Computing minimum cardinality diagnoses by model relaxation
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Computing Minimum-cardinality Diagnoses by Model Relaxation. Sajjad Siddiqi National University of Sciences and Technology (NUST) Islamabad, Pakistan. Consistency-based Diagnosis. NOT AND. C. Abnormal observation : A  B  D. A. X. D. Y. B. System model  :

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Computing Minimum-cardinality Diagnoses by Model Relaxation

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Computing minimum cardinality diagnoses by model relaxation

Computing Minimum-cardinality Diagnoses by Model Relaxation


National University of Sciences and Technology (NUST)

Islamabad, Pakistan

Consistency based diagnosis

Consistency-based Diagnosis



Abnormal observation :

A  B  D






System model :

okX (A  C)

okY (B  C)  D

Health variables: okX, okY

Observables: A, B, D

Nonobservable: C

Consistency based diagnosis1

Consistency-based Diagnosis


Abnormal observation :

A  B  D






System model :

okX (A  C)

okY (B  C)  D

Find values of (okX, okY) consistent with   :

(0, 0), (0, 1), (1, 0) OR

{okX=0, okY=0}, …

Consistency based diagnosis2

Consistency-based Diagnosis

System model  over

health variables(okX, okY, …)



Given observation , diagnosis is assignment to health variables consistent with   

Consider minimum-cardinality diagnoses

Cardinality is the number of failing components in a diagnosis

Decomposable negation normal form dnnf

Decomposable Negation Normal Form (DNNF)


  • DAG of nested and/or

  • Conjuncts share no variable (decomposable)


Min-cardinality as well as min-cardinality diagnoses can be computed efficiently using DNNF






Compilation based approach

Compilation-based Approach








Previous method hierarchical diagnosis

Previous Method: Hierarchical Diagnosis

Significantly reduces number of health variables – through Abstraction

Requires only 160 health variables for c1908; c1908 has 880 gates

Able to compile larger systems

(Siddiqi and Huang, 2007)

Previous method hierarchical diagnosis1

Previous Method: Hierarchical Diagnosis

Without Abstraction: Requires 6 healthvariables:

okU, okV, okE, okB, okJ, okA

Previous method hierarchical diagnosis2

Previous Method: Hierarchical Diagnosis

  • Abstraction:{U,V,E,A}

  • Treats self contained sub-systems (E) as single components (cones):

    Requires 4 healthvariables:

    okU, okV, okE, okA

Previous method hierarchical diagnosis3

Previous Method: Hierarchical Diagnosis

{E,A} is a an abstract min-cardinality diagnosis

{E}, {J}, {B}are min-cardinality diagnoses of coneE.

{J,A}, {B,A} are deduced as more min-cardinality diagnoses

Need to find abnormal observation for coneE

Previous method diagnosis of cones

Previous Method: Diagnosis of Cones

Reorder{E,A} as {A,E}

(deeper gates first)

Propagate normal values in the circuit (input values in given observation)

Propagate faults in the order they appear in diag.

Sets the required abnormal obs for cone E

Previous method

Previous Method

Again Compilation becomes a bottleneck for very large systems – even after abstraction

New method

New Method

Combines abstraction, model relaxation (node splitting), and search to scale up

Compiles the abstraction of a relaxed model instead of the original

Applies two stage branch-and-bound search to compute minimum-cardinality diagnoses

Node splitting

Node Splitting

Splits Y

Y1’ and Y2’ are clones of Y

(Choi et al., 2007)

Node splitting1

Node Splitting

Splits gate B

Node splitting2

Node Splitting

Some components may come out of cones

The components in the abstraction of the split system form a superset of the set of components in the original abstraction

The abstract min-cardinality diagnoses (once computed correctly) of the split system form a superset of the set of abstract min-cardinality diagnoses of the original

Search for minimum cardinality first stage

Search for minimum cardinality (First Stage)

∆ and ∆’ – models of original and split system

e – a given assignment to variables in ∆

e – the compatible assignment to corresponding clones in the split system

For example, if e = {B = b} then e = {B’ = b}

Search for minimum cardinality

Search for minimum cardinality

∆’ provides basis for computing lower bounds on minimum cardinality for B-n-B search

min_card(∆ | e) >= min_card(∆’ | ee)

if e contains a complete assignment to split variables then

min_card(∆ | e) == min_card(∆’ | ee)

Search for minimum cardinality1

Search for minimum cardinality

B-n-B search in the space of assignments s to split variables S

At each node compute min_card(∆’ | eess)

At leaf nodes we get candidate minimum cardinalites

Elsewhere, we get lower bounds to prune search

Search for minimum cardinality2

Search for minimum cardinality

A good Seed for search

In the given observation, if k components output values inconsistent with the normal values then k is the upper bound on the minimum cardinality.

Search for minimum cardinality3

Search for minimum cardinality

Variable and value ordering

Nogood-based scoring heuristic similar to (Siddiqi and Huang, 2009):

Every value of a variable X is associated with a score S(X = x)

Score of X, S(X), is the average of the scores of its values

Vars and values with higher scores are preferred.

Search for minimum cardinality4

Search for minimum cardinality

Variable and value ordering

During search if X is assigned a value x then

S(X = x) += new_bound – cur_bound

cur_bound = bound before the assignment

new_bound = bound after the assignment

Early Backtracking

Search for diagnoses second stage

Search for diagnoses (Second Stage)

First Strategy

if e contains a complete assignment to split variables then

min_card_diags(∆|e) == min_card_diags(∆’|ee)

Search in the space of assignments to split variables; enumerate all min-card diagnoses at those leaf nodes where cardinality is minimum.

Search for diagnoses

Search for Diagnoses

Second Strategy

Search in the space of assignments to health-vars

Partial assignment to h-vars == partial diagnosis

Enumerate all valid min-cardinality diagnoses.

Search for diagnoses1

Search for Diagnoses

First Strategy

Can efficiently compute very large number of diagnoses at leaf nodes by evaluating the DNNF

Often resulted in very large search spaces even when the number of diagnoses was small

Second Strategy

Efficient only when the number of diagnoses was reasonably small

Combined approach benefit from both

Combined Approach; benefit from both

Systematic search in both spaces simultaneously:

Search starts in the space of assignments to health variables

At each search node, another search is performed in the space of assignments to split variables; IF REQUIRED.

Combined approach

Combined Approach

Search on health variables

Validate each partial diagnosish at each node

If h is valid and card(h) < mincard, then

continue search; else backtrack

h is validiff:

h is consistent with the model + observation

h can be extended to a valid min-card diagnosis


Combined approach1

Combined Approach

Validate partial diagnosish:

B-n-B Search for complete assignment to split varsS such that for each partial assignment s

∆’|heessis consistent AND

card (h) + min_card(∆’|heess) <= mincard

If such a complete assignment s exists then his valid; otherwise his invalid

Combined approach2

Combined Approach


At each node where h is valid compute min-card diagnoses as:


Union of all such diagnoses is the complete set of min-card diagnoses [redundancy is an issue]

Combined approach3

Combined Approach

<h1, s1, D1>

okX = true

okX = false

<h2, s2, D2>

<h3, s3, D3>

h2 h1

h3 h1

min-card diagnoses = D1 D2 D3 (may overlap)

Avoiding redundancy 1

Avoiding Redundancy - 1

If s1 == s2 then

D1 D2

<h1, s1, D1>

okX = true

okX = false

<h2, s2, D2>

<h3, s3, D3>

h2 h1

h3 h1

Solution: At each node, keep passing `the assignment

to split variables used’ to the children nodes, AND…

Avoiding redundancy 11

Avoiding Redundancy - 1

At each node with partial diagnosis h:

Let sp = assignment to split vars used at the parent node

First check if h is valid under sp:

YES: Don’t search, don’t enum diagnoses

NO: Search for a new assignment to split vars and enum diagnoses only if found

Avoiding redundancy 2

Avoiding Redundancy - 2

During search treat the all recorded diagnoses (so far) as nogoods

Watch literals scheme (as in Satisfiability):

As soon as all but one broken component in a recorded diagnosis have been assumed as broken, that remaining broken component is forced to be healthy

Combined approach4

Combined Approach

Variable and Value Ordering

Same selection heuristic as in the first stage; scores try to minimize search on split vars

Let search on split vars explored p nodes, when okX was assigned a value okx, then

S(okX=okx) += 1/p

If search is not performed at a node

p = ½ of the value used at the parent

Combined approach5

Combined Approach

Variable and Value Ordering

Initial scores:

S(okX = true) = 0

S(okX = false) = failure probability of X

(Siddiqi & Huang 11)

Effectively orders components according to decreasing value of their failure probabilities

Diagnosis of cones

Diagnosis of Cones

Same as in the previous method with some extra care:

All clones must be assigned the same value during value and fault propagation

When reordering components in abstract diagnosis, original depth values for components must be used despite changes due to splitting



Use ISCAS85 circuits

Observations (inputs/outputs) randomly generated

Multiple instances per circuit



New method solves most of the cases on every circuit (except c6288)

Previous method cannot solve any case beyond c2670

New method is either as fast as the previous, or 4 times faster, or 2 orders of magnitude faster.



New tool to compute minimum-cardinality diagnoses of a faulty system employing model relaxation, abstraction and search

Solves non-trivial diagnostic cases on large systems, not possible before

Significantly faster than the previous on cases solvable by both

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