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when the length of the goal is long, yet [2] analyzes the scalability of large P2P sys-
tems. On the other hand, we send the clause itself without splitting and no recollection
is made. Our second approach can be regarded as a dynamic version of the first ap-
proach, in which messages are sent whenever new clauses are derived, and there is no
presupposed network structures of agents. Such dynamic aspects are not seen in the P2P
setting. Another difference is that [1] can only deal with propositional knowledge bases,
while SOL resolution and SOLAR in our paper can be used for consequence finding in
first-order clausal theories.
Although not in the context of consequence finding, abduction has also been con-
sidered in a distributed setting. Since abduction in clausal theories can be implemented
with consequence finding, such work is somehow related to distributed consequence
finding. Greco [10] considers how to build
joint explanations
from multiple agents in a
P2P setting like [1], but incorporates preference handling to have an agreement between
agents. By extending a blackboard architecture of [5], Ma
et al.
[19] address distribu-
tion of abductive logic programming agents by allowing agents to enter and exit proofs
done by other agents. Those works do not use consequence finding, and communica-
tion between agents are fully guaranteed. More recently, Bourgne
et al.
[4] propose the
learner-critique
approach in which the role of each agent dynamically changes between
a generator and a tester of hypotheses when each agent never knows which symbols are
shared with other agents. In our methods, all agents work uniformly as a reflective in-
ference system that derives consequences upon input of new formulas, although shared
symbols are assumed to be known to both agents. Fisher [9] shows that certain forms
of negotiation can be characterized by distributed theorem proving in which agents act
as theorem-proving components. Analogously, distributed consequence finding might
contribute to extended types of negotiation between agents.
In this work, we have focused on distributed reasoning systems in which a clause set
is partitioned and the common symbols between partitions are associated with links. In
contrast, there is another formalization of distribution in which variables or symbols are
partitioned and clauses containing symbols from different partitions are associated with
links between those partitions. The former formalization is called
clause-set partitioned
distribution
, while the latter is called
variable-set partitioned distribution
. Most works
on
distributed constraint satisfaction problems
(DCSP) are based on the latter formal-
ization [25,11]. It is known that these two formalizations can be converted into each
other in the propositional case (cf., [7]), yet the effect of the latter case is unknown for
consequence finding and the former case can be seen more often in the real situations.
7
Conclusions
In this paper, we have proposed the two complete approaches for distributed conse-
quence finding. The first one extends the method of partition-based theorem proving
in a suitable way, and the second one is a more cooperative method for inherently dis-
tributed systems. This paper rather focuses on completeness of inference systems, and
both approaches have merits and demerits. Partition-based approaches can utilize com-
munication languages to realize restricted consequence finding between the partitions,
while the cooperative approach does not need Cycle Cut algorithm. On the negative
