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 but no real interest about causal information.

3 Introduction Example: we can use it to conclude '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  sw(1) sw(2)  light Trivial case: we had opened sw(1) while sw(2) closed... 

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

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

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

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

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

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

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) causes light if sw(2)sw(1),sw(2) causes light sw(2) causes light if sw(1) in favor of: sw(1)  sw(2) causes light

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 2 A. Examples: {toggle(1)}, {toggle(2)}, {toggle(1), toggle(2)}, Ø Notation: a, b,... = actionsA, B,... = compound actions f, g,... = fluents , ,... = sets of compound actions p, 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  2 A

18 Semantics  = standard truth valuation  : S  { t, f }  F = state  A = performed (compound) action  = causal relevance relation   2 A  S Example: ( {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 that for instance,  A (light) = t iff (A, light)  . Interpretation  ,  

19 Semantics 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 v I : L  { t, f }  2 A A and follows the next rules... Let I=  ,  

20 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø "Short-circuit" behavior when false + persistent

21 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø Truth + persistent = "copy" the other conjunct

22 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø one conjunct false + caused explains whole conjunction, when the other conjunct is true

23 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø both false + caused: any of their causes is also a cause for the conjunciton

24 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø both true + caused: (any) union of cause in  with cause in  is a cause for the conjunciton

25 Semantics  f  t  f   f Ø f  t  t Ø f Ø f  f Øf Ø f (    )f  t  f Øf Ø f  t (    )t  t Ø f Øf Ø f  t  t Ø ,   Ø v I (  )  f Ø Areas for  and .

26 Semantics v I (A  )  t {A} if v I (  ) = t  and A   f {Ø}otherwise We add a pair of restrictions:  (a)  {{a}} if  (a) = t {Ø}otherwise 1 - For any atomic action a, 2 - Axiom: A   a for 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  )   (A 1   A 2  ) (2) A 1  A 2 = A N (   )  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 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).  causes g if  after    causes  g if  after 

31 LP translation Example: switches scenario toggle(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, L 2, (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 L 2. 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 dead causes 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  sw(1)  light  sw(2) Apparently equivalent to: light := sw(1)  sw(2)... but when sw(1) is true (down), sw(2) is irrelevant: light := sw(1)   sw(1)  sw(2)