  This paper presents a logic framework for modeling the interaction among
deductive databases in a P2P (Peer to Peer) environment. Each peer joining a
P2P system provides or imports data from its neighbors by using a set of
mapping rules, i.e. a set of semantic correspondences to a set of peers
belonging to the same environment. Two different types of mapping rules are
defined: mapping rules allowing to import a maximal set of atoms not leading to
inconsistency (called maximal mapping rules) and mapping rules allowing to
import a minimal set of atoms needed to restore consistency (called minimal
mapping rules). Implicitly, the use of maximal mapping rules states it is
preferable to import as long as no inconsistencies arise; whereas the use of
minimal mapping rules states that it is preferable not to import unless a
inconsistency exists. The paper presents three different declarative semantics
of a P2P system: (i) the Max Weak Model Semantics, in which mapping rules are
used to import as much knowledge as possible} from a peer's neighborhood
without violating local integrity constraints; (ii) the Min Weak Model
Semantics, in which the P2P system can be locally inconsistent and the
information provided by the neighbors is used to restore consistency, that is
to only integrate the missing portion of a correct, but incomplete database;
(iii) the Max-Min Weak Model Semantics that unifies the previous two different
perspectives captured by the Max Weak Model Semantics and Min Weak Model
Semantics. This last semantics allows to characterize each peer in the
neighborhood as a resource used either to enrich (integrate) or to fix (repair)
the knowledge, so as to define a kind of integrate-repair strategy for each
peer.
  Under consideration in Theory and Practice of Logic Programming (TPLP).
