An Introduction to Causal Reasoning about Actions and Change

1
An Introduction to CausalReasoning about Actions
and Change

• Pedro Cabalar
• AI Lab., Dept. of Computer Science
• University of Corunna, SPAIN.

2
Outline

• Reasoning about Actions Change
• Causal approaches
• Reasoning about causes
• Conclusions

3
Reasoning about Actions Change (RAC)

• "Programs with common sense" (McCarthy59)(Advice
  Taker) capable of performing tasks like
• prediction
• temporal explanation
• planning
• Main innovation explicit (logical)
  representation of domain knowledge
• (McCarthyHayes69) introduce Situation Calculus
  1st order logic 3 sorts
• Actions
• Fluents
• Situations

4
RAC scenarios

• Dynamic systems usually discrete transition
  systems
• Ex. of typical scenarios
• circuits like
• missionaries and cannibals3 missionaries and 3
  cannibals come to a river and find a boat that
  only holds 2. If the cannibals ever outnumber the
  missionaries in either bank, the missionaries
  will be eaten. How shall they cross?
• We identify
• Fluents system properties like num(can,left,2)
• Actions what the agent can perform to force
  transitionstransport(1,0) (1 cannibal,0
  missionaries)

sw(2)
?sw(1)
?light
5
RAC goals

• Typical goal problems

• Prediction (or temporal projection)
• we know initial state sequence of actions
• we obtain final state.
• Postdiction (or temporal explanation)
• we know partial observations of final and
  initial state we obtain complete initial state
  (even performed actions) that explain the
  observations
• Planning
• we know initial state goal state
• we obtain sequence of actions that guarantee
  reaching goal

6
RAC goals

• But then RAC playing with transition diagrams?
• No ? Main goal quality of knowledge
  representation.We look for a formal
  representation that allows
• clear descriptions
• reliable/efficient inference methods
• flexibility with respect to slight modifications
  in the scenario "elaboration tolerance".
• All this can be done using formal logic.

7
Representational problems

• Elaboration tolerance logic some problems.
• Effect axioms describe the effects of each
  action sw(1) ? toggle(1) ? ?sw(1)'
• Frame axioms describe what the action does not
  modify battery ? toggle(1) ? battery
• "Frame problem" frame axioms are not scalable.
  We needInertia default in absence of evidence
  on the contrary, every fluent remains unchanged

8
Nonmonotonic Reasoning

• AI Journal 13, 1980
• Circumscription (McCarthy)minimize the extent of
  a predicate
• Default logic (Reiter)use special inference
  rules like
• Autoepistemic Logic (Moore)non-monotonic modal
  logic (L ? ? is "believed")
• Important connection (BidoitFroidevaux87)
• Logic Programs under Answer Sets semantics are
  some type of default theories.

? ? ?
9
New problems minimal change

• Minimize changes (use of abnormal predicate)
• Yale Shooting Problem load, wait, shoot

load
wait
shoot
alive ?loaded
alive loaded
alive loaded
? alive
10
Causal minimizations

• (Lifschitz 87) (Haugh 87)Include some type of
  facts like causes(shoot, alive,
  false) causes(load, loaded, true)
• Since we don't have causes(wait, loaded, false)
• the problem is avoided.

11
Ramification problem

• Describing indirect effects avoid explicit
  reference to actions involved
• Example a suitcase with two latches
• up1 ? up2 ? open

toggle1
?up1 up2 ?open
up1 up2 open
12
Causal approaches

• (McCainTurner 95) (McCainTurner 97)conditional
  operator ???
• (Lin 95) special predicate caused(fluent,situation
  )
• Other
• (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.

13
A Preliminary Study on Reasoning About Causes
Pedro Cabalar AI Lab., Dept. of Computer
Science University of Corunna, SPAIN.
14
Outline

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

15
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 ...

16
A motivating example
? sw(1)
sw(2)
? light
?

• Trivial case we had opened sw(1) while sw(2)
  closed ...

17
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.

18
A motivating example
? sw(1)
sw(2)
?
? light

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

19
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).

20
A motivating example
? sw(1)
sw(2)
?
? light

• Interesting case toggling both switches
  simultaneously.

21
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!

22
Another example
sw(1)
sw(2)
?
?
light
?

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

23
Another example
sw(1)
sw(2)
light

• whereas, toggling both switches again...

24
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

25
Outline

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

26
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

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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

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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) ? ?.

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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...

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Semantics
?, ? ? Ø
vI (?) ? f Ø
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Semantics
?, ? ? Ø
vI (?) ? f Ø
Truth persistent "copy" the other conjunct
32
Semantics
?, ? ? Ø
vI (?) ? f Ø
one conjunct false caused explains whole
conjunction, when the other conjunct is true
33
Semantics
?, ? ? Ø
vI (?) ? f Ø
both false caused any of their causes is also
a cause for the conjunciton
34
Semantics
?, ? ? Ø
vI (?) ? f Ø
both true caused (any) union of cause in ?
with cause in ? is a cause for the conjunciton
35
Semantics
?, ? ? Ø
vI (?) ? f Ø
Areas for ? and ?.
36
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)

38
Outline

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

39
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).

40
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).

41
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).

42
Outline

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

43
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), ...

44
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.

45
Outline

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

46
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, ...

47
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)