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 KRFOL 7−→ Semantic-Network 7−→ Conceptual-Graphs 7−→ DLs


Description Logics

Description Logics

Decidable Fragments of FOLBinary (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 onindividuals (Aboxes).


Description Logics

ALC semantics

Given by: I = (∆I , I)

>I = ∆I

⊥I = ∅(¬C)I = ∆I \ CI

(C u D)I = CI ∩ DI

(C t D)I = CI ∪ DI

(∃R.C)I = a ∈ ∆I | ∃b.(a,b) ∈ RI ∧ b ∈ CI(∀R.C)I = a ∈ ∆I | ∀b.(a,b) ∈ RI → b ∈ CI


Description Logics

A T-Box on Family Relationships

Woman ≡ Person u FemaleMan ≡ Person u ¬Woman

Mother ≡ Woman u ∃hasChild .PersonFather ≡ Man u ∃hasChild .PersonParent ≡ Father tMother

Grandmother ≡ Mother u ∃hasChild .ParentMotherWithoutDaughter ≡ Mother u ∀hasChild .¬Woman

MotherInTrouble ≡ Mother u ≥ 10hasChild


Description Logics

Some definitions

DefinitionThe concept description D subsumes the concept description C,written C v D, if and only if CI ⊆ DI for all interpretations I.

DefinitionC is satisfiable if and only if there exists an interpretation I such thatCI 6= ∅.


Description Logics

Some definitions

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

DefinitionC and D are equivalent, written C ≡ D, if and only if C v D and D v C.


The Sequent Calculus SCALC

An ALC Sequent Calculus: main motivation

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



Labeled Formulas

LB → ∀R | ∃RL→ LB,L | ∅φlc → Lφc

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

∃R2.∀Q2.∃R1.∀Q1.α ≡ ∃R2,∀Q2,∃R1,∀Q1α



SCALC

Structural rules

α⇒ α ⊥ ⇒ α

∆⇒ Γ weak-l∆, δ ⇒ Γ

∆⇒ Γ weak-r∆⇒ Γ, γ

∆, δ, δ ⇒ Γcontraction-l

∆, δ ⇒ Γ

∆⇒ Γ, γ, γcontraction-r

∆⇒ Γ, γ

∆1, δ1, δ2,∆2 ⇒ Γperm-l

∆1, δ2, δ1,∆2 ⇒ Γ

∆⇒ Γ1, γ1, γ2, Γ2 perm-r∆⇒ Γ1, γ2, γ1, Γ2

∆1 ⇒ Γ1,Lα Lα,∆2 ⇒ Γ2 cut

∆1,∆2 ⇒ Γ1, Γ2



SCALC

Rules on role quantification

∆, L,∀Rα⇒ Γ∀-l

∆, L(∀R.α)L2 ⇒ Γ

∆⇒ Γ, L,∀Rα∀-r

∆⇒ Γ, L(∀R.α)

∆, L,∃Rα⇒ Γ∃-l

∆, L(∃R.α)⇒ Γ

∆⇒ Γ, L,∃Rα∃-r

∆⇒ Γ, L(∃R.α)



SCALC

Boolean Rules

∆, ∀Lα, ∀Lβ ⇒ Γu-l

∆, ∀L(α u β)⇒ Γ

∆⇒ Γ, ∀Lα ∆⇒ Γ, ∀Lβu-r

∆⇒ Γ, ∀L(α u β)

∆, ∃Lα⇒ Γ ∆, ∃Lβ ⇒ Γt-l

∆, L(α t β)⇒ Γ

∆⇒ Γ, ∃Lα, ∃Lβt-r

∆⇒ Γ, ∃L(α t β)

∆⇒ Γ, ¬Lα¬-l

∆, L¬α⇒ Γ

∆, ¬Lα⇒ Γ¬-r

∆⇒ Γ, L¬α



SCALC

Generalization rules

δ ⇒ Γ prom-∃+∃Rδ ⇒ +∃RΓ

∆⇒ γprom-∀

+∀R∆⇒ +∀Rγ



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∀-l∃child>, ∀child .¬(∃child .¬Doctor)⇒ ∃child .∀child .Doctor∃-l∃child .>,∀child .¬(∃child .¬Doctor)⇒ ∃child .∀child .Doctoru-l∃child .> u ∀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 beany SCALC proof in which sequents from Ω are permitted as initialsequents (in addition to the logical axioms). The soundness of SCALCstates that if a sequent ∆⇒ Γ has an Ω-proof, then ∆⇒ Γ is satisfiedby every interpretation which satisfies Ω. That is,

if Ω `SALC ∆⇒ Γ then Ω |=l

δ∈∆

T (δ) v⊔γ∈Γ

T (γ)

for all interpretation I .



SCALC completeness

ALC sequent calculus deduction rules without labels behaveexactly as sequent calculus rules for classical propositional logic.The derivation of the rule of necessitation

⇒ α prom-∀⇒ ∀Rα ∀-r⇒ ∀R.α

The axiom∀R.(α u β) ≡ ∀R.α u ∀R.β



SCALC Cut elimination

We follow Gentzen’s original proof for cut elimination. Let δ be alabeled formula. An inference of the following form is called mix withrespect to δ:

∆1 ⇒ Γ1 ∆2 ⇒ Γ2∆1,∆

∗2 ⇒ Γ∗1, Γ2

(δ)




Quasi-mixes

Lα⇒ Γ1 ∆2 ⇒ Γ2∃R,Lα,∆∗2 ⇒

+∃RΓ∗1, Γ2(Lα, ∃R,Lα)

∆1 ⇒ Γ1Lα⇒ Γ2

∆1 ⇒ Γ∗1,+∃RΓ2

(∃R,Lα,Lα)

∆1 ⇒ Lα ∆2 ⇒ Γ2+∀R∆1,∆

∗2 ⇒ Γ2

(Lα, ∀R,Lα)∆1 ⇒ Γ1 ∆2 ⇒ Lα

∆1,+∀R∆∗2 ⇒ Γ∗1,

∀R,Lα(∀R,Lα, Lα)




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

Lemma

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




Definition (SCTALC system)

SCALC was defined with initial sequents of the form α⇒ α with α aALC concept definition (logical axiom). However, it is often convenientto allow for other initial sequents. So if T is a set of sequents of theform ∆⇒ Γ, where ∆ and Γ are sequences of ALC conceptdescriptions (non-logical axioms), we define SCTALC to be the proofsystem defined like SCALC but allowing initial sequents to be from Ttoo.




Definition (Free-quasi-mix free proof)Let P be an SC∗TALC-proof. A formula occurring in P is anchored (by anT -sequent) if it is a direct descendent of a formula occurring in aninitial 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 ofthe mix formulas in the upper sequents is anchored, or (ii) the mixformulas are atomic and both of the occurrences of the mix formulas inthe upper sequents are anchored. A quasi-mix inference which is notanchored is said to be free. A proof P is free-quasi-mix free if itcontains no free quasi-mixes.




Theorem (Free-mix Elimination)

Let T be a set of sequents. If SC∗TALC ` ∆⇒ Γ then there is afree-quasi-mix free SC∗TALC-proof of ∆⇒ Γ.

Lemma

If P is a proof of S (in SC∗TALC) 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 thenumber of quasi-free-mix occurring in a proof P.




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


Comparing SCALC with other deduction systems

The Structural Subsumption algorithm

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

C ≡ A1 u . . . u Am u ∀R1.C1 u . . . u ∀Rn.Cn

D ≡ B1 u . . . u Bk u ∀S1.D1 u . . . u ∀Sl .Dl

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



The Tableaux algorithm

Strongly based on the use of individuals and FOL Tableaux.C v D iff C0 ≡ C u ¬D is unsatisfiable. C0 in negation normalform.Try to construct a finite interpretation I such that CI0 6= ∅.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 proofcorresponds to a step towards this matching by means of the SSA.

A1 ⇒ B1

∀R1.C1,A1 ⇒ B1

A1, ∀R1.C1 ⇒ B1

R1C1 ⇒ S1D1R1C1 ⇒ ∀S1.D1

∀R1.C1 ⇒ ∀S1.D1

A1,∀R1.C1 ⇒ ∀S1.D1

A1,∀R1.C1 ⇒ B1 u ∀S1.D1

A1 u ∀R1.C1 ⇒ B1 u ∀S1.D1



Refining SCALC towards Automatic Theorem Proving

SCALC was extended to S[]ALC 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

∀-l∃child>, ∀child .¬(∃child .¬Doctor)⇒ ∃child .∀child .Doctor∃-l∃child .>, ∀child .¬(∃child .¬Doctor)⇒ ∃child .∀child .Doctoru-l∃child .> u ∀child .¬(∃child .¬Doctor)⇒ ∃child .∀child .Doctor




Example

A counter-model I not only has to guarantee BI 6v AI but alsoAI 6v BI .

B ⇒ A B ⇒ B u-rB ⇒ A u B prom-∃

∃RB ⇒ ∃RA u Bweak-l∃RA, ∃RB ⇒ ∃RA u B∃-r∃RA, ∃RB ⇒ ∃R.A u B∃-l∃RA, ∃R.B ⇒ ∃R.A u B∃-l∃R.A, ∃R.B ⇒ ∃R.A u Bu-l∃R.A u ∃R.B ⇒ ∃R.A u B

A⇒ A A⇒ B u-rA⇒ A u B prom-∃

∃RA⇒ ∃RA u Bweak-l∃RA, ∃RB ⇒ ∃RA u B∃-r∃RA, ∃RB ⇒ ∃R.A u B∃-l∃RA, ∃R.B ⇒ ∃R.A u B∃-l∃R.A, ∃R.B ⇒ ∃R.A u Bu-l∃R.A u ∃R.B ⇒ ∃R.A u B




S[]ALC 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 thecontext.

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




∆, δ ⇒ δ, Γ

[∆, δ]k ,∆⇒ Γ, [Γ]kweak-l

∆, δ ⇒ Γ

[∆]k ,∆⇒ Γ, [Γ, γ]kweak-r

∆⇒ Γ, γ




∆, ∀Lα, ∀Lβ ⇒ Γu-l

∆, ∀L(α u β)⇒ Γ

∆⇒ Γ, ∀Lα ∆⇒ Γ, ∀Lβu-r

∆⇒ Γ, ∀L(α u β)

∆, ∃Lα⇒ Γ ∆, ∃Lβ ⇒ Γt-l

∆, L(α t β)⇒ Γ

∆⇒ Γ, ∃Lα, ∃Lβt-r

∆⇒ Γ, ∃L(α t β)

∆⇒ Γ, ¬Lα¬-l

∆, L¬α⇒ Γ

∆, ¬Lα⇒ Γ¬-r

∆⇒ Γ, L¬α




[∆], Lδ ⇒ Γ, [Γ1]prom-∃

[∆], ∃R,Lδ ⇒ +∃RΓ, [Γ1]

[∆1],∆⇒ Lγ, [Γ]prom-∀

[∆1],+∀R∆⇒ ∀R,Lγ, [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 havingformulas 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 tuplesatisfy ∆⇒ 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 andonly if, no interpretation in the tuple satisfy its corresponding context.



S[]ALC properties

Lemma

Consider ∆⇒ Γ a SCALC sequent. If P is a proof of ∆⇒ Γ in SC[]ALC

then it is possible to construct a proof P ′ of ∆⇒ Γ in SCALC .



S[]ALC properties

A fully expanded proof-tree of ∆⇒ Γ is a tree having ∆⇒ Γ as root,each internal node is a premise of a valid SC[]

ALC rule application, andeach leaf is either a SC[]

ALC axiom (initial sequent) or a top-sequentwhich is not an axiom, not necessarily atomic. In other words, asequent is a top-sequent if and only if it does not contain reduciblecontexts.



S[]ALC properties

If we consider a particular strategy of rules application anyfull-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 weakrules once it is possible to detected that they will not be useful toobtain an initial sequent;

4 Weak rules will be used with the unique purpose of enablingpromotion rules applications.



The weak∗ rule

Π′

[∆, δ1, δ2]k ,∆⇒ Γ, [γ1, γ2, Γ]kweak∗

∆, 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



SC∗[]ALC normal proof

[A]2, [. . .]3, B ⇒ A, [. . .]3, [B]2 [A]2, [. . .]3, B ⇒ B, [. . .]3, [B]2

u-r[A]2, [∃RA, ∃RB]3, B ⇒ A u B, [∃R (A u B)]3, [B]2

prom-∃[A]2, [∃RA, ∃RB]3, ∃RB ⇒ ∃R (A u B), [∃R (A u B)]3, [B]2

weak∗[A]2, ∃RA, ∃RB ⇒ ∃R (A u B), [B]2

f-exch[∃RA, ∃RB]1, A⇒ B, [∃R (A u B)]1 [. . .]1, A⇒ A, [. . .]1

u-r[. . .]1, A⇒ A u B, [. . .]1

prom-∃[∃RA, ∃RB]1, ∃RA⇒ ∃R (A u B), [∃R (A u B)]1

weak∗∃RA, ∃RB ⇒ ∃R (A u B)∃-r∃RA, ∃RB ⇒ ∃R.(A u B)∃-l∃RA, ∃R.B ⇒ ∃R.(A u B)∃-l∃R.A, ∃R.B ⇒ ∃R.(A u B)u-l∃R.A u ∃R.B ⇒ ∃R.(A u B)




Π1 ≡

Π2∃RA, ∃RB ⇒ ∃R(A u B)

∃-r∃RA, ∃RB ⇒ ∃R.(A u B)∃-l∃RA,∃R.B ⇒ ∃R.(A u B)∃-l

∃R.A,∃R.B ⇒ ∃R.(A u B)u-l

∃R.A u ∃R.B ⇒ ∃R.(A u B)




Π2 ≡

Π3

[. . .]1,A⇒ B, [. . .]1 [. . .]1,A⇒ A, [. . .]1u-r

[. . .]1,A⇒ A u B, [. . .]1prom-∃

[∃RA, ∃RB]1, ∃RA⇒ ∃R(A u B), [∃R(A u B)]1

Π3 ≡

[A]2, [. . .]3,B ⇒ A, [. . .]3, [B]2 [A]2, [. . .]3,B ⇒ B, [. . .]3, [B]2u-r

[A]2, [. . .]3,B ⇒ A u B, [. . .]3, [B]2prom-∃

[A]2, [∃RA, ∃RB]3, ∃RB ⇒ ∃R(A u B), [∃R(A u B)]3, [B]2



SC∗[]ALC: obtaining counter-models

Theorem

If P is a full-expanded proof-tree in SC[]ALC with sequent S as root

(conclusion) and if P is in the normal form, from any top-sequent notinitial (non-axiom), one can construct a counter-model for S.

To be fixed!




Lemma

If P is a full-expanded proof-tree in SC∗[]ALC with sequent S as root(conclusion) and if P is in the normal form, from any S1 top-sequentnot 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 notinitial and having S as the bottom sequent. That is, a fragment whereno weak rule were applied. If I is a counter-model for one of itstop-sequents, There is I ′ that is a counter-model for S.




Lemma

Given a weak∗ application with a conclusion S, reading top-down, thisapplication has two proof fragments with roots S1 and S2, theirpremise and the context that was frozen. If there are interpretations I1and I2 such that I1 6|= S1 and I2 6|= S2 then there is I such that I 6|= S.


A Natural Deduction forALC

Motivation I

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



The NDALC system

L∀(α u β)

L∀αu-e

L∀(α u β)

L∀βu-e

L∀α L∀βL∀(α u β)

u-i

L∃(α t β)

[L∃α]....γ

[L∃β]....γ

γ t-e

L∃αL∃(α t β)

t-iL∃β

L∃(α t β)t-i

Lα∀R,Lα

Gen



The NDALC system

Lα ¬L¬α⊥ ¬-e

[Lα]....⊥¬L¬α

¬-i

[¬L¬α]....⊥Lα⊥c

L∃R.αL,∃Rα

∃-eL,∃RαL∃R.α

∃-iL∀R.αL,∀Rα

∀-e

L,∀RαL∀R.α

∀-iL1α L1α v L2β

L2βv -e

[L1α]....L2β

L1α v L2βv -i



NDALC semantics

If Φ1,Φ2 ` Ψ is an inference rule involving only concept formulasthen it states that whenever the premises are taken as non-emptycollections of individuals the conclusion is taken as non-empty too.If a is an individual belonging to both interpreted concepts then italso belongs to the interpreted conclusion.A subsumption Φ v 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 v.



NDALC semantics

DefinitionLet Ω = (C,S) be a tuple composed by a set of labeled conceptsC = α1, . . . , αn and a set of subsumptionS = γ1

1 v γ12 , . . . , γ

k1 v γk

2. We say that an interpretation I = (∆I , I)satisfies Ω and write I |= Ω whenever:

1 I |= C, which means⋂α∈C T I(α) 6= ∅; and

2 I |= S, which means that for all γ i1 v γ i

2 ∈ S, we haveT I(γ i

1) ⊆ T I(γ i2).



NDALC soundness

TheoremNDALC 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 |=

(dA∈C A

)v F and

If F is a subsumption A1 v A2: S |=(d

A∈C A)u A1 v A2



NDALC completeness

NDALC is a conservative extension of the classical propositionalcalculus.NDALC has the generalization as a derived rule, and, provesaxiom ∀R.(A uB) ≡ (∀R.A u ∀R.B), we have the completeness forNDALC by a relative completeness to the axiomatic presentation ofALC.



NDALC normalization

Proposition

The NDALC t-rules and ∃-rules are derived in ND−ALC .

Lemma (Moving ⊥c downwards on branches)

If Ω `ND−ALCα, then there is a deduction in ND−ALC of α from Ω where

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 v-formulas)

If Ω `ND−ALCα is a deduction in ND−ALC which contains maximal

v-formulas, that is, maximal formulas with v as principal sign, thenthere is a deduction in ND−ALC of α from Ω without any occurrence ofmaximal v-formulas.



NDALC normalization

Theorem (normalization of NDALC)

If Ω `ND−ALCα, then there is a normal deduction in ND−ALC of α from Ω.


Towards to a proof theory forALCQI

Introduction

Some pratical applications require a more expressive DL. Forinstance, if we want to formalize and reasoning over ER or UMLdiagrams using DL.A sequent calculus and a natural deduction for ALCQI areproposed.Language ALCQI:

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

R ::= P | P−



The system SCALCQI

∆, L,≤nRα⇒ Γ≤-l

∆, L≤ nR.α⇒ Γ

∆⇒ Γ, L,≤nRα≤-r

∆⇒ Γ, L≤ nR.α

∆, L,≥nRα⇒ Γ≥-l

∆, L≥ nR.α⇒ Γ

∆⇒ Γ, L,≥nRα≥-r

∆⇒ Γ, L≥ nR.α



The system SCALCQI

∆, L∀≥α, L∀≥β ⇒ Γu-l

∆, L∀≥ (α u β)⇒ Γ

∆⇒ Γ, L∀≤α ∆⇒ Γ, L∀≤βu-r

∆⇒ Γ, L∀≤ (α u β)

∆, L∃≤α⇒ Γ ∆, L∃≤β ⇒ Γt-l

∆, L∃≤ (α t β)⇒ Γ

∆⇒ Γ, L∃≥α, L∃≥βt-r

∆⇒ Γ, L∃≥ (α t β)

∆⇒ Γ,¬L∀∃α¬-l

∆, L∀∃¬α⇒ Γ

∆,¬L∀∃α⇒ Γ¬-r

∆⇒ Γ, L∀∃¬α



The system SCALCQI

∆,≥nR,Lα⇒ Γn ≤ m shift-≥-l

∆,≥mR,Lα⇒ Γ

∆⇒ ≥nR,Lα, Γn ≥ m shift-≥-r

∆⇒ ≥mR,Lα, Γ

∆,≤nR,Lα⇒ Γn ≥ m shift-≤-l

∆,≤mR,Lα⇒ Γ

∆⇒ ≤nR,Lα, Γn ≤ m shift-≤-r

∆⇒ ≤mR,Lα, Γ



The system SCALCQI

∆,≥1R,Lα⇒ Γ≥ ∃-l

∆, ∃R,Lα⇒ Γ

∆⇒ Γ,≥nR,Lαn ≥ 1 ≥ ∃-r

∆⇒ Γ, ∃R,Lα

∆, ∃R,Lα⇒ Γn ≥ 1 ∃ ≥-l

∆,≥nR,Lα⇒ Γ

∆⇒ Γ, ∃R,Lα∃ ≥-r

∆⇒ Γ,≥1R,Lα



The system SCALCQI

∆, ∃R,L1α⇒ L2β, Γ∃-inv

∆, L1α⇒ ∀R−,L2β, Γ

∆, L1α⇒ ∀R−,L2β, Γinv-∃

∆, ∃R,L1α⇒ L2β, Γ

∆⇒ Γ prom-≥+≥nR∆⇒ +≥nRΓ

δ ⇒ γprom-≤

+≤nRγ ⇒ +≤nRδ

δ ⇒ Γ prom-∃+∃Rδ ⇒ +∃RΓ

∆⇒ γprom-∀

+∀R∆⇒ +∀Rγ



ExampleIn the proof below, Fem is an abbreviation for Female and child forhasChild.

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

≥1child Fem⇒ ∃child .(Male t Fem)

≥ 1child .Fem⇒ ∃child .(Male t Fem)

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

≥1child Male⇒ ∃child .(Male t Fem)

≥ 1child .Male⇒ ∃child .(Male t Fem)

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



SCALCQI soundness

Theorem (SALCQ is sound)

Considering Ω a set of sequents, a theory presentation or a TBox, letan Ω-proof be any SALCQ proof in which sequents from Ω arepermitted as initial sequents (in addition to the logical axioms). Thesoundness of SALCQ states that if a sequent ∆⇒ Γ has an Ω-proof,then ∆⇒ Γ is satisfied by every interpretation which satisfies Ω. Thatis,

if Ω `SCALCQI ∆⇒ Γ then Ω |=l

δ∈∆

T (δ) v⊔γ∈Γ

T (γ)

for all interpretation I.



SCALCQI soundness

Diagram 1 Diagram 2

≤ nR.(A tB) //

))SSSSSSSSSSSSSS≤ nR.A

≤ nR.B // (≤ nR.A) t (≤ nR.B)

≥ nR.(A tB) ≥ nR.A

oo

≥ nR.B //

OO

(≥ nR.A) t (≥ nR.B)

iiSSSSSSSSSSSSSS

Diagram 3 Diagram 4

≤ nR.(A uB) ≤ nR.Aoo

≤ nR.B

OO

(≤ nR.A) u (≤ nR.B)

OO

oo

iiSSSSSSSSSSSSSS

≥ nR.(A uB) //

))SSSSSSSSSSSSSS≥ nR.A

≥ nR.B (≥ nR.A) u (≥ nR.B)

OO

oo



SCALCQI completeness

The proof of SCALCQI completeness should be obtained following thesame strategy used for SCALC . A deterministic version of SCALCQIcan be designed along the same basic idea used on SC[]

ALC .Afterwards, provision of counter-example from fully expanded treesthat are not proofs must be obtained.



The ND system NDALCQI

L∀≥(α u β)

L∀≥αu-e

L∀≥(α u β)

L∀≥βu-e

L∀≤α L∀≤βL∀≤(α u β)

u-i

L∃≤(α t β)

[L∃≤α]....

γ

[L∃≤β]....

γγ t-e

L∃≥αL∃≥(α t β)

t-iL∃≥β

L∃≥(α t β)t-i

L∀∃α ¬L∀∃¬α⊥ ¬-e

[L∀∃α]....⊥

¬L∀∃¬α¬-i

[¬L∀∃¬α]....⊥

L∀∃α⊥c

L∃R.αL,∃Rα

∃-eL,∃RαL∃R.α

∃-iL∀R.αL,∀Rα

∀-e

L,∀RαL∀R.α

∀-iL≤ nR.α

L,≤nRα≤ -e

L,≤nRαL≤ nR.α

≤ -i

L≥ nR.αL,≥nRα

≥ -eL,≥nRα

L≥ nR.α≥ -i



The ND system NDALCQI

∃R,Lα≥1R,Lα

≥ ∃≥nR,Lα∃R,Lα

∃ ≥ (n ≥ 1)

≥mR,Lα≥nR,Lα

− ≥ (m ≥ n)≤mR,Lα≤nR,Lα

+ ≥ (m ≤ n)Lα∀R,Lα

Gen

L1α L1α v L2βL2β

v -e

[L1α]....L2β

L1α v L2βv -i

∃R,L1α v L2βL1α v ∀R−,L2β

inv


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 infinitequantification.


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) 6|= 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) 6|= 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 6|= 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

AnalyticityEvery proof of Γ ` α has only occurrences of sub-formulas of Γand 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 stillunder investigation by the community.The use of anaphoras (linguistic reference to an antecedent pieceof text) and cataphoras (linguistic reference to a posterior piece oftext) 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 abetter text, at least concerning the use of endophoras.


Proof Explanation

Example 1/2

Doctor ⇒ Rich,Doctort-r

Doctor ⇒ (Rich t Doctor)prom-∀

∀child Doctor ⇒ ∀child (Rich t Doctor)weak-l

>, ∀child Doctor ⇒ ∀child (Rich t Doctor)¬-r

> ⇒ ∃child¬Doctor, ∀child (Rich t Doctor)weak-r

> ⇒ ∃child¬Doctor, ∃child Lawyer, ∀child (Rich t Doctor)∃-r

> ⇒ ∃child¬Doctor, ∃child.Lawyer, ∀child (Rich t Doctor)∃-r

> ⇒ ∃child.¬Doctor, ∃child.Lawyer, ∀child (Rich t Doctor)t-r

> ⇒ (∃child.¬Doctor) t (∃child.Lawyer), ∀child (Rich t Doctor)prom-∃

∃child> ⇒ ∃child ((∃child.¬Doctor) t (∃child.Lawyer)), ∃child,∀child (Rich t Doctor)¬-l∃child>, ∀child¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child,∀child (Rich t Doctor)∀-r∃child>, ∀child¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child∀child.(Rich t Doctor)∀-l∃child>, ∀child.¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child∀child.(Rich t Doctor)∃-r∃child>, ∀child.¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child.∀child.(Rich t Doctor)∃-l∃child.>, ∀child.¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child.∀child.(Rich t Doctor)u-l∃child.> u ∀child.¬((∃child.¬Doctor) t (∃child.Lawyer))⇒ ∃child.∀child.(Rich t Doctor)


Proof Explanation

Example 2/2

The explanation below was build from top to bottom, by a procedurethat tries to not repeat conjunctive particles:

1 Doctors are Doctors or Rich2 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 orrich.

6 Thus, whoever has all children not having at least one child not a doctor or at least onechild lawyer has at least one child having every children doctors or rich.


Proof Explanation

Arguments in favour of Natural Deduction as a basisfor theorem explanation

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

Technical reasonsNatural 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 beingunderstanding 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 ModellingDomain

A Case Study in UML

1 Why UML?⇒ It is complex (UML consistency isEXPTIME-Complete), useful and popular.

2 What do we need?I A Logical Language to express properties and their proofs

(ALCQI)I A Good (Normalizable) Natural Deduction for ALCQII 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.

2.4. General constraints

Disjointness and covering constraints are in practice the most commonly used con-straints in UML class diagrams. However, UML allows for other forms of constraints,specifying class identifiers, functional dependencies for associations, and, more generallythrough the use of OCL [8], any form of constraint expressible in FOL. Note that, dueto their expressive power, OCL constraints could in fact be used to express the semanticsof the standard UML class diagram constructs. This is an indication that a liberal use ofOCL constraints can actually compromise the understandability of the diagram. Hence,the use of constraints is typically limited. Also, unrestricted use of OCL constraints makesreasoning on a class diagram undecidable, since it amounts to full FOL reasoning. In thefollowing, we will not consider general constraints.We conclude the section with an example of a full UML class diagram.

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 showsthat a MobileCall is a particular kind of PhoneCall and that the Origin of each PhoneCallis one and only one Phone. Additionally, a Phone can be only of two different kinds: aFixedPhone or a CellPhone. Mobile calls originate (through the association MobileOrigin)from cell phones. The association MobileOrigin is contained in the binary association Ori-gin: hence MobileOrigin inherits the attribute place of association class Origin. Finally, aPhoneCall is referenced in one and only one PhoneBill, whereas a PhoneBill contains atleast one PhoneCall. In FOL, the diagram is represented as shown in Fig. 13.Notice that, in the above diagram, one would like to express that each MobileCall is

related via the association Origin only to instances of CellPhone. Similarly for the otherdirection of the association. This can be expressed in FOL as follows:

!y1, y2, x. MobileCall(y1) "Origin(x) " call(x, y1) " from(x, y2) # CellPhone(y2)

!y1, y2, x. CellPhone(y2) "Origin(x) " call(x, y1) " from(x, y2) #MobileCall(y1)

The association MobileOrigin approximates this, making it explicit in the diagram that Mo-bileCalls and CellPhones are related to each other.

13 This diagram is based on an example provided with I.COM, a prototype design tool for conceptual modelingwith reasoning support [17].

Origin v ∀place.StringOrigin v ∃place.> u (≤ 1 place)Origin v ∃call.PhoneCall u (≤ 1 call) u ∃from.Phone u (≤ 1 from)

MobileOrigin v ∃call.MobileCall u (≤ 1call) u ∃from.CellPhone u (≤ 1 from)

PhoneCall v (≥ 1 call−.Origin) u (≤ 1 call−.Origin)

> v ∀reference−.PhoneBill u ∀reference.PhoneCallPhoneBill v (≥ 1 reference−)

PhoneCall v (≥ 1 reference) u (≤ 1 reference)MobileCall v PhoneCall

MobileOrigin v OriginCellPhone v Phone

FixedPhone v PhoneCellPhone v ¬FixedPhone

Phone v CellPhone t FixedPhone



Example: A Negative Testing

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

Cell v ¬Fixed [Cell]1

¬FixedCell v Fixed [Cell]1

Fixed⊥

1

Cell v ⊥



Example: A False Positive in the new KB

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

[Phone]1 Phone v Cell t Fixed

Cell t Fixed

[Cell] Cell v Fixed

Fixed [Fixed]

Fixed1

Phone v Fixed



Example: A False Positive yielding a refining of KB

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

[≥ 2 c−.MO]2MO v O

≥ 2 c−.MO v ≥ 2 c−.O

≥ 2 c−.O

[MC]1 MC v PC

PC PC v ≥ 1 c−.O u ≤ 1 c−.O

≥ 1 c−.O u ≤ 1 c−.O

≤ 1 c−.O⊥

2

¬ ≥ 2 c−.MO1

MC v ¬ ≥ 2 c−.MO


The prototype theorem prover

The language

fmod SYNTAX isinc 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 sequent calculus

fmod SEQUENT-CALCULUS isinc LALC-SYNTAX .inc SETExpression .inc SETLabel ....sorts Sequent Goal State Proof .subsort Goal State < Proof .

op next : Nat -> State .op goals : SetNat -> State .

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

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

op _|-_ : SetExpression SetExpression ->Sequent [ctor prec 122 gather(e e)] .

...endfm



A proof

goals(empty) next(7)[0 from 0 by ’init is < nil | ALL(R, A & B) > |-

< nil | ALL(R, A) & ALL(R, B) > ]

[1 from 0 by ’forall-l is < al(R) | A & B > |-< nil | ALL(R, A) & ALL(R, B) > ]

[2 from 1 by ’and-l is < al(R) | A >, < al(R) | B > |-< nil | ALL(R, A) & ALL(R, B) > ]

[3 from 2 by ’and-r is < al(R) | A >, < al(R) | B > |-< nil | ALL(R, A) > ]

[4 from 2 by ’and-r is < al(R) | A >, < al(R) | B > |-< nil | ALL(R, B) > ]

[5 from 3 by ’forall-r is < al(R) | A >, < al(R) | B > |-< al(R) | A > ]

[6 from 4 by ’forall-r is < al(R) | A >, < al(R) | B > |-< al(R) | B > ]


Conclusion and Future work

Contributions

cut-free sequent calculi for ALC and ALCQI;Purely propositional (not based on nominals) proof procedure forpropositional DLs;Structural feature of sequent calculus helps the production ofexplanations.We have also developed ND for ALC and ALCQI. A (cut-free) SCproof is related to more than one ND proof. Explanations shouldbe as specific as their proof-theoretical counterparts.We implemented a theorem prover using our deduction systems.We shown how they can be useful to explain formal facts ontheories obtained from UML models.



Contributions

A Sequent Calculus for ALC [Report PUC-Rio, 2007]A Proof Theory for ALC (Sequent Calculus[EBLRadeHaeuPere2008,2009])A Deterministic Sequent Calculus for ALC [RadeHaeuSBIA2008]Maude Implementations of SC Provers for ALC and ALCQI[Rade2009]A Good Natural Deduction for ALC [RadeHaeu2008-9]A Natural Deduction for ALCQI [JUCS-RadeHaeu2009]



Future work

The extension of the calculi in order to deal with strongerDescription Logics, mainly, SHIQ [?];The development of methods for proof explanation extraction fromproofs;A proof of completeness for NDALCQI and SCALCQI ;The development of constructive versions of NDALC and SCALC ;The cut-rule can provide shorter proofs and may not increase thecomplexity of the explanation (it provides structure);...But is hard to implement;There are approaches to include analytical cuts in Tableaux, as faras we know there is no research on how to extend this to ALCTableaux.


On the proof theory for Description Logics

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