  Abduction, first proposed in the setting of classical logics, has been
studied with growing interest in the logic programming area during the last
years.
  In this paper we study {\em abduction with penalization} in logic
programming. This form of abductive reasoning, which has not been previously
analyzed in logic programming, turns out to represent several relevant
problems, including optimization problems, very naturally. We define a formal
model for abduction with penalization from logic programs, which extends the
abductive framework proposed by Kakas and Mancarella. We show the high
expressiveness of this formalism, by encoding a couple of relevant problems,
including the well-know Traveling Salesman Problem from optimization theory, in
this abductive framework. The resulting encodings are very simple and elegant.
We analyze the complexity of the main decisional problems arising in this
framework. An interesting result in this course is that ``negation comes for
free.'' Indeed, the addition of (even unstratified) negation does not cause any
further increase to the complexity of the abductive reasoning tasks (which
remains the same as for not-free programs).
