Affiliations: Department of Pure Mathematics, University of
Calcutta, 35, Ballygunge Circular Road, Kolkata 700 019, India. E-mail:
[email protected] | Department of Mathematics and Statistics, Indian
Institute of Technology, Kanpur 208 016, India. E-mail: [email protected]
Abstract: A dialogue is an 'activity' by a pair of agents to arrive at
some kind of understanding over a concept/belief/piece of information etc.
represented by a subset (the extension) in some universe of discourse. The
universe is partitioned into two different sets of granules (equivalence
classes) representing the perceptions of the agents. So, there are two
approximation spaces at the beginning. A third approximation space arises out
of superimposition of the two partitions. A dialogue is a finite process of
gradual enhancement of the two base subsets of the agents, in their 'common'
approximation space. Through this process, various kinds of overlap may emerge
between the two final subsets. A first introduction of the idea of a dialogue
in rough context was made in [6]. This paper further develops the notion and
focusses upon the study of the above-mentioned overlaps in a systematic manner.
Given two sets A and B in an approximation space, there are nine possible
inclusion relations among the sets lo(A),A, up(A), lo(B),B and up(B) where lo
and up denote the lower and upper approximation operators respectively. There
are five resulting equivalence classes and the quotient set forms a lattice by
implication ordering. That is, of the nine relations, only five are independent
and they form an implication or entailment lattice. Starting with this basic
lattice other implication lattices are formed. Relationship of these lattices
with the various overlap conditions between the final pair of sets arrived at
after a dialogue is studied. Finally, examples are given, one of which relates
dialogues in rough context with rough belief revision [3] – in a line
similar to the approach of [5].
Keywords: Rough sets, Negotiation, Modal system S5, Belief change
