Products of Mealy-type Rough Finite State Machines
Shambhu Sharan
∗
and S.P. Tiwari
†
Department of Applied Mathematics
Indian School of Mines
Dhanbad-826004, India
∗
shambhupuremaths@gmail.com
†
sptiwarimaths@gmail.com
Abstract—The aim of this paper is to introduce several kinds
of products of Mealy-type rough finite state machines (a rough
finite-state machine with output). We establish the relationship
among such products through coverings and investigate some
algebraic properties of these products.
Keywords-Rough finite state machine; Mealy-type rough fi-
nite state machine; Covering; Direct product; Wreath product;
Cascade product.
I. I NTRODUCTION
The concept of a finite-state machine (also known as
finite-state semiautomaton), i.e., a triple (,,), consist-
ing of two (finite) sets (of states) and (of inputs) and
a map : × → (called the transition map), is
well-known (cf. [6]). In [11], the notion of fuzzy finite -
state machine (also called fuzzy finite semiautomaton) has
been studied by Mordeson and Malik. Also, similar, or
closely related, notions have been introduced and studied
subsequently by G´ omez, Lizasoain and Moreno [5], Kim,
Kim and Cho [7] and Li and Pedrycz [9]. In view of
algebraic study of such fuzzy machines, the concept of fuzzy
transformation semigroups, coverings and several kinds of
products e.g., direct products, cascade products and wreath
products were introduced and studied in ([7], [8], [10] and
[11]). After the introduction of rough set theory by Pawlak
[12], Basu [3] introduced the concept of a rough finite state
(semi)automaton and extended the idea further by designing
a ‘recognizer’ that accepts imprecise statements. This finite-
state machine differs from its crisp and fuzzy versions only
in terms of return of transition map; specifically, in case
of rough finite-state machine, the transition map returns a
rough set of states (rather than a single state or a subset of
states or a fuzzy (sub)set of states). Further, for the sake
the algebraic study of rough finite-state machines, in [15],
the concepts of rough transformation semigroup associated
with a rough finite-state machine and coverings of rough
finite-state machines was introduced and studied; in [16],
Tiwari, Sharan and Singh introduced and studied the notion
of several products viz., (full) direct product, restricted direct
product, cascade product and wreath product of rough finite-
state machines and established the relationship among such
products through coverings. The purpose of this paper is
to introduce the concept of Mealy-type rough finite-state
machine, which is a rough finite-state machine with output
and their several kinds of products. These Mealy-type rough
finite-state machines have been introduced here for the sole
purpose of examining how they may be connected together
to produce new Mealy-type rough finite-state machine. Fur-
thermore, we explore the relationship among such products
through coverings and discuss some algebraic properties.
II. PRELIMINARIES
In this section, we recall some concepts associated with
rough sets, rough finite-state machines. Finally, we introduce
the concept of Mealy-type rough finite-state machines and
their coverings.
A. Rough Sets
Over the past three decades, a number of definitions of the
rough set have been appeared in the literature (cf. e.g., [1],
[12], [13], [14]), among those in [2] it has shown that some
are equivalent. For completeness, we recall the following
key notions, for other notions, readers are referred to [2].
Definition 2.1: [12] An approximation space is a pair
(,), where is a nonempty set and is an equivalence
relation on .
If is an equivalence relation on and ∈ , then []
will denote the equivalence class of under and / will
denote the quotient set consisting of equivalence classes of
.
Definition 2.2: [18] For an approximation space (,)
and ⊆ , the lower approximation of and the upper
approximation of are respectively defined as follows:
=
∪
{[] ∈ / ∣ [] ⊆ },
=
∪
{[] ∈ / ∣ [] ∩ ∕= }.
The pair ( , ) is called a rough set. We shall denote it by
A.
For give an approximation space (,) and ⊆ , the
boundary BnA of is defined by − . is called definable
(or exact) in (,) iff = , i.e., = . Equivalently,
a definable set is an union of equivalence classes under .
2012 National Conference on Computing and Communication Systems (NCCCS)
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