# Abduction in ASP

Well, forgive my ignorance about the matter as I have been playing with ASP for the last couple of days.

Consider this simple example

p.
s :- p.


And the corresponding output generated after running the program in clingo:

$./clingo.exe ex.lp --number=0 Answer: 1 p s SATISFIABLE Models : 1 ...  Where possible models are generated having p true and the formula p -> s. But if I want to ask some query that requires abductive reasoning in order to generate some answers; in other words, I need to know the possible solutions to the fact s true. So the "supposed" example should be like the following: s. s :- p.  But unfortunately the answer does not contain p as expected. $ ./clingo.exe  ex.lp --number=0
% warning: p/0 is never defined
s
SATISFIABLE

Models      : 1
...


Could that be done in any way in ASP?

I found out that this could not be done natively in ASP (or at least in the solvers that I am using). So the abduction theory needs to be modeled with the problem in order to derive expected results.

This is an example that demonstrates how it can be done. I haven't had the time to thoroughly test its efficiency but it works for some basic examples.

% abduction.lp

%%%%%%%%%%%%% Preprocessing %%%%%%%%%%

% Remove tautological clauses
taut(C) :- pos(C,X), neg(C,X).
preprocessed_clause(C) :- clause(C), not taut(C).

% which variable is in which clause
var_in_clause(C,X) :- preprocessed_clause(C), pos(C,X).
var_in_clause(C,X) :- preprocessed_clause(C), neg(C,X).

%%%%%%%%%%%%% Guess a Candidate Solution %%%%%%%%%%

% S, a subset of hypotheses  is a solution iff (1) and (2) hold
solution(S) :- hypothesis(S), not nosolution(S).
nosolution(S) :- hypothesis(S), not solution(S).

%%%%%%%%%%%%% (1) background theory is consistent with S %%%%%%%%%%

% guess an assignment for all variables
true_consistency(X) :- variable(X), not false_consistency(X).
false_consistency(X) :- variable(X), not true_consistency(X).

%% Solution must be true
true_consistency(S) :- solution(S).

%% Check for each clause in T whether it is satisfied
sat(C) :- preprocessed_clause(C), pos(C,V), true_consistency(V).
sat(C) :- preprocessed_clause(C), neg(C,V), false_consistency(V).

%% In case a clause is not satisfied, remove AS
notsat :- preprocessed_clause(C), not sat(C).
:- notsat.

%%%%%%%%%%%%%% (2) background theory and solution entail the manifestation%%%%%%%%%%

%% Find assignment, which is a counter-example to entailment
true_entail(X) | false_entail(X) :- variable(X).

% ordering over variables in preprocessed clauses
lowerThan(C,X,Y) :- var_in_clause(C,X), var_in_clause(C,Y), X<Y.
not_successor(C,X,Z) :- lowerThan(C,X,Y), lowerThan(C,Y,Z).
successor(C,X,Y) :- lowerThan(C,X,Y), not not_successor(C,X,Y).
not_infimum(C,X) :- lowerThan(C,Y,X).
not_supremum(C,X) :- lowerThan(C,X,Y).
infimum(C,X) :- not not_infimum(C,X), var_in_clause(C,X).
supremum(C,X) :- not not_supremum(C,X), var_in_clause(C,X).

% check if unsat
unsatupto(C,V) :- infimum(C,V), pos(C,V), false_entail(V).
unsatupto(C,V) :- infimum(C,V), neg(C,V), true_entail(V).
unsatupto(C,V) :- unsatupto(C,PreV), successor(C,PreV,V), pos(C,V), false_entail(V).
unsatupto(C,V) :- unsatupto(C,PreV), successor(C,PreV,V), neg(C,V), true_entail(V).

unsat(C) :- unsatupto(C,V), supremum(C,V).
unsat :- unsat(C).

% make sure that variables in manifestations and solution get the right truth value
false_entail(X) :- manifestation(X).
true_entail(X) :- solution(X).

% saturation
true_entail(X) :- variable(X), unsat.
false_entail(X) :- variable(X), unsat.

:- not unsat.

#show solution/1.
%#show manifestation/1.
%#show hypothesis/1.
%#show variable/1.


And the example provided in this post's question:

% a_ex2.lp

% Variables V
variable("s";"p").

% Theory T over V
clause(1).
pos(1,"s").
neg(1,"p").

hypothesis("p").

manifestation("s").


Output:

\$ ./gringo.exe abduction.lp a_ex2.lp | ./claspD.exe --number 0