Marco Gavanelli – Università di Ferrara, ItalyMarco Alberti – Universidade nova de Lisboa,

PortugalEvelina Lamma – Università di Ferrara, Italy

Abductive Logic ProgrammingALP = < KB, A, IC >

KB = logic program (set of clauses)A = atoms without definitions, can be

assumedIC = Integrity constraints (usually,

implications)

KB ╞═ G KB ╞═ ICReasoning from effects to causes

Diagnosisheadache:- flu.headache:- period.headache:- hangover.flu, vaccine -> false.

hangover -> drank.

period, sex(male) -> false.

?- headache.Yes, flu More? ;Yes, period

Event Calculusholdsat(Fluent,Time):- initially(Fluent),

not(clipped(0,Fluent,Time)).holdsat(Fluent,Time):- initiates(Action,Fluent),

happens(Action,T1), not(clipped(T1,Fluent,Time)).clipped(T1,Fluent,T2):- terminates(Action),

happens(Action,T), T1<T<T2.initially(f1). initially(f2).

happens(a,2) happens(b,4) happens(c,7)

Abductive Event Calculusholdsat(Fluent,Time):- initially(Fluent),

not(clipped(0,Fluent,Time)).holdsat(Fluent,Time):- initiates(Action,Fluent),

happens(Action,T1), not(clipped(T1,Fluent,Time)).clipped(T1,Fluent,T2):- terminates(Action),

happens(Action,T), T1<T<T2.initially(f1). initially(f2).

happens(a,2) happens(b,4) happens(c,7)={ }

Sound negationp(1).

p(f(X)):- q(X).

q(2).

?- not(p(Y)).

yes, Y\=1, Y\=f(2)

Abduction = constraint solving[Kowalski, Toni, Wetzel 98]headache :- flu.headache :- hangover.

flu, vaccine -> false

Constraint Store

?- , headache.vaccine

fluConstraint Solver

fail

Abduction = constraint solving[Kowalski, Toni, Wetzel 98]headache :- flu.headache :- hangover.headache :- period.

flu, vaccine -> false.hangover -> drank.period, sex(male) -> false.

Constraint Store

?- , headache.vaccine

flu

Constraint Solver

fail

Constraint Handling Rules (CHR)Declarative language for defining constraint solversSimplification rules:

c1, c2, ..., cn <=> guard | bodyactivated if some constraints in the store match with c1, c2, ..., cn and guard is true

removes c1, c2, ..., cn from the store and executes body

Propagation rules:c1, c2, ..., cn ==> guard | body

activated if some constraints in the store match with c1, c2, ..., cn and guard is true

executes body

Example: leq (less or equal)reflexivity@ leq(X,X) <=> true.

antisymmetry@ leq(X,Y), leq(Y,X) <=> X=Y.

transitivity@ leq(X,Y), leq(Y,Z) ==> leq(X,Z).

leqleq(A,B), (A,B), leqleq(B,C), (B,C), leqleq(C,A)(C,A)

leqleq(A,B), (A,B), leqleq(B,C), (B,C), leqleq(C,A), (C,A), leqleq(A,C)(A,C)

leqleq(A,B), (A,B), leqleq(B,A), A=C(B,A), A=C

A=B, A=CA=B, A=C

Abduction in CHR [Abdennadher, Christiansen, Dahl]Abducibles mapped to CHR constraintsheadache :- flu.headache :- hangover.

flu, vaccine -> false

Abduction in CHRheadache :- flu.headache :- ...

flu, vaccine ==> false

Constraint Store

?- , headache.vaccine

flufail

Abduction in CHRheadache :- hangover.headache :- ...drank.hangover ==> drank

Constraint Store

?- headache.

hangover

success

drank

Abduction in CHRheadache :- period.headache :- ...sex(male).

Constraint Store

period, sex(male) ==> false

! CHR: invalid syntax "sex(male)"! Undeclared constraint sex/1 in head of rule

Problem:

Operational semanticsPropagation

a(X) a(Y), b -> c (X=Y, b) -> cCase analysis

(X=Y, b) -> c X=Y, (b -> c) \/ X\=Y

Equality rewriting p(A,B,C)=p(D,E,F) A=D, B=E, C=F

Unfoldingp(X) -> Goal p(X):- a. p(X):-b.a -> Goal. b -> Goal

...Constraint solving

Abduction in CHR (SCIFF)headache :- flu.headache :- ...

Constraint Store

?- , headache.vaccine

flufail

ic( )vaccine, flu -> false

ic( flu -> false)

TransitionsPropagation transition (+ case analysis):abd(X), ic([abd(Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,Boolean),

( Boolean=1, ic(Rest’->Head’)

; Boolean=0).

TransitionsPropagation transition (+ case analysis):abd(X), ic([abd(Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,Boolean),

( Boolean=1, ic(Rest’->Head’)

; Boolean=0).

ic([abd(vaccine),abd(flu)]-> false)

ic([abd(period),sex(male)]-> false)

ic([abd(hangover)]-> drank)

abd(hangover)

No hashingDoes not use CHR’s hashingSolution: abducibles are represented with

redundant information:abd(Functor, Arity, Atom)

So to abduce atom X:abd(X):- functor(F, A, X),

abd(F, A, X).E.g., if I abduce atom mother(X,john), the

constraint store containsabd(mother, 2, mother(X,john))

HashingPropagation transition (+ case analysis):abd(F,A,X), ic([abd(F,A,Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,Boolean),

( Boolean=1, ic(Rest’->Head’)

; Boolean=0).

Requires the first two

arguments identical

HashingPropagation transition (+ case analysis):abd(F,A,X), ic([abd(F,A,Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,Boolean),

( Boolean=1, ic(Rest’->Head’)

; Boolean=0).

ic([abd(vaccine,1,vaccine),abd(flu,1,flu)]-> false)

ic([abd(period,1,period),sex(male)]-> false)

ic([abd(hangover,1,hangover)]-> drank)

abd(hangover,1,hangover)

Postpone choicesnew CHR constraint

nondeterministic(Goal)says that Goal can open a choice point, so should be called as late as possible.

Two phases, declared by a CHR constraint phase/1:phase(deterministic): only deterministic goals are executedphase(nondeterministic): exactly ONE nondeterministic goal

can be executed

switch2det @ phase(nondeterministic), nondeterministic(G) <=>

call(G), phase(deterministic).

switch2nondet @ phase(deterministic) <=> phase(nondeterministic)

Postponing nondet.Propagation transition (+ case analysis):abd(F,A,X), ic([abd(F,A,Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,B),

( B=1, ic(Rest’->Head’)

; B=0).

Postponing nondet.Propagation transition (+ case analysis):abd(F,A,X), ic([abd(F,A,Y)|Rest]-> Head)

==>

copy(ic([abd(Y) |Rest ]-> Head) ,

ic([abd(Y’)|Rest’]-> Head’)),

reif_unify(X,Y’,B),

(B == 1 -> ic(Rest’, Head’) ;

B == 0 -> true ;

nondeterministic((B#=1,ic(Rest’,Head’)) ; B#=0))

).

ResultsExperiment SCIFF

2005SCIFF 2011

Auction protocol 2.27 s 0.37 sBlock world 45.0 s 15.7 sAlLoWS Feeble Conformance

84.4 s 36.8 s

AlLoWS non-conformant

3.7 s 3.3 s

ConclusionsCHR implementation of an abductive proof-

procedureSound, complete, sound treatment of

negationWell integrated with constraint solving,

CLP(FD), CLP(R), universally quantified variables, quantifier restrictions, etc.

Easy to extend for other features (see other talk after coffee break)

Thank you for your attention!

Questions?