1.
Is It Important to Explain a Theorem?
A Case Study on UML and ALCQI
Edward Hermann Haeusler Alexandre Rademaker
Departamento de Informática - PUC-Rio - Brasil
Ethecom 2009
2.
Conceptual Modelling from a Logical Point of View
Main Steps
Additional Observations
3.
Conceptual Modelling from a Logical Point of View
Main Steps
1. Observe the “World”.
Additional Observations
4.
Conceptual Modelling from a Logical Point of View
Main Steps
1. Observe the “World”.
2. Determine what is relevant.
Additional Observations
5.
Conceptual Modelling from a Logical Point of View
Main Steps
1. Observe the “World”.
2. Determine what is relevant.
3. Choose/Deﬁne your terminology (non-logical linguistic terms).
Additional Observations
6.
Conceptual Modelling from a Logical Point of View
Main Steps
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).
Additional Observations
7.
Conceptual Modelling from a Logical Point of View
Main Steps
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.
Additional Observations
8.
Conceptual Modelling from a Logical Point of View
Main Steps
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.
Additional Observations
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).
9.
Conceptual Modelling from a Logical Point of View
Main Steps
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.
Additional Observations
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.
10.
Conceptual Modelling from a Logical Point of View
Main Steps
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.
Additional Observations
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.
11.
The Validation Cycle
Figure: Reﬁnements and Cascaded Validation
12.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
13.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
Results/analysis of the philosophy of science are compared to
software validation [Haeberer98, Maibaum01, Cengarle98, C.
George05, etc].
14.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
Results/analysis of the philosophy of science are compared to
software validation [Haeberer98, Maibaum01, Cengarle98, C.
George05, etc].
Formal Speciﬁcations as Scientiﬁc Theories ⇒ Observable
terms, Theoretical terms, Evidences, Refutations, False
Negatives, False positives, etc.
15.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
Results/analysis of the philosophy of science are compared to
software validation [Haeberer98, Maibaum01, Cengarle98, C.
George05, etc].
Formal Speciﬁcations as Scientiﬁc Theories ⇒ Observable
terms, Theoretical terms, Evidences, Refutations, False
Negatives, False positives, etc.
Popper’s Falseability Principle drives (formal) validation analysis.
16.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
Results/analysis of the philosophy of science are compared to
software validation [Haeberer98, Maibaum01, Cengarle98, C.
George05, etc].
Formal Speciﬁcations as Scientiﬁc Theories ⇒ Observable
terms, Theoretical terms, Evidences, Refutations, False
Negatives, False positives, etc.
Popper’s Falseability Principle drives (formal) validation analysis.
Correctness ⇔ Positives and False Positives.
17.
Validation of (Formal?) Speciﬁcations
The Scientiﬁc Basis of our approach
Results/analysis of the philosophy of science are compared to
software validation [Haeberer98, Maibaum01, Cengarle98, C.
George05, etc].
Formal Speciﬁcations as Scientiﬁc Theories ⇒ Observable
terms, Theoretical terms, Evidences, Refutations, False
Negatives, False positives, etc.
Popper’s Falseability Principle drives (formal) validation analysis.
Correctness ⇔ Positives and False Positives.
Completeness ⇔ Negatives and False Negatives.
18.
Positives, False Negatives, False Positives
Is anything true about Truth ??
Is anything wrong with the Truth ??
Is anything true about Falsity ??
19.
Positives, False Negatives, False Positives
Is anything true about Truth ??
M |= φ and Spec(M) φ.
Is anything wrong with the Truth ??
Is anything true about Falsity ??
20.
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.
Is anything wrong with the Truth ??
Is anything true about Falsity ??
21.
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.
Is anything wrong with the Truth ??
M |= φ, but Spec(M) |= φ.
Is anything true about Falsity ??
22.
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.
Is anything wrong with the Truth ??
M |= φ, but Spec(M) |= φ.
A counter-model is found. Why is this a counter-model ??
Is anything true about Falsity ??
23.
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.
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 !!
Is anything true about Falsity ??
24.
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.
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 ??
25.
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.
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.
26.
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.
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.
28.
Conceptual Modelling: Some motivation on explaining a theorem
Consider an ontology/KB containing:
(Quad ∧ PissOnFireHydrant) → Dog
29.
Conceptual Modelling: Some motivation on explaining a theorem
Consider an ontology/KB containing:
(Quad ∧ PissOnFireHydrant) → Dog
This KB draws
(Quad → Dog) ∨ (PissOnFireHidrant → Dog)
30.
Conceptual Modelling: Some motivation on explaining a theorem
Verifying this using Tableaux: V Quad ∧ PoFH → Dog
F (Quad → Dog) ∨ (PoFH → Dog)
F (Quad → Dog)
F PoFH → Dog
V Quad
F Dog
V PoFH
F Dog
F Quad ∧ PoFH V Dog
F Quad F PoFH
31.
Conceptual Modelling: Some motivation on explaining a theorem
Another tableaux proof of Quad ∧ PoFH → Dog (Quad → Dog) ∨ (PoFH → Dog):
V Quad ∧ PoFH → Dog
F Quad ∧ PoFH V Dog
F (Quad → Dog) ∨ (PoFH → Dog) F (Quad → Dog) ∨ (PoFH → Dog)
F (Quad → Dog) F (Quad → Dog)
F PoFH → Dog F PoFH → Dog
V Quad V Quad
F Dog F Dog
V PoFH
F Dog
F Quad F PoFH
32.
Conceptual Modelling: Some motivation on explaining a theorem
One more tableaux proof of Quad ∧ PoFH → Dog (Quad → Dog) ∨ (PoFH → Dog):
V Quad ∧ PoFH → Dog
F Quad ∧ PoFH V Dog
F (Quad → Dog) ∨ (PoFH → Dog)
F (Quad → Dog)
F Quad F PoFH
F PoFH → Dog
F (Quad → Dog) ∨ (PoFH → Dog) F (Quad → Dog) ∨ (PoFH → Dog)
V Quad
F (Quad → Dog) F (Quad → Dog)
F Dog
F PoFH → Dog F PoFH → Dog
V Quad V PoFH
F Dog F Dog
33.
Conceptual Modelling: Some motivation on explaining a theorem
Yet another Tableaux: V Quad ∧ PoFH → Dog and many more.....
F (Quad → Dog) ∨ (PoFH → Dog)
F (Quad → Dog)
F PoFH → Dog
F Quad ∧ PoFH V Dog
V Quad
F Quad F PoFH
V Quad F Dog
V Quad
F Dog
F Dog
V PoFH
F Dog
34.
In Sequent Calculus
A proof that KB (Quad → Dog) ∨ (PoFH → Dog)
Quad ⇒ Quad PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad Quad, PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad ∧ PoFH Dog ⇒ Dog
Quad, PoFH, PoFH ∧ Quad → Dog ⇒ Dog
Quad, PoFH, PoFH ∧ Quad → Dog ⇒ Dog, Dog
KB ⇒ PoFH ∧ Quad → Dog PoFH, PoFH ∧ Quad → Dog ⇒ (Quad → Dog), Dog
PoFH, KB ⇒ (Quad → Dog), Dog
KB ⇒ (Quad → Dog), (PoFH → Dog)
KB ⇒ (Quad → Dog) ∨ (PoFH → Dog)
35.
In Sequent Calculus
Other proof that KB (Quad → Dog) ∨ (PoFH → Dog)
Quad ⇒ Quad PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad Quad, PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad ∧ PoFH Dog ⇒ Dog
KB ⇒ PoFH ∧ Quad → Dog Quad, PoFH, PoFH ∧ Quad → Dog ⇒ Dog
KB, Quad, PoFH ⇒ Dog
KB, Quad, PoFH ⇒ Dog, Dog
KB, PoFH ⇒ (Quad → Dog), Dog
KB ⇒ (Quad → Dog), (PoFH → Dog)
KB ⇒ (Quad → Dog) ∨ (PoFH → Dog)
36.
In Sequent Calculus
One more proof that KB (Quad → Dog) ∨ (PoFH → Dog)
Quad ⇒ Quad PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad Quad, PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad ∧ PoFH Dog ⇒ Dog
KB ⇒ PoFH ∧ Quad → Dog Quad, PoFH, PoFH ∧ Quad → Dog ⇒ Dog
KB, Quad, PoFH ⇒ Dog
KB, PoFH ⇒ (Quad → Dog)
KB, PoFH ⇒ (Quad → Dog), Dog
KB ⇒ (Quad → Dog), (PoFH → Dog)
KB ⇒ (Quad → Dog) ∨ (PoFH → Dog)
37.
In Sequent Calculus
Yet another proof that KB (Quad → Dog) ∨ (PoFH → Dog)
Quad ⇒ Quad PoFH ⇒ PoFH
Quad, PoFH ⇒ Quad Quad, PoFH ⇒ PoFH Dog ⇒ Dog
Quad, PoFH ⇒ Quad ∧ PoFH Dog ⇒ Dog, Dog
KB ⇒ PoFH ∧ Quad → Dog Quad, PoFH, PoFH ∧ Quad → Dog ⇒ Dog, Dog
KB, Quad, PoFH ⇒ Dog, Dog
KB, PoFH ⇒ (Quad → Dog), Dog
KB, PoFH ⇒ (Quad → Dog), Dog
KB ⇒ (Quad → Dog), (PoFH → Dog)
KB ⇒ (Quad → Dog) ∨ (PoFH → Dog)
and many more...
38.
In Natural Deduction
A (normal) proof
[Quad]a [PoFH]b
Quad ∧ PoFH Quad ∧ PoFH → Dog
Dog
b
PoFH → Dog
(Quad → Dog) ∨ (PoFH → Dog) [¬((Quad → Dog) ∨ (PoFH → Dog))]c
⊥
a
[Quad]d ¬Quad
⊥
Dog
d
Quad → Dog
(Quad → Dog) ∨ (PoFH → Dog) [¬((Quad → Dog) ∨ (PoFH → Dog))]c
⊥
c
(Quad → Dog) ∨ (PoFH → Dog)
39.
In Natural Deduction
THE other (normal) proof
[Quad]a [PoFH]b
Quad ∧ PoFH Quad ∧ PoFH → Dog
Dog
b
PoFH → Dog
(Quad → Dog) ∨ (PoFH → Dog) [¬((Quad → Dog) ∨ (PoFH → Dog))]c
⊥
Dog
a
Quad → Dog
(Quad → Dog) ∨ (PoFH → Dog) [¬((Quad → Dog) ∨ (PoFH → Dog))]c
⊥
c
(Quad → Dog) ∨ (PoFH → Dog)
40.
Fundamental facts on Automating S.C. and N.D.
Analyticity
Every proof of Γ α has only occurrences of sub-formulas
of Γ and α (Sub-formula Principle SFP).
41.
Fundamental facts on Automating S.C. and N.D.
Analyticity
Every proof of Γ α has only occurrences of sub-formulas
of Γ and α (Sub-formula Principle SFP).
Cut-Elimination in S.C entails SFP. Haupsatz
42.
Fundamental facts on Automating S.C. and N.D.
Analyticity
Every proof of Γ α has only occurrences of sub-formulas
of Γ and α (Sub-formula Principle SFP).
Cut-Elimination in S.C entails SFP. Haupsatz
Normalization in N.D. entails SFP. Normalization
43.
Fundamental facts on Automating S.C. and N.D.
Analyticity
Every proof of Γ α has only occurrences of sub-formulas
of Γ and α (Sub-formula Principle SFP).
Cut-Elimination in S.C entails SFP. Haupsatz
Normalization in N.D. entails SFP. Normalization
Strongly related to analytic Tableaux based procedures.
44.
Arguments in favour of Natural Deduction as a basis for theorem explanation
Common Sense and Intuitive reasons
Technical reasons
45.
Arguments in favour of Natural Deduction as a basis for theorem explanation
Common Sense and Intuitive reasons
“Fewer” proofs of a proposition when compared to other
Deductive Systems.
Technical reasons
Natural Deduction reveals the computational content of a
proof. CH-Isomorphism
46.
Arguments in favour of Natural Deduction as a basis for theorem explanation
Common Sense and Intuitive reasons
“Fewer” proofs of a proposition when compared to other
Deductive Systems.
“More” structure and existence of speciﬁc patterns to help
paragraph construction in NL.
Technical reasons
Natural Deduction reveals the computational content of a
proof. CH-Isomorphism
The prover can choose the pattern it wants the proof
should have. Seldin Prawitz
47.
Arguments in favour of Natural Deduction as a basis for theorem explanation
Common Sense and Intuitive reasons
“Fewer” proofs of a proposition when compared to other
Deductive Systems.
“More” structure and existence of speciﬁc patterns to help
paragraph construction in NL.
Working hypothesis: “Optimal explanations should be
tailored from well-known proof patterns”
Technical reasons
Natural Deduction reveals the computational content of a
proof. CH-Isomorphism
The prover can choose the pattern it wants the proof
should have. Seldin Prawitz
48.
Conceptual Modelling in UML and ER
The Informal Side
The Logical Side
49.
Conceptual Modelling in UML and ER
The Informal Side
Graphical notations seem to be adequate to the human
being understanding and manipulation.
The Logical Side
50.
Conceptual Modelling in UML and ER
The Informal Side
Graphical notations seem to be adequate to the human
being understanding and manipulation.
Lacking of a formal consistency checking.
The Logical Side
51.
Conceptual Modelling in UML and ER
The Informal Side
Graphical notations seem to be adequate to the human
being understanding and manipulation.
Lacking of a formal consistency checking.
The Logical Side
FOL cannot provide checking of KB consistency.
52.
Conceptual Modelling in UML and ER
The Informal Side
Graphical notations seem to be adequate to the human
being understanding and manipulation.
Lacking of a formal consistency checking.
The Logical Side
FOL cannot provide checking of KB consistency.
Decidable logics seems to be more adequate.
53.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
54.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
1. Why UML ? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
55.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
1. Why UML ? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
2. What do we need ?
56.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
1. Why UML ? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
2. What do we need ?
A Logical Language to express properties and their proofs
(ALCQI)
57.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
1. Why UML ? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
2. What do we need ?
A Logical Language to express properties and their proofs
(ALCQI)
A Good (Normalizable) Natural Deduction for ALCQI
58.
Explaining Theorems on the Conceptual Modelling Domain
A Case Study in UML
1. Why UML ? ⇒ It is complex (UML consistency is
EXPTIME-Complete), useful and popular.
2. What do we need ?
A Logical Language to express properties and their proofs
(ALCQI)
A Good (Normalizable) Natural Deduction for ALCQI
Proof Patterns that yield good explanation (to come...)
59.
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
Origin ∀place.String
Origin ∃place. (≤ 1 place)
Origin ∃call.PhoneCall (≤ 1 call) ∃from.Phone (≤ 1 from)
Disjointness and covering constraints are 1call) ∃from.CellPhone commonly used con-
MobileOrigin ∃call.MobileCall (≤ in practice the most (≤ 1 from)
straints in UML class diagrams. However, UML allows for other forms of constraints,
PhoneCall (≥ 1 call− .Origin) (≤ 1 call− .Origin)
−
specifying class identiﬁers, ∀reference .PhoneBill ∀reference.PhoneCall
functional dependencies for associations, and, more generally
PhoneBill (≥ 1 reference− )
through the use of OCL [8], any form of (≤ 1 reference)
PhoneCall (≥ 1 reference) constraint expressible in FOL. Note that, due
to their expressive power, OCL constraints could in fact be used to express the semantics
MobileCall PhoneCall
of the standard UML class Origin
MobileOrigin
CellPhone
diagram constructs. This is an indication that a liberal use of
Phone
OCL constraints can actually compromise the understandability of the diagram. Hence,
FixedPhone Phone
CellPhone ¬FixedPhone
the use of constraints is typically limited. Also, unrestricted use of OCL constraints makes
Phone CellPhone FixedPhone
reasoning on a class diagram undecidable, since it amounts to full FOL reasoning. In the
following, we will not consider general constraints.
60.
Towards a Natural Deduction for ALCQI
A Sequent Calculus for ALC (EDOC2007, AOW2007, etc)
A Proof Theory for ALC (Sequent Calculus
[RadeHaeuPere2008,2009])
A Deterministic Sequent Calculus for ALC
[RadeHaeuSBIA2008]
Maude Implementations of S.C. Provers for ALC and ALCQI
[Rade2009]
A Good Natural Deduction for ALC [RadeHaeu2008-9]
A Natural Deduction for ALCQI [RadeHaeu2009]
61.
ALC, ALCQI and further DLs
ALC
C ::= ⊥ | | A | ¬C | C1 C2 | C1 C2 | ∃R.C | ∀R.C
ALCQI
C ::= ⊥ | A | ¬C | C1 C2 | C1 C2 | ∃R.C | ∀R.C |≤ nR.C |≥ nR.C
R ::= P | P −
UML with OCL constraints
SecureUML needs ID(C) role for each concept C for specifying
uniqueness of a default in a concept. [BragaHaeu2009]
∀ID( ).A ∃ID( ).A A ≡ (= 1isdefault.(= 1isdefault − .Role))
62.
Labeling formulas of ALC
Labeling Grammar:
LL ::= R, LL | ∅
LR ::= R, LR | R(LL ), LR | ∅
C ::= LL C LR
The ALC formula:
∃R2 .∀Q2 .∃R1 .∀Q1 .α
is represented by the labeled formula:
Q2 ,Q1
αR1 (Q2 ),R2
65.
Main properties of NALC
Theorem
NALCQI is complete regarding the standard semantics of ALC.
Theorem
NALCQI is sound regarding the standard semantics of ALC.
if Ω γ then Ω |= γ.
Fact
The NALCQI -rules and ∃-rules are derived in ALCQI − { , ∃} .
Lemma (Moving ⊥c downwards on branches)
If Ω α in ALCQI − { , ∃} then there is a deduction Π of α from Ω,
such that, each branch in Π has at most one application of ⊥c -rule,
which is the last rule in it.
Theorem (Eliminating maximal -formulas)
If Π is a deduction of α from Ω, in the restricted system, then
reductions
there is a deduction Π of α from Ω without any maximal formulas.
Fact
SFP holds in ALC.
66.
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
Origin ∀place.String
Origin ∃place. (≤ 1 place)
Origin ∃call.PhoneCall (≤ 1 call) ∃from.Phone (≤ 1 from)
Disjointness and covering constraints are 1call) ∃from.CellPhone commonly used con-
MobileOrigin ∃call.MobileCall (≤ in practice the most (≤ 1 from)
straints in UML class diagrams. However, UML allows for other forms of constraints,
PhoneCall (≥ 1 call− .Origin) (≤ 1 call− .Origin)
−
specifying class identiﬁers, ∀reference .PhoneBill ∀reference.PhoneCall
functional dependencies for associations, and, more generally
PhoneBill (≥ 1 reference− )
through the use of OCL [8], any form of (≤ 1 reference)
PhoneCall (≥ 1 reference) constraint expressible in FOL. Note that, due
to their expressive power, OCL constraints could in fact be used to express the semantics
MobileCall PhoneCall
of the standard UML class Origin
MobileOrigin
CellPhone
diagram constructs. This is an indication that a liberal use of
Phone
OCL constraints can actually compromise the understandability of the diagram. Hence,
FixedPhone Phone
CellPhone ¬FixedPhone
the use of constraints is typically limited. Also, unrestricted use of OCL constraints makes
Phone CellPhone FixedPhone
reasoning on a class diagram undecidable, since it amounts to full FOL reasoning. In the
following, we will not consider general constraints.
67.
Example : A Negative Testing
An (incorrect) generalization (a CellPhone is a
FixedPhone) is introduced in the KB.
68.
Example : A Negative Testing
An (incorrect) generalization (a CellPhone is a
FixedPhone) is introduced in the KB.
CellPhone FixedPhone is added to KB.
69.
Example : A Negative Testing
An (incorrect) generalization (a CellPhone is a
FixedPhone) is introduced in the KB.
CellPhone FixedPhone is added to KB.
CellPhone is empty (inconsistent)
.
Cell ¬Fixed [Cell]1 Cell Fixed [Cell]1
¬Fixed Fixed
⊥ 1
Cell ⊥
70.
Example: A False Positive in the new KB
In the modiﬁed diagram, Phone ≡ FixedPhone can be drawn.
This is not directly proved from the inconsistency of CellPhone.
71.
Example: A False Positive in the new KB
In the modiﬁed diagram, Phone ≡ FixedPhone can be drawn.
This is not directly proved from the inconsistency of CellPhone.
It is shown that Phone FixedPhone since
FixedPhone Phone is already an axiom of KB.
72.
Example: A False Positive in the new KB
In the modiﬁed diagram, Phone ≡ FixedPhone can be drawn.
This is not directly proved from the inconsistency of CellPhone.
It is shown that Phone FixedPhone since
FixedPhone Phone is already an axiom of KB.
Proof:
[Phone]1 Phone Cell Fixed [Cell] Cell Fixed
Cell Fixed Fixed [Fixed]
Fixed
1
Phone Fixed
73.
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
Origin ∀place.String
Origin ∃place. (≤ 1 place)
Origin ∃call.PhoneCall (≤ 1 call) ∃from.Phone (≤ 1 from)
Disjointness and covering constraints are 1call) ∃from.CellPhone commonly used con-
MobileOrigin ∃call.MobileCall (≤ in practice the most (≤ 1 from)
straints in UML class diagrams. However, UML allows for other forms of constraints,
PhoneCall (≥ 1 call− .Origin) (≤ 1 call− .Origin)
−
specifying class identiﬁers, ∀reference .PhoneBill ∀reference.PhoneCall
functional dependencies for associations, and, more generally
PhoneBill (≥ 1 reference− )
through the use of OCL [8], any form of (≤ 1 reference)
PhoneCall (≥ 1 reference) constraint expressible in FOL. Note that, due
to their expressive power, OCL constraints could in fact be used to express the semantics
MobileCall PhoneCall
of the standard UML class Origin
MobileOrigin
CellPhone
diagram constructs. This is an indication that a liberal use of
Phone
OCL constraints can actually compromise the understandability of the diagram. Hence,
FixedPhone Phone
CellPhone ¬FixedPhone
the use of constraints is typically limited. Also, unrestricted use of OCL constraints makes
Phone CellPhone FixedPhone
reasoning on a class diagram undecidable, since it amounts to full FOL reasoning. In the
following, we will not consider general constraints.
75.
Example: A False Positive yielding a reﬁning of KB
76.
Example: A False Positive yielding a reﬁning of KB
MobileCall participates on the association MobileOrigin
with multiplicity 0..1, instead of the 0..* presented in the UML
diagram
77.
Example: A False Positive yielding a reﬁning of KB
MobileCall participates on the association MobileOrigin
with multiplicity 0..1, instead of the 0..* presented in the UML
diagram
Proof: [MC]1 MC PC
MO O PC PC ≥ 1 c− .O ≤ 1 c− .O
− 2 − − − −
[≥ 2 c .MO] ≥ 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
Sequent
78.
Conclusions
Yes !! It is Important to explain a theorem !!!
Advices
79.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Advices
80.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
Advices
81.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
N.D. for DLs is an important step towards good explanations in
Conceptual Modeling. NALCQI provides a good basis regarding
UML and ER reasoning explanation.
Advices
82.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
N.D. for DLs is an important step towards good explanations in
Conceptual Modeling. NALCQI provides a good basis regarding
UML and ER reasoning explanation.
Advices
83.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
N.D. for DLs is an important step towards good explanations in
Conceptual Modeling. NALCQI provides a good basis regarding
UML and ER reasoning explanation.
Advices
Conceptual Modeling in UML is not tractable
(EXPTIME-complete)
84.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
N.D. for DLs is an important step towards good explanations in
Conceptual Modeling. NALCQI provides a good basis regarding
UML and ER reasoning explanation.
Advices
Conceptual Modeling in UML is not tractable
(EXPTIME-complete)
Unless CoNP = NP, proofs can be really huge !!! Introducing
Cuts/Maximal formulas cannot reduce always the size of a proof.
85.
Conclusions
Yes !! It is Important to explain a theorem !!!
Proof explanations provide good and adequate support for
formal validation of KB. It is as important as Model Checking
based explanations.
Under our Working Hyp., N.D. provides the adequate basis for
explanation generation from formal proofs.
N.D. for DLs is an important step towards good explanations in
Conceptual Modeling. NALCQI provides a good basis regarding
UML and ER reasoning explanation.
Advices
Conceptual Modeling in UML is not tractable
(EXPTIME-complete)
Unless CoNP = NP, proofs can be really huge !!! Introducing
Cuts/Maximal formulas cannot reduce always the size of a proof.
86.
Curry-Howard Isomorphism
The computational content of Intuitionistic Proofs
Any Proof of α from γ1 , . . . , γn in IL corresponds to an
algorithm that yields values of type α from any list of n values of
types γ1 , . . . , γn , respectively
IntuitionisticLogic
Technically:
Any proof π of α from γ1 , . . . , γn corresponds to a typed λ-term
t(x1 , . . . , xn ) : α[x1 : γ1 , . . . , xn : γn ], such that any evaluation in t
corresponds a normalization step in π, and vice-versa.
return
87.
Seldin’s strategy to normalize Classical Proofs
Moving the Classical Absurdity Rule towards the Conclusion of the
proof
Given any Classical derivation Π of α from Γ, one can transform Π
into a derivation Π1 of α from Γ of the following form:
Γ, [¬α]a
Π1
a ⊥ α
where Π1 is intuitionistic. reductions
return
88.
Prawitz’s strategy to normalize Classical Proofs
Moving the Classical Absurdity Rule towards atomic conclusions in
the proof
Given any Classical derivation Π of α from Γ, one can transform Π
into a derivation Π1 of α from Γ where the Classical-⊥ has only
atomic conclusions
[¬(α ∧ β)]a
Π
⊥
a
α∧β
Transforms into
[α ∧ β]c
[α ∧ β]a
α [¬α] b β [¬β]d
⊥ ⊥
a c
¬(α ∧ β) ¬(α ∧ β)
Π Π
⊥ ⊥
b d
α β
α∧β
return
89.
Example of reduction
[¬α]a [¬β]b
Π1 Π2
⊥ ⊥
a b
α α
α∧β
Transforms into
[α]a [β]b
α∧β [¬(α ∧ β)]c
⊥
a
¬α
Π1
⊥
b
¬β
Π2
⊥
c
α∧β
other
93.
The Haupsatz
The cut rule:
Γ1 ⇒ ∆ 1 , α α, Γ2 ⇒ ∆2
Γ1 , Γ2 ⇒ ∆1 , ∆2
Every proof of Γ ⇒ ∆ can be rewritten without the cut-rule.
return
94.
The Haupsatz
The cut rule:
Γ1 ⇒ ∆ 1 , α α, Γ2 ⇒ ∆2
Γ1 , Γ2 ⇒ ∆1 , ∆2
Every proof of Γ ⇒ ∆ can be rewritten without the cut-rule.
Corollary: Every formula in a proof of Γ ⇒ ∆ is subformula from
at least one formula of Γ ∪ ∆.
return
95.
The Haupsatz
The cut rule:
Γ1 ⇒ ∆ 1 , α α, Γ2 ⇒ ∆2
Γ1 , Γ2 ⇒ ∆1 , ∆2
Every proof of Γ ⇒ ∆ can be rewritten without the cut-rule.
Corollary: Every formula in a proof of Γ ⇒ ∆ is subformula from
at least one formula of Γ ∪ ∆.
Corollary: If the Haupsatz holds for a logic/theory L then this
logic is consistent. (There is no proof of the empty sequent).
return
96.
The Haupsatz
The cut rule:
Γ1 ⇒ ∆ 1 , α α, Γ2 ⇒ ∆2
Γ1 , Γ2 ⇒ ∆1 , ∆2
Every proof of Γ ⇒ ∆ can be rewritten without the cut-rule.
Corollary: Every formula in a proof of Γ ⇒ ∆ is subformula from
at least one formula of Γ ∪ ∆.
Corollary: If the Haupsatz holds for a logic/theory L then this
logic is consistent. (There is no proof of the empty sequent).
Gentzen proved that PA is consistent by means of Haupsatz.
return
97.
Normalization and Normal Proofs
A → B is maximal formula in a ND proof:
[A]
Π2
Π1 B
A A→B
B
reduces to
Π1
[A]
Π2
B
return
98.
Normalization and Normal Proofs
A → B is maximal formula in a ND proof:
[A]
Π2
Π1 B
A A→B
B
reduces to
Π1
[A]
Π2
B
Normalization: Every derivation of α from ∆ can be transformed
into a Normal derivation (without maximal formulas) of α from ∆
(∆ ⊆ ∆)
return
99.
Normalization and Normal Proofs
A → B is maximal formula in a ND proof:
[A]
Π2
Π1 B
A A→B
B
reduces to
Π1
[A]
Π2
B
Normalization: Every derivation of α from ∆ can be transformed
into a Normal derivation (without maximal formulas) of α from ∆
(∆ ⊆ ∆)
Corollary: Every formula in a proof of α from Γ is subformula of Γ
or α.
return
100.
Moving the ⊥ towards the conclusion of a derivation [Seldin1977]
[¬A]a
Π1
Π2
a ⊥
A B
A∧B
reduces to
Π2
a
[A] B
A∧B [¬(A ∧ B)]b
a ⊥
¬A
Π1
⊥
b
A∧B
return
101.
Classical Logic × Intuitionistic Logic
Theorem
There are a, b ∈ R − Q, such that, ab ∈ Q
A Classical Proof (Math Folklore)
√
Consider a = b = 2. Then, either ab ∈ Q or ab ∈ √ In the ﬁrst case
Q.
√ 2 √
we are done. In the second case, consider a = 2 and b = 2,
hence, ab = 2 ∈ Q.
An Intuitionistic (constructive) proof (E. Bishop)
√
Consider a = 2 and b = 2log2 (3). We have a, b ∈ Q and
ab = 3 ∈ Q
return
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