The learning framework of ACL differs from ILP because both the background
knowledge and the learned theory
are abductive theories.
An abductive theory T in Abductive Logic Programming is a
triple < P, A, I >, where P is a definite logic program, A is a set
of predicates called
abducible predicates (or simply abducibles),
and I is a set of rangerestricted clauses called integrity constraints.
As a knowledge representation framework, when we represent a problem
in ALP via an abductive theory T, we generally assume that the abducible
predicates in A carry all the incompleteness of the program P
in modelling the external problem domain in the sense that if we (could)
complete the abducible predicates in P then P
would completely
describe the problem domain. An abductive theory can support abductive
(or hypothetical) reasoning for several purposes such as diagnosis, planning
or default reasoning. The central notion used for this is that of an abductive
explanation for an observation or a given goal. When an observation
O
can be abductively explained in a theory T with explanation Delta
we write
T=_{A}O. with Delta.
We
can now define the learning problem when the language of the background
and target theories is the one of ALP.
A set of rules P' belonging to H and a set of constraints I' belonging to Ysuch that the new abductive theory T'=< P union P', A, I union I'> satisfies the following conditions
In effect, we have replaced the deductive entailment in the ILP problem
with abductive entailment to define the ACL learning problem.
The full ACL problem can be split into two subproblems: (1) learning the rules together with appropriate strong explanations and (2) learning integrity constraints. The solutions of the two subproblems can be combined to obtain a solution for the original problem.
The first subproblem, called ACL1, has the following definition.
A set of rules P' belonging to H such that the new abductive theory T_{ACL1}=< P union P', A, I> satisfies the following conditions
The ACL1 problem is solved by the system ACL1. The input file for the learning problem of example 1 is father.bg. From this input file, the ACL1 system finds the above solution, as can be seen from the father.rules file.
Indeed, the information generated by ACL1 through the abductive explanations for negative examples can be used to provide a solution of the full ACL problem through a second learning phase. From the output of ACL1, i.e., its set of rules and the sets of assumptions and for covering positive examples and uncovering negative ones, a solution to ACL can be found by learning constraints that are consistent with Delta^{+}and inconsistent with the complement of every abducible in Delta^{} .
The definition of the second subproblem, called ACL2, can be given as follows.
A set of constraints I' Î Ysuch that the new abductive theory satisfies the following condition
The ACL2 problem is solved by means of the ICL system. The input for
ICL is generated by the ACL1 system, for the case of example 1 the input
files for ICL are again father.bg, containing the
background knowledge, and father.kb, containing
the interpretations, that is generated by the ACL1 system.
The theory, obtained by combining the solutions of the two subproblems,
gives a solution to the full ACL problem.
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