Understanding conditionals through conditional structures - A ...pfn23853/rffcw/ki_e.pdfUnderstanding conditionals through conditional structures A tutorial Gabriele Kern-Isberner,

Department ofComputer Science

Understanding conditionalsthrough conditional structures

A tutorial

Gabriele Kern-Isberner, Christian EichhornAugust 20, 2012

Kern-Isberner / Eichhorn Conditional Structures August 20, 2012 1/58


Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?

1 Motivation

2 Some formal preliminaries

3 Conditional Structures

4 Reasoning with conditional structures

5 Properties of structural inference

6 Going beyond structural inference – c-representations

7 Conditional structures and probabilities



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?What conditionals are, and why we should care about them

1 Motivation










Conditionals

A conditional (B|A) is understood as the uncertain rule“If A, then usually B”.

A probabilistic conditional (B|A)[x] is understood as the rule“If A, then the probability that B holds is expected to be x”.

Example (flying birds)Let b = “bird”, f = “(being able to) fly” , x ∈ [0, 1]

(f |b) “Birds usually fly.”(f |b) [0.9] “Birds fly with a probability of 90 %.”




What are conditionals used for?

Conditionals are rules with exceptions:(Usually,) Philosophers are wise.(However, usually,) Philosophers from the StrangeUniversityare stupid.(But,) Philosopher WiseMan from the StrangeUniversity isvery wise.

A more up-to-date and (maybe very) relevant example:(Usually,) European countries are economically strong.. . .




Why conditionals?

Indeed, some (most?) of the most complicated problems makeuse of conditionals, and it can be of crucial importance to evaluatesituations with respect to such conditionals, e.g., for comparingsituations where the Euro crashes to those where it does not.

But conditionals are also important for our everyday lives:Usually, for buying food I go to the supermarket. But for buyingmeat, I go to the butcher’s, except when it is late in the evening.




Why is classical logic not enough, . . .

Classical logic just knows true and false, not plausible vs.implausible, most vs. some, etc..

So, if you say

All philosphers are wise.philosopher ⇒ 1 wise

∀x philosopher(x) ⇒ wise(x),

you really have to mean that – there is no way to allow exceptions!

1⇒ means material implication, i.e., A⇒ B ≡ ¬A ∨ B.Kern-Isberner / Eichhorn Conditional Structures August 20, 2012 7/58



. . . and what makes conditionals so special?

A conditional (B|A) focusses on cases where the premise A isfulfilled but does not say anything about cases when A does nothold.

It leaves semantical room for modelling acceptance in case itsconfirmation A ∧ B is more plausible than its refutationA ∧ ¬B.




What are the fallacies of conditional reasoning? I

Conditional reasoning sometimes appears to be fallacious sincemost inference rules (like syllogism, modus ponens, contrapositionetc.) do not hold for conditionals:

Example (Syllogism, classical)A⇒ B, B ⇒ C

A⇒ Cpenguins ⇒ birds, birds ⇒ animals

penguins ⇒ animalsGiven this, we evaluate the statement/situation A ∧ B ∧ C to bemore compliant with reality than A ∧ B ∧ ¬C .




What are the fallacies of conditional reasoning? II

Example (Syllogism, conditionals)However,

penguins are (usually) birds, birds (usually) flypenguins (usually) fly

does not hold!

On the basis of the (uncertain) knowledge about penguins, birds,and flying, how can we evaluate the situationpenguin ∧ bird ∧ ¬flying to be more plausible thanpenguin ∧ bird ∧ flying?

N.B.: The fallacies are not due to the conditionals but to theirimproper usage in classical schemata.



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?Logic and Conditionals Knowledge bases

1 Motivation

2 Some formal preliminariesLogic and ConditionalsKnowledge bases








Logic and possible worlds

We use a propositional alphabet Σ generating the propositionallanguage L with the connectors ∧,∨,¬ in the usual way.

AbbreviationsA ∧ B ≡ AB ¬A ≡ A

Possible worlds ω ∈ Ω correspond to interpretations and aredefined in the usual way.

Example

For Σ = a, b, c, Ω =

abc, abc, abc, ab c, abc, abc, a bc, a b c

.




Evaluation of conditionalsA conditional (B|A) is understood as the uncertain rule“If A, then usually B”.

Example (flying birds)Let b = “bird”, f = “(being able to) fly”,

(f |b) “Birds usually fly.”

J(B|A)Kω =

1 if ω |= AB (ω satisfies A and B) (verification)0 if ω |= AB (ω satisfies A and not B) (falsification)u if ω |= A (ω satisfies not A)

Definition (Language of conditionals)The language of conditionals is defined as (L | L)




Evaluation of conditionals Examples

Let Σ = p, b, f .

J(b|p)Kpbf = 1 J(b|p)Kpbf = 1 J(b|p)Kpbf = 0 J(b|p)Kpb f = 0J(b|p)Kpbf = u J(b|p)Kpbf = u J(b|p)Kp bf = u J(b|p)Kp b f = u

J(f |b)Kpbf = 1 J(f |b)Kpbf = 0 J(f |b)Kpbf = u J(f |b)Kpb f = uJ(f |b)Kpbf = 1 J(f |b)Kpbf = 0 J(f |b)Kp bf = u J(f |b)Kp b f = u

J(f |p)Kpbf = 0 J(f |p)Kpbf = 1 J(f |p)Kpbf = 0 J(f |p)Kpb f = 1J(f |p)Kpbf = u J(f |p)Kpbf = u J(f |p)Kp bf = u J(f |p)Kp b f = u




Knowledge base

A knowledge base is a (here: finite) set of conditionals

∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L).

Idea:∆ shall represent (all of) the reasoner’s knowledge.Knowlege is “rule-based”.Reasoning shall be (solely) based on ∆.




Knowledge base Simple penguin example

The knowlege base ∆ =

(f |b), (f |p), (b|p)

shall represent ourknowledge about penguins, birds and the capability of flying. Weknow, that. . .

(f |b) . . . birds usually fly.(f |p) . . . penguins usually don’t fly.(b|p) . . . penguins are usually birds.




Motivation Examples of exceptions

Penguins are usually Birds (?)(b|p)

Penguins do not fly (?)(f |p)



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?Basic ideas The formal machinery The complete definition

1 Motivation


3 Conditional StructuresBasic ideasThe formal machineryThe complete definition







Conditional structures Basic ideas

Let ∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L).

The conditional structure of a world ω shall represent theeffect of each conditional in ∆ to ω.This effect shall be independent of the context in which aconditional is applied, i.e., it should not matter which otherconditionals in ∆ are applied or not.This effect shall also be independent of the order in whichthe conditionals are applied to ω.Conditional structures should be comparable to serve as basefor a preference relation on worlds; this preference relationshall express which world is more plausible than another.




Introducing symbols a+, a−

Let ∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L). For each conditional(Bi |Ai) ∈ ∆ we define two abstract symbols, a+

i and a−i such thata+

i is linked to the verification AiBi of (Bi |Ai).a−i is linked to the falsification AiBi of (Bi |Ai).

We express this by the following function2:

Definition (σi for each (Bi |Ai))

σi : Ω→ a+i ,a−i , 1 σi(ω) =

a+

i iff ω |= AiBia−i iff ω |= AiBi1 iff ω |= Ai

2“iff” means “if and only if”Kern-Isberner / Eichhorn Conditional Structures August 20, 2012 20/58



ExamplesLet ∆ =

(f |b)︸︷︷︸

1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

.

σ1(pbf ) = a+1 σ1(pbf ) = a−1 σ1(pbf ) = 1 σ1(pb f ) = 1

σ1(pbf ) = a+1 σ1(pbf ) = a−1 σ1(p bf ) = 1 σ1(p b f ) = 1

σ2(pbf ) = a−2 σ2(pbf ) = a+2 σ2(pbf ) = a−2 σ2(pb f ) = a+

2

σ2(pbf ) = 1 σ2(pbf ) = 1 σ2(p bf ) = 1 σ2(p b f ) = 1

σ3(pbf ) = a+3 σ3(pbf ) = a+

3 σ3(pbf ) = a−3 σ3(pb f ) = a−3σ3(pbf ) = 1 σ3(pbf ) = 1 σ3(p bf ) = 1 σ3(p b f ) = 1




Free abelian3 group

In order to be able to “calculate” with the a−1 ,a+1 , we define

(F∆, ·) as the free abelian group for∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L) with

a−1 ,a+1 , . . . ,a−n ,a+

n

as a basis,

· resp. juxtaposition as multiplication, and1 as the neutral element

F∆ is simply the set of all products of the a−i ,a+i without

cancellations, i.e., no effect of one conditional, either positive ornegative, can be simulated by the effect of another conditional.

3“abelian” means “commutative”Kern-Isberner / Eichhorn Conditional Structures August 20, 2012 22/58



Conditional structureConditional structure shall be defined as the combination of alleffects of all conditionals in ∆:

Definition (conditional structure σ(ω))Let ∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L). The conditionalstructure of a world ω ∈ Ω is defined asa

σ(ω) =n∏

i=1σi(ω) =

∏ω|=AiBi

a+i ·

∏ω|=AiBi

a−i

a∏ is the product symbol, i.e.,∏n

i=1 ai = a1 · . . . · an

Of cause every σ(ω) ∈ F; due to the properties of an abeliangroup, the order of the symbols a+/−

i is irrelevant, and factors 1can be ignored.




Example for conditional structures

Let ∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

.

σ(p b f ) = a+1 a−2 a+

3 σ(p b f ) = a+1 · 1 · 1

σ(p b f ) = a−1 a+2 a+

3 σ(p b f ) = a−1 · 1 · 1σ(p b f ) = 1 · a−2 a−3 σ(p b f ) = 1 · 1 · 11σ(p b f ) = 1 · a+

2 a−3 σ(p b f ) = 1 · 1 · 11



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?An idea of preference Preference relation Inference relation



4 Reasoning with conditional structuresAn idea of preferencePreference relationInference relation







Conditional structures Review of basic ideas

Let ∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L).

X The conditional structure of a world ω shall represent theeffect of each conditional in ∆ to ω.

X This effect shall be independent of the context in which aconditional is applied, i.e., it should not matter which otherconditionals in ∆ are applied or not.

X This effect shall also be independent of the order in whichthe conditionals are applied to ω.

? Conditional structures should be comparable to serve as basefor a preference relation on worlds; this preference relationshall express which world is more plausible than another (forevaluating situations).




Structural preference Basic ideas

Reminder: A conditional (B|A) is interpreted as the defeasible rule“if A then usually B”.

A world falsifying a conditional defies the associated rule.Defying rules is bad.Ceteris paribus, we want to prefer worlds that do not defy agiven rule to those worlds that do.




Structural preference The definition

Let ∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L), ω, ω′ possible worlds

A conditional structure σ(ω) is preferred to a conditional structureσ(ω′) (written as σ(ω) ≺σ σ(ω′)) iff

σ(ω′) falsifies every conditional falsified by σ(ω) andσ(ω′) falsifies at least one conditional more than σ(ω).

Definition (σ(ω) ≺σ σ(ω′))

σ(ω) ≺σ σ(ω′) iffσi(ω) = a−i implies σi(ω′) = a−i for all 1 ≤ i ≤ nthere is an i such that σi(ω) ∈

a+

i , 1

and σi(ω′) = a−i




Example for structural preference

∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

simple penguins knowledge base.

σ(p b f ) = a+1 a−2 a+

3 σ(p b f ) = a−1 a+2 a+

3 σ(p b f ) = a−2 a−3 σ(p b f ) = a+2 a−3

σ(p b f ) = a+1 σ(p b f ) = a−1 σ(p b f ) = 1 σ(p b f ) = 1

E.g., we obtain the following preferences

σ(pb f ) = a+2 a−3 ≺σ a−2 a−3 = σ(pbf )

σ(pbf ) = a+1 ≺σ a+

1 a−2 a+3 = σ(pbf )

σ(pbf ) = 1 ≺σ a+2 a−3 = σ(pb f )

σ(p b f ) = 1 ≺σ a−2 a−3 = σ(pbf )




Preference relation on worlds

The preference relation on conditional structures induces apreference relation on worlds:

Definition (Preference relation on worlds)We prefer a world ω to a world ω′ iff the conditional structure of ωis preferred to the conditional structure of ω′, i.e.

ω <σ ω′ iff σ(ω) ≺σ σ(ω′).




Inference relation based on conditional structures

Now, we use this preference relation for inference, i.e., to entail(new) plausible knowledge:

Let A,B ∈ L. We structurally infer B from A (A|∼σ∆B) iff forevery world that falsifies (B|A) there is a preferred world thatverifies (B|A). Formally:

Definition (Structural inference by preferential semantics, A|∼σ∆B)

A|∼σ∆B iff for all ω′ |= AB there is an ω |= AB with ω <σ ω′.

In this case, the conditional (B|A) is accepted by |∼σ∆.

Structural inference follows the lines of preferential entailment[Makinson 1994].




Examples for structural inference

Let ∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

.

σ(p b f ) = a+1 a−2 a+

3 σ(p b f ) = a−1 a+2 a+

3 σ(p b f ) = a−2 a−3 σ(p b f ) = a+2 a−3


E.g., we may infer:pb|∼σ∆f since pb f <σ pbfbf |∼σ∆p since pbf <σ pbfb|∼σ∆f since pbf <σ pbf and pbf <σ pbf



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?Handling of irrelevant information Limitations of structural inference




5 Properties of structural inferenceHandling of irrelevant informationLimitations of structural inference






Irrelevant information Example

Which influence has information that is obviously irrelevant since itis not mentioned in the knowledge base?

We enlarge our alphabet with r for being red to Σ′ = p, b, f , r.

ω σ(ω) ω σ(ω) ω σ(ω) ω σ(ω)p b f r a+

1 a−2 a+3 p b f r a−2 a−3 p b f r a+

1 p b f r 1p b f r a+

1 a−2 a+3 p b f r a−2 a−3 p b f r a+

1 p b f r 1p b f r a−1 a+

2 a+3 p b f r a+

2 a−3 p b f r a−1 p b f r 1p b f r a−1 a+

2 a+3 p b f r a+

2 a−3 p b f r a−1 p b f r 1

Obviously, r (being true or false) has no influence on theconditional structures of worlds.




Irrelevant information

Let’s look at inferences – two possibilities:Can the additional information r invalidate inferences, e.g.,we could conclude b|∼σ∆f , but can we also conclude that redbirds fly – br |∼σ∆f ?Can the additional information r induce new inferences, e.g.,are we now able to conclude that birds are red – b|∼σ∆r?

N.B.: The first problem touches monotony but |∼σ∆ is known tobe nonmonotonic. However, one of the most crucial questions ofnonmonotonic reasoning is when exactly should we expect amonotonic behaviour even for nonmonotonic inference relations ?




Irrelevant information Example, continued

Are red birds capable of flying? Formally: does br |∼σ∆f hold?

ω σ(ω) ω σ(ω) ω σ(ω) ω σ(ω)p b f r a+

1 a−2 a+3 p b f r a−2 a−3 p b f r a+

1 p b f r 1p b f r a+

1 a−2 a+3 p b f r a−2 a−3 p b f r a+

1 p b f r 1p b f r a−1 a+

2 a+3 p b f r a+

2 a−3 p b f r a−1 p b f r 1p b f r a−1 a+

2 a+3 p b f r a+

2 a−3 p b f r a−1 p b f r 1

σ(p b f r) = a+1 ≺σ a−1 a+

2 a+3 = σ(p b f r) X

σ(p b f r) = a+1 ≺σ a−1 = σ(p b f r) X

br |∼σ∆f holds, red birds can (also) fly.

So, information r proves to be truely irrelevant (as expected).




Irrelevant information Example, continued

Are birds usually red? Formally: does b|∼σ∆r hold?ω σ(ω) ω σ(ω) ω σ(ω) ω σ(ω)

p b f r a+1 a−2 a+

3 p b f r a−2 a−3 p b f r a+1 p b f r 1

p b f r a+1 a−2 a+

3 p b f r a−2 a−3 p b f r a+1 p b f r 1

p b f r a−1 a+2 a+

3 p b f r a+2 a−3 p b f r a−1 p b f r 1

p b f r a−1 a+2 a+

3 p b f r a+2 a−3 p b f r a−1 p b f r 1

σ(p b f r) = a+1 ≺σ a+

1 a−2 a+3 = σ(p b f r) X

σ(p b f r) = a+1 ≺σ a−1 a+

2 a+3 = σ(p b f r) X

σ(p b f r) = a+1 ≺σ a−1 = σ(p b f r) X

∅ = ≺σ a+1 = σ(p b f r)

Does b|∼σ∆r hold? – NO; for the same reasons b 6|∼σ∆r , too, so:Are birds usually red? Unknown X.




Limitations of (purely) structural inference

∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

simple penguin knowledge base

σ(p b f ) = a+1 a−2 a+

3 σ(p b f ) = a−1 a+2 a+

3 σ(p b f ) = a−2 a−3 σ(p b f ) = a+2 a−3


<σ is no total ordering: e.g. pbf and pbf are incomparable⇒ We are not able to conclude p 6|∼σ∆f !!

I.e. A|∼σ∆B does not hold for all (B|A) ∈ ∆ !!




Drowning problem (example)

The drowning problem addresses the problem that exceptionalsubclasses (like penguins) are treated as exceptions throughout,i.e., they may not inherit other typical properties of the superclass.In our simple penguin example, we add the rule that birds usuallyhave wings (w|b) to ∆,

so let ∆′ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

, (w|b)︸︷︷︸4

.

The question now is:

Can we conclude that (also) penguins have wings?

N.B.: This addresses transitivity resp. the validity of syllogism.




Drowning problem (example, continued)

ω σ(ω) ω σ(ω) ω σ(ω) ω σ(ω)p b f w a+

1 a−2 a+3 a+

4 p b f w a−2 a−3 p b f w a+1 a+

4 p b f w 1p b f w a+

1 a−2 a+3 a−4 p b f w a−2 a−3 p b f w a+

1 a−4 p b f w 1p b f w a−1 a+

2 a+3 a+

4 p b f w a+2 a−3 p b f w a−1 a+

4 p b f w 1p b f w a−1 a+

2 a+3 a−4 p b f w a+

2 a−3 p b f w a−1 a−4 p b f w 1

Do penguins have wings?

σ(p b f w) = a+1 a−2 a+

3 a+4 ≺σ a+

1 a−2 a+3 a−4 = σ(p b f w) X

σ(p b f w) = a−1 a+2 a+

3 a+4 ≺σ a−1 a+

2 a+3 a−4 = σ(p b f w) X

σ(p b f w) = a+2 a−3 ≺σ a−2 a−3 = σ(p b f w) X

∅ ≺σ a+2 a−3 = σ(p b f w)




Drowning problem (example, continued 2)

Why are there no worlds ω |= pw with ω <σ p b f w?Remember: σ(p b f w) = a+

2 a−3 .

ω |= pw why ω 6<σ p b f wp b f w a−2 vs. a+

2p b f w a−1 vs. 1 (= σ1(p b f w))p b f w a−2 a−3 >σ a+

2 a−3p b f w a+

2 a−3 6<σ a+2 a−3

Do penguins have wings?

unknown

So, penguins do not inherit the flying-property of their superclass“birds”.




Drowning problem vs. transitivity

Maybe structural preference is too weak to handle transitiveconclusions by chaining rules?We extend our penguin-example by the conditional (a|f ) – flyingobjects are (usually) airborne:

∆′′ =

(f |b), (f |p), (b|p), (w|b), (a|f ).

Here, we find:b|∼σ∆′′ a,

i.e., (f |b), (a|f ) can be chained successfully.N.B.: This does not involve penguins explicitly!



Motivation Preliminaries Conditional Structures Reasoning with C.S. Inference-Properties C-Representations Probabilities?Motivation The link to OCFs C-representations solve existing problems




6 Going beyond structural inference – c-representationsMotivationThe link to OCFsC-representations solve existing problems


8 Conclusion




Motivation

Conditional structures and structural inference prove to beadequate to capture formally the essence of conditionals. However,problems arise because we are not able to compare falsifications ofdifferent conditionals.

In order to overcome the limitations of structural inference we needto weigh the “severeness of falsification” of conditionals, encodingthe idea that the less plausible an exception to a conditional is, thehigher should be the penalty for falsifying the conditional.




Motivation

Example(f |b): exceptions are easy to imagine, non-flying birds are e.g.penguins, dodos, kiwis, ostriches,. . .(b|p): penguins which aren’t birds are quite far-fetched(f |p): flying penguins seem to be very implausible

. . . but we saw such exceptions . . .




Ranking Functions

Conditional structures can be easily combined with OCFs:An Ordinal Conditional Function (OCF) or ranking function κ[Spohn 1988] is a function that assigns a degree ofdisbelief/implausibility to any world ω ∈ Ω.

Definition (κ)κ := Ω→ N∞0 such that:

κ−1(0) 6= ∅κ(A) = min

ω|=Aκ(ω)

κ(B|A) = κ(AB)− κ(A)

Example (ranked flyers)

κ(ω) = 0

κ(ω) = 1

κ(ω) = 2

κ(ω) = 4

p bf p b f p b f

pbf p bf

pbf pb f

pb f




C-representationsA c-representation of ∆ is an OCF with ranks calculated on thebasis of conditional structures [GKI 2001].

Definition (c-representation)

κc∆ : Ω→ N∞0 κc

∆(ω) =∑

1≤i≤nω|=AiB i

κ−i

(i.e., a−i ∼ κ−i , a+i ∼ 0), with κ−i chosen in such a way that

κc∆ |= ∆, i.e.,

κ−i > minω|=AiBi

∑

ω|=AjBji 6=j

κ−j

− minω|=AiBi

∑

ω|=AjBji 6=j

κ−j




c-representations Example: minimal c-representation

c-representation of ∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

(κ−1 = 1, κ−2 = κ−3 = 2)

ωfalsified

κc∆(ω)conditionals

p b f 2 κ−2 = 2p b f 1 κ−1 = 1p b f 2, 3 κ−2 + κ−3 = 4p b f 3 κ−3 = 2p b f 1 κ−1 = 1p b f — 0p b f — 0p b f — 0

κc∆(ω)

0

1

2

4

p bf p b f p b f

pbf p bf

pbf pb f

pb f




Inference with c-representations

Definition (Inference relation |∼c∆)

A|∼c∆ B iff. κc

∆(AB) < κc∆(AB ) iff. κc

∆ |= (B|A)

Example (are not flying penguins birds in c-representation?)

pf |∼c∆ b

κ(pbf ) = 1 < 2 = κ(pb f )

0

1

2

4

p bf p b f p b f

pbf p bf

pbf pb f

pb f




Drowning problem (example)

We add the rule that birds usually have wings (w|b) to ∆, so let

∆ =

(f |b)︸︷︷︸1

, (f |p)︸︷︷︸2

, (b|p)︸︷︷︸3

, (w|b)︸︷︷︸4

, (κ−1 = 1, κ−2 = 2, κ−3 = 2, κ−4 = 1)

ω κc∆(ω) ω κc

∆(ω) ω κc∆(ω) ω κc

∆(ω)p b f w κ−2 = 2 p b f w κ−2 + κ−3 = 4 p b f w 0 p b f w 0p b f w κ−2 + κ−4 = 1 p b f w κ−2 + κ−3 = 4 p b f w κ−4 = 1 p b f w 0p b f w κ−1 = 1 p b f w κ−3 = 2 p b f w κ−1 = 2 p b f w 0p b f w κ−1 + κ−4 = 2 p b f w κ−3 = 2 p b f w κ−1 + κ−4 = 2 p b f w 0

Do penguins have wings? Yes, because

κ(pw) = minω|=pw

κc∆(ω) = 1 < 2 = min

ω|=pwκc

∆(ω) = κ(pw)









8 Conclusion




Maximum Entropy

MaxEnt Principle: Maximise indeterminacy (i.e. entropy)

H (P) = −∑ω∈Ω

P(ω) · log2 P(ω)

of a probability distribution P givenR = (B1|A1)[x1], . . . , (Bn |An)[xn ],i.e., solve the optimisation problem

(arg) maxP|=R

H (P) = −∑ω∈Ω

P(ω) · log2 P(ω)

to get the probabilistic model P of R which adds as littleinformation as possible.




Conditional structures and maximum entropy

Lagrangre-function of the (above) optimisation problem

−→ ME(R)(ω) = α0 ·∏

1≤i≤nω|=AiBi

α1−xii ·

∏1≤i≤nω|=AiBi

α−xii

Compare this to conditional structures:

σ(ω) =∏

1≤i≤nω|=AiBi

a+i ·

∏1≤i≤nω|=AiBi

a−i

−→ ME(R) is a probabilistic c-representation of R !




Conclusion

Conditional structuresMake the effect of conditionals in a knowledge base explicitand transparentStructural inference and inductive reasoning from knowledgebasesHowever, have limitations since falsifications of differentconditionals cannot be handled

C-representations are based on conditional structures:Allow to associate conditionals with “severeness offalsification”Solve drowning problemComply with high-standard inference properties (system P, Retc.)Idea can also be used for belief revision




Bibliography

Gabriele Kern-Isberner. Conditionals in Nonmonotonic Reasoning and Belief Revision – ConsideringConditionals as Agents, volume 2087. Springer, 2001.

David Makinson. General patterns in nonmonotonic reasoning. In Dov M. Gabbay, C. J. Hogger, and J. A.Robinson, editors, Handbook of logic in artificial intelligence and logic programming, volume 3, pages35–110. Oxford University Press, Inc., New York, NY, USA, 1994. ISBN 0-19-853747-6.

Wolfgang Spohn. Ordinal conditional functions: A dynamic theory of epistemic states. Springer, August31, 1988. ISBN 9027726345. 105–134 pp.



Group theory

(Mathematical) Groups

A Group (G, ∗) consists of a set of group-elements G and a binaryoperation ∗ satisfying the following axioms:

Associativity: For all a, b, c ∈ G it is a ∗ (b ∗ c) = (a ∗ b) ∗ c.

Neutral element:There is an e ∈ G such that for all a ∈ G: a ∗ e = a = e ∗ a.

Inverse element:For all a ∈ G there is an b ∈ G such that a ∗ b = e = b ∗ a



Group theory

(Mathematical) Groups Examples

The set of integers Z and addition (+) build a group:

Associativity: 2 + (3 + 4) = (2 + 3) + 4Neutral Element (0): 5 + 0 = 5 = 0 + 5Inverse Element: 2 + (−2) = 0 = (−2) + 2

The set of rational numbers Q and multiplication (·) build a group:

Associativity: 2 · (3 · 4) = (2 · 3) · 4Neutral Element (1): 5 · 1 = 5 = 1 · 5Inverse Element: 2 · 1/2 = 1 = 1/2 · 2



Group theory

Further properties of groups

Definition (Abelian groups)A group (G, ∗) is called abelian iff ∗ is commutative, i.e. for alla, b ∈ G it is a ∗ b = b ∗ a.

Definition (Free abelian groups)A abelian group (G, ∗) is called a free abelian group iff the grouphas a “basis”, i.e. each element of G can be written as linearcombination of the basis (base elements are also called generators).

Example (Free abelian groups)(Z,+) is a free abelian group with base 1.(Q, ·) is a free abelian group with the prime numbers as base.



Group theory

Free abelian group

We define (F, ·)∆ as the free abelian group for∆ = (B1|A1), . . . , (Bn |An) ⊆ (L | L) with

a−1 ,a

+1 , . . . ,a−n ,a+

n

as base,

· (“usual” multiplication ) as binary relation and1 as neutral element(F is the set of all linear combinations of the base).


Documents

Understanding conditionals through conditional structures - A ...pfn23853/rffcw/ki_e.pdfUnderstanding conditionals through conditional structures A tutorial Gabriele Kern-Isberner,