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A Preliminary Study on Reasoning About Causes

A Preliminary Study on Reasoning About Causes. Pedro Cabalar AI Lab., Dept. of Computer Science University of Corunna, SPAIN. Introduction. Causality in Reasoning about Actions : causal assertions (McCarthy 69) . Yale Shooting Problem: causal minimizations (Lifschitz 87) (Haugh 87) .

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A Preliminary Study on Reasoning About Causes

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  1. A Preliminary Study on Reasoning About Causes Pedro Cabalar AI Lab., Dept. of Computer Science University of Corunna, SPAIN.

  2. Introduction • Causality in Reasoning about Actions: • causal assertions (McCarthy 69). • Yale Shooting Problem:causal minimizations (Lifschitz 87) (Haugh 87). • Ramification Problem:(McCain&Turner 95) (Lin 95) (Thielscher 97) (Denecker et al. 98) (Schwind 99) (Shanahan 99) (Giunchiglia et al. 02). • Causality = technical solution to ramif. problem butno real interest about causal information.

  3. Introduction • Example: we can use it toconclude 'dead' after 'shoot' but not to express that the shot was the cause for 'dead'. • Facts like this not trivial: indirect effects, concurrence, etc. They should be derived from our causal rules. • We present a mechanism to obtain the causes of each derived formula in terms of subsets of the performed actions.

  4. Outline • A motivating example • Syntax and Semantics • LP translation • Related work • Conclusions

  5. A motivating example  sw(1) sw(2) light • How did we reach this (successor) state? • "Who was responsible" of turning off the light? • Let us study some possible performed actions ...

  6. A motivating example • Trivial case: we had opened sw(1) while sw(2) closed ...  sw(1) sw(2) light 

  7.  A motivating example • Trivial case: we had opened sw(1) while sw(2) closed ...  sw(1) sw(2) light  • Toggling sw(1) has caused light.

  8. A motivating example • 2nd case: we had closed sw(2) while sw(1) open ...  sw(1)  sw(2) light

  9. A motivating example • 2nd case: we closed sw(2) while sw(1) open ...  sw(1) sw(2) light • The light persists off (no cause for light).

  10. A motivating example  sw(1)  sw(2) light • Interesting case: toggling both switches simultaneously.

  11. A motivating example  sw(1) sw(2) light • Interesting case: toggling both switches simultaneously. • Toggling sw(1) has caused light (after all, sw(2) has been closed). Note that light remains off, but caused!

  12. both actions together cause light to be on. Another example  sw(1)  sw(2) light  • Consider now this state. If we close both switches...

  13.   • any of the actions alone is a cause for light off. Another example sw(1) sw(2) light • whereas, toggling both switches again...

  14. Summary • Any change of value is due to causation. However, the opposite does not hold. • An effect may be equally due to different causes, and each cause can be the concurrent combination of several actions. • Our goal: obtain causal facts, avoiding:sw(1)causeslightifsw(2)sw(1),sw(2)causeslightsw(2)causeslightifsw(1) • in favor of:sw(1) sw(2)causeslight

  15. Outline • A motivating example • Syntax and Semantics • LP translation • Related work • Conclusions

  16. Syntax • Symbols S = A F • Actions A ={toggle(1), toggle(2)} • Fluents F ={sw(1),sw(2)} • Compound actions 2A. Examples: {toggle(1)}, {toggle(2)}, {toggle(1), toggle(2)}, Ø • Notation:a, b, ... = actions A, B, ... = compound actions f, g, ... = fluents , , ... = sets of compound actionsp, q, ... = symbols

  17. Syntax • Formulas: L denotes the language formed with, p, , , , A  A   "compound action A has caused  to hold" • Usual derived operators , , , , plus:C   A  N    C A  2A

  18. Semantics Interpretation ,  •  = standard truth valuation : S { t, f}F= state A = performed (compound) action •  = causal relevance relation 2A  SExample: ( {toggle(1), toggle(2)}, light ) means: {toggle(1), toggle(2)} has caused truth value (light). •  can be seen as a set of functions A: S { t, f} so thatfor instance, A(light) = t iff (A, light)  .

  19. Semantics Let I=,  • Truth  ()for propositional connectives is standard • () will be a set of comp. actions pointing out:A  () iff A() = t • The valuation w.r.t. I is defined as vI : L  { t, f}  2A Aand follows the next rules...

  20. "Short-circuit" behavior when false + persistent Semantics ,   Ø vI () f Ø

  21. Semantics ,   Ø vI () f Ø Truth + persistent = "copy" the other conjunct

  22. Semantics ,   Ø vI () f Ø one conjunct false + caused explains whole conjunction, when the other conjunct is true

  23. Semantics ,   Ø vI () f Ø both false + caused: any of their causes is also a cause for the conjunciton

  24. Semantics ,   Ø vI () f Ø both true + caused: (any) union of cause in  with cause in  is a cause for the conjunciton

  25. Semantics ,   Ø vI () f Ø Areas for  and .

  26. 1 - For any atomic action a, t{A} if vI () = t  and A  f {Ø} otherwise {{a}} if (a) = t {Ø} otherwise vI (A)  (a)  Semantics We add a pair of restrictions: 2 - Axiom: A afor any comp. action A, and any a  A.

  27. Some properties • Disjunction table: change t by f and vice versa. • Relevance in tautologies: p p cannot be just replaced by . • "Unfolding" properties:A ( )  (A   N  )  (A  N ) (1)A ( )  (A   N  )  (A  N )  (A1  A2)(2) A1A2 = AN( )  N N (3)N( )  NN(4)

  28. Outline • A motivating example • Syntax and Semantics • LP translation • Related work • Conclusions

  29. LP translation • Dynamic action domains introduce new requirements: • NMR for inertia default, • directional behavior for causal rules. • A simple solution: we follow (Gelfond&Lifschitz93) methodology: • high level action language, plus • translation into Logic Programming (answer sets).

  30. causes gif after causes gif after  Action Language • Causal rules: causes  if after classical formula,  fluent literal,  and  fluent formulas. • Intuitive meaning: once  and  proved true, check whether A holds for some A. If so, derive A. • Abbreviation: g:=if after   • Translation into LP use properties (1)-(4) to "unfold" causal dependences (details in the paper).

  31. LP translation • Example: switches scenariotoggle(N)causes sw(N)after sw(N)toggle(N)causes sw(N)after sw(N)light := sw(1)  sw(2) • some generated program rules:c({t(1)},light) :- c({t1},sw(1)), n(sw(2)).c({t(2)},light) :- c({t2},sw(2)), n(sw(1)).c({t(1),t(2)},light) :- c({t1},sw(1)), c({t2},sw(2)).c({t(1)},-light) :- c({t1},-sw(1)), -n(-sw(2)).c({t(2)},-light) :- c({t2},-sw(2)), -n(-sw(1)). • other axioms:c(Lit) :- c(A,Lit). g :- g', not c(-g). Lit :- c(Lit). -g :- -g', not c(g). n(Lit) :- Lit, not c(Lit).

  32. Outline • A motivating example • Syntax and Semantics • LP translation • Related work • Conclusions

  33. Related work • Transformation of causal expressions: Event Calculus (Shanahan 99), inductive causation (Denecker et al.98). • Use of influence relations (which action may affect which fluent value): • (Thielscher 97) constraints+influence = causal rules. • (Castilho et al.99) use influence relations as primitive information (problem of elaboration tolerance). • Use of a "caused" flag: caused predicate (Lin 95), occlusion (Sandewall 94), ...

  34. Related work • But the most related approach is Pertinence Logic, L2,(Otero97), which has been used as a starting point. • Two valuation functions: truth {t, f} + pertinence {p, n}.Pertinence = flag caused/non-caused, regardless the actions responsible for that. • When limiting to unique action, current approach degenerates into L2.Exception:  and  become pertinent when any of their operands are so, regardless their truth.

  35. Outline • A motivating example • Syntax and Semantics • LP translation • Related work • Conclusions

  36. Conclusions • Causal "introspection": derive the reasons for each effect. • We could even go further, and use this in rule conditions: A deadcauses jail(peter) if perfomed(peter, A) • Allows characterizing causally different domains apparently equivalent w.r.t. truth-value transitions (see Pearl's circuit example (Pearl00) in the paper). • A lot of topics for future work: causes minimization, nesting of causal operators, delayed effects, ...

  37. Pearl's circuit Apparently equivalent to: light := sw(1)  sw(2)  sw(2)  sw(1) light ... but when sw(1) is true (down), sw(2) is irrelevant:light := sw(1)   sw(1) sw(2)

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