Reasoning about controllable and uncontrollable variables

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Reasoning about controllable and uncontrollable
variables
Souhila KACI CRIL-CNRS Lens
Leendert van der Torre ILIAS Luxembourg
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Preference reasoning

• Logics of preferences attract much attention in
  KR
• Application qualitative decision making
• Algorithms used in some non-monotonic preference
  logics are too simple to be used in KR and
  reasoning applications
• min/max specificity principles

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Reasoning about preferences
? gt ?
I prefer ? to ?
Our aim to compute a total pre-order ? on ?
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Example
?2,?3,?6,?7gt?0,?1,?4,?5 ?1,?3gt?0,?2

• P b gt ?b, ?s ? c gt ?s ? ?c
• ? ?0 ?s?b?c, ?1 ?s?bc, ?2 ?sb?c, ?3 ?sbc,
  ?4 s?b?c, ?5 s?bc, ?6 sb?c, ?7 sbc

opt. reas.
strong reas.
pess. reas.
?

• Different preference relations may be consistent
  with preferences

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min/max specificity principles

• both compute the most compact preference relation

min
max
an alternative is considered to be satisfactory
as much as there is no other alternatives that
are considered to be better
an alternative is considered to be unsatisfactory
as much as there is no other alternatives that
are considered to be worse
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Opt./Pess. preferences controllable/uncontrollab
le variables

• minimal specificity principle ? gravitation
  towards the ideal ? the best will hold for the
  alternatives ? optimistic reasoning on
  preferences ? controllable variables
• maximal specificity principle ? gravitation
  towards the worst ? the worst will hold for the
  alternatives ? pessimistic reasoning on
  preferences ? uncontrollable variables

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Contr./Uncontr. variables Qualitative decision
theory

• states, actions, consequences
• state variables
• observable variables ? controllable variables
• unobservable variables ? uncontrollable variables
• actions controllable variables

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Preferences in qualitative decision theory

• hypothesis all state variables are unobservable
  ? uncontrollables
• preferences on states, actions
• preferences on consequences

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How can we use min/max specificity algorithms?

• ? the set of worlds on contr./uncontr. variables
• minimal specificity principle on preferences
  based on controllable variables ? optimistic
  reasoning ? ?c
• maximal specificity principle on preferences
  based on uncontrollable variables ? pessimistic
  reasoning ? ?u
• merging ?c and ?u

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Merging optimistic and pessimistic preferences
O xgty, ygtz,
P pgtq, qgtr,
step 1
step 2
x
p
q
y
Distinguished Pre-orders
z
r
step 3
xp
xq , yp
xr , yq , zp
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Some merging operators

• ?c (mp , ?mp , m?p , ?m?p)
• ?u (mp , m?p , ?mp , ?m?p)
• Symmetric mergers
• ?c (E1, , En), ?u (E'1, , E'm)
• ? (E''1, , E''nm-1) (mp , ?mp, m?p ,
  ?m?p)
• Dictators
• minmax ?1gt?2 iff ?1gtc?2 or (?1?c?2 and ?1gtu?2)
• ? (mp , ?mp , m?p , ?m?p)
• maxmin ?1gt?2 iff ?1gtu?2 or (?1?u?2 and ?1gtc?2)
• ? (mp , m?p , ?mp , ?m?p)

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Is this merging process satisfactory?

• Not really
• interaction between controllable and
  uncontrollable variables is not possible
• Example
• If my boss accepts to pay the conference fee then
  I will work hard to finish the paper
• conditional preferences

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Optimistic conditional preference specification

• qi ? LU, xi, yi ? LC
• O? qi ?( xi ?yi),
• q ? (x?y) (q ? x) ?(q ? y)
• O? (qi ? xi) ?(qi ? yi)
• ?o following the minimal specificity principle

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Pessimistic conditional preference specification

• xi ? LC, qi , ri ? LU,
• O? xi ?( qi ?ri),
• x ? (q?r) (x ? q) ?(x ? r)
• O? (xi ? qi) ?(xi ? ri)
• ?p following the maximal specificity principle

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Example

• O money?(work gt ?work), ?money?(?work gt work),
  ?money ? (project gt ?project)
• ?o (?m?wp, mwp, mw?p , m?w?p, m?wp, ?mwp ,
  ?m?w?p, ?mw?p)
• P ?project?(moneygt?money), ?work?(?moneygt
  money)
• ?p (mw?p, m?w?p , ?m?w?p, ?m?wp , ?mw?p,
  ?mwp, m?wp, mwp)
• Symmetric merger
• ? (mw?p , ?m?wp, m?w?p , mwp , m?wp,
  ?mwp, ?m?w?p , ?mw?p)

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Application to qualitative decisionExample
(Savage'54)

• An agent is preparing an omelette.
• 5 fresh eggs are already in the omelette.
• There is one more egg.
• The agent does not know whether this egg is fresh
  or rotten.
• She can
• add it to the omelette the whole omelette may be
  wasted,
• throw it away one egg may be wasted, or
• put it in a cup, check whether it is ok or not
  and add it to the omelette in the former case,
  throw it in the latter. A cup has to be washed.

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Example (Savage'54, Brewka05)

• A controllable variable in_omelette, in_cup,
  throw_away
• An uncontrollable variable fresh, rotten
• Consequences of cont./uncont. variables
• 5_omelette ? throw_away
• 6_omelette ? fresh, in_omelette
• 0_omelette ? rotten, in_omelette
• 6_omelette ? fresh, in_cup
• 5_omelette ? rotten, in_cup
• ?wash ? not in_cup
• wash ? in_cup
• Agent's desires
• ?wash ? wash
• 6_omelette ? 5_omelette ? 0_omelette

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Example (Savage'54, Brewka05)

• S1 6_omelette, ?wash, fresh, in_omelette
• S2 0_omelette, ?wash, rotten, in_omelette
• S3 6_omelette, wash, fresh, in_cup
• S4 5_omelette, wash, rotten, in_cup
• S5 5_omelette, ?wash, fresh, throw_away
• S6 5_omelette, ?wash, rotten, throw_away

S1
S5 , S6
S3
S4
S2
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Our approach Extension of the example

• Preferences over consequences Preferences over
  alternatives

fresh ? in_omelette gt in_cup fresh ? in_cup gt
throw_away rotten ? throw_away gt in_cup rotten ?
in_cup gt in_omelette
in_omelette ? fresh gt rotten in_cup ? fresh gt
rotten throw_away ? rotten gt fresh
O
P
?1 fresh ? in_omelette, ?2 rotten ?
in_omelette, ?3 fresh ? in_cup, ?4 rotten ?
in_cup, ?5 fresh ? throw_away, ?6 rotten ?
throw_away

• ?o (?1, ?6 , ?3, ?4 , ?2, ?5)
• ?p (?1, ?3, ?6 , ?2, ?4, ?5)
• Symmetric merger ? (?1, ?6 , ?3 , ?4 ,
  ?2, ?5)

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Example
?1 , ?6

• S1 6_omelette, ?wash, fresh, in_omelette
• S2 0_omelette, ?wash, rotten, in_omelette
• S3 6_omelette, wash, fresh, in_cup
• S4 5_omelette, wash, rotten, in_cup
• S5 5_omelette, ?wash, fresh, throw_away
• S6 5_omelette, ?wash, rotten, throw_away
• ?wash ? wash
• 6_omelette ? 5_omelette ? 0_omelette

?3
?4
?2 , ?5
S1 gt S6 gt S3 gt S4 gt S5 gt S2
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To summarize
preferences on controllables
preferences on uncontrollables
?p
?o
? pref. on contr./uncontr.
preferences on consequences P
refine ? with P
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Another way

• Preference statements involving consequence
  variables only
• P ?wash gt wash,
• 6_omelette gt 5_omelette gt 0_omelette,
• 5_omelette ? wash gt 0_omelette ?
  ?wash
• ? in_omelette?throw_away gt in_cup,
• fresh?(in_omelette?in_cup) gt
  throw_away?(in_cup?rotten),
• throw_away?(in_cup?rotten) gt
  rotten?in_omelette,
• in_cup?rotten gt in_omelette?rotten

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Conclusion

• non-monotonic logic of preferences distinction
  between controllable and uncontrollable variables
• Future research
• related works
• more complex merging tasks social and group
  decision making