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5
NARROW PLACES IN LOGICAL SYSTEMS WITH ANRELIABLE ELEMENTS
Gurami Tsitsiashvili •
e-mail: [email protected] 640041, Vladivostok, Radio str. 7, IAM, FEB RAS
In this paper models of logical systems with anreliable elements are considered [1], [2]. Definitions of narrow places in these systems are made, algorithms of narrow places constructions are built. An asymptotic analysis of work probability (or failure probability) of logical systems in appropriate asymptotic conditions for a work probability (or a failure probability) of their elements is made. All main definitions and algorithms are based on an idea of a recursive construction of logical models with anreliable elements.
Preliminaries
Suppose that Z is a set which consists of |Z| logical variables z . Define the recursive class Q of logical expressions of variables z e Z :
z g Z ^ z G g , A g g, Ä2 G g Al a 4 )g£ , ( Al v A2 )g£ .
(1)
Denote 2Z = {zf, i g I = {l,...,2Z}} the family of all subsets of the set Z . Define the disjunctive normal form of the logical expression A g G : for z g Z, A1 g G, A2 g G, I\,I2 c I
D (z ) = z, D (A )= v
igl
f \
a z
V zGZi J
D (Ä2 )= v
igl2
f >
a z
VzGZi J
D (Ä1 v Ä2 )= v
igli U12
f ^
a z
V zGZi J
D (a1 a ä2 )= v
igl1,jgl 2
f
\
(2)
a z
zgZ,. UZ
j J
Analogously define the conjunctive normal form K (A), A eQ : for z e Z, A1 eQ, A2 eQ, I1,12 e I
f \
k (z) = z, k (Ai ) = a v z , k (Ä2 ) = a
V
igl,
V zGZi J
igl 2
f V v z
VzGZi J
K (Al v Ä2 )= a
iglijglj
v z
zgz. UZ
j J
K (Al a Ä2 )= a
igl, uI2
f V v z
V zGZi J
(3)
For the families of sets X = {X} c 2Z , Y = {Y} c 2Z put
x ® Y = {X U Y : X g x, Y g Y} , Z (x)= U X, N (x) = min (|X| : X g x) . If Z (x)1 Z (y) = 0, then
X gX
N (X ® Y ) = N (X ) + N (y ), N ( x U Y ) = min ( N ( x ), N (y )).
(4)
Suppose that pz = P (z = 1), qz = P (z = 0), pz + qz = 1, and the random variables z g Z are independent. The logical function A with random z g Z call the logical system A .
Lowreliable elements
Suppose that 3 d > 0 so that for Vz g Z 3 the natural number c (z):
Pz = Pz (h) ~ exp( -h^M ), h ^ 0 .
(5)
Denote by t(z) variables which equal to lifetimes of logical elementsz g Z, and P(t(z) > t) = pz (h). If h = h (t) is monotonically decreasing and continuous function and h ^ 0, t ^^, then the formula (5) may be transformed to the form
P (t(z )> t ) ~ exp(-h (t )-dc( z}),
t ,
characteristic of Weibull asymptotic used in lifetime models of systems which consist of lowreliable elements [3], [4].
f \
Define C (A) = minmax c (z) by known D (A) = V A z . Then from (2) obtain
IgI zgZ
IgI.
V ZGZi
C (( a A2 ) = max (C (( ), C (A2 C ( A. v A2 ) = min (C ( A. ), C ( A2 )). (6)
Correspond to the logical function A the families of sets S ( A) œ 2z , t ( A) œ 2z by recursive formulas: s (z) = {z}, T (z) = {z},
s (A. a A2 ) =
s (A. v A2 ) =
s (A. ), C (A. )> C (A2 ), s (A ), C (A. )< C (A2 ), T (( a A2 ) = s (A. )® s (A2 ), C (A. ) = C (A2 ), s (A. ), C (A. )< C (A2 ), s (a2 ), c (A )> c (a2 ), T (( v A2 ) = s (A. )U s (A ), C (A. ) = C (A2 ),
t ( A. ), C (A. )> C ( A2 ), t (A2 ), C (A. )< C (A2 ), t (A. )U T (A2 ), C (A. ) = C (A2 ), t ( A. ), C (A. )< C (A2 ), T (A2 ), C (A. )> C (A2 ), t (A. )® t (A2 ), C (A. ) = C (A2 ).
Put I' = \ i e I : maxc (z) = C (A) I, then from the formulas (2), (6) obtain
I zeZi J
S (A) = {{z e Zt : c (z) = C (A)}: i e I'} . (7)
The formulas (1), (2), (6), (7) lead to the statements.
Theorem 1. In conditions (5) the formula - ln P (A = 1) ~ N (S (A)) h~C (A), h ^ 0, is true. Theorem 2. In conditions (5) the following statements take place:
1. for each S e S (A) the following formula is true
(c (z) ^ c (z)- s, z e S) => (C (A) ^ C (A) - s), 0 < s < 1; (8)
2. if a set S e Z and satisfies (8), then 3 S* e S (A) : S* e S;
3. for each T e T (A) the following formula is true
(c(z) ^ c(z) + s, z e T) => (C(A) ^C(A) + s), 0 <s < 1; (9)
4. if T e Z S e Z and satisfies the formula (9), then 3 T* e T (A) : T* e T.
The theorem 2 allows to call sets from the families S (A), T (A) by narrow places in the logical system A with lowreliable elements. For any S e S (A) (for any T e T(A)) an increase of elements
z e S (a decrease of elements z e T) reliabilities leads to an increase (to a decrease) of the logical system A reliability. The formulas (6) and the theorem 2 allow to calculate recursively the numbers C (A), N(s(A)) and to define the families S(A), T (A) and their subfamilies
S'(A) = {S e S(A) : |S| = N(S(A))}, T'(A) = {T e T(A) : |T| = N(T(a))} .
Highreliable elements
Suppose that 3 d > 0 so that for V z e Z 3 the natural number c (z):
qz = qz (h) ~exp(-h"dc(z>), h ^ 0. (10)
Denote P (t(z) < t) = qz (h). If h is monotonically increasing and continuous function and h ^ 0, t ^ 0, then the formula (10) may be transformed to the form
P(t(z)< t) ~ exp(-h(t)-dc(z}) , t ^ 0 ,
characteristic of Weibull asymptotic used in lifetime models of systems which consist of highreliable elements.
f >
Redefine C (A) = minmax c (z) by known K (A) = a v z . From the formula (3) obtain
g zgz,. igi,
V zGZi
C(Ai a A2) = min(C(Ai),C(()), C(A v A2) = max(C(A),C(A2)). (11) Redefine I' = \i gI :minc(z) = C(A)f, then the formulas (3), (11) lead to the formula (7) for
I zGZ,- J
highreliable elements also. The formulas (3), (7), (11) lead to the statements.
Theorem 3. In conditions (10) the formula - ln P (A = 0) ~ N (S (A)) h~C (A), h — 0 , is true.
Theorem 4. In conditions (10) the following statements take place:
1. for any S g S (A) the following formula is true
(c (z) —^ c (z) + e, z g S)=>(C (A ) — C (A) + e), 0 <e<1; (12)
2. if a set S e Z and satisfies (12), then 3 S* g S (A) : S* e S ;
3. for any T g T (A ) the following formula is true
(c (z) — c (z)-e, z g T)(C (A) — C (A)-e), 0 <e<1; (13)
4. if a set T e Z and satisfies (13), then 3 T* g T (A) : T* e T .
The theorem 4 allows to call sets from the families S (A), T (A) by narrow places in the logical system A with highreliable elements. For any S g S (A) (T g T (A)) an increase of elements z g S (a
decrease of elements z g T) reliabilities leads to an increase (to a decrease) of the logical system A reliability. The formulas (12) and the theorem 4 allow to calculate recursively the numbers C (A), N(s (A)) and to define the families S(A), T (A) and their subfamilies S'(A), T'(A).
Remark 1. Denote X1 ={Zi,i g I1}, x2 = {Zi,i gI2}, suppose that Z(X1 )1 Z(X2) = 0. Then the
formula (4) allow to simplify significantly calculations of N(S(A1)®S(A2)), N(s(A1 )US(A2)),
which are necessary to find recursively N (S ( A)) for the asymptotic formulas of the theorems 1, 3.
References
[1] Riabinin I.A. Logic-probability calculus as method of reliability and safety investigation in complex systems with complicated structure// Automatics and remote control, 2003, № 7, p. 178186 (in Russian).
[2] Solojentsev E.D. Specifics of logical-probability risk theory with groups of antithetical events// Automatics and remote control, 2003, № 7, p. 187-203 (in Russian).
[3] Rocchi P. Boltzmann-like Entropy in Reliability Theory// Entropy, 2002, vol. 4, p. 142-150.
[4] Rocchi P. The Notion of Reversibility and Irreversibility at the Base of the Reliability Theory// Proccedings of the International Symposium on Stochastic Models in Reliability, Safety, Security and Logistics. Sami Shamoon College of Engineering. Beer Sheva, February 15-17, 2005, p. 287291.
