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8/2/2019 L-Valued Topology PPT
1/21
Departmental Seminar, 2011
-valued automata and associated -valued
topologies
Shambhu SharanDeptt. of Applied Maths
ISM, Dhanbad
Friday, April 8, 2011
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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Departmental Seminar, 2011
Outline
1 Preliminaries
Complete orthomodular lattice-valued subset-valued topology and -valued closure operator
2 -valued approximation operators and associated -valuedtopologies
-valued relation-valued approximation space-valued approximation operator
3 -valued topologies for -valued automata
-valued automaton-valued source and -valued successor-valued subautomaton and -valued separatedsubautomaton
4 Conclusion
5 ReferencesShambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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Departmental Seminar, 2011
Complete orthomodular lattice
Definition
7-tuple = (L, , , , , 0, 1), where,
1 (L, , , , , 0, 1) is complete lattice,
2 0 and 1 are respectively the least and greatest elements of
L; is the partial ordering in L,
3 A L, A and A are respectively the greatest lowerbound and the least upper bound of A,
4 is a uninary operator (called orthocomplement ) on L,such that a, b L,
a a = 0, a a = 1,
a = a,
a b b a,
a (a (a b)) b.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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-valued subsets
Definition
Let X be a nonempty set.
1 A mapping A : X L is called an -valued subset of X.
2
LX
will denote the set of all -valued subsets of X.3 A, B LX, |= A B, if A(x) B(x), x X.
4 For given -valued sets (Ai)iI, the -valued sets (
iI Ai)and (
iI Ai) are respectively given by
(
iI Ai)(x)def
=
iI Ai(x), x X,(
iI Ai)(x)def=
iI Ai(x), x X.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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-valued closure operator
Definition
A mapping c : LX LX is called an -valued closure operator if,
A, B
LX,
1 c(0) = 0,
2 |= A c(A),
3 |= c(A B) c(A) c(B),
4
|=
c(c(A)) c(A).
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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-valued topology
Definition
An -valued topology on a nonempty set X is a family of
-valued subsets in X, which is closed under arbitrary union
and finite intersection and which contains and X.
The pair (X, ) is called an -valued topological space and-valued subset of X in are called -valued open sets. The
complement of an
-valued open set is called
-valued closedset.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
S
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-valued relation
Definition
An -valued relation R on a set X is a map R : X X L.
DefinitionAn -valued relation R is called
1 -valued reflexive if R(x, x) = 1, x X,
2 -valued symmetric if R(x, y) R(y, x), x, y X, and
3 -valued transitive ifR(x, z)
{R(x, y) R(y, z) : y X} , x, z X.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
D t t l S i 2011
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Departmental Seminar, 2011
-valued approximation space and -valued
approximation operator
Definition
An -valued approximation space is a pair (X, R), where X is anonempty set and R is a -valued relation on X.
DefinitionFor an approximation space (X, R), c : LX LX, an -valuedapproximation operator on X is defined as,
c(A)(x)def=
{R(x, y) A(y) : y X}, A LX, x X
A natural generalization of lower approximation operator to
-valued lower approximation operator can also define.
However, our interest is only on -valued upper approximation
operator. So, we call it an -valued approximation operator.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar 2011
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-valued approximation operator and associated
-valued topologies
Theorem
An-valued relation R on a set X is-valued reflexive and
-valued transitive if and only if (the associated) -valued
approximation operator is a Kuratowski saturated-valuedclosure operator on X.
As a consequence, the -valued approximation operator, say c
on X associated with an -valued approximation space (X, R),induces a saturated -valued topology on X, which we shall
denote as (X).
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar 2011
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Results
Theorem
Let R be another-valued ralation on X such that
R(x, y) R(y, x). Then R is also an-valued reflexive and-valued transitive relation on X.
It will induce another -valued approximation operator, say c
,on X. This will induce another -valued topology, say (X), onX.
Theorem
The following statements are equivalent:
(i) L satisfies the distributive law:
a (b c) = (a b) (a c), a, b, c L
(ii) a, b L, b (b a) a.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar 2011
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Results
The relationship between the topologies (X) and (X) aregiven by the following Theorem.
Theorem
If L is a distributive lattice then the topologies(X) and(X)are dual, i.e., A LX is(X)-open if and only if A is(X)-closed.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar 2011
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-valued automata
Definition
M = (Q, X, )
1 Q is a nonempty set (of states of M)
2 X is a monoid (the input monoid of M) with identity e3 : Q X Q L, such that q, p Q, x, y X,
(q, e, p) =
1 if q = p0 if q = p
and (q, xy, p) = {(q, x, r) (r, y, p) : r Q}.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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-valued source and -valued successor
Definition
Let (Q, X, ) be an -valued automaton and A LQ, the
-valued source and -valued successor of A are respectivelythe sets
(A)(q)def= {A(p) (q, x, p) : p Q, x X}, and
s(A)(q)def= {A(p) (p, y, q) : p Q, y X}.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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p ,
-valued approximation operator and -valued topology
Result
Let (Q, X, ) be an -valued automaton. Consider an -valuedrelation R on Q given by
R(p, q)def= s(1{p})(q), p, q Q.
This -valued relation is also -valued reflexive and -valued
transitive. So, as in the previous, we can also define another
-valued approximation operator on Q given by
c(A)(q)
def
=
{s(1{p}(q) A(p) : p X}, A LQ
, q Q.This operator c must be a Kuratowski saturated -valued
closure operator on Q. It will induce a saturated -valued
topology on Q, say (Q).
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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p
-valued topologies for -valued automata
Result
Similar to above, if we define another -valued relation R on Q,
given by R(p, q) = (1{p})(q), p, q Q. ThusR(p, q) R(q, p) and so, R is also an -valued reflexive and
-valued transitive relation on Q, and hence it will induceanother -valued approximation operator, say c, on Q and it
will induce a -valued topology on Q, say (Q).
Remark
The -valued topologies (Q) and (Q) on Q are precisely the-valued topologies S and R respectively, introduced by Qiu.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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-valued automata theoretic concepts
Definition
A LQ is called an -valued subautomaton of -valuedautomaton (Q, X, ) if
A(q)
(q, x, p)
((q, x, p)
A(p)) : q
Q, x
X,q Q.
Definition
An -valued subautomaton A LQ is called -valued separated
subautomaton of -valued automaton (Q, X, ) if
A(p) {A(q) (q, x, p) : q Q, x X}, p Q.
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Departmental Seminar, 2011
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Results
Theorem
A LQ is an-valued subautomaton of-valued automaton(Q, X, ) iff|= s(A) A, i.e., A is-valued(Q)-open.
Theorem
A LQ is a-valued separated subautomaton of-valuedautomaton M = (Q, X, ) if and only if it is(Q)-clopen i.e.,
(Q)-open as well as(Q)-closed.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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Conclusion
1 The relationships among -valued approximation operator,
-valued topology, and -valued automata may offer some
new insights in quantum computation.
2 It may possible to introduce the -valued product topologyon the state-set of product of two -valued automata.
3 The decompositions of an -valued automaton can be
proposed and it will be interesting to see that up to which
extent these concepts depend on the distributivity ofassociated lattice.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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References
M.L.D. Chiara, Quantum logic, in: Handbook of
Philosophical Logic, III: Alternative to Classical Logic,Reidal, Dordrecht, 1986, 427-469.
D. Qiu, Automata theory based on quantum logic: Some
characterizations, Information and Computation, 190
(2004) 179-195.
Y.H. She, G.J. Wang, An axiomatic approach of fuzzyrough sets based on residuated lattices, Computer and
Mathematics with Applications, 58 (2009) 189-201.
A.K. Srivastava, S.P. Tiwari, A topology for automata, in:
Proc. AFSS Internat. Conf. on Fuzzy System, Lecture
Notes in Artificial Intelligence, Springer, Berlin, 2275
(2002) 484-490.
A.K. Srivastava, S.P. Tiwari, On relationships among fuzzy
approximation operators, fuzzy topology, and fuzzy
automata, Fuzzy Sets and Systems 138 (2003) 197-204.Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
Departmental Seminar, 2011
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References
S.P. Tiwari, A.K. Srivastava, On a decomposition of fuzzy
automata, Fuzzy Sets and Systems 151 (2005) 503-511.
M.S. Ying, Automata theory based on quantum logic (I),
International Journal of Theoretical Physics, 39 (2000)
981-991.M.S. Ying, Automata theory based on quantum logic (II),
International Journal of Theoretical Physics, 39 (2000)
2545-2557.
Y.Y. Yao, Two views of the theory of rough sets in finiteuniverses, International Journal of Approximate
Reasoning, 15 (1996) 291-317.
Shambhu Sharan Deptt. of Applied Maths ISM, Dhanbad -valued automata and associated -valued topologies
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