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The “Logic” of Reachability David E. Smith Ari K. Jónsson

The “Logic” of Reachability David E. Smith Ari K. Jónsson. Apologies. No results ideas & formalism Adverse reactions “Logic”. Outline. Background & Motivation Simple Reachability Mutual Exclusion “Practical Matters”. Expand plan graph Derive mutex relationships

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The “Logic” of Reachability David E. Smith Ari K. Jónsson

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  1. The “Logic” of ReachabilityDavid E. SmithAri K. Jónsson

  2. Apologies • No results • ideas & formalism • Adverse reactions • “Logic”

  3. Outline Background & Motivation Simple Reachability Mutual Exclusion “Practical Matters”

  4. Expand plan graph • Derive mutex relationships • If goals are present & consistent • search for a solution Graphplan

  5. Expand plan graph • Derive mutex relationships • If goals are present & consistent • search for a solution Graphplan Reachability! (optimistic achivability)

  6. Why Reachability? • Pruning • ¬reachable  ¬achievable • Guidance • distance

  7. TGP • Actions • Real duration • Concurrent Heater Thrust closevalve comlink

  8. TGP Limitations • Actions • Preconditions hold throughout • Effects occur at end • Affected propositions undefined during • No exogenous conditions pre1 pre2 A eff1 eff2

  9. Monotonicity of Reachability 0 1 2 3 x p q ¬x r … x p q ¬x … x p q ¬x … A A A B B B ¬x … C Propositions & actions monotonically increase

  10. Monotonicity of Mutex 0 1 2 3 x p q ¬x r … x p q ¬x … x p q ¬x … A A A B B B ¬x … C Mutex relationships monotonically decrease

  11. Cyclic Plan Graph Propositions Actions x1 p1 q1 ¬x0 r3 … A0 B0 C2 Earliest start times

  12. Cyclic Plan Graph Propositions Actions x1 p1 q1 ¬x0 r3 … A0 B0 C2 Earliest end time 2 2

  13. ≥5A pre1 pre2 cond3 A eff A –5A +5A Impact? • Actions • Preconditions hold throughout • Effects occur at end • Affected propositions undefined during • Exogenous Conditions t=0600z t=1300z Closed(SJC)

  14. Windows of Reachability Actions Propositions A[0,3],[6,9] B[11,] C[…]… … p[0,5],[8.1,16] q[2,17]… r[3,]… …

  15. Windows of Mutex Actions Propositions [3,4]x[11,] A[0,3],[6,9] B[11,] C[…]… … p[0,5],[8.1,16] q[2,17]… r[3,]… … [0,3]x[11,] [0,3]x[3,4]

  16. ≥5A r Action Model cond1 Duration Parallel (pre) Conditions over intervals Effects over intervals cond2 cond3 A eff A +5A –5A • A • cond: r;0, p;[0,2] • eff: r;(0,2), r;2, e;2 p A ¬ r r e

  17. r Semantics P stops holding • A • cond: r;0, p;[0,2] • eff: r;(0,2), r;2, e;2 p A r ¬ r e

  18. r Semantics p stops holding • A • cond: r;0, p;[0,2] • eff: r;(0,2), r;2, e;2 p ??? A ??? ¬ r r ??? e ??? Incomplete

  19. t=0517z t=0642z Visible(NGC132) t=0600z t=1300z Closed(SJC) Exogenous Conditions Inititial Conditions • X • cond: • eff: At(Pkg1, BOS-PO);0 • At(Truck1, BOS);0 • Closed(SJC);[0600,1300] • Visible(NGC132);[0517,0642] • … t=0 At(Pkg1, BOS-PO) At(Truck1, BOS)

  20. Outline Motivation Simple Reachability Mutual Exclusion Practical Matters

  21. Possibility & Reachability (p;t)  p is logically possible at t ∆(p;t)  p is reachable at t (rich;tomorrow) ¬∆(rich;tomorrow)

  22. Possibility & Reachability (p;t)  p is logically possible at t ∆(p;t)  p is reachable at t Extend to Intervals (p;i)   t i (p;t) ∆(p;i)   t i ∆ (p;t)

  23. Basic Axioms Facts are possible & reachable p;i  (p;i) p;i  ∆(p;i) Negations are not … p;i   t i ¬(¬p;t) p;i   t i ¬∆(¬p;t) Transitivity ∆(p;t) (p;t  q;t’)  ∆(q;t’)

  24. Basic Axioms Actions a;t  Cond(a;t)  Eff(a;t) Exogenous conditions X;0 Closure of X (Eff(x;0) = ¬p;t)— (p;i) | |\

  25. 0 1 2 3 4 5 6 Example r X;0 p p

  26. 0 1 2 3 4 5 6 Closure r X;0 p p  p  p closure  r

  27. 0 1 2 3 4 5 6 Basic ∆ r basic ∆p ∆ p r X;0 p p  p  p closure  r

  28. 0 1 2 3 4 5 6 Persistence ∆(p;i) meets(i,j) (p;j) ∆(p;i||j) ∆ r basic ∆p ∆ p r X;0 p p  p  p closure  r

  29. 0 1 2 3 4 5 6 Persistence ∆(p;i) meets(i,j) (p;j) ∆(p;i||j) basic & persist ∆ r ∆p ∆p r X;0 p p  p  p closure  r

  30. Actions Reachability ∆Cond(a;t)Eff(a;t) ∆(a;t) Conjunctive optimism ∆p1;i1 … ∆pn;in  ∆(p1;i1 … pn;in)

  31. p A ¬ r r e r 0 1 2 3 4 5 6 Action Application ∆Cond(a;t)Eff(a;t) ∆(a;t) • A • cond: r;0, p;[0,2] • eff: r;(0,2), r;2, e;2 ∆A ∆ r ∆p ∆p

  32. p A ¬ r r e r 0 1 2 3 4 5 6 Action Application ∆Cond(a;t)Eff(a;t) ∆(a;t) • A • cond: r;0, p;[0,2] • eff: r;(0,2), r;2, e;2 ∆ e ∆ ¬ r ∆A ∆ r ∆p ∆p

  33. p A ¬ r r e r 0 1 2 3 4 5 6 Persistence Again ∆(p;i) meets(i,j) (p;i) ∆(p;i||j) ∆ e ∆ ¬ r ∆A ∆ r ∆p ∆p

  34. p A ¬ r r e r Persistence (revised) ∆(p;i) meets(i,j) (p;i) ∆(p;i||j) a;t ∆(a;t) p;i  PersistEff(a;t)  meets(i,j) (p;i) ∆(p;i||j)

  35. 0 1 2 3 4 5 6 Persistence a;t ∆(a;t) p;i  PersistEff(a;t)  meets(i,j) (p;i) ∆(p;i||j) ∆ e ∆ ¬ r ∆A ∆ r ∆p ∆p

  36. Outline Motivation Simple Reachability Mutual Exclusion Practical Matters

  37. Mutual Exclusion M(p1;t1, …, pn;tn) Intervals M(p1;i1, …, pn;nn)   t1 i1, …, tn in M(p1;t1, …, pn;tn) Conjunctive optimism (∆p1;i1 … ∆pn;in ) ¬M(p1;i1, …, pn;nn)  ∆(p1;i1 … pn;in)

  38. Logical Mutex ¬(1  … n)  M(1, …, n) Consequences M(p;t, ¬p;t)

  39. Consequences ¬(1  … n)  M(1, …, n) Consequences • A • cond: p; … • eff: e; … A;t  p;t+ A;t  e;t+e M(A;t, ¬p;t+) M(A;t, ¬e;t+)

  40. Consequences ¬(1  … n)  M(1, …, n) Consequences • A • cond: p; … A;t  p;t+ B;t  ¬p;t+e • B • cond: ¬p; … M(A;t, B;t+–)

  41. Implication Mutex M(1, …, n)  ( 1)  M(, …, n)

  42. Implication Mutex Example M(1, …, n)  (1 1)  M(1, …, n) M(1, …, n)  ( 1)  M(, …, n) p;1 q;1 A;1 B;1 e;2 f;2 Example M(p;1,q;1) • A • cond: p;0 • eff: e;1 • B • cond: q;0 • eff: f;1

  43. Implication Mutex Example M(1, …, n)  (1 1)  M(1, …, n) M(1, …, n)  ( 1)  M(, …, n) p;1 q;1 A;1 B;1 e;2 f;2 Example M(p;1,q;1) • A • cond: p;0 • eff: e;1 A;t  p;t • B • cond: q;0 • eff: f;1 B;t  q;t

  44. Implication Mutex Example M(1, …, n)  (1 1)  M(1, …, n) M(1, …, n)  ( 1)  M(, …, n) p;1 q;1 A;1 B;1 e;2 f;2 Example M(p;1,q;1) • A • cond: p;0 • eff: e;1 A;t  p;t M(A;1,q;1) • B • cond: q;0 • eff: f;1 B;t  q;t M(p;1,B;1)

  45. Implication Mutex Example M(1, …, n)  (1 1)  M(1, …, n) M(1, …, n)  ( 1)  M(, …, n) p;1 q;1 A;1 B;1 e;2 f;2 Example M(p;1,q;1) • A • cond: p;0 • eff: e;1 A;t  p;t M(A;1,q;1) • B • cond: q;0 • eff: f;1 B;t  q;t M(p;1,B;1) M(A;1,B;1)

  46. Implication Mutex for Intervals M(1, …, n)  ( 1)  M(, …, n) M(1;i1, …, n;in)  j= {t: ;t  t1 i1 1;t1}  M(;j, …, n;in) p;[1,3) q;[2,3) A;[1,3) B;[2,3) e;… f;…

  47. Explanatory Mutex  {( 1)  M(, …, n)}  M(1, …, n) If “all ways of proving” 1 are mutex with 2, …, n  M(1, …, n) A p B p;1 q;1 A;1 B;1 e;2 f;2 A p 

  48. Outline Motivation Simple Reachability Mutual Exclusion Practical Matters

  49. Limiting Mutex Reachable propositions Time spread M(p;2, q;238) [0,2] [236,240] p q A Mutex spread theorem ?

  50. CSP? Actions Propositions A[0,3],[6,9] B[11,] C[…]… … p[0,5],[8.1,16] q[2,17]… r[3,]… …

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