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Relative Expressiveness of Defeasible Logics II

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:

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Relative Expressiveness of Defeasible Logics II

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  1. Relative Expressiveness ofDefeasible Logics II Michael Maher

  2. Relative Expressiveness ofDefeasible Logics II Michael Maher

  3. Outline • Defeasible Reasoning • Defeasible Logics • Accrual • Ambiguity • Relative Expressiveness • Results • Tricks

  4. 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

  5. Colin Colin is a cassowary

  6. Colin Colin is a cassowary All cassowaries are birds Birds fly

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

  8. Colin Colin is a cassowary Cassowaries do not fly

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

  10. Colin Colin is a cassowary All cassowaries are birds Birds fly Colin flies Cassowaries do not fly Colin does not fly

  11. Colin Colin is a cassowary All cassowaries are birds Birds fly, typically Colin flies Cassowaries do not fly, typically Colin does not fly

  12. 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)

  13. 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

  14. 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) <

  15. Colin cassowary(colin) cassowary(X) → bird(X) bird(X)  flies(X) cassowary(X)  flies(X) <

  16. Colin < cassowary(colin) cassowary(X) → bird(X) bird(X)  flies(X) cassowary(X)  flies(X) injured(X)  flies(X)

  17. Two orthogonal design choices for defeasible logics Accrual Ambiguity Defeasible Logics

  18. 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

  19. 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

  20. 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

  21. Ambiguity pacifist Quaker Republican Nixon

  22. Ambiguity protests war pacifist middle-aged Quaker Republican Nixon

  23. 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 Ambiguity - dueling principles

  24. Inference Strength For any single theory Infers more Ambiguity Blocks Team Defeat Ambiguity Blocks Individual Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Infers less

  25. Relative Expressiveness

  26. Relative Expressiveness Relative Expressiveness identifies: Similar logics Similar languages Similar behaviours One logic can imitate the other Preserving reasoning structure

  27. 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

  28. 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

  29. Relative Expressiveness Preserving the reasoning structure - indirectly For every defeasible theory A in a class C such 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)

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

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

  32. 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 

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

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

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

  36. Simulation wrt addition of rules

  37. 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

  38. 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 :s4r4: B4 pp5  C4 :s4 r5: B5 pp3  C5 :s5 r5: B5 p5p5  C5 :s5

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

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

  41.  p  ¬p  p  p1  ¬p1  p2  ¬p2 p1  p ¬ p2  ¬ p  p TD simulates ID wrt rules (AB) Concludes nothing Concludes p

  42. 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)

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

  44. 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.

  45. 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

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

  47. Inference Strength For any single theory Infers more Ambiguity Blocks Team Defeat Ambiguity Blocks Individual Defeat Ambiguity Propagates Team Defeat Ambiguity Propagates Individual Defeat Infers less

  48. 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

  49. 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

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