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
Diagnosis
headache:- flu.
headache:- period.
headache:- hangover.
flu, vaccine -> false.
hangover -> drank.
period, sex(male) -> false.
?- headache.
Yes, flu
Yes, period
More? ;
Event Calculus
holdsat(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 Calculus
holdsat(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 negation
p(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.
Constraint Store
 headache :- hangover.
 flu, vaccine -> false
?- vaccine , headache.
Constraint Solver
flu
fail
Abduction = constraint solving
[Kowalski, Toni, Wetzel 98]
 headache :- flu.
Constraint Store
 headache :- hangover.
 headache :- period.
 flu, vaccine -> false.
 hangover -> drank.
 period, sex(male) -> false.
?- vaccine , headache.
flu
Constraint Solver
fail
Constraint Handling Rules (CHR)
 Declarative language for defining constraint solvers
 Simplification rules:
c1, c2, ..., cn <=> guard | body
 activated 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).
leq(A,B), leq(B,C), leq(C,A)
leq(A,B), leq(B,C), leq(C,A), leq(A,C)
leq(A,B), leq(B,A), A=C
A=B, A=C
Abduction in CHR
[Abdennadher, Christiansen, Dahl]
 Abducibles mapped to CHR constraints
 headache :- flu.
 headache :- hangover.
 flu, vaccine -> false
Abduction in CHR
 headache :- flu.
 headache :- ...
 flu, vaccine ==> false
?-vaccine, headache.
flu
fail
Constraint Store
Abduction in CHR
 headache :- hangover.
 headache :- ...
 drank.
 hangover ==> drank
?-
headache.
hangover
drank
success
Constraint Store
Abduction in CHR
 headache :- period.
Constraint Store
 headache :- ...
 sex(male).
 period, sex(male) ==> false
! CHR: invalid syntax "sex(male)"
! Undeclared constraint sex/1 in head of rule
Problem:
Syntax
Implementation
(CHR)
Declarative
Semantics
Syntax
Operational
Semantics
Implementation
Operational semantics
 Propagation
 a(X)
a(Y), b -> c
 Case analysis
 (X=Y, b) -> c
 Equality rewriting
 p(A,B,C)=p(D,E,F)
 Unfolding
 p(X) -> Goal
p(X):- a.
 a -> Goal.
b -> Goal
 ...
 Constraint solving
(X=Y, b) -> c
X=Y, (b -> c)
\/
A=D, B=E, C=F
p(X):-b.
X\=Y
Abduction in CHR (SCIFF)
Constraint Store
 headache :- flu.
 headache :- ...
ic( vaccine, flu -> false )
?-vaccine, headache.
flu
fail
ic( flu -> false)
Transitions
 Propagation 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).
Transitions
 Propagation transition (+ case analysis):
abd(X), ic([abd(Y)|Rest]-> Head)
==>
copy(ic([abd(Y) |Rest ]-> Head) ,
ic([abd(Y’)|Rest’]->
Head’)),
ic([abd(vaccine),abd(flu)]->
reif_unify(X,Y’,Boolean),
ic([abd(period),sex(male)]->
(
Boolean=1, ic(Rest’->Head’)
ic([abd(hangover)]-> drank)
; Boolean=0).
abd(hangover)
false)
false)
No hashing
 Does not use CHR’s hashing
 Solution: 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 contains
abd(mother, 2, mother(X,john))
Hashing
 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’,Boolean),
(
Boolean=1, ic(Rest’->Head’)
; Boolean=0).
Requires the first
two arguments
identical
Hashing
 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’)),
ic([abd(vaccine,1,vaccine),abd(flu,1
reif_unify(X,Y’,Boolean),
ic([abd(period,1,period),sex(male)](
Boolean=1, ic(Rest’->Head’)
ic([abd(hangover,1,hangover)]-> dran
; Boolean=0).
abd(hangover,1,hangover)
Postpone choices
 new 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 executed
 phase(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))
).
Results
Experiment
Auction protocol
Block world
AlLoWS Feeble
Conformance
AlLoWS nonconformant
SCIFF
2005
2.27 s
45.0 s
84.4 s
SCIFF 2011
3.7 s
3.3 s
0.37 s
15.7 s
36.8 s
Conclusions
 CHR implementation of an abductive proof-procedure
 Sound, complete, sound treatment of negation
 Well 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?