Explanation and DiagnosesThomas Thomas & Jansen Orfan

Previously...

Deduction P → Q P Therefore Q.

Induction Every P seen comes with a Q Qs sometimes occur without P. Maybe P → Q

And now...

Abduction P → Q Q Posit: P?

Example. If it rained, the grass is wet. The grass is wet. I bet it rained.

Diagnosis

Might have: DiseaseA ^ someQualifiers → SymptomsA

The following is unlikely: SymptomsA ^ someQualifiers → DiseaseA

We want an explanation for symtoms, we won't be able to know for certain a disease.

Explanation

So, what do we want in an explanation? A is sufficient to account for B.

KB U {A} |= B or KB |= (A → B) A is consistent with the KB A is as simple and parsimonious as possible A is in the the appropriate vocabulary

The abducibles

Prime Implicates

So how do we get an explanation out of a KB? A clauce c is a prime implicate of a KB iff

KB |= c For every proper subset c' of c, it is not the

case that KB |= c' We can use these implicates to form

explanations for facts.

Prime Implicates Continued

KB: [(p ^ q ^ r > g), (~p ^ q > g), (~q ^ r > g)] Convert to CNF...

[(~p v ~q v ~r v g), (p v ~q v g), (q v ~r v g)] Resolve to get

(p v ~q v g), and (~r v g) These are entailed by the KB but no subset of

them are entailed. So they are among our prime implicants.

Prime Implicants Continued.

So if we want an explanation for g. Find all the prime implicates containing g.

(p v ~q v g), (~r v g), (g v ~g) Remove the atom g.

(p v ~q), (~r), (~g) Negate the remaining clauses.

(~p ^ q), (r), (g)

Unfortunately

Exponentially many prime implicants. Worst case it's every possible combination of

terms. (a v b) (a v c) (a v d) (b v c) (b v d) (c v d) Could all be prime implicants.

So we can't realistically precompute them without knowing what we're trying to explain.

Drinks Example 1

KB

Vodka_Tonic → Vodka ^ Tonic Black_Russian → Vodka ^ Coffee_Liqueur White_Russian → Vodka ^ Coffee_Liqueur ^ Cream Whiskey_Sour → Whiskey ^ Lemon Irish_Coffee → Whiskey ^ Coffee Rum&Coke → Rum ^ Cola

I know I will be drinking a mixed drink tonight and my bar is currently stocked with Whiskey, Lemon, Coffee, and Rum. Given only these recopies, what might I be able to make?

Drinks Example 1

β= Whiskey V Lime V Lemon V Coffee

I will be drinking something

Start by finding the relevant prime implicants

~Irish_Coffee V (Whiskey ^ Coffee)

~Whiskey_Sour V (Whiskey ^ Lemon)

~Rum&Coke V (Rum ^ Cola)

{Whiskey V ~Whiskey}, {Rum V ~Rum} etc.

Remove anything that implies β

~Irish_Coffee

~Whiskey_Sour

~Rum&Coke V Cola

{~Whiskey}, {~Rum}, etc

Drinks Example 1

Negate

Irish_Coffee

Whiskey_Sour

Rum&Coke ^ ~Cola

Contradiction! (Rum&Coke → Cola) {Whiskey}, {Rum}, etc

Remove any explanation not conforming to the vocabulary H

Since I plan on drinking a mixed drink, H is all of the mixed drinks

Irish_Coffee

Whiskey_Sour

Drinks Example 2

KB Vodka_Tonic → Vodka ^ Tonic Black_Russian → Vodka ^ Coffee_Liqueur White_Russian → Vodka ^ Coffee_Liqueur ^

Cream Whiskey_Sour → Whiskey ^ Lemon Irish_Coffee → Whiskey ^ Coffee Rum&Coke → Rum ^ Cola

β= Vodka V Coffee_Liqueur V Cream V Coffee

Drinks Example 2

Start by finding the relevant prime implicants

Vodka_Tonic → Vodka ^ Tonic

Black_Russian → Vodka ^ Coffee_Liqueur

White_Russian → Vodka ^ Coffee_Liqueur ^ Cream

Irish_Coffee → Whiskey ^ Coffee

{Vodka V ~Vodka}, etc.

Remove anything that implies β

~Irish_Coffee V Whiskey

~Vodka_Tonic V Tonic

~Black_Russian

~White_Russian

~Irish_Coffee V Whiskey

{~Vodka}, etc.

Drinks Example 2

Negate

Irish_Coffee ^ ~Whiskey

Vodka_Tonic ^ ~Tonic

Black_Russian

White_Russian

Irish_Coffee ^ ~Whiskey

{Vodka}, etc.

Remove any explanation not consistent with KB or conforming to the vocabulary H

Black_Russian

White_Russian

Circuit Example

Circuits in Logic

Components AndGate(a1), XorGate(b1), OrGate(o1) fullAdder(f)

Connectives In1(b1) = in1(f), in1(b2) = out(b1)

Normal behavior AndGate(x) ^ ~Abnormal(X) =>

out(x) == and(in1(x), in2(x))

Fault Model

Then we might have a model for abnormal behavior.

Short Circuit: [OrGate(x) v XorGate(x)] ^ Abnormal(x) =>

out(x) = in2(x)

Abductive Diagnosis

So we're given. The previous KB Some input settings

in1(f) = 1, in2(f) = 0, in3(f) = 1 Some output observations

out1(f) = 1, out2(f) = 0

We want an A such that KB U settings U {A} |= observations

Consistency-Based Diagnosis

As before, we're given... The KB, some input settings, and some output

observations This time we want a set of components D such

that the following two sets are consistent. {ab(c) | c is in D} U {~ab(c) | c is not in D}

What does this mean? KB U settings U observations

Differences

Abductive diagnosis requires a fault model Only finds those errors that are in the fault

model. Consistency based can use a fault model, but

doesn't require it. Checks every fault model that is consistent

with the given one. Abductive diagnosis gives a much stronger

claim. But how to calculate?

Calculation

x2 a1 a2 o1 entailed? consistent?1 1 1 1 no yes1 1 1 0 no yes1 1 0 1 no yes1 1 0 0 no yes1 0 1 1 yes yes1 0 1 0 no yes1 0 0 1 yes yes1 0 0 0 yes yes0 1 1 1 no yes

Calculation Continued.

All gates behaving abnormally is always consistent with the output but not helpful.

Remember that we're interested in minimal sets. By default, nothing is wrong.

So we would prefer just b2 being abnormal As given, we only have a fault model for short

circuits on 'or' and 'xor' gates. So we cannot entail anything about other faults or about other gates.

Possible Extensions

Variables and Quantifiers Could we return variables or predicates rather

than atoms? ie. All canaries rather than Tweety Bird.

Propabilities Could simpler be a part that is more likely to

fail than some other part? Could X make Y likely?

Extensions Continued

Negative Evidence So far we've insisted that explanations entail

the thing to be explained. Could we accept an explanation that does not

entail fever as long as it didn't entail ~fever. IE bird(x) explains flies(x)

Other Applications

Object Recognition Model ^ camera → image. Would like to recover model.

Hypothetical Reasoning What would I have to be told to believe that B

is true? What would I learn if I were told that A is true?

Exercise

Probem 3, pg 283. Consider the binary circuit for logical AND.

Figure 13.3 on page 284.

Exersise

Write sentences describing this circuit – its coomponents, connectivity, and normal behavior.

Write a sentence for a fauilt model saying that a faulty inverter has it's output the same as its input.

Assuming the fault model and that the output is 1 given inputs of 0 and 1, generate the three abductive explanations for this behavior.

Generate the three consistency based diagnoses for thie circuit under the same conditions.

Compare the abductive and consistency based diagnoses and explain informally why they are different.