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On the proof theory for Description Logics

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My phd thesis. The slides are about proof theory results for descriptions logics.

My phd thesis. The slides are about proof theory results for descriptions logics.

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• 1. On the Proof Theory for Description Logics Alexandre Rademaker March 30, 2010Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 1 / 58
• 2. Description LogicsDescription LogicsNotations and Formalism for KR FOL −→ Semantic-Network −→ Conceptual-Graphs −→ DLs Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 2 / 58
• 3. Description LogicsDescription LogicsDecidable 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 2 / 58
• 4. Description LogicsALC is the core of DLs Syntax: φc ::= ⊥ | A | ¬φc | φc φc | φc φc | ∃R.φc | ∀R.φc φf ::= φc φc | φc ≡ φc Axiomatic Presentation: (1) from C propositional taut, ∀R.C; (2) ∀R.(A B) ≡ ∀R.A ∀R.B; Interpreting ALC into K modal-logic. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 3 / 58
• 5. Description LogicsALC semanticsGiven 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 } Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 4 / 58
• 6. Description LogicsA 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 5 / 58
• 7. Description LogicsSome deﬁnitionsDeﬁnitionThe concept description D subsumes the concept description C,written C D, if and only if C I ⊆ D I for all interpretations I.DeﬁnitionC is satisﬁable if and only if there exists an interpretation I such thatC I = ∅. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 6 / 58
• 8. Description LogicsSome deﬁnitionsDeﬁnitionC is valid or a tautology if and only if for all interpretation I, C I ≡ ∆I .DeﬁnitionC and D are equivalent, written C ≡ D, if and only if C D and D C. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 6 / 58
• 9. The Sequent Calculus SCALCAn 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 7 / 58
• 10. The Sequent Calculus SCALCLabeled Formulas LB → ∀R | ∃R L → LB, L | ∅ φlc → L φcEach labeled ALC concept has an straightforward ALC conceptequivalent. For example: ∃R2 .∀Q2 .∃R1 .∀Q1 .α ≡ ∃R2 ,∀Q2 ,∃R1 ,∀Q1 α Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 8 / 58
• 11. The Sequent Calculus SCALCSCALCStructural 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 9 / 58
• 12. The Sequent Calculus SCALCSCALCRules 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) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 9 / 58
• 13. The Sequent Calculus SCALCSCALCBoolean 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 9 / 58
• 14. The Sequent Calculus SCALCSCALCGeneralization rules δ⇒ Γ ∆⇒ γ +∃R prom-∃ prom-∀ δ ⇒ +∃R Γ +∀R ∆⇒ +∀R γ Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 9 / 58
• 15. The Sequent Calculus SCALCSCALC 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 10 / 58
• 16. The Sequent Calculus SCALCSCALC soundnessTheorem (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 |= T (δ) T (γ) δ∈∆ γ∈Γfor all interpretation I. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 11 / 58
• 17. The Sequent Calculus SCALCSCALC 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 12 / 58
• 18. The Sequent Calculus SCALCSCALC Cut eliminationWe 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 (δ) ∆1 , ∆∗ ⇒ Γ∗ , Γ2 2 1 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 19. The Sequent Calculus SCALCSCALC Cut eliminationQuasi-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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 20. The Sequent Calculus SCALCSCALC Cut elimination ∗Deﬁnition (The SCALC system) ∗We call SCALC the new system obtained from SCALC by replacing thecut 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 21. The Sequent Calculus SCALCSCALC Cut elimination TDeﬁnition (SCALC 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 concept Tdescriptions (non-logical axioms), we deﬁne SCALC to be the proofsystem deﬁned like SCALC but allowing initial sequents to be from Ttoo. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 22. The Sequent Calculus SCALCSCALC Cut eliminationDeﬁnition (Free-quasi-mix free proof) ∗TLet P be an SCALC -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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 23. The Sequent Calculus SCALCSCALC Cut eliminationTheorem (Free-mix Elimination) ∗TLet T be a set of sequents. If SCALC ∆ ⇒ Γ then there is a ∗Tfree-quasi-mix free SCALC -proof of ∆ ⇒ Γ.Lemma ∗TIf 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 thenumber of quasi-free-mix occurring in a proof P. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 24. The Sequent Calculus SCALCSCALC Cut eliminationWe 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 13 / 58
• 25. Comparing SCALC with other deduction systemsThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 14 / 58
• 26. Comparing SCALC with other deduction systemsThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 15 / 58
• 27. Comparing SCALC with other deduction systemsComparing with SSAEach step taken by a bottom-up construction of a SCALC proofcorresponds 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 16 / 58
• 28. Comparing SCALC with other deduction systemsReﬁning 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 ); Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 29. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem ProvingExampleAn 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 30. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem ProvingExampleA counter-model I not only has to guarantee B I AI but alsoAI 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 31. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem Proving [] 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 32. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem Proving ∆, δ ⇒ δ, Γ [∆, δ]k , ∆ ⇒ Γ, [Γ]k [∆]k , ∆ ⇒ Γ, [Γ, γ]k weak-l weak-r ∆, δ ⇒ Γ ∆ ⇒ Γ, γ Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 33. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem Proving ∆, ∀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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 34. Comparing SCALC with other deduction systemsReﬁning SCALC towards Automatic Theorem Proving [∆], 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 17 / 58
• 35. Comparing SCALC with other deduction systemsSatisfability of frozen labeled sequentsLet ∆ ⇒ 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−1Let (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 ⇒ ΓnThe sequent ∆ ⇒ Γ is not satisﬁable by a tuple of interpretations, if andonly if, no interpretation in the tuple satisfy its corresponding context. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 18 / 58
• 36. Comparing SCALC with other deduction systems []SALC propertiesLemma []Consider ∆ ⇒ Γ a SCALC sequent. If P is a proof of ∆ ⇒ Γ in SCALCthen it is possible to construct a proof P of ∆ ⇒ Γ in SCALC . Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 19 / 58
• 37. Comparing SCALC with other deduction systems []SALC propertiesA 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-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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 19 / 58
• 38. Comparing SCALC with other deduction systems []SALC propertiesIf 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 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 19 / 58
• 39. Comparing SCALC with other deduction systemsThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 20 / 58
• 40. Comparing SCALC with other deduction systems ∗[]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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 21 / 58
• 41. Comparing SCALC with other deduction systems ∗[]SCALC normal proofΠ1 ≡ 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) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 21 / 58
• 42. Comparing SCALC with other deduction systems ∗[]SCALC normal proofΠ2 ≡ 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)]1Π3 ≡ [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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 21 / 58
• 43. Comparing SCALC with other deduction systems ∗[]SCALC : obtaining counter-modelsTheorem []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 notinitial (non-axiom), one can construct a counter-model for S.To be ﬁxed! Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 22 / 58
• 44. Comparing SCALC with other deduction systems ∗[]SCALC : obtaining counter-modelsLemma ∗[]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-sequentnot initial (non-axiom), we can construct a counter-model of S1 .LemmaIf 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 22 / 58
• 45. Comparing SCALC with other deduction systems ∗[]SCALC : obtaining counter-modelsLemmaGiven 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 |= S1 and I2 |= S2 then there is I such that I |= S. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 22 / 58
• 46. A Natural Deduction for ALCMotivation 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 23 / 58
• 47. A Natural Deduction for ALCThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 24 / 58
• 48. A Natural Deduction for ALCThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 24 / 58
• 49. A Natural Deduction for ALCNDALC 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 . Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 25 / 58
• 50. A Natural Deduction for ALCNDALC semanticsDeﬁnitionLet Ω = (C, S) be a tuple composed by a set of labeled conceptsC = {α1 , . . . , αn } and a set of subsumptionS = {γ1 γ2 , . . . , γ1 γ2 }. We say that an interpretation I = (∆I , I ) 1 1 k ksatisﬁes 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 25 / 58
• 51. A Natural Deduction for ALCNDALC soundnessTheoremNDALC is sound regarding the standard semantics of ALC. if Ω NDALC γ then Ω |= γLemmaLet 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 andIf F is a subsumption A1 A2 : S |= A∈C A A1 A2 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 26 / 58
• 52. A Natural Deduction for ALCNDALC 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 27 / 58
• 53. A Natural Deduction for ALCNDALC normalizationProposition −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 ALCeach 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 28 / 58
• 54. A Natural Deduction for ALCNDALC normalizationLemma (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 ofmaximal -formulas. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 28 / 58
• 55. A Natural Deduction for ALCNDALC normalizationTheorem (normalization of NDALC ) −If Ω − NDALC α, then there is a normal deduction in NDALC of α from Ω. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 28 / 58
• 56. Towards to a proof theory for ALCQIIntroduction 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 − Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 29 / 58
• 57. Towards to a proof theory for ALCQIALCQI 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} Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 30 / 58
• 58. Towards to a proof theory for ALCQIThe system SCALCQI LB ::= ∀R | ∃R |≤ nR |≥ nR R ::= P | P − L ::= LB, L | ∅ φlc ::= L φc Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 59. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 60. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 61. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 62. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 63. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 31 / 58
• 64. Towards to a proof theory for ALCQIExampleIn the proof below, Fem is an abbreviation for Female and child forhasChild. 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) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 32 / 58
• 65. Towards to a proof theory for ALCQISCALCQI soundnessTheorem (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 |= T (δ) T (γ) δ∈∆ γ∈Γfor all interpretation I. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 33 / 58
• 66. Towards to a proof theory for ALCQISCALCQI 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) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 33 / 58
• 67. Towards to a proof theory for ALCQISCALCQI completenessThe proof of SCALCQI completeness should be obtained following thesame 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 treesthat are not proofs must be obtained. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 34 / 58
• 68. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 35 / 58
• 69. Towards to a proof theory for ALCQIThe 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 35 / 58
• 70. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 71. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 72. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 73. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 74. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 75. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 76. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 77. Proof ExplanationConceptual 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 36 / 58
• 78. Proof ExplanationPositives, False Negatives, False PositivesIs anything true about Truth? M |= φ and Spec(M) φ. Why is φ truth? Provide me a proof of φ. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 37 / 58
• 79. Proof ExplanationPositives, False Negatives, False PositivesIs anything true about Truth? M |= φ and Spec(M) φ. Why is φ truth? Provide me a proof of φ.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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 37 / 58
• 80. Proof ExplanationPositives, False Negatives, False PositivesIs anything true about Truth? M |= φ and Spec(M) φ. Why is φ truth? Provide me a proof of φ.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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 37 / 58
• 81. Proof ExplanationExisting 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) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 38 / 58
• 82. Proof ExplanationFundamental facts on Automating SC and NDAnalyticity 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 39 / 58
• 83. Proof ExplanationFundamental facts on Automating SC and NDAnalyticity 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 39 / 58
• 84. Proof ExplanationFundamental facts on Automating SC and NDAnalyticity 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 39 / 58
• 85. Proof ExplanationFundamental facts on Automating SC and NDAnalyticity 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 39 / 58
• 86. Proof ExplanationProof 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 40 / 58
• 87. Proof ExplanationExample 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 41 / 58
• 88. Proof ExplanationExample 2/2The explanation below was build from top to bottom, by a procedurethat 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 42 / 58
• 89. Proof ExplanationArguments in favour of Natural Deduction as a basisfor theorem explanationCommon 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 43 / 58
• 90. Proof ExplanationArguments in favour of Natural Deduction as a basisfor theorem explanationCommon 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 43 / 58
• 91. Proof ExplanationArguments in favour of Natural Deduction as a basisfor theorem explanationCommon 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 43 / 58
• 92. Proof ExplanationArguments in favour of Natural Deduction as a basisfor theorem explanationCommon 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). Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 43 / 58
• 93. Reasoining UML in NDALCQIConceptual Modelling in UML and ERThe 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 44 / 58
• 94. Reasoining UML in NDALCQIConceptual Modelling in UML and ERThe 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 44 / 58
• 95. Reasoining UML in NDALCQIConceptual Modelling in UML and ERThe 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 44 / 58
• 96. Reasoining UML in NDALCQIConceptual Modelling in UML and ERThe 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 44 / 58
• 97. Reasoining UML in NDALCQIConceptual Modelling in UML and ERThe 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. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 44 / 58
• 98. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 99. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 100. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 101. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 102. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 103. Reasoining UML in NDALCQIExplaining Theorems on the Conceptual ModellingDomainA 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...) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 45 / 58
• 104. Reasoining UML in NDALCQIALCQI 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
• 105. Reasoining UML in NDALCQIExample: 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 ⊥ Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 47 / 58
• 106. Reasoining UML in NDALCQIExample: 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 48 / 58
• 107. Reasoining UML in NDALCQIExample: A False Positive yielding a reﬁning of KBMobileCall participates on the association MobileOrigin withmultiplicity 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 49 / 58
• 108. The prototype theorem proverThe languagefmod 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 50 / 58
• 109. The prototype theorem proverThe sequent calculusfmod 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 Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 51 / 58
• 110. The prototype theorem proverA proofgoals(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 ] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 52 / 58
• 111. The prototype theorem proverThe rulescrl [and-r] : [ X from Y by Q is ALFA |- GAMMA, L | A B ] next(N) goals((X, XS)) = next(N + 2) goals((XS, N, N + 1)) [ X from Y by Q is ALFA |- GAMMA, L | A B ] [ N from X by ’and-r is ALFA |- GAMMA, L | A ] [ N + 1 from X by ’and-r is ALFA |- GAMMA, L | B ]if not has-ex(L) .crl [prom-exist] : [ X from Y by Q is ex(R) L | A |- GAMMA ] next(N) goals((X, XS)) = next(N + 1) goals((XS, N)) [ X from Y by Q is ex(R) L | A |- GAMMA ] [ N from X by ’prom-exist is L | A |- GAMMA’ ]if all-label(GAMMA, ex(R)) = true / GAMMA’ := remove-label(GAMMA, ex(R), empty) . Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 53 / 58
• 112. The prototype theorem proverSearch for a proofMaude search [1,20] th nil | EXIST(R, CTRUE) ALL(R, ~ EXIST(R, ~ A)) |- nil | EXIST(R, ALL(R, A)) end =! P:Proof goals(empty) .P:Proof -- next(10)[0 from 0 by ’init is nil | EXIST(R, CTRUE) ALL(R, ~ EXIST(R, ~ A)) |- nil | EXIST(R, ALL(R, A)) ][1 from 0 by ’and-l is nil | ALL(R, ~ EXIST(R, ~ A)) , nil | EXIST(R, CTRUE) |- nil | EXIST(R, ALL(R, A)) ][2 from 1 by ’forall-l is nil | EXIST(R, CTRUE) , al(R) | ~ EXIST(R, ~ A) |- nil | EXIST(R, ALL(R, A)) ][3 from 2 by ’neg-l is nil | EXIST(R, CTRUE) |- nil | EXIST(R, ALL(R, A)) , ex(R) | EXIST(R, ~ A) ][4 from 3 by ’exist-r is nil | EXIST(R, CTRUE) |- ex(R) | ALL(R, A) , ex(R) | EXIST(R, ~ A) ][5 from 4 by ’forall-r is nil | EXIST(R, CTRUE) |- ex(R) | EXIST(R, ~ A) , ex(R) al(R) | A ][6 from 5 by ’exist-r is nil | EXIST(R, CTRUE) |- ex(R) ex(R) | ~ A , ex(R) al(R) | A ][7 from 6 by ’exist-l is ex(R) | CTRUE |- ex(R) ex(R) | ~ A , ex(R) al(R) | A ][8 from 7 by ’prom-exist is nil | CTRUE |- ex(R) | ~ A , al(R) | A ][9 from 8 by ’neg-r is nil | CTRUE , al(R) | A |- al(R) | A ] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 54 / 58
• 113. The prototype theorem proverStrategies(smod BACKTRACKING-STRAT is strat solve : @ Proof . strat expand : @ Proof . var P : Proof . sd expand := (((try(prom-exist | prom-all) ; (and-l | and-r | or-l | or-r | forall-l | forall-r | exist-l | exist-r | neg-l | neg-r)) orelse (weak-l | weak-r)) orelse frozen-exchange) . sd solve := if (match P s.t. (is-solution(P))) then idle else expand ; if (match P s.t. (is-ok(P))) then solve else idle fi fi .endsm) Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 55 / 58
• 114. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 115. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 116. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 117. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 118. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 119. Conclusion and Future workContributions cut-free sequent calculi for ALC and ALCQI; Purely propositional (not based on nominals) proof procedure for propositional DLs; Structural feature of sequent calculus helps the production of explanations. We have also developed ND for ALC and ALCQI. A (cut-free) SC proof is related to more than one ND proof. Explanations should be 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 on theories obtained from UML models. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 56 / 58
• 120. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 121. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 122. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 123. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 124. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 125. Conclusion and Future workContributions 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] Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 57 / 58
• 126. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 127. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 128. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 129. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 130. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 131. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58
• 132. Conclusion and Future workFuture work The extension of the calculi in order to deal with stronger Description Logics, mainly, SHIQ [?]; The development of methods for proof explanation extraction from proofs; 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 the complexity of the explanation (it provides structure); ...But is hard to implement; There are approaches to include analytical cuts in Tableaux, as far as we know there is no research on how to extend this to ALC Tableaux. Alexandre Rademaker () On the Proof Theory for Description Logics March 30, 2010 58 / 58