Integration of abductive reasoning and constraint optimization in SCIFF 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)
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) ok! X≤3 X≤3 X≤3 X=3
Implementation ok! • Depends on the implementation of ALP • CHR [Abdennadher&Christiansen, 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
Semantics of optimization • Semantics of optimization in CLP [Fages, Marriot&Stuckey] min( X : p(X,Y) ) • Naïve abductive semantics: min( X : p(X,Y) ) p(X,Y) not( p(X1,Y1), X1 < X ) 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 ) KBÈ{a(1), a(2)} |= p(1) Ùnot(p(X1),X1<1) M=1 KBÈ{a(1)}|= p(1) Ùnot(p(X1),X1<1) M=1 KBÈ{a(2)} |= p(2) Ùnot(p(X1),X1<2) M=2
What is wrong? • We are not optimizing w.r.t. the set : • We want the minimum to be among all the possible sets . • Let us consider programs stratified both w.r.t. negation and optimization KB ╞═min( X : p(X,Y) )
3-valued completion H 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). False not True
3-valued completion H 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). False not True
3-valued completion H 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). False not True
3-valued completion for abducibles 2 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). p(1) :- not q(1). 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’<vs.t. p(v’) is true with the same • there is no v’’<vs.t. p(v’’) is true with another . • (false otherwise)
Example: Game Theory • [Gavanelli et al. ECAI 2008]
Interactions? a(Ma,Xa),b(Mb,Xb)->Xa=Xb=f(Ma,Mb). max(Vb : a(Ma,Xa), min(Xb : b(Mb,Xb), ) = Vb ) max a(0) a(1) min b(0) b(1) b(0) b(1) 5 10 4 3 • 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!
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’<vs.t. p(v’) is true with the same • there is no v’’<vs.t. p(v’’) is true with another ’’ ∆’’ ∩ (A \ ∆ ) ⊆ Am , ∆’’ ∩ (A \ Am ) = ∆ ∩ (A \ Am )
Interactions? max a(0) a(1) min max{a,b}(Vb : a(Ma,Xa), min{b}(Xb : b(Mb,Xb), ) = Vb ) • Now min{b} is • true in {a(0),b(o)} and {a(1),b(1)} • false in {a(1), b(0)} {a(0),b(1)} b(0) b(1) b(0) b(1) 5 10 4 3 Am={b} Am={b} b(0) b(1) a(0) a(1) b(1) b(0) ’’ ’’
Interactions? max a(0) a(1) min max{a,b}(Vb : a(Ma,Xa), min{b}(Xb : b(Mb,Xb), ) = Vb ) • Now min{b} is • 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) b(1) 5 3 Am={b} Am={b} b(0) b(1) a(0) a(1)
Soundness • Soundness of success: if P├─∆G with abductive answer (∆, σ), then P |= ∆σGσ • Soundness of failure: if the SCIFFopt derivation for a goal G finitely fails, then PG • 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?
