Relative Expressiveness of Defeasible Logics II Michael Maher

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

Colin Colin is a cassowary All cassowaries are birds Birds fly

Colin Colin is a cassowary All cassowaries are birds Birds fly Colin flies

Colin Colin is a cassowary Cassowaries do not fly

Colin Colin is a cassowary Cassowaries do not fly Colin does not fly

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) <

Colin cassowary(colin) cassowary(X) → bird(X) bird(X)  flies(X) cassowary(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 Republican Quaker pacifist

Ambiguity Nixon Republican Quaker 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 Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual 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 L 1 is more (or equal) expressive than L 2 iff: Inference from any theory D 2 in L 2 can be simulated by inference from another theory D 1 in L 1 D 1 preserves the reasoning structure of D 2, and D 1 can be computed from D 2 in polynomial time

Relative Expressiveness Logic L 1 is more (or equal) expressive than L 2 iff: Inference from any theory D 2 in L 2 can be simulated by inference from another theory D 1 in L 1 D 1 preserves the reasoning structure of D 2, and D 1 can be computed from D 2 in polynomial time Simulation consists of a correspondence between conclusions c 1 of L 1 and conclusions c 2 of L 2 so that D 1 |- c 1 if and only if D 2 |- c 2

Relative Expressiveness Preserving the reasoning structure - indirectly For every defeasible theory A in a class C such that  (D 1 )   (A)   (D 2 )  (D 1 )   (A) = Ø  (D 2 )   (A) = Ø, D 2 +A is simulated by D 1 +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)

Some results Blocked ambiguity infers more than propagated ambiguity But blocked ambguity cannot simulate propagated ambiguity (and vice versa) Team defeat (sometimes) infers more than individual defeat But individual defeat can simulate team defeat (and vice versa)

Relative Expressiveness Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual Defeat Simulation wrt addition of facts Maher 2012

Relative Expressiveness Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual 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 Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual Defeat Simulation wrt addition of facts

Relative Expressiveness Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual Defeat Simulation wrt addition of facts

Relative Expressiveness Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual Defeat Simulation wrt addition of facts

Simulation wrt addition of rules

TD simulates ID wrt rules r 1 : B 1  p  p  C 1 :s 1 r 2 : B 2  p  p  C 2 :s 2 r 3 : B 3  p  p  C 3 :s 3 r 4 : B 4  p  p  C 4 :s 4 r 5 : B 5  p  p  C 5 :s 5

TD simulates ID wrt rules r 1 : B 1  p1  p1  C 1 :s 1 r 1 : B 1  p  p4  C 1 :s 1 r 2 : B 2  p  p1  C 2 :s 2 r 2 : B 2  p  p4  C 2 :s 2 r 3 : B 3  p  p1  C 3 :s 3 r 3 : B 3  p  p4  C 3 :s 3 r 4 : B 4  p  p1  C 4 :s 4 r 4 : B 4  p4  p4  C 4 :s 4 r 5 : B 5  p  p1  C 5 :s 5 r 5 : B 5  p  p4  C 5 :s 5 r 1 : B 1  p  p2  C 1 :s 1 r 2 : B 2  p2  p2  C 2 :s 2 r 3 : B 3  p  p2  C 3 :s 3 r 4 : B 4  p  p2  C 4 :s 4 r 5 : B 5  p  p2  C 5 :s 5 r 1 : B 1  p  p3  C 1 :s 1 r 1 : B 1  p  p5  C 1 :s 1 r 2 : B 2  p  p3  C 2 :s 2 r 2 : B 2  p  p5  C 2 :s 2 r 3 : B 3  p3  p3  C 3 :s 3 r 3 : B 3  p  p5  C 3 :s 3 r 4 : B 4  p  p3  C 4 :s 4 r 4 : B 4  p  p5  C 4 :s 4 r 5 : B 5  p  p3  C 5 :s 5 r 5 : B 5  p5  p5  C 5 :s 5

TD simulates ID wrt rules (AB)  p  ¬p  p1  ¬p1  p2  ¬p2 p1  p ¬ p2  ¬ p

TD simulates ID wrt rules (AB)  p  ¬p  p  p1  ¬p1  p2  ¬p2 p1  p ¬ p2  ¬ p  p

TD simulates ID wrt rules (AB)  p  ¬p  p  p1  ¬p1  p2  ¬p2 p1  p ¬ p2  ¬ p  p Concludes nothingConcludes p

TD simulates ID wrt rules (AB) To patch this problem we add extra rules: For every literal q one(q)  q This rule has lower priority than the rule for ~q So it does not interfere with existing conclusions For every rule B  q B  one(q)

Relative Expressiveness Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual 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 Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual Defeat Simulation wrt addition of rules ≠

Inference Strength Ambiguity Blocks Team Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Ambiguity Blocks Individual 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

Questions?