Action Theory Contraction and Minimal Change

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Action Theory Contraction and Minimal Change

  1. 1. Action Theory Contraction and Minimal Change Ivan Jos´ Varzinczak e Knowledge Systems Group Meraka Institute CSIR Pretoria, South Africa KR’2008 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 1 / 24
  2. 2. Motivation Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  3. 3. Motivation Knowledge Base ‘A coffee is a hot drink’ ‘With a token I can buy coffee’ ‘After buying I have a hot drink” ... Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  4. 4. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  5. 5. Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  6. 6. Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  7. 7. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  8. 8. Motivation ¬t, c, h c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  9. 9. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  10. 10. Motivation ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  11. 11. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  12. 12. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  13. 13. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  14. 14. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  15. 15. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  16. 16. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 4 / 24
  17. 17. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  18. 18. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  19. 19. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  20. 20. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  21. 21. Action Theories in Dynamic Logic Possible worlds semantics: Transition Systems M = W , R W : possible worlds R : accessibility relation a1 p1 , ¬p2 p1 , p2 a2 M : a2 a1 ¬p1 , p2 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 6 / 24
  22. 22. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  23. 23. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  24. 24. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  25. 25. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  26. 26. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ ± Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  27. 27. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  28. 28. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  29. 29. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  30. 30. Action Theories in Dynamic Logic One model of our scenario example ¬t, c, h b b M : t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 9 / 24
  31. 31. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 10 / 24
  32. 32. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  33. 33. Intuitions About Contraction Contracting coffee → hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  34. 34. Intuitions About Contraction Contracting coffee → hot ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  35. 35. Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  36. 36. Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  37. 37. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  38. 38. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  39. 39. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  40. 40. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  41. 41. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  42. 42. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  43. 43. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  44. 44. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  45. 45. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  46. 46. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  47. 47. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  48. 48. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  49. 49. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  50. 50. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  51. 51. Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
  52. 52. Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
  53. 53. Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h t, c, ¬h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
  54. 54. Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
  55. 55. Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
  56. 56. Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
  57. 57. Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b t, c, h t, ¬c, h t, c, h t, ¬c, h M b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
  58. 58. Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
  59. 59. Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
  60. 60. Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
  61. 61. Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
  62. 62. Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
  63. 63. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  64. 64. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  65. 65. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  66. 66. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 20 / 24
  67. 67. Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  68. 68. Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  69. 69. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  70. 70. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  71. 71. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  72. 72. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  73. 73. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  74. 74. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  75. 75. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  76. 76. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  77. 77. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  78. 78. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  79. 79. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  80. 80. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  81. 81. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  82. 82. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
  83. 83. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
  84. 84. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24

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