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Non-minimal Diagnoses

Philippe Dague and Yuhong YanNRC-IIT

Philippe.dague@lipn.univ-paris13.frYuhong.yan@nrc.gc.ca

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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

{A, B, C}

{A, B} {A, C} {B, C}

{A} {B} {C}

{}

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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).

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Counter Example(1):exhaustive fault modes

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)=0Short(x) out(x) = in(x)

Diagnoses = minimal diagnoses = {I1} (stuck at 0 or shorted) {I2} (shorted)But the superset {I1,I2} is not a diagnosisReason: 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

0I1 I2

1Example 1.a)

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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

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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 0Example 2.a)

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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.

B-1S

B-2 B-3

W-1

W-2

W-3

W-4

W-5

W-6

Example 2.b)

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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

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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)

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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)

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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

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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

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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

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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

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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 .

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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).

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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

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the Ignorance of Abnormal Behaviour (IAB) Def: the Ignorance of Abnormal Behaviour

(IAB) condition holds iff in the clause form of SDOBS every occurrence of an AB-predicate is positive

Theorem: If (SD, Components, OBS) satisfies the IAB condition, then MDH holds

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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)

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Limited Knowledge of Abnormal Behaviour (LKAB) Def: the Limited Knowledge of Abnormal

Behaviour (LKAB) condition holds iff (Cp, Cn, c), CpComponents, Cn Components, CpCn =, cComponents, cCp,cCn,

SDOBS{[AB(x)|xCp] [AB(x)|xCn]} satisfiable,

SDOBS{AB(c)} satisfiable SDOBS{[AB(x)|

xCp{c}] [AB(x)|xCn]}

Remark: IAB LKAB

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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.

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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, SDOBS {AB(c)} is satisfiable

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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)

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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, SDOBS {P’} is satisfiable

Remark: if C, of size k, is the set of all

components mentioned in P, the P [K(c)|cComponents\C] is a diagnosis, where each K(c) is AB(c) or AB(c). So P codes 2n-k diagnoses

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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

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Kernel Diagnoses (2): Examples1.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)

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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)|cK] [AB(c)|cK] [AB(c)| cComponents\K]

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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=0x=1), find kernel diagnosis