Logics for Data and Knowledge Representation

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Logics for Data and Knowledge Representation

• Exercises DL

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DL family of languages
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• AL
• ltAtomicgt A B ... P Q ... ? ?
• ltwffgt ltAtomicgt ltAtomicgt ltwffgt ? ltwffgt
  ?R.C ?R.?
• ALU ltwffgt ? ltwffgt
• ALE ?R.C
• ALN nR nR
• ALC ltwffgt
• FL- is AL with the elimination of ?, ? and ?
• FL0 is FL- with the elimination of ?R.?

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DL family of languages
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 ?
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Formalization of a semantic network
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Person ? ?Drives.Car
• Person ? ?HasHobby.SportCar
• Person ? ?HasHobby.Opera
• Student ? Person
• SportCar ? Car
• Student(Ralf)
• Opera(DonCarlos)

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Modeling a problem
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 top
  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
• Doctor (Peter)
• Patient (Smith)
• cure(Peter, Smith)

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Venn diagrams
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Provide the Venn diagram for A ? B ? C

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Proofs in DL semantics
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Verify the following equivalences hold for all
  interpretations (?,I). Use definitions or Venn
  diagrams.
• 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)

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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 ?i in
  the input proposition P ?1 ? ? ?n
• DL sentences must to be translated in PL (via
  TBox and ABox elimination)
• Model checking Does ? satisfy P? (? ? P?)
• Check if ?(P) true
• Satisfiability Is there any ? such that ? ? P?
• Check that DPLL(P) succeeds and returns a ?
• Unsatisfiability Is it true that there are no ?
  satisfying P?
• Check that DPLL(P) fails
• Validity Is P a tautology? (true for all ?)
• Check that DPLL(?P) fails

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Modeling the 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

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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 either draw a Venn diagram or
  apply the tableaux calculus and provide the ABox
  built

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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
• does 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)

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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(XFile) D2 I(Restricted) D1 I(Read)
(B, D2) I(Policeman) B
I(FederalAgent) A I(TopSecretDocument)
D2 I(Access) (A, D1), (A, D2)
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Reduce to PL reasoning
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Consider the TBox T A ? B ? C, C ? D ? E.
  Determine if A ? E by elimination of T.
• T C (B ? C) ? X, C D ? E ? Y
• A (B ? (D ? E ? Y)) ? X
• E E
• A (B ? (D ? E ? Y)) ? X
• E E
• P CNF(B ? (D ? E ? Y)) ? X ? E)
• Call DPLL(P)

The steps T Normalize(T) A Expand(A,
T) E Expand(E, T) A RewriteInPL(A) E
RewriteInPL(E) return DPLL(CNF(A ? E))
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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)

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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

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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)

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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)

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Reasoning using Tableau calculus (a)
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Given the ABox A hasParent(Speedy, Furia),
• prove with Tableau algorithm the satisfiability
  of the following formula
• ?Parent.Horse ? ? (Horse ? Mule)
• Both ?Parent.Horse and ? (Horse ? Mule) have to
  be satisfied ?-rule
• (i) ?hasParent.Horse ? A A ? Horse(Furia)
  ?-rule
• (ii) ? (Horse ? Mule) ? ? Horse ? ? Mule ?
• It is not consistent for A A ? ?
  Horse(Furia) ?-rule
• (backtracking)
• It is consistent for A A ? ? Mule(Furia)
  ?-rule

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Reasoning using Tableau calculus (b)
SYNTAX SEMANTICS TBOX REASONING ABOX
REASONING TABLEAU CALCULUS

• Using the tableau calculus, say whether the
  formula (?R.A ? ?R.A) ? ?R.(A ? B) is
  satisfiable given the TBox T A ? ?B. Motivate
  your response with a proof.
• By ?-rule we split into (i) (?R.A ? ?R.A) (x)
  and (ii) ?R.(A ? B) (x).
• We need one of the two to be satisfiable.
• (i) By ?-rule we put into the ABox both ?R.A (x)
  and ?R.A (x)
• (i.1) For ?R.A (x), by ?-rule we add into the
  ABox both R(x, y) and A(y)
• (i.2) For ?R.A (x), by ?-rule we add into the
  ABox both R(x, y) and A(y)
• i.1. and i.2 clearly contradict each other
• (backtracking)
• (ii) It is clearly in contradiction with the
  axiom in T. In fact A and B are disjoint
• Since none of the two is satisfiable, then the
  formula is NOT satisfiable.