A Preliminary Study on Reasoning About Causes

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 Problemcausal minimizations
  (Lifschitz 87) (Haugh 87).
• Ramification Problem(McCainTurner 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 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)
sw(2)
?
? light

• 2nd case we had closed sw(2) while sw(1) open ...

9
A motivating example
? sw(1)
sw(2)
? light

• 2nd case we closed sw(2) while sw(1) open ...

• 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
Another example
sw(1)
sw(2)
?
?
light
?

• Consider now this state. If we close both
  switches...

13
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)
  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 2A. Examples toggle(1),
  toggle(2), toggle(1), toggle(2), Ø
• Notationa, b, ... actions A, 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 ?, ?, ?, ?, plusC? ?
  ? A ? N? ? ? ?? C? A ? 2A

18
Semantics
Interpretation ??, ??

• ? standard truth valuation ? S ? t, f ?F
  state ?A performed (compound) action
• ? causal relevance relation ? ? 2A ? 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 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
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
Semantics

• We add a pair of restrictions

2 - Axiom A ? ? a for any comp. action A, and
any a ? A.
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Some properties

• Disjunction table change t by f and vice versa.
• Relevance in tautologies p ??p cannot be just
  replaced by ?.
• "Unfolding" propertiesA (? ?? ) ? (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 (GelfondLifschitz93)
  methodology
• high level action language, plus
• translation into Logic Programming (answer sets).

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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 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 rulesc(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 axiomsc(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) constraintsinfluence 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 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
Apparently equivalent to light sw(1) ?
sw(2)
? sw(2)
? sw(1)
? light
... but when sw(1) is true (down), sw(2) is
irrelevantlight sw(1) ? ? sw(1) ? sw(2)