Marco Gavanelli – Università di Ferrara, Italy
Marco Alberti – Universidade nova de Lisboa, Portugal
Evelina Lamma – Università di Ferrara, Italy
Abductive Logic Programming
 ALP = < KB, A, IC >
 KB = logic program (set of clauses)
 A = atoms without definitions, can be assumed
 IC = Integrity constraints (usually, implications)
KB   ╞═ G
KB   ╞═ IC
 Reasoning from effects to causes
Abductive Constraint Logic
Programming
 E.g., ALP+CLP(FD),
 KB = { p(X) :- a(X), 1 ≤ X ≤ 2. }
G = p(X)
0={a(1)}
1={a(2)}
2={a(1), a(2)}
KB |= p(1)
KB |= p(2)
KB |= p(1)
KB |= p(2)
Optimization in ALP
 Find the best explanation, plan, … w.r.t. some given
objective function
 Syntax:
min( X : p(X,Y) ) = M
means
M = min{ X : p(X,Y) }
Use the minimize predicate of
CLP(FD)
 Operational semantics
p(X,Y)
X≤3
X≤3
X≤3
X=3
ok!
Implementation
 Depends on the implementation of ALP
 CHR [Abdennadher&Christiansen,
ok!
Christiansen&Dahl, Alberti et al., Badea et al.]
 can call all the predicates of the host (CLP) language ->
also minimize!
 p(X) :- a(X), 1 ≤ X ≤ 2.
min( X : p(X)) = M
0={a(1)}
1={a(2)}
2={a(1), a(2)}
KB |= p(1)
KB |= p(2)
KB |= p(1)
KB |= p(2)
Semantics of optimization
 Semantics of optimization in CLP [Fages,
Marriot&Stuckey]
min( X : p(X,Y) )
p(X,Y)  not( p(X1,Y1), X1 < X )
 Naïve abductive semantics:
min( X : p(X,Y) )
KB   ╞═ p(X,Y)  not( p(X1,Y1), X1 < X )
Naïve declarative semantics
 KB = { p(X) :- a(X), 1 ≤ X ≤ 2. }
?- min(X : p(X)) = M
KB   |= p(X)  not( p(X1), X1 < X )
KB{a(1), a(2)} |=
p(1)  not(p(X1),X1<1)
KB{a(1)} |=
p(1)  not(p(X1),X1<1)
M=1
KB{a(2)} |=
p(2)  not(p(X1),X1<2)
M=2
M=1
What is wrong?
 We are not optimizing w.r.t. the set :
KB   ╞═ min( X : p(X,Y) )
 We want the minimum to be among all the possible
sets .
 Let us consider programs stratified both w.r.t. negation
and optimization
3-valued completion
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
H
not
False
True
3-valued completion
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
H
not
False
True
3-valued completion
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
H
not
False
True
3-valued completion for
abducibles
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
p(1) :- not q(1).
1
2
not
3
4
Tp for optimization (simplified)
 Given the atom
min( X : p(X) ) = M
 Suppose that all atoms from the grounding of p(X) are
either true or false
 then min( X : p(X) )=M is true iff:
 there exists some value v such that
 p(v) is true
 there is no v’<v s.t. p(v’) is true with the same 
 there is no v’’<v s.t. p(v’’) is true with another .
 (false otherwise)
Example: Game Theory
   
 [Gavanelli et al. ECAI 2008]
Interactions?
max
a(0)
a(Ma,Xa),b(Mb,Xb)->Xa=Xb=f(Ma,Mb).
max(Vb : a(Ma,Xa),
min(Xb : b(Mb,Xb),
) = Vb
)
a(1)
min
b(0)
b(1) b(0)
5
10
b(1)
4
• true iff for each possible , there is no better value
• Thus, the only option is  ={a(1),b(1)}
• Second player plays also the move of the first!
3

Solution
Am
 Add the set of literals Am that can be
’’
abduced “inside” min in the syntax:
minAm ( X : p(X) )
 Suppose that all atoms from the grounding of p(X)
are either true or false
 then minAm ( X : p(X) )=M is true iff:
 there exists some value v such that
 p(v) is true
 there is no v’<v s.t. p(v’) is true with the same 
 there is no v’’<v s.t. p(v’’) is true with another ’’
∆’’ ∩ (A \ ∆ ) ⊆ A , ∆’’ ∩ (A \ A ) = ∆ ∩ (A \ A )
max
Interactions?
a(0)
max{a,b}(Vb : a(Ma,Xa),
b(0)
min{b}(Xb : b(Mb,Xb),
) = Vb
5
)
 Now min{b} is
Am={b}
 true in {a(0),b(o)} and
{a(1),b(1)}
 false in {a(1), b(0)}
{a(0),b(1)}
a(1)
min
b(1) b(0)
10
 b(0)
4
’’
3
Am={b}
 b(1)
a(0)
b(1)
b(1)
a(1)
b(0)
’’
max
Interactions?
a(0)
max{a,b}(Vb : a(Ma,Xa),
b(0)
min{b}(Xb : b(Mb,Xb),
) = Vb
5
)
 Now min{b} is
Am={b}
 true in {a(0),b(o)} and
{a(1),b(1)}
 false in {a(1), b(0)}
{a(0),b(1)}
 max{a,b} is true for
{a(0),b(o)}
 b(0)
a(1)
min
b(1)
3
Am={b}
 b(1)
a(0)
a(1)
Soundness
 Soundness of success: if
P ├─∆ G
with abductive answer (∆, σ), then
P |= ∆σ Gσ
 Soundness of failure: if the SCIFFopt derivation for a
goal G finitely fails, then
P
G
 Proved for SCIFF, but should hold for most (sound)
abductive procedures
Limitations
We do not currently consider the following cases
 Optimization goals in the conditions of integrity
constraints:
min( X : p(X,Y) ) -> …
 Conjunctions of optimization atoms
min( X : p(X) ), max( X : p(X) )
min( X : p(X),X>0 ), min( Y : q(Y),Y>0 ) p(X),q(X) -> false
 Protected variables
Conclusions
 First declarative semantics for abductive logic
programming (ALP) with optimization (for finding the
optimal )
 Proof of soundness of ALP+optimization
 Proved for SCIFF, should hold for all (sound)
abductive procedures
 Directly usable in abductive proof-procedures
implemented in CHR
Thank you for your attention!
Questions?