Relative Expressiveness of Defeasible Logics II

Relative Expressiveness ofDefeasible Logics II

Michael Maher

Outline

Defeasible Reasoning Defeasible Logics

Accrual Ambiguity

Relative Expressiveness Results

Tricks

Defeasible Reasoning Drawing conclusions when:

Arguments conflict Statements are inconsistent Statements are not certain - perhaps rule-of-thumb

Computational formalizations of regulations business rules contracts high-school biology

Colin

Colin is a cassowary

Colin


All cassowaries are birds

Birds fly

Colin



Birds fly

Colin flies

Colin


Cassowaries do not fly

Colin



Colin does not fly

Colin



Birds fly

Colin flies


Colin does not fly

Colin



Birds fly, typically

Colin flies

Cassowaries do not fly,

typically

Colin does not fly

Colin





typically

Colin does not fly

cassowary(colin)

cassowary(X) → bird(X)

bird(X) flies(X)

cassowary(X) flies(X)

Defeasible Reasoning There are several principles that permit an inference to

over-rule another specificity recent law over-rules an older law Constitution over-rules legislation ...

We use a partial ordering > on rules as a general mechanism to express that one rule can over-ride another

Colin





typically

Colin does not fly

cassowary(colin)


bird(X) flies(X)


<

Colin

cassowary(colin)


bird(X) flies(X)


<

Colin

cassowary(colin)


bird(X) flies(X)


injured(X) flies(X)

<

Defeasible Logics Two orthogonal

design choices for defeasible logics Accrual Ambiguity

Accrual When one argument over-rides all competing

arguments, it should win But what should happen when there are multiple

arguments on both sides, without a single argument winning? Complicated

A simple case: If every argument on one side is over-ridden by an argument on

the other side, then

the other side, considered as a team, wins

Accrual: team defeat

R1: monotreme mammal

R2: has_fur mammal

R3: lays_eggs ¬ mammal

R4: webbed_feet ¬ mammal

R1 > R3

R2 > R4

monotreme.

has_fur.

lays_eggs.

webbed_feet.

Accrual: team defeat

R1: monotreme mammal

R2: has_fur mammal

R3: lays_eggs ¬ mammal

R4: webbed_feet ¬ mammal

R1 > R3

R2 > R4

monotreme.

has_fur.

lays_eggs.

webbed_feet.A platypus is a mammal

Ambiguity

Nixon

RepublicanQuaker

pacifist

Ambiguity

Nixon

RepublicanQuaker

pacifist

middle-aged

protests war

Ambiguity - dueling principles Block ambiguity

We already agreed that we cannot conclude that Nixon is a pacifist

So, the argument that Nixon protests war is invalid So, there is no competition to the argument that Nixon does not

protest war

Propagate ambiguity There is a possibility that Nixon is a pacifist So the argument that Nixon protests war cannot be discounted So we draw no conclusion about Nixon protesting war

Inference Strength

Ambiguity BlocksTeam Defeat

Ambiguity PropagatesTeam Defeat

Ambiguity PropagatesIndividual Defeat

Ambiguity BlocksIndividual Defeat

Infers more

Infers less

For any single theory

Relative Expressiveness


Relative Expressiveness identifies: Similar logics

• Similar languages• Similar behaviours

One logic can imitate the other• Preserving reasoning structure


Logic L1 is more (or equal) expressive than L2 iff: Inference from any theory D2 in L2 can be simulated by

inference from another theory D1 in L1

D1 preserves the reasoning structure of D2, and

D1 can be computed from D2 in polynomial time


Logic L1 is more (or equal) expressive than L2 iff: Inference from any theory D2 in L2 can be simulated by

inference from another theory D1 in L1

D1 preserves the reasoning structure of D2, and

D1 can be computed from D2 in polynomial time

Simulation consists of a correspondence between conclusions c1 of L1 and conclusions c2 of L2 so that D1 |- c1 if and only if D2 |- c2

Relative Expressiveness Preserving the reasoning structure - indirectly

For every defeasible theory A in a class Csuch that (D1) (A) (D2)

(D1) (A) = Ø

(D2) (A) = Ø,

D2+A is simulated by D1+A

C can be: Set of all defeasible theories Defeasible theories consisting only of rules Defeasible theories consisting only of facts The empty theory (equivalence)






Simulation wrt addition of facts Maher 2012






Simulation wrt addition of facts

AB simulates AP The ambiguity propagating logics employ three inference levels

definite conclusions defeasible conclusions support: a very weak evidence for a conclusion

The simulating theory derives three kinds of conclusions strict(q) q supp(q)

which are all reasoned with as defeasible knowledge

The simulating theory reflects the inference rules for , and






Simulation wrt addition of facts

Simulation wrt addition of rules

TD simulates ID wrt rules

r1: B1 p p C1 :s1

r2: B2 p p C2 :s2

r3: B3 p p C3 :s3

r4: B4 p p C4 :s4

r5: B5 p p C5 :s5

TD simulates ID wrt rules r1: B1 p1 p1 C1 :s1 r1: B1 p p4 C1 :s1

r2: B2 p p1 C2 :s2 r2: B2 p p4 C2 :s2

r3: B3 p p1 C3 :s3 r3: B3 p p4 C3 :s3

r4: B4 p p1 C4 :s4 r4: B4 p4 p4 C4 :s4

r5: B5 p p1 C5 :s5 r5: B5 p p4 C5 :s5

r1: B1 p p2 C1 :s1

r2: B2 p2 p2 C2 :s2

r3: B3 p p2 C3 :s3

r4: B4 p p2 C4 :s4

r5: B5 p p2 C5 :s5

r1: B1 p p3 C1 :s1 r1: B1 p p5 C1 :s1

r2: B2 p p3 C2 :s2 r2: B2 p p5 C2 :s2

r3: B3 p3 p3 C3 :s3 r3: B3 p p5 C3 :s3

r4: B4 p p3 C4 :s4 r4: B4 p p5 C4 :s4

r5: B5 p p3 C5 :s5 r5: B5 p5 p5 C5 :s5

TD simulates ID wrt rules (AB)

p

¬p

p1

¬p1

p2

¬p2

p1 p

¬ p2 ¬ p


p

¬p

p

p1

¬p1

p2

¬p2

p1 p

¬ p2 ¬ p

p


p

¬p

p

p1

¬p1

p2

¬p2

p1 p

¬ p2 ¬ p

p

Concludes nothing Concludes p

TD simulates ID wrt rules (AB) To patch this problem we add extra rules:

For every literal qone(q) q

This rule has lower priority than the rule for ~q So it does not interfere with existing conclusions

For every rule B qB one(q)







≠

ID simulates TD wrt rules (AP)

Ambiguity propagating logics employ both and Rules in simulating theory are used by both inference rules Devise rules only useful for

….., g, ¬g q

g

¬g inferences are also valid inferences [Billington etal, 2010]

so no rules only useful for are needed.

ID simulates TD wrt rules (AP) Ambiguity propagating logics employ both and

Need to identify strict conclusions

q strict(q)

<

¬ strict(q)

strict(q) true(q)

>

¬ true(q)

Infers true(q) iff infers true(q) iff infers q







≠

Inference Strength





Infers more

Infers less

For any single theory

Conclusions Simulation wrt addition of facts is less discriminating

than it first appears But it confirms the similarity of the logics

Team defeat is no more expressive than individual defeat Probably doesn’t extend to other forms of accrual

Different treatments of ambiguity have different expressiveness Probably extends to other defeasible reasoning formalisms

Relative expressiveness is only weakly related to inference strength

Future work Relative expressiveness is a tool for comparing the

many defeasible reasoning formalisms Nute and Maier’s defeasible logics Plausible logics Courteous logic programs Ordered logic programs LP without NAF Argumentation systems

Questions?

Relative Expressiveness of Defeasible Logics II

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