L-Valued Topology PPT

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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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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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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.

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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.

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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).

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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.


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.


D t t l S i 2011
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-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.


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).




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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.




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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.




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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}.


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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}.


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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).


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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.


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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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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.


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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.


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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

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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.




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