Relative Expressiveness of Defeasible Logics II Michael Maher.
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Transcript of Relative Expressiveness of Defeasible 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
All cassowaries are birds
Birds fly
Colin flies
Cassowaries do not fly
Colin does not fly
Colin
Colin is a cassowary
All cassowaries are birds
Birds fly, typically
Colin flies
Cassowaries do not fly,
typically
Colin does not fly
Colin
Colin is a cassowary
All cassowaries are birds
Birds fly, typically
Cassowaries do not fly,
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
Colin is a cassowary
All cassowaries are birds
Birds fly, typically
Cassowaries do not fly,
typically
Colin does not fly
cassowary(colin)
cassowary(X) → bird(X)
bird(X) flies(X)
cassowary(X) flies(X)
<
Colin
cassowary(colin)
cassowary(X) → bird(X)
bird(X) flies(X)
cassowary(X) flies(X)
injured(X) flies(X)
<
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 - 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
Relative Expressiveness
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
Relative Expressiveness
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)
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of facts Maher 2012
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
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
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of facts
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of facts
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of facts
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) 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)
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of rules
≠
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
Relative Expressiveness
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
Simulation wrt addition of rules
≠
Inference Strength
Ambiguity BlocksTeam Defeat
Ambiguity PropagatesTeam Defeat
Ambiguity PropagatesIndividual Defeat
Ambiguity BlocksIndividual Defeat
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