Logics for Data and Knowledge Representation Exercises: DL

DL family of languages AL SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS AL <Atomic> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic> | ¬<Atomic> | <wff> ⊓ <wff> | ∀R.C | ∃R.⊤ ALU <wff> ⊔ <wff> ALE ∃R.C ALN ≥nR | ≤nR ALC ¬ <wff> FL- is AL with the elimination of ⊤, ⊥ and  FL0 is FL- with the elimination of ∃R.⊤

DL family of languages ∃R.C ≥2R Formula AL ALU ALE ALN ALC A  A⊔B SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Examples of formulas in each of the languages: Formula AL ALU ALE ALN ALC A  A⊔B ∃R.C ≥2R (A⊓B) A⊔B 3

Formalization of a semantic network SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS instance-of Person ⊑ ∃Drives.Car Person ⊑ ∃HasHobby.SportCar Person ⊑ ∃HasHobby.Opera Student ⊑ Person SportCar ⊑ Car Student(Ralf) Opera(DonCarlos)

Modeling a problem Formalize the following problem in DL: SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Formalize the following problem in DL: In a hospital patients, doctors and computers are equipped with proximity sensors able to detect whether doctors curated a patient or worked at their computer. The system detected that doctor Peter curated the patient Smith. Doctor ⊑ cure.Patient ⊔ work.Computer cure ⊑ near work ⊑ near Doctor (Peter) Patient (Smith) cure(Peter, Smith)

Venn diagrams Provide the Venn diagram for: A ⊑ B ⊓ ¬C SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Provide the Venn diagram for: A ⊑ B ⊓ ¬C Note: Venn Diagrams can be used only for propositional DL B A C

Interpretation of Value Restriction I(∀R.C) = {a ∈ ∆ | for all b, if (a,b)∈I(R) then b∈I(C)} Those a that have only values b in C with role R. b I(C) a if (a,b) ∈I(R) b' b'' if (a,b') ∈I(R) if (a,b'') ∈I(R)

Interpretation of Existential Quantifier I(∃R.C) = {a ∈ ∆ | exists b s.t. (a,b) ∈ I(R), b ∈ I(C)} Those a that have some value b in C with role R. b I(C) a (a,b)∈I(R)

Proofs in DL semantics SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Verify the following equivalences hold for all interpretations (∆,I). Use definitions or graphical representation. I(¬(C ⊓ D)) = I(¬C ⊔ ¬D) I(¬(C ⊔ D)) = I(¬C ⊓ ¬D) I(¬∀R.C) = I(∃R.¬C) I(¬∃R.C) = I(∀R.¬C) Let us prove the last one: I(¬∃R.C)= {a ∈ ∆ | not exists b s.t. (a,b) ∈ I(R), b ∈ I(C)} = {a ∈ ∆ | not ∃b. R(a,b) and C(b)} = {a ∈ ∆ | ∀b. not (R(a,b) and C(b))} = {a ∈ ∆ | ∀b. if R(a,b) then ¬C(b)} = I(∀R.¬C) 9

TBox and ABox reasoning TBox reasoning requires in general the usage of the Tableaux algorithm (subject of the next lecture) Now, we provide a list of cases that can be addressed WITHOUT it 10

Using DPLL for reasoning tasks SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS DPLL solves the CNFSAT-problem by searching a truth-assignment that satisfies all clauses θi in the input proposition P = θ1  …  θn NOTE: DL sentences must to be translated in PL (via TBox and ABox elimination) and therefore it can only be used for propositional DL. Model checking Does ν satisfy P? (ν ⊨ P?) Check if ν(P) = true Satisfiability Is there any ν such that ν ⊨ P? Check that DPLL(P) succeeds and returns a ν Unsatisfiability Is it true that there are no ν satisfying P? Check that DPLL(P) fails Validity Is P a tautology? (true for all ν) Check that DPLL(P) fails 11

Modeling the problem Codify in DL the following problem: SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Codify in DL the following problem: “A federal agent can access top secret documents. An X file is a top secret document with no restricted access that can be read by policemen.” FederalAgent ⊑ ∃Access.TopSecretDocument XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman

Satisfiability of a TBox (a set of formulas) SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Say if the following TBox is satisfiable and provide a model in the case: FederalAgent ⊑ ∃Access.TopSecretDocument XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman I(FederalAgent) = {A} We have that I ⊨ T I(TopSecretDocument) = {D1, D2} I(Access) = {(A, D1), (A, D2)} I(XFile) = {D1} I(Restricted) = {D2} I(Read) = {(B, D1)} I(Policeman) = {B} Otherwise you can apply the tableaux calculus and provide the ABox built

Satisfiability w.r.t. a TBox SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Consider the TBox T: FederalAgent ⊑ ∃Access.TopSecretDocument XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman and the formula P: TopSecretDocument ⊓ Restricted Is there any model such that T ⊨ P ? Yes, if fact these is an interpretation I (e.g. the one of the previous exercise) such that I ⊨ T and I ⊨ P I(TopSecretDocument) = {D1, D2} I(Restricted) = {D1} Notice that T does not affect P at all (i.e. we cannot further expand P w.r.t. T)

I(Access) = {(A, D1), (A, D2)} Subsumption SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Consider the TBox T: FederalAgent ⊑ ∃Access.TopSecretDocument XFile ⊑ TopSecretDocument ⊓ Restricted ⊓ ∀Read-1.Policeman does T ⊨ TopSecretDocument ⊑ Restricted in all models? By definition, it must be I(TopSecretDocument )  I(Restricted ) for every model I of T. Even if this is true for the I of the previous exercise, this is not true in general. It is enough to provide a counterexample: I(FederalAgent) = {A} I(TopSecretDocument) = {D2} I(Access) = {(A, D1), (A, D2)} I(XFile) = {D2} I(Restricted) = {D1} I(Read) = {(B, D2)} I(Policeman) = {B}

Expansion of an ABox SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Provide the expansion of A w.r.t. T (without normalization), where: TBox T = {Student ⊑ Faculty, Professor ⊑ Faculty ⊓ Teach} ABox A = {Professor(Bob), Faculty(Rui)} Professor(Bob), Faculty(Bob), Teach(Bob) Faculty(Rui)

ABox Reasoning services: Consistency SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS An ABox A is consistent with respect to a TBox T if there is an interpretation I which is a model of both A and T. T = {Parent⊑≤1hasChild} A = {hasChild(mary, bob), hasChild(mary, cate), Parent(mary)} A is consistent ALONE but is not consistent with respect T. In fact, from A mary has two children while T imposes maximum one 17

Drawing consequences (I) SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Given the TBox and ABox below T: Female⊑Human Male⊑Human Mother⊑Female Father⊑Male Child≡∃has.Mother⊓ ∃has.Father Male⊓Female⊑⊥ A: Mother(Anna) Father(Bob) has(Cate,Anna) has(Cate,Bob) Prove: Human(Anna) ¬Female(Bob) Child(Cate) 18

Drawing consequences (II) SYNTAX :: SEMANTICS :: TBOX REASONING :: ABOX REASONING :: TABLEAU CALCULUS Expand A w.r.t. T: A: Mother(Anna)  Female(Anna)  Human(Anna) Father(Bob)  Male(Bob)  Human(Bob) , ¬Female(Bob) has(Cate,Anna)  Child(Cate) has(Cate,Bob)  Child(Cate) 19