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Non-minimal Diagnoses

Non-minimal Diagnoses. Philippe Dague and Yuhong Yan NRC-IIT Philippe.dague@lipn.univ-paris13.fr Yuhong.yan@nrc.gc.ca. {A, B, C}. {A, B}. {A, C}. {B, C}. {A}. {B}. {C}. {}. Diagnosis.

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Non-minimal Diagnoses

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  1. Non-minimal Diagnoses Philippe Dague and Yuhong Yan NRC-IIT Philippe.dague@lipn.univ-paris13.fr Yuhong.yan@nrc.gc.ca

  2. {A, B, C} {A, B} {A, C} {B, C} {A} {B} {C} {} Diagnosis • Consider only assignment AB(c) and ¬AB(c) for diagnoses, the size of diagnostic space is 2n, n= number of components • Diagnostic space is structure by set inclusion as a lattice

  3. A principle of parsimony has been adopted by Reiter: considering only minimal (for set inclusion) diagnoses • Question: Do these minimal diagnoses characterize all diagnoses? • Expected answer: yes, any superset of a diagnosis is a diagnosis as well (Minimal Diagnosis Hypothesis) • This is verified for the polybox with correct mode, and the 3-inverter with correct and faulty modes (but with the unknown mode).

  4. Counter Example(1):exhaustive fault modes I1 I2 • Assume the only fault modes are stuck at 0 and shorted (no unknown mode): Inverter(x)AB(x)  S0(x)  Short(x) S0(x)  out(x)=0 Short(x)  out(x) = in(x) Diagnoses = minimal diagnoses = {I1} (stuck at 0 or shorted) {I2} (shorted) But the superset {I1,I2} is not a diagnosis Reason: I2 can’t be stuck at 0, so it should be shorted, but in this case out(I1)=1 and I1 can’t be stuck at 0 nor shorted 1 0 Example 1.a)

  5. Counter Example(1):exhaustive fault modes • Suppose that in addition to correct modes, we have AB(adder)  adder acts as multiplier • Same observation as before {F=10, G=12} • {M1} is still a minimal diagnosis but the superset {M1, A2} is not any more Example 1,b) Polybox

  6. Counter example (2): Exoneration • Exoneration: correct mode expressed as necessary and sufficient condition of correctness • 2-inverter: Inverter(x)  (¬AB(x)  [In(x) = 0  Out(x)=1][In(x) = 1  Out(x)=0]) • Minimal diagnosis = {} • But the supersets {I1} and {I2} are not diagnoses. Each inverter exonerates the other (is an alibi for the other) 0 0 Example 2.a)

  7. Counter example (2): Exoneration • 3 light bulbs Bulb(x)  voltage(x, on) [¬AB(x)  lit(x)] • Observation: only B3 is lit • {B1, B2} is a minimal diagnosis. The superset {B1, B2, B3} is not • Reason: B3 can’t be faulty, as it is lit. Example 2.b)

  8. Conclusion: • The minimal diagnosis hypothesis is not satisfied in general, as soon as exhaustive fault modes or sufficient condition of correctness exists • So in the diagnostic space lattice, diagnoses are not characterized by minimal diagnoses • Questions: does a logical characterization of the diagnoses in the general case exist? • Answer: yes. • For this, the notion of conflict has to be generalized

  9. Recall: • Notation: for Components, D() = [AB(c)|c ] [AB(c)|c  Components\] • Definition: a diagnosis is a D() such that SD  OBS  {D()} is satisfiable • Definition: minimal diagnosis is a diagnosis D() such that for no proper subset ’ of  is D(’) a diagnosis • Definition: a conflict as defined by Reiter (named from now a R-conflict) is a subset C of Components such that SD  OBS  {AB(c)|c  C} |=  • Logically it is equivalent to SD  OBS |= {AB(c)|c  C} ( a disjunct of AB(c) is entailed by SD  OBS)

  10. What appears in the counter example? 1.a (2-inverter) SD  OBS |= AB(I1)AB(I2) But also SD  OBS |= ¬AB(I1)AB(I2) 1.b (polybox) SD  OBS |= AB(M1)AB(M2) and SD  OBS |= AB(M1)AB(M3) But also SD  OBS |= AB(M2)AB(M3)¬AB(A2) 2.a SD  OBS doesn't entail disjunct of AB but SD  OBS |= AB(I1)AB(I2) SD  OBS |= AB(I1)AB(I2) 2.b SD  OBS |= AB(B1) and SD  OBS |= AB(B2) but also SD  OBS |= AB(B3)

  11. Extension: conflict • So the idea is to extend a conflict to any conjunct of AB(c) and ¬AB(c) entailed by SD  OBS . • Definition: An AB-literal is AB(c) or ¬AB(c) for some c Components. • An AB-clause is a disjunction of AB-literals containing no complementary pair of AB-literals. • A positive AB-clause is an AB-clause all of its literals are positive • Definition: A conflict of (SD, Components, OBS) is an AB-clause entailed by SD  OBS. • A positive conflict is a conflict which is a positive AB-clause • Remark: one can identify a positive conflict with an R-conflict

  12. Extension: conflict (2) • Definition: a minimal conflict is a conflict no proper sub-clause of which is a conflict • Example: see 1.a) 1.b) 2.a) 2.b) (the right side formulas in slide 10 are the minimal conflicts) • Remark: one can identify a minimal positive conflict with a minimal R-conflict

  13. Extension: conflict (3) • Suppose  is a set of first order sentences, a ground clause is an implicate of  iff  entails c. c is a prime implicate of  iff no proper sub-clause of c in entailed by  • Minimal conflicts are AB-clauses which are prime implicates of SD  OBS. • Minimal conflicts can be computed by theorem prover or ATMS

  14. Extension: conflict (4) • Reiter’s property relating minimal diagnosis to minimal R-conflict can be reformulated. • Property: let + be the set of positive minimal conflicts of (SD, Components, OBS) and Components, then D() is a minimal diagnosis iff  is a minimal subset such that +{D()} is satisfiable • This property generalizes as Property: let  be the set of minimal conflict of (SD, Components, OBS) and Components, then D() is a diagnosis iff {D()} is satisfiable

  15. Characterizing minimal diagnoses from positive minimal conflicts • Def: Suppose  is a set of propositional formulas, a conjunction of literals  (containing no pair of complementary literals) is an implicant of  iff  entails each formula of .  is a prime implicant of  iff no proper sub conjunction of  is an implicant of .

  16. Characterizing minimal diagnoses from positive minimal conflicts (2) • The Reiter’s characteristics of minimal diagnoses as minimal hitting sets of the collection of minimal R-conflicts can be reformulated as: • Theorem: D() is a minimal diagnosis of (SD, Components, OBS) iff [AB(c)|c ] is a prime implicant of the set of the positive minimal conflicts of (SD, Components, OBS).

  17. When minimal diagnoses are enough to characterizing all diagnoses? • Theorem: Minimal diagnosis hypothesis holds (i.e. D(’) is a diagnosis iff ’ with D() a minimal diagnosis) iff all minimal conflicts are positive • Unfortunately there is no equivalent condition on the syntactic form of SD and OBS. But it exists sufficient conditions. We consider 2 of them

  18. the Ignorance of Abnormal Behaviour (IAB) • Def: the Ignorance of Abnormal Behaviour (IAB) condition holds iff in the clause form of SDOBS every occurrence of an AB-predicate is positive • Theorem: If (SD, Components, OBS) satisfies the IAB condition, then MDH holds

  19. IAB(2) • IAB is ensured, for example, if all sentence of SD where AB appears follow the schema: AB(x)P1(x)P2(x)… Pn(x)G1(x)… Gm(x) Where literals Pi(x) and Gj(x) do not mention AB • i.e. when only necessary condition of correct behaviour are expressed • Example: AB(x)transistor(x)On(x)off(x)saturated(x) AB(x)resistor(x)ports(x,[a b])resistance(x)=r v(x, a, b) = r * i(x,a)

  20. Limited Knowledge of Abnormal Behaviour (LKAB) • Def: the Limited Knowledge of Abnormal Behaviour (LKAB) condition holds iff (Cp, Cn, c), CpComponents, Cn  Components, CpCn =, cComponents, cCp,cCn, SDOBS{[AB(x)|xCp]  [AB(x)|xCn]} satisfiable, SDOBS{AB(c)} satisfiable  SDOBS{[AB(x)|xCp{c}] [AB(x)|xCn]} • Remark: IAB  LKAB

  21. LKAB(2) • LKAB is ensured, for example, if all sentences of SD where AB appears have one of the following two forms: AB(x)P1(x)P2(x)… Pn(x)G1(x)… Gm(x) AB(x)P1(x)P2(x)… Pn(x)F1(x)… Fm(x)U(x) Where Gi(x) describes a possible correct behaviour for x, Fi(x) describes a possible faulty behaviour for x, U(x) an unknown behaviour (Gi(x), Fi(x), U(x) only occur negatively in other clauses and U(x) only occurs in clauses expressing it is distinct of any Gi(x) and any Fi(x).) • i.e. when only necessary conditions of correct behaviours and necessary condition of non-exhaustive faulty behaviours (with unknown mode) are expressed.

  22. LKAB(3) • (see example in lecture “diagnoses with fault modes”). • Theorem: if (SD, Components, OBS) satisfies the LKAB condition and D() is a diagnosis, then D(’) is a diagnosis for every ’  , such that for each c, SDOBS {AB(c)} is satisfiable

  23. Charactering Diagnoses from Minimal Conflicts • Compact representation of diagnoses • Example: 1.b) AB(M1)  AB(A2)  K1(M2)  K2(M3)  K3(A1), where Ki={AB or AB} they can be coded as AB(M1)  AB(A2)

  24. Compact representation of diagnoses • Definition: A partial diagnosis for (SD, Components, OBS) is a satisfiable conjunction P of AB-literals such that for every satisfiable conjunction P’ of AB-literals containing P as sub-conjunction, SDOBS {P’} is satisfiable • Remark: if C, of size k, is the set of all components mentioned in P, the P [K(c)|cComponents\C] is a diagnosis, where each K(c) is AB(c) or AB(c). So P codes 2n-k diagnoses

  25. Kernel diagnosis • It is natural to consider the minimal such partial diagnoses: • Definition: A kernel diagnosis is a partial diagnosis whose no proper sub-conjunction is a diagnosis • Property (Characterization of Diagnoses) D() is a diagnosis iff there is a kernel diagnosis which is a sub-conjunction of it

  26. Kernel Diagnoses (2): Examples 1.a) 2 kernel diagnoses AB(I1)AB(I2) and AB(I1)AB(I2) 1.b) 4 kernel diagnoses: AB(M1)AB(A2) AB(M1)AB(M2) AB(M1)AB(M3) AB(M2)AB(M3) 2.a) 2 kernel diagnoses AB(I1)AB(I2) AB(I1)AB(I2) 2.b) 1 kernel diagnosis AB(B1)AB(B2)AB(B3)

  27. Theorem • Theorem (Characterization of partial and kernel diagnoses from minimal conflicts) • The partial diagnoses of (SD, Components, OBS) are the implicants of the minimal conflicts of (SD, Components, OBS) • The kernel diagnoses of (SD, Components, OBS) are the prime implicants of the minimal conflicts of (SD, Components, OBS) • The minimal diagnoses are the prime impliants of positive minimal conflicts • Remark: if all minimal conflicts are positive, there is a 1 to 1 correspondence between kernel diagnoses and minimal diagnoses [AB(c)|cK]  [AB(c)|cK]  [AB(c)| cComponents\K]

  28. Exercise • Full adder in Reiter’s paper (figure 1). • Use kernel diagnosis to find diagnosis • Use two-direction imply () in the model to find kernel diagnosis • Add the axiom that all variables are Boolean (x=0x=1), find kernel diagnosis

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