Possible and Necessary Answer Sets
of Possibilistic Answer Set Programs
Kim Bauters
Steven Schockaert
Martine De Cock
Dirk Vermeir
University of Ghent, Belgium
Department of Applied Mathematics
and Computer Science
November 8, 2012
Possible and Necessary Answer Sets of PASP
talk overview
overview of this talk :
◾ recall ASP, possibilistic logic and possibilistic ASP
◾ uncertain conclusions versus uncertain rules
◾ new results presented in this paper:
◾ semantics of uncertain rules
◾ main reasoning tasks
◾ complexity results
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Possible and Necessary Answer Sets of PASP
what is ASP?
ASP program is a set of rules of the form
a0 ← a1 , ..., am , not am+1 , ..., not an
head
positive
body
negative
body
not is default negation
not a is true when no proof for a
different types of rules
◾ definite rule : single atom in head and positive body
◾ normal rule : as above; definite rule with negative body
form of non-monotonic reasoning
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Possible and Necessary Answer Sets of PASP
what is possibility theory?
main concepts in possibility theory
interpretatio
ns
set of possible worlds Ω not possible entirely pos
sible
possibility given by π(ω) = i, 0 ≤ i ≤ 1, ω ∈ Ω
possibility measure Π(p) = max{π(ω) | ω |= p}
necessity measure N (p) = 1 − Π(¬p)
example
π({p, q}) = 1
π({p, ¬q}) = 0
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π({¬p, q}) = 0
π({¬p, ¬q}) = 0
hence N (p) = 1
and N (q) = 1
Possible and Necessary Answer Sets of PASP
possibilistic answer set programming
possibilistic answer set programming (PASP)
◾ combines ASP (non-monotonicity, declarative)
◾ with possibility theory (reason about uncertainty)
different semantics exist, all combine a weight with a rule:
c : a0 ← a1 , ..., am , not am+1 , ..., not an
how do we interpret this weight?
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Possible and Necessary Answer Sets of PASP
existing semantics for PASP : uncertain conclusions
earliest semantics from Nicolas, 2006
unintuitive results when using negation-as-failure
because certainty is ignored when dealing with e.g. not a
alternative semantics from Bauters, 2010
N (head) ≥ min(N (body), c)
(atom in the) head is necessary when body/rule is necessary
example
0.1 : strike ←
0.4 : Coral ← strike
0.9 : bus ← not strike
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!(Nicolas, 2006)
strike 0.1 , Coral 0.1
!(Bauters, 2010)
strike 0.1 , Coral 0.1 , bus0.8
Possible and Necessary Answer Sets of PASP
new semantics for PASP : uncertain rules
new semantics for PASP
c : a0 ← a1 , ..., am , not am+1 , ..., not an
degree to which we believe that the rule is valid
rather than limiting certainty of conclusion
e.g. rule obtained from unreliable source;
uncertainty as to whether information
encoded in the rule is valid; …
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Possible and Necessary Answer Sets of PASP
how do we deal with valid / invalid rules?
invalid rules in possibilistic logic
considering invalid rules
erroneous conclusions
α-cut : omit the least certain rules
invalid rules in PASP
considering invalid rules
but omit valid rules
erroneous conclusions
erroneous conclusions
due to non-monotonicity
example
include and you cannot deduce safe
collapse ←
omit and you deduce safe
safe ← not collapse
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Possible and Necessary Answer Sets of PASP
formal semantics of uncertain rules
semantics defined as possibility distribution πP
over subsets P 0 of PASP program P
classic ASP program
omit a rule results in ‘penalty’
by ignoring weights
(
0
0⇤
consistent
1
−
max
{c
|
(c
:
r)
∈
P
\
P
}
when
P
0
πP (P ) =
0
otherwise
example
(r1) 0.2 : collapse ←
(r2) 0.9 : safe ← not collapse
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πP ({r1 , r2 }) = 1
πP ({¬r1 , r2 }) = 0.8
πP ({r1 , ¬r2 }) = 0.1
πP ({¬r1 , ¬r2 }) = 0.1
Possible and Necessary Answer Sets of PASP
semantics of uncertain rules : intuition
example
higher ‘penalty’
if omitted
safe to omit
0 : RomneyWins ←
0.1 : strike ←
0.4 : Coral ← strike
0.9 : bus ← not strike
1 : Thursday ←
must be included
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Possible and Necessary Answer Sets of PASP
uncertain rules : main reasoning tasks
what are the main reasoning tasks?
for each subprogram, we can determine whether an atom a :
is a brave conclusion
P 0 |=b a
is a cautious conclusion
P 0 |=c a
true in at least one answer set
true in every answer set
example
P 0 = {(k ← not m), (m ← not k), (z ←)}
has two answer sets {k, z} and {m, z}
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P 0 |=b k
P 0 6|=c k
P 0 |=c z
Possible and Necessary Answer Sets of PASP
uncertain rules : main reasoning tasks (continued)
what are the main reasoning tasks? (continued)
for each subprogram, we can determine whether an atom a :
is a brave/cautious conclusion
in some subprogram P 0 such
that πP (P 0 ) ≥ λ
Π(P |=... a) ≥ λ
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dual problem:
1 − Π(P 6|=... a) ≥ λ
Possible and Necessary Answer Sets of PASP
uncertain rules : main reasoning tasks overview
4 main reasoning tasks :
!
"
#
⇤
b
Π P |= l = max πP (P 0 ) | P 0 ⊆ P and P 0 |=b l ≥ λ
!
⇤
c
Π (P |= l) = max πP (P 0 ) | P 0 ⊆ P and P 0 |=c l ≥ λ
!
"
#
⇤
b
N P |= l = 1 − max πP (P 0 ) | P 0 ✓ P and P 0 6|=b l ≥ λ
!
⇤
c
N (P |= l) = 1 − max πP (P 0 ) | P 0 ✓ P and P 0 6|=c l ≥ λ
with various complexity results for determining :
!
"
b
Π P |= l ≥ λ which is NP -complete
Π!(P |=c l)" ≥ λ which is ΣP2-complete
N P |=b l ≥ λ which is ΠP2-complete
N (P |=c l) ≥ λ which is coNP-complete
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results f
or
normal
program
s!
Possible and Necessary Answer Sets of PASP
uncertain rules : conclusions
introduced new semantics for PASP :
◾ intuition of uncertain rules
◾ intuitive results with negation-as-failure (as in Bauters, 2010)
◾ proper extension of ASP (as in Nicolas, 2006)
◾ allows explicit choice (i.e. choose either A or B)
◾ better for reasoning in inconsistent setting
◾ higher complexity
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Possible and Necessary Answer Sets of PASP
Questions?
contact: [email protected]