Commonsense Reasoning 0809 HC 13: Abduction

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Commonsense Reasoning 08/09HC 13 Abduction

• Henry Prakken
• 07-01-2009
• (with thanks to Annette Ten Teije)

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Reasoning with causal defaults

• Prediction sprinkler on what will happen?
• Modelling deduction (M.P.)
• Explanation grass wet why?
• Modelling abduction
• Invalid!

Sprinkler on ? Grass wet
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Abduction what is it?

• Finding the best explanation for a set of
  observations
• What is the best explanation for the wet grass?

Sprinkler on ? Grass wet Rain ? Grass wet
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Abduction application areas

• Commonsense reasoning
• Diagnosis
• Legal proof
• Planning
• Scientific theory formation

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Abduction in AI

• Part of Model-based diagnosis
• Model based reasoning
• Build a formal model of a system
• Reason about the systems behaviour by reasoning
  with the model
• Applied to diagnosis
• Build causal model of the systems abnormal
  behaviour
• Observe behaviour
• Find explanation of abnormal behaviour by
  reasoning with the causal model

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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing headache
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing headache
Abduction is nonmonotonic!
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing headache
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing headache
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Causal network (1)
flu
cold
hangover
smoke allergy
fever
coughing
headache
observed coughing headache
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(No Transcript)
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piston-rings used
oil-cup holed
old-spark-plugs
?1
oil-below-car
lubric-oil burning
oil loss
spark-plugs used-up
irreg-oil consumpt
oil lack
stack smoke
?3
dirty-spark-plugs
high-engine temp
burnout
irreg- ignition
ignition problems
?2
?4
temp-indic red
power decrease
coolant evaporation
mumbling engine
vapour
?5
lack-of-accel
melting
melted pistons
?6
smoke-from engine
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piston-rings used
oil-cup holed
old-spark-plugs
?1
oil-below-car
lubric-oil burning
oil loss
spark-plugs used-up
irreg-oil consumpt
oil lack
stack smoke
?3
dirty-spark-plugs
high-engine temp
burnout
irreg- ignition
ignition problems
?2
?4
temp-indic red
power decrease
coolant evaporation
mumbling engine
vapour
?5
lack-of-accel
melting
melted pistons
?6
smoke-from engine
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piston-rings used
oil-cup holed
old-spark-plugs
?1
oil-below-car
lubric-oil burning
oil loss
spark-plugs used-up
irreg-oil consumpt
oil lack
stack smoke
?3
dirty-spark-plugs
high-engine temp
burnout
irreg- ignition
ignition problems
?2
?4
temp-indic red
power decrease
coolant evaporation
mumbling engine
vapour
?5
lack-of-accel
melting
melted pistons
?6
smoke-from engine
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Logical model of abduction idea

• Given
• a causal model CM
• a set of observations O
• Find explanations for O, i.e. hypotheses H such
  that
• H ? CM - O
• H ? CM is consistent
• Compare the explanations

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Logical model of abduction(with strict causality)

• Causal specification (DFS,OBS,CM)
• DFS d1, , dn (di literals)
• possible defects
• OBS o1, , om (oi literals)
• possible observations
• CM set of causal rules
• d1 ? ? dm ? dn
• d1 ? ? dj ? ok
• Abductive Causal Problem (C,O)
• C is a causal specification
• O ? OBS
• Solution an explanation for O in terms of C
• with H ? DFS

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Example theory

• H1 ? S1
• H2 ? S2
• H3 ? S3
• S1 ? Obs1
• S2 ? Obs1
• S2 ? Obs2
• S3 ? S4
• S4 ? Obs2
• observed behaviour Obs1 ? Obs2

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Negative observations
- observed coughing, headache, ?fever. - An
explanation only needs to be consistent with
the negative observations!
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Revised definition solution

• H ? DFS is a solution of (C,O) iff
• H ? CM - O
• H ? CM ? Oc is consistent
• Oc is the set of negative observations
• Oc ? ?oo ? OBS/O
• (Further constraints are possible)

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Weak causality
- coughing ? O ?headache ? Oc - Problem smoke
allergy is no explanation! - Solution allow
weak causal rules
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Weak causality logically

• DFS also contains assumption literals ?i
• A ? ?i ? B A may cause B
• Definitions remain unchanged
• Explanations can now also contain assumption
  literals.

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flu ? fever flu ? ?1 ? coughing flu ?
headache cold ? coughing hangover ?
headache smoke-all. ? ?2 ? coughing smoke-all. ?
?3 ? headache
O coughing Oc ?headache
CM U smoke-all., ?2 -- coughing CM U
smoke-all., ?2 U Oc is consistent
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Preference criteria for explanations

• subset minimal
• number minimal (cardinality)
• Variations
• Ignore assumption literals
• Only consider initial causes
• Consider only designated defect literals
• Add probabilities

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Abstract model abduction(Bylander et al.)

• Abduction problem ?Dall,Hall,e,pl ?
• Dall data to-be explained
• Hall individual hypotheses
• e map from subsets of Hall to subsets of Dall
• h explains e(h)
• pl partial ordering of subsets of Hall
• plausibility

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e(h1)d1 e(h2)d1,d2 .
Hallh1,h2,h3,h4,h5 Dalld1,d2,d3,d4
pl(h1,h2) pl(h2,h3) pl(h1,h2) lt
pl(h4,h5) ..
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Terminology

• H is a hypothesis iff H ? Hall
• H is complete iff e(H) Dall
• H is parsimonious iffno proper subset of H
  explains the same data
• Hypothesis H is an explanation iffH is complete
  and H is parsimonious
• H is a "best explanation" iff there is no
  explation H' such that pl(H') gt pl(H)

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Example

• Is h1,h2,h3,h4 an explanation?

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Logical model in terms of abstract model

• Dall O
• Hall DFS
• e(H)
• o ? O H ? CM - o (if H ? CM ? Oc is
  consistent)
• ? otherwise
• pl (for example)
• pl(H) ? pl(H) iff H ? H

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Types of abduction problems

• (1) Independent abd-problem
• (2) Monotonic abd-problem
• (3) Incompatibility abd-problem
• (4) Cancellation abd-problem
• Type of abduction problem
• depends on domain properties!!
• Does not depend on representation method

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1. Independent abduction problem

• Composite hypothesis explains a datum iff one of
  its elements explains that datum
• ? H ? Hall (e(H) ?h?H e(h))

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Example

• h1 and h2 together explain d1, but not
  independently
• ? no independent abduction problem!!

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2. Monotonic abduction problem (MAP)

• Composite hypothesis can explain more than its
  parts
• ? H, H ? Hall (H ? H ? e(H) ? e(H))
• - composite hypothesis does not lose explained
  data
• - composite hypothesis possibly explains
  additional data

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Example
h1
h2
h3
d2
d4
d3
d1
e(h1) ? e(h2) ? e(h3) ? e(h1,h2,h3)
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3. Incompatibility abduction problem

• Some hypotheses exclude each other
• I hi,hj, hk,hl, set of incompatible
  hypothesis pairs
• H ? Hall ((? I ? I (I ? H)) ? e(H) ?)
• independent incompatibility problem
• ? H?Hall ((?? I?I (I?H)) ? e(H) ?h?He(h))

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Cancellation problems

• Sometimes extending a hypothesis causes loss of
  previously explained data
• pneumonia causes fever and breathing problems
• increased adrenal glands function causes moonface
• pneumonia with increased adrenal glands function
  causes breathing problems and moonface (so no
  fever any more)

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4. Cancellation abduction problem

• Abduction problem ?Dall,Hall,e,pl,eproc,econs?
• eproc, econs, maps of Hall to subsets of Dall
• for resp. producing and consuming
• d ? e(H) ?
• ?h h ? H ? d ? eprod(h) ?gt
• ?h h ? H ? d ? econs(h) ?

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Example
h1
h2
h3
h4
d3
d1
d2
d2 ? e(h2) d2 ? e(h1,h2)