessful proofs;
Determinism to avoid backtracking in proof search;
Alternative cut-elimination and completeness (for SCALCQI );




Example
An unsuccessful proof of a valid sequent in SCALC :

⇒ ∀child Doctor
prom-∃
∃child
⇒ ∃child,∀child Doctor
weak-l
∃child
, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child,∀child Doctor
∀-r
∃child
child, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child ∀child.Doctor
∃-r
∃child
, ∀child ¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
∃child
∀-l
, ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
∃-l
∃child. , ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor
-l
∃child. ∀child.¬(∃child.¬Doctor ) ⇒ ∃child.∀child.Doctor




Example
A counter-model I not only has to guarantee B I AI but also
AI B I .
B⇒ A B⇒ B A⇒ A A⇒ B
-r -r
B⇒ A B A⇒ A B
prom-∃ prom-∃
∃R
B ⇒ ∃R A B ∃R
A ⇒ ∃R A B
∃R
weak-l weak-l
A, B ⇒ ∃R A B
∃R ∃R
A, B ⇒ ∃R A B
∃R

∃R
∃-r ∃-r
A, ∃R B ⇒ ∃R.A B ∃R
A, ∃R B ⇒ ∃R.A B
∃R
∃-l ∃R
∃-l
A, ∃R.B ⇒ ∃R.A B A, ∃R.B ⇒ ∃R.A B
∃-l ∃-l
∃R.A, ∃R.B ⇒ ∃R.A B ∃R.A, ∃R.B ⇒ ∃R.A B
-l -l
∃R.A ∃R.B ⇒ ∃R.A B ∃R.A ∃R.B ⇒ ∃R.A B




[]
SALC sequents are expressions of the form ∆ ⇒ Γ that range over
labeled concepts and indexed-frozen labeled concept ([α]n , [∆]n ).
In frozen-exchange all formulas in ∆2 and Γ2 must be atomic;
Reading bottom-up, weak rules freeze all formulas saving the
context.
∆1 , [∆2 ]1 , . . . , [∆n ]n−1 ⇒ Γ1 , [Γ2 ]1 , . . . , [Γn ]n−1




∆, δ ⇒ δ, Γ

[∆, δ]k , ∆ ⇒ Γ, [Γ]k [∆]k , ∆ ⇒ Γ, [Γ, γ]k
weak-l weak-r
∆, δ ⇒ Γ ∆ ⇒ Γ, γ




∆, ∀L α, ∀L β ⇒ Γ ∆ ⇒ Γ, ∀L α ∆ ⇒ Γ, ∀L β
-l -r
∆, ∀L (α β) ⇒ Γ ∆ ⇒ Γ, ∀L (α β)
∆, ∃L α ⇒ Γ ∆, ∃L β ⇒ Γ ∆ ⇒ Γ, ∃L α, ∃L β
-l -r
∆, L (α β) ⇒ Γ ∆ ⇒ Γ, ∃L (α β)
∆ ⇒ Γ, ¬L α ∆, ¬L α ⇒ Γ
¬-l ¬-r
∆, L ¬α ⇒ Γ ∆ ⇒ Γ, L ¬α




[∆], L δ ⇒ Γ, [Γ1 ] [∆1 ], ∆ ⇒ L γ, [Γ]
prom-∃ prom-∀
[∆], ∃R,L δ ⇒ +∃R
Γ, [Γ1 ] [∆1 ], +∀R ∆ ⇒ ∀R,L γ, [Γ]

[∆], [∆2 ]k , ∆1 ⇒ Γ1 , [Γ2 ]k , [Γ]
frozen-exchange
[∆], ∆2 , [∆1 ]n ⇒ [Γ1 ]n , Γ2 , [Γ]



Satisfability of frozen labeled sequents

Let ∆ ⇒ Γ be a sequent with its succedent and antecedent having
formulas that range over labeled concepts and frozen labeled concept.
This sequent has the general form of

∆1 , [∆2 ]1 , . . . , [∆n ]n−1 ⇒ Γ1 , [Γ2 ]1 , . . . , [Γn ]n−1

Let (I1 , . . . , In ) be a tuple of interpretations. We say that this tuple
satisfy ∆ ⇒ Γ, if and only if, one of the following clauses holds:

I1 |= ∆1 ⇒ Γ1 I2 |= ∆2 ⇒ Γ2 ... In |= ∆n ⇒ Γn

The sequent ∆ ⇒ Γ is not satisﬁable by a tuple of interpretations, if and
only if, no interpretation in the tuple satisfy its corresponding context.



[]
SALC properties

Lemma
[]
Consider ∆ ⇒ Γ a SCALC sequent. If P is a proof of ∆ ⇒ Γ in SCALC
then it is possible to construct a proof P of ∆ ⇒ Γ in SCALC .



[]
SALC properties

A fully expanded proof-tree of ∆ ⇒ Γ is a tree having ∆ ⇒ Γ as root,
[]
each internal node is a premise of a valid SCALC rule application, and
[]
each leaf is either a SCALC axiom (initial sequent) or a top-sequent
which is not an axiom, not necessarily atomic. In other words, a
sequent is a top-sequent if and only if it does not contain reducible
contexts.



[]
SALC properties

If we consider a particular strategy of rules application any
full-expanded proof tree will have a special form called normal form.
1 only fair strategies of rules applications that avoid inﬁnite loops;
2 Promotional rules will be applied whenever possible;
3 The strategy will discard contexts created by successive
applications of weak rules and avoid further applications of weak
rules once it is possible to detected that they will not be useful to
obtain an initial sequent;
4 Weak rules will be used with the unique purpose of enabling
promotion rules applications.



The weak ∗ rule

Π
[∆, δ1 , δ2 ]k , ∆
⇒ Γ, [γ1 , γ2 , Γ]k
weak ∗
∆, δ1 , δ2 ⇒ γ1 , γ2 , Γ

Π
[∆, δ1 , δ2 ]k , [∆, δ2 ]k +1 , [∆]k +2 , [∆]k +3 , ∆ ⇒ Γ, [γ1 , γ2 , Γ]k , [γ1 , γ2 , Γ]k +1 , [γ1 , γ2 , Γ]k +2 , [γ2 , Γ]k +3
weak-r
[∆, δ1 , δ2 ]k , [∆, δ2 ]k +1 , [∆]k +2 , ∆ ⇒ γ2 , Γ, [γ1 , γ2 , Γ]k , [γ1 , γ2 , Γ]k +1 , [γ1 , γ2 , Γ]k +2
weak-r
[∆, δ1 , δ2 ]k , [∆, δ2 ]k +1 , ∆ ⇒ γ1 , γ2 , Γ, [γ1 , γ2 , Γ]k , [γ1 , γ2 , Γ]k +1
weak-l
[∆, δ1 , δ2 ]k , ∆, δ2 ⇒ γ1 , γ2 , Γ, [γ1 , γ2 , Γ]k
weak-l
∆, δ1 , δ2 ⇒ γ1 , γ2 , Γ



∗[]
SCALC normal proof

[A]2 , [. . .]3 , B ⇒ A, [. . .]3 , [B]2 [A]2 , [. . .]3 , B ⇒ B, [. . .]3 , [B]2
-r
[A]2 , [∃R A, ∃R B]3 , B ⇒ A B, [∃R (A B)]3 , [B]2
prom-∃
[A]2 , [∃R A, ∃R B]3 , ∃R B ⇒ ∃R (A B), [∃R (A B)]3 , [B]2
weak ∗
[A]2 , ∃R A, ∃R B ⇒ ∃R (A B), [B]2
f-exch
[∃R A, ∃R B]1 , A ⇒ B, [∃R (A B)]1 [. . .]1 , A ⇒ A, [. . .]1
-r
[. . .]1 , A ⇒ A B, [. . .]1
prom-∃
[∃R A, ∃R B]1 , ∃R A ⇒ ∃R (A B), [∃R (A B)]1
∃R
weak ∗
A, ∃R B ⇒ ∃R (A B)
∃R
∃-r
A, ∃R B ⇒ ∃R.(A B)
∃R
∃-l
A, ∃R.B ⇒ ∃R.(A B)
∃R.A, ∃R.B ⇒ ∃R.(A B)
∃-l
∃R.A ∃R.B ⇒ ∃R.(A B)
-l



∗[]
SCALC normal proof

Π1 ≡

Π2
∃R
A, ∃R
B ⇒ ∃R (A B)
∃-r
∃R
A, ∃R B ⇒ ∃R.(A B)
∃-l
∃R
A, ∃R.B ⇒ ∃R.(A B)
∃-l
∃R.A, ∃R.B ⇒ ∃R.(A B)
-l
∃R.A ∃R.B ⇒ ∃R.(A B)



∗[]
SCALC normal proof

Π2 ≡

Π3
[. . .]1 , A ⇒ B, [. . .]1 [. . .]1 , A ⇒ A, [. . .]1
-r
[. . .]1 , A ⇒ A B, [. . .]1
prom-∃
[∃R A, ∃R B]1 , ∃R A ⇒ ∃R (A B), [∃R (A B)]1
Π3 ≡

[A]2 , [. . .]3 , B ⇒ A, [. . .]3 , [B]2 [A]2 , [. . .]3 , B ⇒ B, [. . .]3 , [B]2
-r
[A]2 , [. . .]3 , B ⇒ A B, [. . .]3 , [B]2
prom-∃
[A]2 , [∃R A, ∃R B]3 , ∃R B ⇒ ∃R (A B), [∃R (A B)]3 , [B]2



∗[]
SCALC : obtaining counter-models

Theorem
[]
If P is a full-expanded proof-tree in SCALC with sequent S as root
(conclusion) and if P is in the normal form, from any top-sequent not
initial (non-axiom), one can construct a counter-model for S.

To be ﬁxed!



∗[]

Lemma
∗[]
If P is a full-expanded proof-tree in SCALC with sequent S as root
(conclusion) and if P is in the normal form, from any S1 top-sequent
not initial (non-axiom), we can construct a counter-model of S1 .

Lemma
If P is a weak ∗ -free proof fragment with at least one top-sequent not
initial and having S as the bottom sequent. That is, a fragment where
no weak rule were applied. If I is a counter-model for one of its
top-sequents, There is I that is a counter-model for S.



∗[]

Lemma
Given a weak ∗ application with a conclusion S, reading top-down, this
application has two proof fragments with roots S1 and S2 , their
premise and the context that was frozen. If there are interpretations I1
and I2 such that I1 |= S1 and I2 |= S2 then there is I such that I |= S.


A Natural Deduction for ALC

Motivation I

Natural Deduction (ND) proofs in intuitionistic logic (IL) have
computational content: Curry-Howard isomorphism.
Computational content of a proof should provide good structures
to explanation extraction from proofs.
An algorithm is one of the most precise arguments to explain how
to obtain a result out of some inputs.
ND is single-conclusion and provides, in this way, a direct chain of
inferences linking the propositions in the proof.
There is more than one ND normal proof related to the same
cut-free SC proof.
We believe that explanations should be as speciﬁc as their
proof-theoretical counterparts.



The NDALC system

L∀ L∀ L∀ α L∀
(α β) (α β) β
-e -e L∀
-i
L∀ L∀
α β (α β)

∃ ∃
[L α] [L β]
.
. .
.
.
. .
. L∃ α L∃
β
L∃
(α β) γ γ -i -i
-e L∃ L∃
γ (α β) (α β)

Lα
∀R,L α Gen



The NDALC system

[L α] [¬L ¬α]
.
. .
.
.
. .
.
Lα ¬L ¬α ⊥ ⊥ ⊥
⊥
¬-e ¬L ¬α ¬-i Lα c

L L,∃R α L
∃R.α ∀R.α
L,∃R α ∃-e L ∃-i L,∀R α ∀-e
∃R.α

[L1 α]
.
.
.
.
L,∀R α L1 α L1 α L2 L2
β β
L ∀-i L2
-e L1 α L2
-i
∀R.α β β



NDALC semantics

If Φ1 , Φ2 Ψ is an inference rule involving only concept formulas
then it states that whenever the premises are taken as non-empty
collections of individuals the conclusion is taken as non-empty too.
If a is an individual belonging to both interpreted concepts then it
also belongs to the interpreted conclusion.
A subsumption Φ Ψ has no concept associate to it. It states,
instead, a truth-value statement.
In terms of a logical system, DL has no concept internalizing .



NDALC semantics

Deﬁnition
Let Ω = (C, S) be a tuple composed by a set of labeled concepts
C = {α1 , . . . , αn } and a set of subsumption
S = {γ1 γ2 , . . . , γ1 γ2 }. We say that an interpretation I = (∆I , I )
1 1 k k

satisﬁes Ω and write I |= Ω whenever:
1 I |= C, which means α∈C T I (α) = ∅; and
2 i
I |= S, which means that for all γ1 i
γ2 ∈ S, we have
T I (γ1 ) ⊆ T I (γ2 ).
i i



NDALC soundness

Theorem
NDALC is sound regarding the standard semantics of ALC.

if Ω NDALC γ then Ω |= γ

Lemma
Let Π be a deduction in NDALC of F with all hypothesis in Ω = (C, S),
then:
If F is a concept: S |= A∈C A F and
If F is a subsumption A1 A2 : S |= A∈C A A1 A2



NDALC completeness

NDALC is a conservative extension of the classical propositional
calculus.
NDALC has the generalization as a derived rule, and, proves
axiom ∀R.(A B) ≡ (∀R.A ∀R.B), we have the completeness for
NDALC by a relative completeness to the axiomatic presentation of
ALC.



NDALC normalization

Proposition
−
The NDALC -rules and ∃-rules are derived in NDALC .

Lemma (Moving ⊥c downwards on branches)
−
If Ω ND − α, then there is a deduction in NDALC of α from Ω where
ALC
each branch in Π has at most one application of ⊥c -rule and,
whenever it has one, it is the last rule applied in the branch.



NDALC normalization

Lemma (Eliminating maximal -formulas)
−
If Ω ND − α is a deduction in NDALC which contains maximal
ALC
-formulas, that is, maximal formulas with as principal sign, then
−
there is a deduction in NDALC of α from Ω without any occurrence of
maximal -formulas.



NDALC normalization

Theorem (normalization of NDALC )
−
If Ω −
NDALC α, then there is a normal deduction in NDALC of α from Ω.


Towards to a proof theory for ALCQI

Introduction

Some pratical applications require a more expressive DL. For
instance, if we want to formalize and reasoning over ER or UML
diagrams using DL.
A sequent calculus and a natural deduction for ALCQI are
proposed.
Language ALCQI:

C ::= ⊥ | A | ¬C | C1 C2 | C1 C2 | ∃R.C | ∀R.C |
≤ nR.C | ≥ nR.C
R ::= P | P −



ALCQI semantics

(P − )I = {(a, a ) ∈ ∆I × ∆I | (a , a) ∈ P I }
(≤ nR)I = {a ∈ ∆I | |{b | (a, b) ∈ R I }| ≤ n}
(≥ nR)I = {a ∈ ∆I | |{b | (a, b) ∈ R I }| ≥ n}
(≤ nR.C)I = {a ∈ ∆I | |{b | (a, b) ∈ R I ∧ b ∈ C I }| ≤ n}
(≥ nR.C)I = {a ∈ ∆I | |{b | (a, b) ∈ R I ∧ b ∈ C I }| ≥ n}



The system SCALCQI

∆, L,≤nR α ⇒ Γ ∆ ⇒ Γ, L,≤nR α
≤-l ≤-r
∆, L ≤ nR.α ⇒ Γ ∆ ⇒ Γ, L ≤ nR.α

∆, L,≥nR α ⇒ Γ ∆ ⇒ Γ, L,≥nR α
≥-l ≥-r
∆, L ≥ nR.α ⇒ Γ ∆ ⇒ Γ, L ≥ nR.α



The system SCALCQI

∀≥ ∀≥ ∀≤ ∀≤
∆, L α, L β⇒ Γ ∆ ⇒ Γ, L α ∆ ⇒ Γ, L β
∀≥
-l ∀≤
-r
L L
∆, (α β) ⇒ Γ ∆ ⇒ Γ, (α β)
∃≤ ∃≤ ∃≥ ∃≥
∆, L α⇒ Γ ∆, L β⇒ Γ ∆ ⇒ Γ, L α, L β
-l -r
L∃≤ L∃≥
∆, (α β) ⇒ Γ ∆ ⇒ Γ, (α β)
∀∃ ∀∃
∆ ⇒ Γ, ¬L α ∆, ¬L α⇒ Γ
∀∃
¬-l ∀∃
¬-r
∆, L ¬α ⇒ Γ ∆ ⇒ Γ, L ¬α



The system SCALCQI

∆, ≥nR,L α ⇒ Γ ∆⇒ ≥nR,L α, Γ
n≤m shift-≥-l n≥m shift-≥-r
∆, ≥mR,L α ⇒ Γ ∆⇒ ≥mR,L α, Γ

∆, ≤nR,L α ⇒ Γ ∆⇒ ≤nR,L α, Γ
n≥m shift-≤-l n≤m shift-≤-r
∆, ≤mR,L α ⇒ Γ ∆⇒ ≤mR,L α, Γ



The system SCALCQI

∆, ≥1R,L α ⇒ Γ ∆ ⇒ Γ, ≥nR,L α
≥ ∃-l n≥1 ≥ ∃-r
∆, ∃R,L α ⇒ Γ ∆ ⇒ Γ, ∃R,L α

∆, ∃R,L α ⇒ Γ ∆ ⇒ Γ, ∃R,L α
n≥1 ∃ ≥-l ∃ ≥-r
∆, ≥nR,L α ⇒ Γ ∆ ⇒ Γ, ≥1R,L α



The system SCALCQI

∆, ∃R,L1 α ⇒ L2
β, Γ ∀R − ,L2
∆, L1 α ⇒ β, Γ
−
∃-inv inv-∃
∀R L2
∆, L1 α ⇒ ,L2
β, Γ ∆, ∃R,L1 α ⇒ β, Γ

∆⇒ Γ δ⇒ γ
prom-≥ prom-≤
+≥nR
∆ ⇒ +≥nR Γ +≤nR γ ⇒ +≤nR
δ

δ⇒ Γ ∆⇒ γ
prom-∃ prom-∀
+∃R
δ ⇒ +∃R Γ +∀R
∆⇒ +∀R γ



Example
In the proof below, Fem is an abbreviation for Female and child for
hasChild.

Fem ⇒ Fem Male ⇒ Male
∃child
Fem ⇒ ∃child Fem ∃child
Male ⇒ ∃child Male
≥1child
Fem ⇒ ∃child Fem ≥1child
Male ⇒ ∃child Male
≥1child
Fem ⇒ ∃child Male, ∃child Fem ≥1child
Male ⇒ ∃child Male, ∃child Fem
≥1child
Fem ⇒ ∃child (Male Fem) ≥1child
Male ⇒ ∃child (Male Fem)
≥1child ≥1child
Fem ⇒ ∃child.(Male Fem) Male ⇒ ∃child.(Male Fem)
≥ 1child.Fem ⇒ ∃child.(Male Fem) ≥ 1child.Male ⇒ ∃child.(Male Fem)
≥ 1child.Male ≥ 1child.Fem ⇒ ∃child.(Male Fem)



SCALCQI soundness

Theorem (SALCQ is sound)
Considering Ω a set of sequents, a theory presentation or a TBox, let
an Ω-proof be any SALCQ proof in which sequents from Ω are
permitted as initial sequents (in addition to the logical axioms). The
soundness of SALCQ states that if a sequent ∆ ⇒ Γ has an Ω-proof,
then ∆ ⇒ Γ is satisﬁed by every interpretation which satisﬁes Ω. That
is,

if Ω SCALCQI ∆⇒Γ then Ω |= T (δ) T (γ)
δ∈∆ γ∈Γ

for all interpretation I.



SCALCQI soundness

Diagram 1 Diagram 2
≤ nR.(A B) / ≤ nR.A ≥ nR.(A B) o ≥ nR.A
SSS O iSSS
SSS SSS
SSS SSS
SSS SSS
SS) SS
≤ nR.B / (≤ nR.A) (≤ nR.B) ≥ nR.B / (≥ nR.A) (≥ nR.B)

Diagram 3 Diagram 4
≤ nR.(A B) o ≤ nR.A
O ≥ nR.(A B) / ≥ nR.A
O
O iSSS SSS
SSS SSS
SSS SSS
SSS SSS
SS SS)
≤ nR.B o (≤ nR.A) (≤ nR.B) ≥ nR.B o (≥ nR.A) (≥ nR.B)



SCALCQI completeness

The proof of SCALCQI completeness should be obtained following the
same strategy used for SCALC . A deterministic version of SCALCQI
[]
can be designed along the same basic idea used on SCALC .
Afterwards, provision of counter-example from fully expanded trees
that are not proofs must be obtained.



The ND system NDALCQI
L∀≥ L∀≥ L∀≤ α L∀≤
(α β) (α β) β
-e -e L∀≤
-i
L∀≥ L∀≥
α β (α β)

∃≤ ∃≤
[L α] [L β]
.
. .
.
.
. .
. L∃≥ α L∃≥
β
L∃≤
(α β) γ γ -i -i
-e L∃≥ L∃≥
γ (α β) (α β)
∀∃ ∀∃
[L α] [¬L ¬α]
.
. .
.
.
. .
.
L∀∃ α ¬L∀∃ ¬α ⊥ ⊥ ⊥
¬-e ¬L∀∃ ¬α
¬-i L∀∃ α
c
⊥

L L,∃R α L
∃R.α ∀R.α
L,∃R α ∃-e L ∃-i L,∀R α ∀-e
∃R.α


The ND system NDALCQI

∃R,L α ≥nR,L α
≥1R,L α
≥∃ ∃R,L α
∃ ≥ (n ≥ 1)

≥mR,L α ≤mR,L α Lα
≥nR,L α
− ≥ (m ≥ n) ≤nR,L α
+ ≥ (m ≤ n) ∀R,L α Gen

[L1 α]
.
.
.
.
L1 α L1 α L2 L2 ∃R,L1 α L2
β β β
L2
-e L1 α L2
-i L1 α ∀R − ,L2
inv
β β β


Proof Explanation

Conceptual Modelling from a Logical Point of View

1 Observe the “World”.
2 Determine what is relevant.
3 Choose/Define your terminology (non-logical linguistic terms).
4 Write down the main laws governing your “World” (Axioms).
5 Verify the correctness (sometimes completeness too) of your set of Laws.

Steps 1 and 2 may be facilitated by the use of an informal notation (UML, ER, FlowCharts,
etc) and their respective methodology, but it is essentially “Black Art” (cf. Maibaum).
Step 5 full-filling demands quite a lot of knowledge of the Model.
Step 5 essentially provides finitely many tests as support for the correctness of an infinite
quantification.


Proof Explanation

Positives, False Negatives, False Positives

Is anything true about Truth?
M |= φ and Spec(M) φ.
Why is φ truth? Provide me a proof of φ.


Proof Explanation



Is anything wrong with the Truth?
M |= φ, but Spec(M) |= φ.
A counter-model is found. Why is this a counter-model?
Model-Checking based reasoning is of great help!
Explanations from counter-examples.


Proof Explanation



Is anything wrong with the Truth?
M |= φ, but Spec(M) |= φ.
A counter-model is found. Why is this a counter-model?
Model-Checking based reasoning is of great help!
Explanations from counter-examples.

Is anything true about Falsity?
M |= φ, but Spec(M) φ.
Why does this false proposition hold? Provide me a proof of φ.


Proof Explanation

Existing Deductive Systems Paradigms

1 Aristotle’s Syllogisms (300 B.C.)
2 Axiomatic (Frege1879, Hilbert, Russell).
3 Natural Deduction (Jaskowski1929,Gentzen1934-5, Prawitz1965)
4 Sequent Calculus (Gentzen1934-5)
5 Tableaux (Beth 1955, Smullyan1964)
6 Resolution-Based (A.Robinson1965)


Proof Explanation

Fundamental facts on Automating SC and ND

Analyticity
Every proof of Γ α has only occurrences of sub-formulas of Γ
and α (Sub-formula Principle SFP).
Cut-Elimination in SC entails SFP.
Normalization in ND entails SFP.
Strongly related to analytic Tableaux based procedures.


Proof Explanation

Proof Explanation: general ideas

The generation of an explanatory text from a formal proof is still
under investigation by the community.
The use of anaphoras (linguistic reference to an antecedent piece
of text) and cataphoras (linguistic reference to a posterior piece of
text) in producing explanations is a must.
Unstructured nesting of endophoras is hard to follow.
As more structured the proof is, as easier the generation of a
better text, at least concerning the use of endophoras.


Proof Explanation

Example 1/2

Doctor ⇒ Rich, Doctor
-r
Doctor ⇒ (Rich Doctor )
∀child ∀child
prom-∀
Doctor ⇒ (Rich Doctor )
weak-l
, ∀child Doctor ⇒ ∀child (Rich Doctor )
∃child ∀child
¬-r
⇒ ¬Doctor , (Rich Doctor )
weak-r
⇒ ∃child ¬Doctor , ∃child Lawyer , ∀child (Rich Doctor )
∃child ∀child
∃-r
⇒ ¬Doctor , ∃child.Lawyer , (Rich Doctor )
∃-r
⇒ ∃child.¬Doctor , ∃child.Lawyer , ∀child (Rich Doctor )
-r
⇒ (∃child.¬Doctor ) (∃child.Lawyer ), ∀child (Rich Doctor )
∃child
prom-∃
⇒ ∃child ((∃child.¬Doctor ) (∃child.Lawyer )), ∃child,∀child (Rich Doctor )
∃child
¬-l
, ∀child ¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child,∀child (Rich Doctor )
∃child
∀-r
, ∀child ¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child ∀child.(Rich Doctor )
∃child
∀-l
, ∀child.¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child ∀child.(Rich Doctor )
∃child
∃-r
, ∀child.¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child.∀child.(Rich Doctor )
∃child. , ∀child.¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child.∀child.(Rich Doctor )
∃-l
∃child. ∀child.¬((∃child.¬Doctor ) (∃child.Lawyer )) ⇒ ∃child.∀child.(Rich Doctor )
-l


Proof Explanation

Example 2/2

The explanation below was build from top to bottom, by a procedure
that tries to not repeat conjunctive particles:
1 Doctors are Doctors or Rich
2 So, Everyone having all children Doctors has all children Doctors or Rich.
3 Hence, everyone either has at least a child that is not a doctor or every children is a doctor
or rich.
4 Moreover, everyone is of the kind above, or, alternatively, have at least one child that is a
lawyer.
5 In other words, if everyone has at least one child, then it has one child that has at least one
child that is a lawyer, or at least one child that is not a doctor, or have all children doctors or
rich.
6 Thus, whoever has all children not having at least one child not a doctor or at least one
child lawyer has at least one child having every children doctors or rich.


Proof Explanation

Arguments in favour of Natural Deduction as a basis
for theorem explanation

Common Sense and Intuitive reasons
“Fewer” proofs of a proposition when compared to other Deductive
Systems.
“More” structure and existence of speciﬁc patterns to help paragraph
construction in NL.
Working hypothesis: “Optimal explanations should be tailored from
well-known proof patterns”

Technical reasons
Natural Deduction reveals the computational content of a proof.
The prover can choose the pattern it wants the proof should have
(Seldin, Prawitz).


Reasoining UML in NDALCQI

Conceptual Modelling in UML and ER

The Informal Side
Graphical notations seem to be adequate to the human being
understanding and manipulation.
Lacking of a formal consistency checking.

The Logical Side
FOL cannot provide checking of KB consistency.
Decidable logics seems to be more adequate.



Explaining Theorems on the Conceptual Modelling
Domain

A Case Study in UML
1 Why UML? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
2 What do we need?
A Logical Language to express properties and their proofs
(ALCQI)
A Good (Normalizable) Natural Deduction for ALCQI
Proof Patterns that yield good explanation (to come...)



ALCQI KB related to UML Class Diagram
[BerCalvGiac2005]
D. Berardi et al. / Artiﬁcial Intelligence 168 (2005) 70–118 81

Fig. 12. UML class diagram of Example 2.5.

Origin ∀place.String
2.4. Origin constraints
General ∃place. (≤ 1 place)
Origin ∃call.PhoneCall (≤ 1 call) ∃from.Phone (≤ 1 from)
Disjointness and covering constraints (≤ 1call) ∃from.CellPhone used 1 from)
MobileOrigin ∃call.MobileCall are in practice the most commonly (≤ con-
straints in UML class diagrams. However, UML allows for other forms of constraints,
PhoneCall (≥ 1 call− .Origin) (≤ 1 call− .Origin)
specifying class identiﬁers, functional dependencies for associations, and, more generally
−
through the use of ∀reference .PhoneBill ∀reference.PhoneCall
OCL [8], any form of constraint expressible in FOL. Note that, due
−
to their expressive (≥ 1 reference ) could in fact be used to express the semantics
PhoneBill power, OCL constraints
PhoneCall
of the standard UML class diagram constructs. reference)
(≥ 1 reference) (≤ 1 This is an indication that a liberal use of
MobileCall PhoneCall
OCL constraints can actually compromise the understandability of the diagram. Hence,
MobileOrigin Origin
the use of constraints is typically limited. Also, unrestricted use of OCL constraints makes
CellPhone a class diagram undecidable, since it amounts to full FOL reasoning. In the
reasoning on Phone
FixedPhonewe will not consider general constraints.
following, Phone
¬FixedPhone
We conclude the section with an example of a full UML class diagram.
CellPhone
Phone CellPhone FixedPhone
Example 2.5. Fig. 12 shows a complete UML class diagram that models phone calls origi-
nating from different kinds of phones, and phone bills they belong to.13 The diagram shows
that a MobileCall is a particular kind of PhoneCall and that the Origin of each PhoneCall
is one and only one Phone. Additionally, a Phone can be only of two different kinds: a
Alexandre Rademaker ()
FixedPhone or a CellPhone. Proof Theory for Description Logics
On the Mobile calls originate (through the association MobileOrigin)March 30, 2010 46 / 58


Example: A Negative Testing

An (incorrect) generalization (a CellPhone is a FixedPhone) is
introduced in the KB.
CellPhone FixedPhone is added to KB.
CellPhone is empty (inconsistent)

Cell ¬Fixed [Cell]1 Cell Fixed [Cell]1
¬Fixed Fixed
⊥ 1
Cell ⊥



Example: A False Positive in the new KB

In the modiﬁed diagram, Phone ≡ FixedPhone can be drawn. This
is not directly proved from the inconsistency of CellPhone.
It is shown that Phone FixedPhone since FixedPhone Phone
is already an axiom of KB.

[Phone]1 Phone Cell Fixed [Cell] Cell Fixed
Cell Fixed Fixed [Fixed]
Fixed 1
Phone Fixed



Example: A False Positive yielding a reﬁning of KB

MobileCall participates on the association MobileOrigin with
multiplicity 0..1, instead of the 0..* presented in the UML diagram.

[MC]1 MC PC
MO O PC PC ≥ 1 c− .O ≤ 1 c− .O
−
[≥ 2 c .MO] 2
≥ 2 c− .MO ≥ 2 c− .O ≥ 1 c− .O ≤ 1 c− .O
≥ 2 c− .O ≤ 1 c− .O
⊥ 2
¬ ≥ 2 c− .MO
1
MC ¬ ≥ 2 c− .MO


The prototype theorem prover

The language

fmod SYNTAX is
inc NAT .

sorts AConcept Concept ARole Role .
subsort AConcept Concept .
subsort ARole Role .

ops ALL EXIST : Role Concept - Concept .
ops CTRUE CFALSE : - AConcept .
op __ : Concept Concept - Concept [ctor gather (e E) prec 31] .
op _|_ : Concept Concept - Concept [ctor gather (e E) prec 32] .
op ~_ : Concept - Concept [ctor prec 30] .

eq ~ CTRUE = CFALSE .
eq ~ CFALSE = CTRUE .
endfm


The prototype theorem prover

The sequent calculus
fmod SEQUENT-CALCULUS is
inc LALC-SYNTAX .
inc SET{Expression} .
inc SET{Label} .
...
sorts Sequent Goal State Proof .
subsort Goal State Proof .

op next : Nat - State .
op goals : Set{Nat} - State .

op [_from_by_is_] : Nat Nat Qid Sequent - Goal [ctor] .

op nil : - Proof [ctor] .
op __ : Proof Proof - Proof [ctor comm assoc] .

op _|-_ : Set{Expression} Set{Expression} -
Sequent [ctor prec 122 gather(e e)] .
...
endfm


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