ROUGH SETS ON STATE SPACES OF AUTOMATA
David Staněk
Doctoral Degree Programme (1), FEEC BUT
E-mail: [email protected]
Supervised by: Jan Chvalina
E-mail: [email protected]
Abstract: This paper discuss subclass of finite automata, which have ordering on the state sets created
by a transition (or next-state) function. Hence, there do not exist cycles of more than one element.
We discuss relation of equality of upper closure on the systems of all subsets of state systems of
quasi-automata, which creates an equivalence.
Keywords: Iterations, Kuratowski-Whyburn equivalence, equivalence, monoid, quasi-automaton,
upper closure.
1
INTRODUCTION
Rough sets theory was established in the 1982 by professor Z. Pawlak. This theory is highly usable
when working with vagueness and indiscernibility in the inaccurate data field. Further, in fields such
as data analysis, determination of recognizing of images theory, electrical (machine) learning, expert
systems, using knowledge basis etc.
Original Pawlak rough sets theory was based on equivalence relation concept. This was generalized
for arbitrary relations and universum covering. Further, this theory was developed in the way of
topological spaces. Study of rough sets and its connection with algebraic structures and ambiguous
mappings and multioperations, has its own importance.
Note, that upper and lower approximations of planar field defined by one variable function, occur in
the Riemann definite integral definition.
2
USED CONCEPTS
Let f be mapping of set a X into itself (transformation of set a X). Then for an arbitrary non negative
integer n ∈ N0 , f n denotes the n−th iteration of the transformation f : X → X. Moreover, f 0 = idX is
identical transformation of the set X and f n+1 = f ◦ f n , if f n : X → X, n ∈ N is defined.
Denote by ∼ f relation on a set X such that x ∼ f y if and only if there exists a pair of non negative
integers m, n ∈ N0 such that f m (x) = f n (y).
The relation ∼ f is obviously reflexive and symmetric on the algebra (X, f ). Moreover, if for x, y, z ∈ X
x ∼ f y, y ∼ f z holds, then there exist elements m, n, k, l ∈ N0 , such that f m (x) = f n (y), f k (y) = f l (z).
Hence the f m+k (x) = f n+k (y) = f k+n (y) = f l+n (z). The relation ∼ f is then the equivalence relation
on X.
Just mentioned relation is called Kuratowski-Whyburn equivalence or shortly K-W equivalence.
Blocks S ∈ X| ∼ f are called f −orbits. A subalgebra (S, f |S) (here symbols f |S mean the restriction
of the function f on the set S) of the mono-unary algebra (X, f ) is said to be a component of the
algebra (X, f ), where the inclusion f (S) ⊂ S holds.
If (Sα , fα ); α ∈ I is the system of all components of the algebra (X, f ) (here fα = f |Sα ), we write
(X, f ) =
∑ (Sα , fα )
α∈I
and this sum is termed as an orbital decomposition of the mono-unary algebra (X, f ).
By monoid we mean an ordered pair (M, ·), where M is non empty set and · is a binary operation with
properties:
• ∀a, b, c ∈ M; a · (b · c) = (a · b) · c,
• ∃e ∈ M such that ∀a ∈ M; a · e = a = e · a.
Just presented properties are called the associativity condition and the existence of a neutral element,
in this order.
By a quasi-automaton we mean a triad A = (S, M, δ), where S is a state set or phase set, M is an input
alphabet or a phase semigroup and δ : S × M → S transition function (or next-state function) satisfying
axioms:
• δ(s, e) = s for every element s ∈ S (neutral element axiom),
• δ(δ(s, m), n) = δ(s, m · n) for every element s ∈ S and m, n ∈ M (mixed associativity axiom).
(This is also a discrete dynamical system or an action of monoid M on the set S.)
Illustrative example. Let us consider transformation f (~x) : {0, 1}4 → {0, 1}4 defined by the rule
f (~x) = A~x +~b, where

0
0
A=
1
0
0
0
0
0
1
0
1
1

1
0
,
1
0
 
0
1

b=
0 .
0
Let us mention, that within the calculations of functional values of function f (~x) the Boolean arithmetic is used. The table below summarizes the sum and the product of elements x, y in Boolean
arithmetic.
x
0
0
1
1
y
0
1
0
1
x+y
0
1
1
1
x·y
0
0
0
1
Orbital structure of function f (~x) with two fixed points x3 , x16 is
3 ; •x16Z c kig
•xO 3
•x1
6 < •x9
•x2
•x6
5 <•xO 12 c
•x5
•x8
•x10
•x15
•x11
•x4
•x7
•x13
•x14
where xi are column vectors listed in the table below.
x1
0
0
0
0
x2
1
0
0
0
x3
0
1
0
0
x4
0
0
1
0
x5
0
0
0
1
x6
1
1
0
0
x7
1
0
1
0
x8
1
0
0
1
x9
0
1
1
0
x10
0
1
0
1
x11
0
0
1
1
x12
1
1
1
0
x13
0
1
1
1
x14
1
0
1
1
x15
1
1
0
1
x16
1
1
1
1
Now we show, that for n−th iteration of function f (~x) we have f n (~x) = An ·~x +~b. For n = 1 the
equality is obvious. Induction assumption is, that f n (~x) = An ·~x +~b holds for all positive integers up
to n ∈ N. Hence
f n+1 (~x) = f ◦ f n = f An (~x) +~b = A An (~x) +~b +~b,
 
0
0

because A ·~b = 
0 , then
0
A An (~x) +~b +~b = AAn~x +~b = An+1~x +~b = f n+1 (~x) +~b. Let us define a function δA as there follows:
δA (~x, n) = An~x +~b
for every vector ~x ∈ {0, 1}4 and every positive integer n ∈ N0 . Then the triad A = {0, 1}4 , N0 , δA is
a quasi-automaton with the state set {0, 1}4 , input alphabet N0 and transition function δA .
With respect to the above considered function, the elements s,t ∈ S, are in relation ∼ f if and only if
for certain a, b ∈ M we have δ(s, a) = δ(t, b), where δ(s, a) = f a (s).
/ and Θ is an equivalence relation on U - we call
A pair (U, Θ) - where U is universal set (U 6= 0)
an approximation space. A mapping Apr:P(U) → P(U) × P(U) defined for X ∈ P(U) by the rule
/ then
Apr(X) = (Ap(X), Ap(X)), where Ap(X) = {x ∈ U; [x]Θ ⊂ X}, Ap(X) = {x ∈ U; [x]Θ ∩ X 6= 0},
Ap(X) (Ap(X)) is called the lower rough approximation of X (the upper rough approximation of X)
in (U, Θ). A pair (A, B) ∈ P(U) × P(U) is called a rough set in (U, Θ) if there exists X ∈ P(U) with
the property (A, B) = Apr(X). (In some literature by a rough set only the upper approximation Ap(X)
is meant. )
For x ∈ X a symbol [x]Θ denotes a block (class) of equivalence Θ containing the element x, i.e.
[x]Θ = {y ∈ X; [x, y] ∈ Θ}.
We define upper closure u+
δ of set Q ⊆ P(S) as follows
u+
δ (Q) = {δ(q, m)|q ∈ Q; m ∈ M}.
3
CHARACTERIZATION OF THE EQUIVALENCE RELATION IN THE UNIVERSAL SET
P(S)
Now, let us define a binary relation ρ on the set P(S) as it follows: elements (sets) A, B ∈ P(S) are in
+
relation ρ if and only if u+
δ (A) = uδ (B). It is easy to show, that the given relation is an equivalence,
since reflexivity, symmetry and transitivity conditions evidently hold.
With respect to the illustrating example we focus on the class of finite automata, which have ordering is on state sets created by transition functions. Hence, for every pair [s, a], [t, b] with property
δ(s, a) = [t, b], δ(t, b) = [s, a], we have [s, a] = [t, b]. Hence, there do not exist cycles with more than
one element. Accurately, relation σ ⊂ S × S defined as sσδ ≡ ∃a ∈ M with property δ(s, a) = t, is
antisymmetric. Moreover, since the relation ρ is reflexive and transitive, it is an ordering, i.e. (S, σδ )
is ordered set.
We are going to prove the following theorem.
Theorem.
Let A = (S, M, δ) be quasi-automaton, for A, B ⊂ S we have
AρB if and only if either A = B or for state s ∈ A4B there exists a state t ∈ A ∩ B such that δ(t, m) = s
for suitable element m ∈ M (Condition C).
+
Proof. 1. Suppose A, B ⊂ S such that AρB, i.e. u+
δ (A) = uδ (B).
/ i.e. A = B and clearly AρB.
If A4B = 0/ we have (A\B) ∪ (B\A) = 0,
Thus we suppose A4B 6= 0/ and s ∈ A4B. Assuming A ∩ B = 0/ we obtain that there exists a pair of
states x ∈ A, y ∈ B with the property δ(x, m) = y, δ(y, n) = x for a suitable pair of elements m, n ∈ M.
In this case states x, y are elements of an at least two-element cycle within S. But this is excluded by
/
the supposition concerning the structure of the quasi-automaton, in question. Thus A ∩ B 6= 0.
+
+
If s ∈ A\B, since s ∈ uδ (A) = uδ (B) there exists t1 ∈ B and m ∈ M with the property s = δ(t1 , m).
But t1 ∈ u+
δ (A). Hence there exists a sequence t2 ,t3 , . . . ,tk ∈ A ∪ B with the property tk ∈ A ∩ B (since
there are excluded cycles), δ(ti , mi ) = ti−1 for suitable finite sequence m2 , m3 , . . . , mk−1 . Denote n =
m1 · m2 · · · · · mk−1 . Then δ(tk , n) = t1 and δ(tk , n · m) = δ(δ(tk , n), m) = s. Denoting t = tk we obtain
δ(t, n · m) = s for t ∈ A ∩ B.
The consideration starting with s ∈ B\A is quite similar to the just described. Hence the above condition (C) is satisfied.
2. Now suppose the condition (C) is satisfied. Let s ∈ u+
δ (A) be an arbitrary state. There exist a
state s1 ∈ A and element m ∈ M such that s = δ(s1 , m). Two cases are possible:
(i) s1 ∈ A\B, (ii) s1 ∈ A ∩ B.
Consider the case (i): Since
A\B ⊂ (A\B) ∪ (B\A) = A4B,
we have s1 ∈ A4B. By the condition (C) there exists a pair [t, n] ∈ (A ∩ B) × M such that s1 = δ(t, n).
Then
t ∈ B, s = δ(s1 , m) = δ(δ(t, n), m) = δ(t, n · m),
thus s ∈ u+
δ (B).
+
+
In the second case (ii) we have also s1 ∈ B and thus s ∈ u+
δ (B) again, hence uδ (A) ⊂ uδ (B).
+
+
Supposing s ∈ uδ (B), we obtain in a similar way as above that s ∈ uδ (A), thus the opposite inclusion
+
+
+
u+
δ (B) ⊂ uδ (A) holds, as well. Consequently uδ (A) = uδ (B), i.e. AρB. The proof is complete. Remark. The above considered necessary and sufficient condition for he relationship AρB can be
reformulated using the ordering ≤ into the form:
For A, B ⊂ S, AρB if and only if either A = B or for any s ∈ A4B there exists t ∈ A ∩ B such that t ≤ s.
4
CONCLUSIONS
Rough set theory, proposed by Pawlak [9] is an extension of the set theory by study of information
systems characterized by inexact and uncertain information. Thus, rough sets theory is an powerful
mathematical tool for uncertain data while modeling the problems in computer science, medical science, data analysis and many other diverse fields.
In this contribution there are explained basic terms of the automata theory motivated by an illustrative
example. Using closures determined by the input monoid and by the transition function we define an
equivalence relation ρ of the state set of an quasi-automaton. There is characterized pair of ρ−related
subsets of the considered state sets, where the equivalence ρ creates an approximation space on P(S).
The considered topic can be extended in several ways.
ACKNOWLEDGEMENT
The author was supported by the Norway Grants mechanism under the project “Mathematical Education Through Modelling Authentic Situations - METMAS”, registration number NF-CZ07-ICP-3201-2015. This support is gratefully acknowledged.
REFERENCES
[1] ABO-TABL, El-Sayed. Rough sets and topological spaces based on similarity. International
Journal of Machine Learning and Cybernetics. 2012, 5(4), 451-458.
[2] BAGIRMAZ, Nurettin a Abdulah F. ÖZCAN. Rough Semigroups on Approximation
Spaces. International Journal of Algebra. 2015, 9(7), 339-350.
[3] BAVEL, Zamir. The Source as a Tool in Automata. Information and Control. 1971, 18(2),
140-155.
[4] BERÁNEK, Jaroslav a Jan CHVALINA. Extensions of Cascades Created by Certain Function Systems. Acta Mathematica Nitriensia. 2015, 1(1), 50-56. ISSN 2453-6083.
[5] CHVALINA, Jan. Diskrétní orbitální struktura zobrazení a funkcí. In Sborník VIII. brněnské
konference o vyučování matematice. Brno: MU Brno, 1992, (8),23-28.
[6] CHVALINA, Jan. Funkcionální grafy, kvaziuspořádané množiny a komutativní hypergrupy.
1. vyd. Brno: Masarykova univerzita, 1995. ISBN 80-210-1148-3.
[7] CHVALINA, Jan. Stars of Subsets within Set Partitions and Isomorphic Closure Operators.
International Didactic Conference. 2015, 9, 14-18.
[8] NOVÁK, M. n-ary hyperstructures constructed from binary quasi- ordered semigroups.
Analele Stiintifice Ale Universitatii Ovidius Constanta, Seria Matematica, 2014, roč. 2014
(22), č. 3, s. 147-168. ISSN: 1224- 1784.
[9] PAWLAK, Zdzisław. Rough Sets. International Journal of Computer and Information Sciences. 1982, 2(5), 341-356.
[10] ZHU, William. Topological approaches to covering rough sets. Information Sciences. 2007,
6(177), 1499-1508.