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0 because i n the o p p o s i t e case, t h e GH E i s not v a l i d . ) | W
= nW ;
22 11 W = "c(l , TIME) > 1"; 3 1 W w
= = nW iw :;
4
3
D D
= M il ■ i1 ); 11 44 1 6
11
r
= 1 2
I 1
5 1
I 11 I 11 I
W 5 W 5 5
where
p
r
W = " ( 3 Z € p r p r x P 1 (pr Z = ir ), W5 = " ( 3 Z € p r 1p r 1x cu 1 (pr3 Z = i cu ), 5 1 1 cu 3 cu 0D =- v ( l , 1 ); 2 5 11
NEW RESULTS IN THE THEORY OF GENERALIZED NETS 1 7 1 I 6 I r 3
= 1 2 1
1 8
true
1 9
false
false
I false 1
W
W
I false
W
6
7 w
12 I 6 7 where where W = "in the last characteristic of the token from 1 there is at W 6 = "in the last characteristic of the token from 1e there is at 6
6
least one GN-control token"
least one GN-control token" (This is possible because the token a has the highest priority. ); (This is possible because the token a has the highest priority. ); 7 6
W
: 1W ;
7
6
D 1 1 I 7 i
10
W 7
3 1
= M l , v(l , 1 )); 6 2 12 1 1 1 11 12 13
15
W a
false
false
false
false
W
W
false
false
false
false ,
w
w 8
1 8
i false 1
false
r = 1 4 9
I false 1
false
W
1 3
| false 1
false
false
1
|
false
false
1 14
8 8
7 W
7
false
7 false
15 I
false
W 7
\t*iere r 9 0 W - "x < pr pr x + pr pr x ", 8 cu 1 3 0 3 3 0
w = -m , 7
8 D -. A ( 1 , v ( l , 1 ), v ( l , 1 >); 4 7 8 9 3 15
w 8
23
24
CHAPTER 1
r 5 5
= 1 I 10 I 10 I
1 17 17
1
W
W 9 9
1 I false 16 I
18 18 10 10 true
where W
= "c(l 9
W 10
, TIME) > 0" 16
= nW , 9 D
= Mv(l 5
, 1 ), 1 ). 10 17 16
Let an arbitrary SHGN E with a set of tokens K = |a, a 1 2
K he given, where
a ). I]
« Let the tokens of E enter place 1 of E 1
(with the
E-charac-
teristics, i.e. with their initial E-characteristics). Let the token a with the ira-gimm
possible priority
(e.g.
n (a) --co) be also in 1 K
without initial characteristic. tic
"the description of
1
Token p with an initial characteris
E"
enters place 1 and the token r with an 2 initial characteristic pr pr E (= T - the initial time of the functi1 3 oning of E) enters place 1 . All E-tokens from 1 enter sequentially 3 1 in place 1 without characteristics. Token a enters place 1 and the16 4 re it receives sequentially as
a characteristic: "the characteristics
of all E-tokens and their priorities" (in their priority-order). After that, the token a enters place 1 without a characteristic. The E-to5 * kens wait for the end of the process of functioning of E in place 1 16
NEW RESULTS IN THE THEORY OF GENERALIZED NETS
Fig. 1. 5
25
26
CHAPTER 1
There they will receive their final characteristics. The priorities of the three tokens (a, p and r) satisfy the condition n (a) > n n , 1 2 (see App. 1) and l e t
r = [L J , L", ||r
II]
( s e e App. 2 ) .
V, »
From
the Theorem for
the completeness of
(see § 5 of App. 1), it follows that every GH can
the GH
transitions
be represented by a
union, a composition and an iteration of its transitions, and therefo re it is sufficient to show a method for representation of an arbitra ry GH's transition and this can also be transformed in all other tran sitions. Hie new GH will have transitions with
S-fonns.
It is conveni
ent that this GH be a GH with a global memory (see § 3 in App. 1). Let this new component contain
as parameters the lists of current capaci
ties of the transition's arcs and of the numbers in the output places: for a transition
2
of the free location
these lists are:
H Z
= [L', Z
L", llm II], where m are the current capacities of the transition's Z i,j i, j arcs between places 1 and 1 ; and (/l € L"], where i J Z c is a function giving the number of tokens in place
1 at the current
time-moment TIME. The GH transitions can be ordered according to
their prioriti
es. Let a transition Z of GH E be given. Our algorithm is as follows: By
"//...//" we shall denote some
comments. A01. Order the input and output places of Z by their priorities, where the order in the first case is descending and in the second
case
in ascending series. A02. Check the inequality |pr Z I i ipr 2| 1 i 1
(1)
for every i (1 i i i sj, where pr X is the j-th projection of the j
NEW RESULTS IN THE THEORY OF GENERALIZED NETS
33
n-dimensional set X (1 i J i n) and S - [Z , Z 1 If the set of the S-transitlons
Z1 2
s
which satisfy (1) is empty, 90
to A07. A03. Check the inequality ipr S 1 z ipr Z| ipr2S 1 1 z ipr2Z| 2 i 2
(2) (2)
for every i (l >. Z «« Z z z
z
z
z
GO t o A20. //Obviously !D
iff !D , Z
it
z because the tokens do not enter in the places of the set L' »
- h'. Z
z The method
of checking
the functioning of
a transition
structed in such a way is similar to the one shown below
and that
con is
why it will be emitted here. // R
A06. Mark the first Ipr Z| 1
input places of Z
with the corresponding
place's identifiers of the input places of Z, and the other input K
places of
Z
with other (different)
are elements of the set (L' «
- L'), Z
identifiers - these places
Go to A10.
Z A07. Check the inequality (2) and if it is not valid, go to
Aoa, else
« mark the first Ipr Zi output places of Z 1
with the corresponding
NEW RESULTS IN THE THEORY OF GENERALIZED NETS
35
place's identifiers of the output places of Z, and the other outK
put places of Z
with other (different)
ces are elements of the set (L"
* z
identifiers - these pla
- L"). Go to A08.
z
A08. Let Z ; Z. Let i ~- 1. 0 A09. Determine this transition of places whose number
is
S with minimal number of input pla-
greater than
|Z
1-1
or,
if such a
i-1 transition does not exist, the S-transltion with the maximm num ber of input places
(let it be
Z'). Construct a transition Z i i from a transition Z by removing its final Ipr Z'I - 1 from the i-1 1i number of places. Mark the first input place of Z' by y' (an 1 identifier
which is not found among the other GN's
i
identifiers)
and the other input places of Z' with the identifiers of the cori responding input places of Z The final output place of the i-1 transition Z' is marKed by y' (it is the first input place of i i-1 the transition Z' , if it exists). i-1
MarK all other output places
of Z' with other identifiers which are not found among the other i GN's identifiers: x' , x' i, 1 i, 2
x'
. Define i, IL' 1-1 Z' i
c(x' > - O, o, i, J j w <x' ) = 0 (for 1 i j i |L' I -1), L i, j Z' i c(y' ) = c(y' > = oo, > i-1 i
36
CHAPTER 1
n (y') = max |n (1) / (1 € pr 2') 8 (1 * y')]. L i L 2 i i If the -transition
Z
has no input places
(i. e. if the second
i case mentioned above is valid), is valid for Z,
then finish the procedure, and if (2)
determine the priorities
and capacities of the other
input places of Z'-transitions, in relation to the corresponding prio rities
and capacities of the places of
Z and
go to A12,
else go to
A10; else i:= i * 1 and execute A09 again. A10. Let Z = Z. Let i r 1. 0 All. Determine this transition of
S
which has the minimum
output places whose number is greater than |Z
number of
1-1
or, if such
S-transition with
the ""'nim
i-1 a transition does not exist,
the
number of output places (let it be Z"). Construct a transition Z i i from a transition Z by removing its final Ipr Z" I - 1 for the i-1 1i number of places. Hark the first output place of Z" by y" (an i identifier, which is not found among the other GH's
i identifiers)
and the other output places of Z" - with the identifiers of the i corresponding output places of Z . The final input place of the i-1 transition Z", is marked by y" (the first output place of the i i-1 transition Z" , if it exists). Hark all other input places of Z" i-1 i with other identifiers
which are not found among the
identifiers: x" , x" ,..., x" . Define i, 1 1,2 i, IL" |-1 Z" i c(x"
) = 0,
other GHJ s
NEW RESULTS IN THE THEORY OF GENERALIZED NETS
37
f (x" ) |L" I I --1), IT T (x" ) = 0 (for ( f o r 1 i
when in
a certain
L' c L there exist at least two of its tokens, which
are generated by one
token "predecessor"
independently from the fact
that (a) there can exist other "kindred" tokens somewhere (i.e. tokens with the same "predecessor"), (b) the tokens can be generated
in different moments and
from diffe
rent, but kindred, "predecessors", then these tokens are united in one token by analogy with the procedu re from
§ 8 in App. 1. The new token
will also be a kindred token of
the tokens which are kindred tokens of its generated ones. Let us call this new net
"a GH allowing
a particular
token's
uniting" (GH-APTU). THEOREM 1.3. 1: I
c I. GH-APTU
Proof: Obviously, every GH-APTU for which
L' = $,
is not defined 0
, otherwise (see above),
I tr tr where t W
is the time for the calculation of the function
W 1
and
1 t t r r
M M
7
= (TIME), t =V V (TIME), t 1 1
+ t , , + t r r 5
K
where t is the time for the calculation of the characteristic toKen in place m (for t see above). 4 r 5
of the
» The GN E
functions in a similar way as GN E. The tokens of the
NN enter E* in place 1 . These neurons which are generated at the time 1 of the functioning of E* GN E
appear in place
m . A control token 4
from the above transfers from transitions 2
of the
Z , Z and Z and is 4 6 7
"a mother" of the born neurons. In the other places
it functions only
66
CHAPTER 2
as a control token for the modelling process.
« The place capacities of GH E The other models
are infinite.
for functioning of HHs
can be represented by
similar GHs. The graphical structures of these GHs will be the same or simpler than the above ones. For example,
the GH which represents the
HH-back propagation method [5] will not have the transition
Z
from 3
the
first GH
functioning form
described above.
of HHs
On the other
band some ideas for the
(e.g. [6-9]) will force
some corrections in the
of the transition condition predicates
tions in this GH.
Finally,
and characteristic func
some relations between the Petri nets and
some types of the HHs are discussed in [10-12]. This paper is based on [13, 14]. REFERENCES: [1] Jun-Vtei Wong,
Recognition of
general patterns using neural net
works, Biol. Cybern. , Vol. 58 (1988), Ho. 6, 361-372. [2] H. Cottrell, Stability and atractivity in associative memory net works, Biol. Cybern., Vol. 58 (1988), Ho. 2, 129-139. [3] A. Fukushima,
A historical
neural network model for associative
memory. Biol. Cybern., Vol. 50, 1984, Ho. 2, 105-113. [4] A. Guez, V. Protopopsescu and J. Brahnen, On the stability stora ge capacity and design of continuous neural networks, Trans. SMC, Vol. 18 (1988), Ho. 1, 80-90. [5] B. Rumelhart and J. HcClellond,
Parallel distributed processing,
explorations in the microstructure of cognitron, HIT Press,
Cam
bridge, Mass. , 1986. [6] J. Hopfield,
Heural networks phisical systems with emergent col
lective computational abilities, 2554-2558.
Proc.
Hat.
Acad. Sci. , 1982,
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
[7] T. Kohonen,
Self-organization and associative memory,
67
Springer-
Verlag, Berlin, 1984. [8] L. Hadjyisky, Parallel processing
in frequency
dependent neural
network, Parallel and Distributed Processing, Elsevier Sci. Publ. , 1991, 59-74. [9] C. Lee, Intelligent
control based
theory. Proc, of Internat. Conf.
on fuzzy logic and neural net
on Fuzzy Logic
and Neural Net
works, Japan, 1990, 759-764. [10] A. Negro, R. Tagliaferri and S. Tagliaferri, Some remarks on Petri nets and neuron networks. Proc. of the IASTE3) Int. Symp. App lied Informatics,
Grindelwald,
(M Hamza, Ed. ),
Anaheim,
Acta
Press, 1987, 28-29. [11] T. Kristian, J. Pavlasek and P. Sapaty,
Modeling of neuronal-ac-
tivity using modified Petri nets, Physiologia-Bohemoslovaca, Vol. 36, No. 6, 1987, 541-551. [12] 14 Zargham,
Parallel processing neural networks,
Proceedings of
the First Annual Meeting of the nternational Neural Network Soci ety, 1988, Boston; Neural Networks Vol. 1, No. 1, 1988, 558. [13] L. Hadjyisky and K. Atanassov,
Theorem for representation of the
neuronal networks by generalized nets. AMSE Review, Vol. 12, No. 3, 1990, 47-54. [14] L. Hadjyisky and K. Atanassov,
A generalized net,
representing
the elements of one neuron network set. AMSE Review Vol. 14, No, 4, 1990, 55-59.
68
CHAPTER 2
§ 2. 2: GENERALIZED HETS REPRESENTING THE TRAVELLING SALESMAN FROELEM Krassimir T. Atanassov
The Generalized Nets (GNs) have been used as a means for model ling real processes and theoretical problems.
For example the process
of solution of the transportation problem was described using a GH. Initially, ire shall construct a GH the process of
(see Fig. 2.3),
solution of a variant of
problem (see e.g. [1-7]).
After this,
describing
the Travelling Salesman (TS)
we shall define the TS problem
in a more extended form. The temporal components of this GN will be
o « T = 0, t = 0 , t =
1. The GN E consists of two subnets E 1 has only one transition Z
GH
"a formal description of the graph of connections
The predicate
one) is "true".
2
1
between points which must be visited by the TS exists in it. The transition Z is activated at every time-moment with 1 ration 1.
E
with one place 1 . One token a with an ini1
tial characteristic
and E . The second 2
of the transition condition
(there is
Toe token a receives, during its transfer from
a duonly 1 to 1
a 1 , the characteristics x , which we shall describe below. 1 i The token p enters GH E in place 1
with an
initial characte-
2 ristic "a name of vertex, where the TS has been initially". The transition condition
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
r 2 2
=
1 3 3
1 4 4
1
1 I 2 2 I I
W
W
W 3 3
1 I I 3 I 3 I
W
1 1
2 2 W
1 1
1 I W I 1 11 5 |
69
5 5
W 2 2
3 3
2
W 3
W
where W
= "From the vertex of the graph, described as
an initial characte-
1 ristic of a,
where
a
is the TS,
more than one path goes
out
which is not passed by the TS", W
= "From the vertex of the graph, described as an
initial characte-
2 ristic of a,
where
a
is the TS,
at least
one path goes out
which is not passed by the TS", W
= "From the vertex of the graph,
described as an initial characte-
3 ristic of or, where a is the TS,
no path goes out
which
is not
passed by the TS". If a token goes in the place 1
(i. e. the predicate W
5 lid), vertex,
then it receives as
the next characteristic:
where the TS is going",
is va-
3 " j i j i j i j / C € j
<MT , TIHE] J, i
wtoere p(T , TIME) is the vertex V i k
where the TS
T
is located. i
Si-
SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
multaneously with the above transfers,
the token
7T
p enters place 1 2
with an initial characteristic "(V , M(V , TIME!), [ = c(l ) = C(l ) = c(l ) = 1. 2 6 10 11 (g) the priorities of all places
and transitions
are equal,
without
Places 1 , 1 and 1 for which it is valid: 0 14 12 n (1 ) = n (1 ) > Tt (1 ). L 6 L 14 L 12 (h) o 1
= A(V(1 , 1 , 1 ), v(l , 1 >), 1 3 9 2 11
(i) D = A(1 , 1 ). 4 6 7 The other transitions are of a disjunctive type. Thus a GN in frames of which the ETTSP1
is solved
is construc
ted. This GN has some faults and the reason for them is
that the num
ber of oc-tokens can grow un-1 .unitedly. Below we problems and
shall introduce some other modifications
of the TS-
discuss the possibilities of their descriptions with the
GNs: (ETSP2) Trace out the locality-optimal (for a criterion LMG: ty-maximal gain") TSs T , T , . . . , T 1 2 t
route of the i-th TS
(l which mist move in the built-
up areas V , V . . . , V 1 2 v .. . , C c
when
"locali
among the independent
carrying some of the products
the different TSs
do not Know
C , C, 1 2
the plans of the
others. (ETSP3) Trace out the locality-optimal (for a criterion LMG): route of a in number groups of TSs:
(T ,T 1,1 1,2
T
), l,t
(T 2,1
1 T
, . . . , T 2, 2 2, t 2
),...,
move in the built-up areas
IT ,T a, 1 a, 2
T ) Which a, t a
V , V , . .. , V , 1 2 v
must
carrying some of
80
CHAPTER 2
the products
C , C ,..., C ; 1 2 c
the TSs
from d i f f e r e n t s e t s do
not concur. The last problem can also be defined for deterministic
and for
non-deterministic cases. Variants of the above problems are those where
the TS
into all the V-places or go to some of these places only, TS must
have exactly determined qualities
must go
where every
of products or where every
TS must travel certain exactly determined distance. The above TS problems can also be described by QHs as above. This paper is based on [8,9]•
REFERENCES: [1] E. Lawler,
J. Lenstra,
A. Rinnooy Kan and D. Sbmoys
(Eds.), The
traveling salesman problem. A guided tour of combinatorial optimi zation, H. Y. : Wiley X Sons, 1985. [2] I. Helamed, S. Sergeev and I. Sigal,
The traveling salesman prob
lem Issues in theory. Avtomatika i telemecbaniKa,
1989,
9, 3-33
(in Russian). [3] I. Helamed, S. Sergeev and I. Sigal,
The traveling salesman prob
lem Exact methods. Avtomatika i telemechanika, 1989, 10, 3-29 (in Russian). [4] I. Helamed, S. Sergeev and I. Sigal,
The traveling salesman prob
lem Approximate algorithms. Avtomatika i telemechanika, 1989, 11, 3-26 (in Russian). [5] J. Little, K. Hurty, D. Sweeney and C. Karel, An algorithm for the traveling salesman problem
Operations Research,
Vol. 11,
1963,
972-989. [6] T. Volgenant and R. Jonker, On some generalizations of the travel ling salesman problem,
Journal of Operational
Vol. 38, Ho. 11, 1987, 1073-1079.
Research
Society,
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
[7] R. Ackoff
and M. Sasieni,
Fundamentals of operations
81
research.
John Wiley and Sons, Inc. , New York, 1968. [8] K. Atanassov,
A generalized net, representing
the travelling sa
lesman problem. AMSE Review Vol. 14, No. 4, 1990, 61-64. [9] K. Atanassov, Generalized net and extensions of the travelling sa lesman problem press).
AMSE
Review
Vol. 21,
No. 2,
1992,
19-26
(in
82
CHAPTER 2
§ 2. 3: GENERALIZED NETS AND LABYRINTHS Erassimir Atanassov and Rumen Cbristov
Different researches on the methods for exist (see e.g., [1-3]).
motion
in a labyrinth
Here we shall describe one of them, which is
based on the GNs. Let the labyrinth L be given. It can have the form of
(m x n) -
dimensional (0, 1)-matrix, the elements "0" and "1" of which correspond to the labyrinth corridors and walls, respectively. We construct
the GN from Fig. 2. 5. It has two transitions with
transition conditions
l1
3 3
m I false II m |false 2 1 r = I 1 1 I W 1 1 3 1 2 1 55
where
|I 1 I ||
W 3 W 33
l1
l1
4 4
false false
5 5
mm
false false
m m
2 2
3
W
W 1 2 W 1 2
W
w 4
W w 5
false
false
w 4 W 44
w W 5 W 55
false
false
false
false
W = "c(l , TIME] + c ( l , TIHE] > 0" 8 " c ( l , TIHE) = 0"; 1 3 5 3 W
= nW ;
2 W
1 = "end of the (unique) path or a meeting with another
token on the
4 path or along this path another token has already passed"; - IV ;
W
3 W
4 = "there is more than one path
5
and along this path
has not passed";
and
1 66 r
1 2
1
22
| W W W W 3 | 6 7
another token
SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
83
where W
= "the token arrives to an exit"; 6
w = -m .
7
Fig. 2. 5 First,
toKens a and p enter the net through places 1 and m , 1
respectively and
p has an initial characteristic in the form
1 of the
above mentioned matrix. If there exists exactly one path for motion, it goes to place 1 with a characteristic "nurber of this path, nurabe3 red
according to the hour hand".
If the number of the paths
is more
84
CHAPTER 2
than one, the token a splits sequentially to the tokens a , a , . . 1 2
(s is the number
time one token enters place 1 3
with the above characteristic
one (which will generate the next tokens) enters place 1
s
Each
of the different paths in front of token a ) .
another
without cha-
5
racteristic.
when there are no paths for motion
a igiven
in front of
token, it goes into place 1 with the characteristic "unsuccessful end 4 of .a motion". place 1 6
If there is an exit in front of
with the characteristic
a token,
it goes into else
"successful end of a motion";
the token enters place 1 without a characteristic. 2
All 1-places have equal priorities,
which are bigger
than the
priorities (equal) of all m places. Both transitions have equal rities. All places have equal (oo) capacities,
prio-
the temporal components
are standard and thus not given. The problem of aspects.
motion in
a labyrinth can
be extended in some
We shall show one extension and its GH-representation:
the labyrinth has some floors GH, we shall
(i.e. it is not planar),
when
in the ;above
change only the characteristic function of place 1 3
"number of the path, numbered as first underneath - uphill and
with
as se-
cond along the hour hand" and the initial characteristic of token P is (m :K n x k) -dimensional (0, 1) -matrix. REFERENCES:
tl] F. George, The foundations of cybernetics,
Gordon and Breach Sci.
Publishers, London, 1977.
[2] H, Gardner,
Mathematical puzzles
and diversions,
Bell and ,Sons,
London, 1965.
[3] K. Shannon,
Research on
theories of informatics and cybernetics,
Inostrannoi Literatury, Moscow, 1963 (in Russian).
SCHB APPLICATIONS OF GENERALIZED HETS IN SCIENCE
85
5 2. 4: FORMAL COMMUNICATION IN SCIENCE: A MODEL BASED ON GENERALIZED HETS Fadosvet Todorov
§ 2. 4. 1
and
Krassimir Atanassov
Introduction
Various studies
of the
been carried out (see [1-5]). plain the informal and/or
connunication network in science have Different models have been used
formal connunication
processes.
to ex
Models of
forma] connunication in science are predominantly linear and emphasize various media (articles, journals, books, etc.), participants (indivi duals and institutions) offers two views
or functions
(activities).
In [1], Atherton
of the connunication network of scientific and tech
nical information, one of which stresses for example
the place of the
publisher. Garvey presents (see [2]) a schematic overview of
the sci
entific communication system which includes all events that occur from the time nal)
the scientist begins his research,
publishes it (in a jour
until its appearance in reviews, treatises, etc. Also Garvey de
scribes in [3] features of the current dissemination system in science and suggests some innovations in it. Some linear models of the formal connunication process are pre sented in a recent review article by Hills ([4]). The review "the scholarly
connunication process in terms
looks at
of the interaction and
concerns of the scholar, the learned society, the publisher, the pro duct, the librarian, and the new technology". King
et al. offer (see [5]) a schematic overview of the disse
mination of scientific and technical research results by different me ans and of the general flow of information among scientists and neers.
The authors also supply a stochastic model of the
engi
transfer of
article manuscripts between authors and publishers in order to descri be the system of journal publishing in terms of cost and flow of mate rials.
86
CHAPTER 2
Here we shall present at outline of a formal communication pro cess by stressing
some
for the traditional
specific functions and participants required
transfer of
(originator) to the reader
article manuscripts
from the author
(consumer). The purpose of the study is to
show a possible description of this process by means of a mathematical model with GHs.
§ 2. 4. 2 Schematic overview o± formal comami cation Here we choose
a simplified
part of
in
science
the formal communication
networK in order to apply in a comprehensible manner the GH (having in mind the
unacquainted reader).
under consideration,
The schematic overview of the process
namely the publishing of article manuscripts and
distribution of journal articles, preprints in Fig. 2.6,
or reprints, is presented
The author (A) submits an article manuscript to the edi
tor (s) of a given journal. "Hie flow of manuscripts between authors and editors is represented by arrow 1 reject obvious
a paper whose subject matter is inappropriate
editor then
This is
symbolized by arrow 2.
transmits the remaining manusrcipts
they believe to be experts this
or which is "an
non-starter" [6]. A recommendation may be made to submit the
manuscript elsewhere. the
(see Fig. 2.6). The editor (E) may
in the particular
activity is represented by arrow 3.
(F) returns the evaluated paper (arrow 4) commendations for acceptance not need re-examination),
"to people whom
topics involved"
The expert or
[6],
the referee
to the editor with his re
(without change),
improvements,
Traditionally,
corrections
modifications or
(that do
rejection.
Only in the first case (direct acceptance) is the manuscript transfer red to the
publisher
(arrow 5),
who in our diagram is to be related
with recording and reproducing activities rather than being corporated in the publishing process. If the manuscript is rejected or a revision is recommended, then the author is informed (arrow 2). The author then
SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
87
decides whether to revise the manuscript, to send it to another jour nal
(usually
without
modifications)
or
to give up
publication of
(withdraw) his paper. After manuscript recording, the publisher sends proof-copies to the author (arrow 6).
The corrected proofs are returned for reproduc
tion and issue preparation (arrow 7). The published articles reach the user (U) through the
distributor (D), who incorporates the functions
of such participants in the formal communication process as publisher, librarian, information services, etc. Fig. 2. 6
also shows
the direct contact between the author and
the user. The significance of arrow 9 is a reprint request and that of arrow 10 correspondingly the satisfaction of the request.
Fig. 2. 6 Schematic overview of the formal conmunication by means of journal articles (A: Author, D: Di stributor, E: Editor, P: Publisher, R: Referee, U: User)
88
CHAPTER 2
§2.4.3
A GS model Fig. 2.7 presents the graphical structure of
a GH which models
toe relations discussed above (see also Fig. 2.6). The circled capital letters are the places Where different activities are performed ting, reviewing, etc.). f
are "fictitious"
(edi
places which are introdu-
i ced
to conform to
the graphical
representation of the GH.
They are
used to determine the length of time needed hy the tokens to pass from a "real" to a "real" place. The f are "fictitious" because the toKens 1 do not receive new characteristics as they transit through the f . i The token (author)
(article manuscripts)
enter the net through place A
with the following initial characteristic "name,
the author,
manuscript's title".
Place A
address of
is the entry point for all
tokens which represent the manuscripts ready for a submission to a jo urnal. Transition condition r
has the trivial form 1 l ff 88
ff II true true 7 II rr
=
R R
II true true II A I I true true A
1
ff II true true 4 II Transition condition r
has the form 2
E r
= 2
f
| W 8 | 1 I R I true
f f 2
f f 33
W W
ff
W 2
W 3
W 4
W 5
false
false
false
false
4
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
89
Fig. 2. 7 where z "A takes the decision to submit a manuscript";
W i W
= "A is ready to return the proof-copies to P"; 2
W
= "A is willing to send a reprint to U"; 3
W = "A decides not to publish or to withdraw his manuscript" 4 (The place W is an output place of the model. This implies that A will take no further steps to publish the paper under conside ration);
90
W
CHAPTER 2
= "A agrees to make changes in order
to publish
the manuscript in
5 the same journal or decides to resufamit it without
improvements
to another journal". When the GH's token
(which models some manuscripts] arrives in
place E, it receives the characteristic "journal title". Transition condition r
r
:
E
has the form 3 R f i 6 1
I W 1 6
3
W 7
W 8
where W
= "The editor E takes the decision to inform the author A" 6 (The notice
could include
a rejection of
the submitted manu
script, a reconmendation for revision, etc. In other words, the token arriving in place A
(coming from E)
receives the charac
teristic "information from E to A"); - "E decides to send the manuscript to be assessed by an expert R";
W 7 W
e
= "E takes the decision to transfer the manuscript to the publisher F in order to be published (after peer reviewing]". Transition conditions r
and r 4 U
f
--
r 1
I true 3| _ I D I true f
I true 5 I
and P r
=
f
5 f
I true 1I
I true 2 I
have the trivial forms 5
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
91
Transition condition r has the form 6 b
r = e e
P U
f 7
D
| i i
W
W
I I I
W
f 5
9 9
to 10 W
12 12
W it li W
13 13
14 14
where W
= "P is ready to send manuscript copies to A" 9 (These are the proof-copies or the reprints according
to the mo
del time), = "P is ready to send the issue containing the article in question
W lO
to D", z "P is ready to send the Journal issue directly to U",
W 11 W
= "U wants to send a letter
(with a comment)
or a request for a
12 reprint to A", W
s "U is willing to receive information or the original from D", 13
W
= "U
is willing to be disseminator of
the article
for
another
14 user". Entering f
from P, the token has the characteristics
"proof-
7 copies" or "reprints". Arriving in D from P, it receives the characte ristic "bibliographic data". When the token enters f from U (through 6 f ) it has the characteristic "Kind of information or requested mate7 rial". K
Let the GN function between the tune -moments T and T + t . Ini tially (at the moment T) the tokens enter the net in ve for E. The predicate W
place A and lea
has the current value "true" and the remal1 ning predicates in the transition condition r have zero-values. From 2
9E
CHAPTER 2
place E
the tokens move to places A or F according to the editor's
judgement. Visiting places A and E the toKens receive the above menti oned characteristics. When a token returns to f , it could enter place f (modifica8 4 tion of the manuscript), W (withdraw) or f (sending a reprint). If 3 the token enters R coining from E it returns obligatorily to E, and ac cording to the characteristic
received in R (referee opinion) arrives
at A or P. From P it moves to A with the characteristic "proof-copies" or "reprints". The token with the characteristic "proof-copy" goes from f to P through f . With a new characteristic (bibliographic da8 2 ta) the token arrives in D or f coming from P. The ultimate aim of 5 the tokens is to get to X). notice that in our case, the place f sym8 bolizes the author's activity and U that of the reader, i.e. we do not consider
(for the sake of simplicity]
the possible manifestation of
the user as an author and correspondingly that of the author as a rea der of publications. § 2.4. 4
Discussion
The proposed diagram of formal conmunication 2.6)
does not show all possible
the process. as well,
in science
(Fig.
interactions among participants
in
In the scheme, there could be represented other contacts
such as a possible
direct link between the author and
the
referee: the author could send an accompanying note to the referee and the latter could openly publish (or answer individually) his recomendations for manuscript Improvement. All the simplifications as well as some artificial divisions are made here to perceive more easily the GH as a modelling tool appropriate
to conmunication network.
The model
based on the GHs is sufficiently powerful to be used for the descrip-
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
93
tion of the whole (ccnplex) structure and dynamic picture of knowledge generation, representation,
distribution and use, or for detailed re
presentation
part of
of a specific
minimum modifications
the conminication process.
With
(simplifications), the model in Fig. 2. 7 can be
used for example, for the description of booK publishing and distribu tion. Fig. 2. 6
can be also presented
by means
of
a block diagram,
which would reflect the interactions among participants lopment of the
and the deve
process under consideration. But a block diagram
can
not describe the simultaneous movement even of two or more manuscripts submitted at the same time to one, two or more journals. neous
transfer of two (or more) manuscripts in the network can be de
scribed by means of ons,
The simulta
e. g. , the Petri nets and their other modificati
but with the increase
in the number of manuscripts, it would be
impossible to follow the individual fate of the manuscripts at various places of the net. The transfer of a given number sented by
of manuscripts
can
be
repre
the stochastic model of King [5]. However, in this case one
cannot follow
the dynamic development of the process or "predict with
certainty the time
required to perform any of the article preparation
functions, nor whether the articles will be rejected, accepted, or mo dified" [5]. Only by means of GNs it is possible to follow manuscript (on the basis of the different ved by the
the history of a
GN's characteristics recei
tokens at each GN's place) and to trace its propagation in
a time-frameworks.
§ 2. 4. 5 Possible
applications
of the GNs
The simulation of a formal comninication process will on
the
values of the transition condition predicates,
be based
which can
be
94
CHAPTER 2
determined by the token's characteristics or on the basis
of experts'
assessment. Using the results of the program realization and simulati on,
a wide variety of questions can be answered and various conclusi
ons can be drawn. The quantitative results may ticles
include: published ar
(on a certain topic or in a given Journal)
all the submitted manuscripts; without revision]
as a percentage of
or directly accepted manuscripts (i.e.
as a percentage of all
the accepted article manu
scripts; the number of resubmitted manuscripts that are accepted; number
of
submitted
etc. The qualitative how
does
the number
manuscripts by authors with
of
submitted manuscripts
change; is there any relation characteristics
the same
answers are to questions such as on
is the flow of
a specific topic and initial
submitted manu
scripts
sufficient for the "safe"
Through
this model many time-related questions can also be
One could
appreciate
address,
the following:
between the rejection rate
of the manuscript;
the
existence of a given journal, etc. answered.
the mean model time needed for the transfer
the manuscripts from place E to place D
of
(publication time-lag) or the
time required to pass a particular place, etc. This paper is based on [7],
REFERENCES: [1] P. Atherton,
Views of the communication network of scientific and
technical information. International Forum on Information
and Do
cumentation, Vol. 1 (1975), Ho. 1, 10. [2] W. Garvey et al. , Research studies in patterns of scientific com munication, Inf. Stor. Retr. Vol. 8 (1972), 111-122, 159-169, 207221, 265-276. [3] W. Garvey, S.
Gottfredson, banging the system: Innovations in the
interactive social system of scientific conmunlcation. Information Processing and Management, Vol. 12 (1976), Bo. 3, 165-176. [4] P. Hills,
The scholarly conmuni cation process.
Annual Revies for
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
Information Science
95
and Technology, Vol. 18 (1903), 99.
[5] D. King, D. McDonald, N. Roderer, The journal in scientific conminication:
The roles of authors, publishers, libraries and readers
in a vital
systen.
National
Science Foundation Contract NSF-C-
DS175-0694-2. RocKville, Mary lands, King Research, Inc. , May 1979. [6] A. Meadows,
The problem of refereeing. The Scientific Journal (A.
Meadows, Ed. ), Aslib Reader Series, Vol. 12 (1979), London, ASLIB, 104-111. [7] Todorov R. , Atanassov K. Formal communication in Science: based on Generalized nets, 177-185.
A model
Scientometrics, Vol.9 (1986), No. 3-4,
96
CHAPTER H
§ 2. 5: GENERALIZED HETS AND EXPERT SYSTEMS. V Krassimir Atanassov,
Peter Georgiev
The main problem in designing, Systems (ESs) is to select lism (see e.g. [1,2]). i.e.
and Martin Tetev
Implementing and
using
Expert
the proper Knowledge representation forma
Host of the ESs apply the rule-based approach,
they represent the knowledge in "IF situation THEN action" man
ner. The papers [3-6] deal with another means of representation, main ly with a generalization of Petri nets
and discuss how the rule-based
approach can be viewed through the new formalism with all the ensuring consequences.
On the other hand the GHs are an
extension of the well
Known Petri nets allowing a detailed description of parallel
(or con
current) processes. The GN described in the paper can be considered as an extension of the nets from
[3-6).
All notations of [3-6] are saved here.
This
gives the possibility to trace the development of the idea for the GHmodelling of the ESs. On
the other hand
we introduce new idea related to
the ESs,
based on the GH-description possibilities. The main components of a production (DB)
containing
system are:
the Data Base
the facts about the problem which are
to be solved,
the Knowledge Base (KB)
containing the rules which are
to be used in
the reasoning process, and the inference engine which operates through the KB using the DB for proving or rejecting rules in form,
the KB
are supposed
to be just
a given hypothesis. in a positive
The
conjunctive
since otherwise the rule can be split into a number of
"subru-
les" of that type. We shall describe the ES J s facts with ties.
How we can compare
priorities.
new components - priori
two arbitrary facts on the basis
This possibility is very
useful
of
their
in the particular cases
when both facts coincide or when both facts have contrary senses.
SCHE APPLICATIONS OF GENERALIZED NETS IN SCIENCE
97
Let to every fact A of KB he Juxtaposed by a natural nuntoer u A which corresponds to tbe priority of A. Let tbe new fact B with a pri ority \i be generated by some way in a certain time-moment when tbe B ES functions.
If both
facts
are not related,
ters the DB. In the ordinary ESs, fact
A, when
function
the new fact
then the new fact en B substitutes the old
B coincides with, or contradicts to A. Now the
in another way, basing on tbe new component,
ES will
when tbe facts
A and B coincide, tbeir representative (in particularly - A or B) stay in tbe DB, but with a new priority - tbe maximm of \i and \> . On tbe A B
other band, tbe fact with tbe mcnrimm priority between \i and \> stays A
B
in tbe DB when tbe facts A and B are in a contradiction. The constructed here GN (see Fig. 2. 8) represents the ESs with data's priorities in tbe above sense. There have been several attempts to describe production systems by some Kinds of nets (see cf. [7-15]). Following [3-6] some definiti ons
will be given here in order to describe
different components
of
tbe GN appearing below. Let a token
a enters place 1 of the GN with initial cbaracte1 a a ristic x {- "a hypothesis") and let z denotes its current cbaracO last teristic. Let a token
b b enters place 1 with initial characteristic x (= 2 O
"DB"). Let tbe token
c enters tbe place 1 with initial cbaracterisric 3
c x
(= "R"), where 8 : IF , I 1 I is tbe list of tbe rules. Each 0 1 2 n rule E has the form , where C is tbe consequi i 1, 1 i, s i i
98
CHAPTER 2
ent and A
A 1,1
are Uhe elements of the conjunction
presen-
i, s i
ting the antecedent,
z
denotes the i-th characteristic of
the token
i with the highest priority in a given place. Finally, let toKens d , d , ..., d 1 E s
(s z 0) enter place 1 wuth 40
d cu initial characteristics x
(= (pr x , x ) i- 0"; 5 1 0 last c a W = "mnf>4pr x , x ) - 0" 6 1 0 last
W v W W vW 5 6
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
where nun*>(Y, y) the ordered set
101
is the number of appearances of the element y in the Y.
(In [3-6] there are other predicates related with K
places 1 , 1 and 1 4 10 12
Here place
the GN from [6]. This subnet
1 10
is an input to the subnet of
realizes the process of checking of the
hypothesis and it is aside of the present model); Z = 6 13 31 29 31 6 13 where 1 29 29 rr 6
= 11 II W W 13 13 II 16 16 1 II W W 31 II 31 16 16
where the predicates W
and W 16
W
1 31 W W 17 17 W W 17
have the form 17
= "there are no tokens in the subnet" 16
(the form of this predicate in [6] is similar to the above); W
= "a new fact is generated in the subnet" & "there are some tokens 17 in the subnet";
Z = 7 29 15 32 33 34 35 7 7 where
rr 7 7
l1 32 32
33 33
= 11 ||W W vvW W 29 II 18 18 29
W W vvW W 19 19
1 30 I 30 I where the predicates W
and W 1«
W
= "the 18
W = nW . 19 ia
ffalse alse
ii
ll
ll 35
3434-
ffalse alse
ffalse alse W W vv W W 18 18
false false W W vv W W 19 19
have the form 19
GN-modelled ES is not a self-addapted system";
102
CHAPTER 2
Fig. 2. 8
SC*E APPLICATIONS OF GENERALIZED NETS IN SCIENCE
103
The characteristic functions f of the tokens associated to the i corresponding place 1 are as follows: i $ 5 $
a a — > "!x " (the hypothesis x is valid); 0 0
a a — > ""Mix )" (the hypothesis x is not valid); 12 o 0
»
$ 31
— > "x U fx), where last which
(The place
x is the last characteristic of the token
has entered place 1 " 20
1 20
is an element of the subnet.
In it enter tokens with
current characteristic "new fact", which is received in the process of a checking of the given hypothesis x ); 0 o d d d / p cu cu cu / (x - (<pr x , Q(pr x )>) U {x i 0 last 10 10 , if the token d is in place 1 cu 41 I •* I
$
46
— —>>
(
\ I
, if the token d is in places 1 or 1 CU 42 44 d
d d cu cu cu (x - (<pr x , Q(npr x )>! V (x ) last 10 10 0 p
, if the token d is in place 1 cu 43 p
cu
x U (x V last 0
1 , if the token d is in place 1 cu 45
(the other functions are not defined). The GN considered in [3-6] ting production systems
shows the possibility
of represen
(their functioning and their results) by GNs,
This possibility follows also from the fact that a GN can describe the functioning of systems were
Predicate/Transition nets,
and in
[7-15]
production
represented by means of Petri nets, Predicate/Transition
10*
CHAPTER 2
nets and other Petrl net modifications. [3-6]
have the following
The GNs described here and in
universal property:
it is not dependent on
the particular modelled production system These GNs are not connected with the description of
the parti
cular ES. It should be mentioned that existing
(Known from the literatu
re) nets representing production systems are the particular system they describe and
rigidly
associated with
cannot be changed (deformed).
On the other hand the two GNs from [121, the second GN from [3, 4] and the GNs from
[5, 6]
they represent.
are maximally
independent of the particular
ES
Finally, the described here GN includes as particular
cases all previous constructed GNs describing ESs. REFERENCES: [1] Building
expert
systems,
F. Hayes,
D. Waterman
and
D. Lenat
(Eds. ) Addi son-Wesley Publ. Co. , Reading, Mass. , 1963. [2] D. Waterman, A guide to expert systems, Addison-Wesley Publ. Co. , Reading, Mass. , 1906. [3] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M
Marinov, and S. Petkov, Generalized nets
tems. I. 12-th
Methods of Operations Research,
Symposium on
Mathematik,
I. Kazalar-
and expert sys
Vol. 59.
Proc. of the
Operations Research of the Gesellschaft
OKonomie
und
Operations
Research,
1987,
fur
Passau,
Frankfurt a.M: Athenaeum, 1989, 301-310. [4] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M
Marinov, and S. Petkov, Generalized nets
tems. II. IFIP Symp.
"Network
Information
I. Kazalar-
and expert sys
Processing Systems",
Sofia, May 1988, Vol. 2, 54-67. [5] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M
Marinov, S. Petkov and M
nets and
expert systems. III.
Stefanova-Pavlova,
Methods of
Operations
I. KazalarGeneralized Research,
SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE
Vol. 63.
Proc. of the
14-th
105
Synposiim on Operations
Research,
Ulm, Sept. 1989, 417-423. [6] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M
Marinov, and S. Petkov, Generalized nets
tems. IV.
Proc. of the XIX
Spring Conf. of
I. Kazalar-
and expert sys
the Union of Bulg.
Math. , Sunny Beach, April 1990, 155-161. [7] M. Bonacina, Petri nets for Knowledge representation,
Petri Nets
Newsletter 27, 1987, 28-36. [8] J. Duggan and J. Browne, ESPNET: Expert-system-based simulator of Petri nets, IEE Proc. D
(Control Theory and Applications),
Vol.
135, No. 4, 1988, 239-247. [9] A. Giordana and L. Saitta,
Modelling production
rules by
means
of predicate/transition networks, Inf. Sciences 35, 1965, 1-41. [10] U. Mainz, Netztheoretische reprasentation pradikatenlogischer begriffe und methoden, Diplomarbeit, Univ. Bonn, 1985. [11] A. Sinachopoulos,
Derivation of
a contradiction
by
resolution
using Petri nets, Petri Nets Newsletter 26, 1987, 16-29. [12] S. Stoeva
and
K. Atanassov,
Generalized net
production systems interpreters,
Proc. of the
representation of Fifteenth
Spring
Conf. of the Union of Bulg. Math. , Sunny Beach, 1986, 456-464. [13] R. Valette,
Nets in production
systems,
Proc.
of an Advanced
Course "Petri nets: Applications and Relationships to dels
of Concurrency",
Bad Honnef,
Other Mo
in Lecture Notes in Computer
Science, 255, 1986, 191-217. [14] K. Voss, Nets in Data Bases, Proc. of an nets:
Applications and Relationships to
rency" , Bad Honnef,
in
Advanced Course
"Petri
Other Models of Concur
Lecture Notes in Computer Science, 255,
1986, 97-134. [15] D. Koutanis, R. Rasshidi, expert systems. 1987, 143-152.
Petri net representation of rule based
First Annual BSD/SMI
Expert Systems Conference,
106
CHAPTER 2
§ 2.6: GENERALIZED NETS RESEARCHING THE BASIC PROPERTIES GP KNOWLEDGE BASE Ewgeni Dimitrov,
U y a KazalarsKy
and Krasslmir Atanassov
One of the basic elements of every
Expert System
(ES)
is the
Data Base (KB). The structure and correctness of the DB determine the properties of the corresponding ES. The Knowledge in the ES
is repre
sented as Production Rules (FRs). The possibility
to check the correctness of
the KB
is a very
important condition for the functioning of the ES. We shall construct a reduced the existence
GN
of at least two rules in
(see App. 5) which determines a given EB,
having identical
antecedents (see Fig. 2.9). The capacities of its places and arcs are
Fig. 2.9 infinite. I t s transitions are as follows: Z = 1 1 2 3 4 1
SOJE APPLICATIONS OF GENERALIZED NETS IN SCIENCE
107
where 1l 3 1 II W W r = 11 II ll r 1 || 1 II W W 2 11 11
1 4 W W 2 W W 2
where W = "the place 1 i s empty" or "in place 1 there e x i s t s at least one 1
3
3
token with the characteristic - the same
antecedent
as in
the
present token",
w = -m ; 2
1 Z 2
= 3 5 6 2
where 1 5 5 r 2 2
= 1 I W 3 1 3 3 1 3
1 6 6 W 4 4
where W
= "the transitions Z 3
and Z are not active" and "at the moment of 1 3 activation of the transition Z in place 1 there exists only 2 3 one token"
(this token enters place 1
and leaves the net;
if it is
the unique
5 token of the net, it receives as
a final characteristic
"there exist
no two PRs with identical antecedents"); W = "the transitions Z and Z are not active" and "at the moment of 4 1 3 activation of the transition Z in place 1 there exist at le2
3
ast two tokens" (these tokens enter place 1 6
and leave the net with a final characte-
CHAPTER 2
loa
ristic "the PRs which are Initial characteristics of these tokens have identical antecedents"); Z 3
= 4 2 3
where
1 2 r
: 1 3
I W 4 | 5
where W = 5
'the places 1 and 1 are empty". 1 2 All PRs of a given KB can be described as initial
characteris-
tics of the tokens of the above GN and thus an answer for tic question of the existence of identical antecedents in PRs can be obtained. The GN
which checks the set of al1
PRs
of a given
the existence of at least two rules frcni the form has the form as in Fig 2. 10.
KB
about
A I- C and A & B I- c
The transitions Z and 1
Z 3
in both nets
coincide with those from Fig. 2. 9 without
w - • the 1
place 1 is empty" or "in place 1 there exists at least one 3 3
token with the characteristic - the same
consequent,
as in the
present token". The transition
z = 2
1 6
= 1 | W 3 1 6
W
1 1 W 9 1 6
W
7 7
where W
6
= ' the place 1 is empty" or "in place 1 there exists at least one 5 5
SOJE APPLICATIONS OF GENERALIZED NETS IN SCIENCE
109
Fig. 2. 10 token with
the characteristic - an antecedent,
being a subtenu
of the antecedent which is an initial characteristic of the present token",
w = nw . 7
6 The t r a n s i t i o n z Z -: , 4
access
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
= »S - ln
W
& "T i X
5
+ t a
2
+ t ", 3
where 0 is an indication for branching at the transition Z . 4 Z = , 5 9 10 5 where 1 10 r
= 1 I true ; 5 9 5 9I I Z = 6 11 12 6 6 11 12 6 where
r = 1 I 6 11 I
1 12 12 W 6
where W
= "in 1 6
there are no tokens with the same characteristic m"; 12 Z
= , 7 7
where
r = 1 I 7 12 I
1 13 13 W 7
where = "T i x
w 7
a
+ t ": 4 Z
= , 14 8 14 8
where 1 14 r = 1 I 8 13 8 13 I I where
W 8 8
14-5
146
W
CHAPTER 3
= "T i t +t + X ". z fl 1 15 5 a Z = >, 2 3 4 where
1"', 1"". 1"" i 1 2
1""!, *. *, r", i
152
CHAPTER 3
a) place 1 is related to passengers boarding the vehicle 4
(the tokens
receive in it the number of these passengers); e) the transition condition is
I I r" =
1"' 1
...
1"' j
1"" 1
...
1"" J
w
...
W
false
...
false
2 1 3 I 1 I false 3 1
3
1 | false 4 |
...
false
W 3
...
W 3
...
false
W 3
...
W 3
■where W
= "the next route station of the vehicle". 3 All transitions have equal priorities. The GN described above is a reduced GN of the type
A,A,A,Tl,e,9,b 3 4 6 A 1 2 T. We conclude that within the framework of the model we can
spe
cify the following main characteristics: priority of the stations (for the places of type
1 ),
capacity of the places of type 1
2
(number of
2
vehicles stopping at a given station) and the priority of the vehicles (tokens of the first kind). The fornullation of an analytical model of the considered tran sport system, particularly the analytical
part of the given GN model,
is related to the influence of factors of various nature. Some of them are:
settings and movements of the population,
presence of different
social and age groups with the appropriate requirements
for security,
comfort, costs, and travel time. These factors affect the determinati on of the initial
parameters of the second kind of tokens
passengers waiting for a vehicle at a given station,
(number of
with preferences
regarding the type of vehicles, current number of passengers, etc. ).
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
153
Different characteristics of the transport system can be obtai ned
by simulating the model built by means of a (34.
It makes it pos
sible to use the GN model as a tool for transport process
management.
Furthermore, it enables optimization of the vehicle routes the
traffic process and passenger streams.
simulating
It is easy to obtain
the
average and maximum time duration of a given route. In the case of any changes, the time schedule and the correspondingly adjusted streams
can be fixed.
On the basis of the obtained results,
passenger strati
fication of the transport net and appropriate estimation of the travel costs can be determined
154
CHAPTER 3
§ 3. 4: GENERALIZED HEX MODEL OF ACTIVITIES OF A BUS-DEPOT Favlin Gyurov
An example for using tion
toe Generalized Hets (GHs) for a descrip
of some transport situations is introduced.
It illustrates
the
solving of the following problem Suppose a public coach agency uses seats are Known,
n
coaches whose numbers of
when the coaches are not in use, they are in a gara
ge. There is a service for maintenance work. The coaches leave for the routes they
from a bus station.
take their own daily
Every morning, before leaving the garage, time table.
At the end of
their workday,
they go back to the garage. This process
will be described below by means of a GH and
the
loading of the lines and coaches will be reported. The GH has a graphical structure as shown in Fig. 3.6. Every coach is represented by
a token in the GH.
when a coach
is on a route,
the token corresponding to it goes into the GH. At the
time -moment T,
all coaches are
bus-tokens BT i
in the garage,
so at that moment all
(1 . Initially, it is in place 1 . When it comes into place 1 , 12 4
it gets a
new characteristic whose seats component is the number of sites of the damaged coach, its waiting-time component travels
component is
is the previous one and the
the set of travels which
when OT comes into place 1 , 9
this coach
would do.
it doesn't change its characteristic. In
SCME APPLICATIONS OF GNs IN ECONOMICS , INDUSTRY AND TRANSPORT
place 1 it gets new characteristics - the previous seats 16 components
157
and travel
and the waiting-time characteristic is the duration of
is
stay in place 1 . 4 Every bus-token gets as
a first characteristic
its
number of
seats and its daily time table. The maintenance component of its first characteristic is
.
When it goes to place
characteristic. Travel component reduces to
1 it changes 13
its
the set of travel fulfil
led. The current characteristic of the token includes information aboit gets
ut it. When a token comes in place 1
a characteristic - the
1 previous last
of
The nunfcer of seats
travel component.
can be changed
the repair component, and the duration
the repairs join
The of
these repairs adds to the current durat ion of maintenance. When
a token comes into place
1 it changes 14
the passengers
(TIME) characteristic. A bus-token that comes into place 1 can append 3 new travels to the travel component of its characteristic. Z
1 1 = >■ 1 2 1 1 2 3 1 2 1 1
where 1 3 r
= 1
1 l 11 l 1 l 21
true true
1 3 M
= 1
1 I 11 1 1 1 21
n
' n
15&
CHAPTER 3
2 2 Z = , 2 3 4 15 5 16 1 2 2 2 3 3 4 where
r = 2
1 3 1 4
1 15
1 5
1 16
I W I 1 I I false I
W 2
false
false
W 3
where W
= "the coach is out of order"; 1
W 2
= iW & ("the moment for going to the station has come" or 1 is an OT
in 1 " 4
and
("the coach has no other travel"
("there or
"the
travel of OT can be done before the travel of the bus-token") and "this coach is optimal about the number of seats"); W
= "the order is taken up by the bus-token"; 3 1 15 1 M
= 2
Z
= , 15 10 11 13 1 2 3 3 15 10 11
where 1 13 r r
3 3
= =
1 I 15 15 I I I I 1 I 10 |
true
1 I 111 1 I
true
true
SCME APPLICATIONS OF GNs IN ECONOMICS,
INDUSTRY AND TRANSPORT
1 13 1 M = 3
I 15 I I 1 I 10 I
n
1
n
n
I
11 I Z = >, 1 1 2 4 4 13
where 1 r
=
1
4
| 13 I
1 true 1 13 M = 4. 4 Z = , 7 8 1 2 5 5 5 6 16
where
1I r
= 5
| 5 I I 1 | 6 I 1 | 16 I
1 M = M 5
5 1 66
1
I 1 I I II
1 16 16 II
0
1 7
1
true
false
true
false
false
true
1 7
I
n
0
n
0
8
a
1
159
160
CHAPTER 3
6 6 Z : = c|) (|) I, II !1 , 1 , 1 ), t , t , r , M , v(l )>, 6 7 10 2 11 1 2 6 6 7 where
r = 6 6
1 7 7
I 1 1
1 10 10
1 14 14
1 2 2
W
W
W
1 1
4 4
5 5
where = iV
W 4 W 5
& "it is time corresponding to the token coach to travel", 1
= iW & 1
("there is nuch time before the next travel" ach's work day is finished" ),
M
: 6
- , 14 11 6 1 2 7 7 14
where
r
= 7 7
1 14 14
l I I I
1 11 11
1 6 6
W 1 1
w W 6 6
1 11
1 6
n
n
where W 6 6
- nW & "it is the end of travel", 1 1
M
= 7
1
I 14 I
;
Z = , 12 8 88 88 9 9 12 11 22 88 88 8 9 9 where
or
"the co-
SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
161
1 12 1 r
=
a
a 1 9
I I i I I
true true
1 12 M, M, = = a a
1 I 8 I a i i i 1 I l i 9 I 9 I
1
l1
9 9 Z --- , 9 12 12 2 9 1 2 9 9 12 where 1 2 r
=
1
9 9
l 12 I 12 I
1 9
W
true 7 7
where W
= "there is a coach which is damaged"
&
"this coach
7 vels",
M
= 9
1
l 12 I
1 2
1
1
1
9 .
For t h e GN-components t o be v a l i d , i A
|Z | = i f o r i
n(l) L 4 12
S 9,
= n ( l ) = l; for all other places: TT (1) = 2, 9 L
) - 2;
c(l
I $i
for all other places: c(l) - n,
has to tra-
162
CHAPTER 3
i t
o + 2. t , if i £ (1, 2, 3, 5, 6, 7)
1 i i 6 (t ) ) = s 1 1
i o t + C. t , if i = 4 (C = const 2* 2), 2), 1 i t
o + t , if i € [», Eft, 9)
1 It means that the private tokens go faster than others.
In or
der to maintain the works character it is possible that the transition Z works rarely. 4 i i i o © (t ) = t = t (1 . 2 i1 2 2 i The numbers t (1 i i £ 9) must be large enough so that all to2 kens can go through transitions K = (BT / 1 i i "waiting-time for place 1 ", 2 -> "general time for processing of the workpiece", "difference between the real time and the time which characteristic",
is given in
the initial
§ -> "name of the processing machine for small workpieces", 5 $ -> "name of the processing machine for medium-size workpieces", 6 $ -> "name of the processing machine for big workpieces", 7 $ -> "time for processing in 15 or 16 or 17, respectively", 9 $ 10 $
-> "waiting time in 1 ", 9 -> "value of the consuming labour",
11 $
->"time for processing separate steps". 12 Tbe places represent the following process components:
1 -> initial status of the process, 1 1 -> a workpiece is waiting in the store for processing, 2 1 -> the workpiece is going for processing, 3 1 -> the workpiece is going out of the system after processing, 4 1 -> the workpiece is processed on a machine for small workpieces, 5 1 -> the workpiece is processed on a machine for medium-size workpie6
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
187
ces, 1 -> the workpiece is processed on a machine for big workpieces, 7 1 -> the workpiece is going back to the store, 8 1 -> the workpiece is waiting for releasing of the washing machines, 9 1 -> the workpiece 10 ne,
is washed and measured from the measuring machi-
1 -> the size of the workpiece does not meet the requirements, 11 workpiece is shelved and going out of the system, 1 -> the workpiece is going for further processing, 12 is going to the store.
the
if necessary it
M *
§ 3. 6. ■*. Third
W-aode]
K
of FMSs
The system to be controlled is part of a flexible workshop the Renault company.
of
The purpose of the system is to mastic seal the
body work of automobiles. For this purpose it has six workstations (or simply stations),
each consisting of one work place P, sufficient for
one car body, and also one transfer bench with two roller tables,
one
designed to load the station (TL) and the other to unload it (TU). The bench, which is mobile,
can adopt two positions:
left,
to load, or
right to unload the car bodies. A system of transport, made up of six roller tables RT ) has 6
the task of transporting all
car bodies which
(RT 1 enter
the
system (via table RT ) to the sealing station, and of later unloading 1 and removing them to the exit via
RT6.
Each table
(TL, TU, RT ) can i
carry only one car body, and at any s t a t i o n there can be a maxirrum two (P-TU, P-TL, or TU-TL) at a given time [27],
of
188
CHAPTER 3
A coloured Petri net model delling the autonomous
coloured
of this system is Petri net
applied.
In mo
model, we can distinguish
three stages: (a) independent modelling of different subsystems, which form
part of
the system under consideration, using a coloured Petri net. In the above-mentioned system, we can distinguish two subsystems:
trans
port and stations; (b) fusion
of the modules by fusing
the transitions which
represent
synchronization between the subsystems. In the system, the loading and unloading of the stations requires synchronization between the transport and the station subsystems; (c) the eventual simplification of the coloured Petri net, obtained by eliminating implicit places. These are places whose marking can be expressed as a positive linear combination of
a set of
the net and whose elimination preserves the liveness
places of
and bounded-
ness of the net. With each place, we associate a set of colours such that we can distinguish all the tasks and resources associated with it. The urs can be simple or compound (n-tuples) depending on whether sociated
tasks or resources are defined
by one or
colo the as
several parameter
With each transition, we associate a set of colours such that its car dinal is equal
to the number of different possible
evolutions of the
state of the system which are defined by their possible firings [27]. Below we
shall construct a GN which describes the above
pro
cess modelled by a coloured Petri net. The graphical structure of this net is given on Fig 3. 10. The GN has no activated
local temporal
components.
Every transition is
when there is at least one token in its input places,
all transition types are of a disjunctive type. The capacities of the places are (cf. [27]):
i. e.
SCME APPLICATIONS OF GNs IN ECONCMICS, INDUSTRY AND TRANSPORT
189
Fig. 3. 10 c(l ) = c(l ) = oo, c(l ) = . . . = c(l ) = 6. 1 8 2 7 The priorities of the places are TI ( 1 ) = TT ( 1 ) = TI ( 1 ) >,
L
3
L
5
L
7
TI ( 1 ) = TI ( 1 ) = TI ( 1 ) = TI ( 1 > = TI ( 1 ) .
LI
L2
L4
L6
L8
The priorities of the transitions are equal. The capacities of the arcs are "a>". Every token enters the net at a certain time-moment (determi ned by a function 0 ) according to some fixed time-scale (with initial K o * time-moment T, elementary time-step t receive
an undetermined number
and time-duration
of characteristics
an initial characteristic "type of the car others)".
and
t ).
It can
enters with
body, current number
(and
190
CHAPTER 3
The GN contains 4 transitions and following
transition conditions
and to
8 places.
They contain the
the GN places, the following
characteristic functions are associated. 1 2 2 r 1
W
1 3 3
= 1 I W 1 1 1
W 2
1 I W 3| |1 3 1
w 2 2
= "there is an empty loaded table" & "there is an enpty roller tab1 le";
W 2
= iW ; 1
$ -> "" 2 2
(the function
3(1)
gives the capacity of the place
1 and the func
tion d(l, TIME) gives the nuntoer of tokens in the place 1); 1 4
1 5
r = 1 I W 2 2 1 2 1
F
IF
W
3 I
2
where W ; "3(1 , TIME) < d ( l )" & "the necessary time 2 2 2 p l a c e 1 i s run out". 2 1 6 1 44
r 3
I II
W 33
1
1 7
1 8
false
false
for
processing in
1 9 false
10 false
1 i false 7 1
w 3
nw & &W W 3 4
nW nW 8nw Snw & &w w 3 4 5
iw i w 8nw s n w 8nw 8nw &(W &(W vw » 3 4 5 6 7
= l1 |i f a l s e 1122 I
w W 3
iv nW & &W W 3 4
n nW W snw 8nw & &W w 3 4 5
nW -m ssnw n w snw snw &(W &(W vw ) 3 4 5 6 7
1
I false 14 14 I
W 3
nW n W& &W W 3 4
nW nW 8nW 8nW & &W W 3 4 5
nW nW 8nW 8nW 8nw 8nw &(W &(W vw ) 3 4 5 6 7
1
| false 17 I
W 3
-m & &W W 3 4
-m SnW 8nW & &W W 3 4 5
nW nW 8nW 8nW 8nw 8nw &(W &(W vW ) 3 4 5 6 7
where W = "d(l , TIME) < d ( l )", 3 6 6
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
W = "3(1 , TIME) < e(l )" & "the necessary time * 11 11
for processing
195
in
place 1 is run out", 6 W
= "d(l 5
W
, TIME) < 6(1 )", 13 13
= "the workpiece is good", 6
W
= "the workpiece is ruined". 7 1 11 11 rr = 11 || 66 | | 4
1 12 12
w w
false false 4
1 II ffalse alse 8 II
W W 4
1 13 13 r r 5
= 11 || 11 11 || 1 9
r r
W W 5
conclusion
W W 55
1 15 15
11 16 16
W W 66
W W 77
false false
false
W 7
1 I false 10 I In
false false
II ffalse alse II
:: 11 II 6 13 II 6 13
1 14 14
1 17 17
we shall note that the above constructed GN-mo-
del has important advantages in relation to the other Petri net models (a) it is considerably more universal; (b) as a result of
simulation
this model, more
of a flexible
manufacturing cell with
information on the process can be received (the
tokens
which enter the net
obtain
new
with certain initial
characteristics while moving
characteristics
through
the net by the
characterizing functions associated with the places); (c) the process (if it proceeds sufficiently slowly)
can be
control-
196
CHAPTER 3
led on the basis of the constructed model; (d) the process can be optimized on the basis of the model, K
K
3. 6. 6
Queue models Scheduling is
K
and GN-models in flexible
manufacturing
one of the important problems
in
the functio
ning of the FMSs. The classical scheduling has
formulated
tactical
literature dating back to the 1960s,
scheduling
as
the following
problem: a job shop has n jobs and m machines. composed of a sequence of m operations,
optimization
Each job has a routing
a unique machine being assig
ned to each. The goal is to find a schedule (the start time of im ope rations),
which minimizes some criteria,
e.g. makespan or tardiness.
A simple approximation of the computational complexity can be calcula ted as follows: a job has m operations, one on each machine, and since each machine can be scheduled in n! ways, m In!|
schedules.
it follows that
there are
A method for finding the optimal schedule would
be
the enumeration of all these schedules and choosing
the best one. The
search space, consisting of the possible number of
valid schedules is
very large. phase of
Traditionally,
scheduling has been
the planning process,
considered
the last
which gives a solution of the problem
as when to allocate a particular resource to an operation [59]. In practice this problem is solved in the following way: (a) the priority of the job is defined; (b) the jobs are ordered according to their priorities; (c) if the planned resource is not available,
the corresponding
jobs
with a lower priority must be transferred. In flexible manufacturing
the machines are ordered
way that they can meet the requirements
in such
a
of ful1 processing of a work-
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
197
piece in one place. The FMSs can be
considered
as one
capacity unit. Because of
centralizing according to objects, but not according to functions, the material flow can be
easily
observed.
In this way the transfer of a
job does not attach to the other capacity units. To solve the scheduling problems in the FMSs, some methods are used: (a) queue models; (b) methods of mathematical programming; (c) methods with priority rules, The GNs
are also used
for modelling
the functioning of
the
FMSs. We shall try to compare the queue models with the GN-models ac cording to the method of solving scheduling problems in FMSs. The queue model consists of
knots
workpieces in front of them The time
(machines) with a queue of
for processing
is random
The
workpieces are processed according to the order of their entrance into the system (first in, first out) or in a random way. After processing, the workpiece goes to another machine with some probability. cal
Analyti
results are as follows: middle grade of loading up the machines,
middle quantity of workpieces in the system,
middle quantity
of wai
ting workpieces [60]. One of the main disadvantages of the queue models for modelling FMSs
is the
value of
the analytical
results.
These results do not
match the real situation. It is due to several factors: (a) the often-used exponential distribution for the processing time is not close account
to the real distribution;
it is possible
deterministic processing time-moments only
cessing times are identical; (b) the processing sequence is strictly determined;
to take into when the pro
i9fl
CHAPTER 3
(c> for the queue models an unlimited
capacity
is implied; this fact
does not match with the capacities of the machine buffer and
sto
rage; (d) only one priority rule for scheduling
is used
(FIFO - first
in,
first out); (e) the workpieces
cannot be individually traced, the description be
ing for the whole system; (f) the disturbances
(machine breakdown or lack of
tools)
cannot be
taken in account. The GN-model of FMSs removes the drawbacks of queue models: (a) we can use an arbitrary distribution of
the processing
times so
that in future extensions of the GNs one distribution can be ex changed with another; (b) the processing sequence is not strictly determined;
it depends on
the optimal choice at a particular time-moment; (c) the capacities correspond to the real system (machine buffer, sto rage ); (d) different priority rules can be used - rules for the optimal choi ce of the workpieces; (e) the workpieces can be individually traced; (f) the disturbances can be taken into account in the model. Comparing the de
two approaches of modelling FMSs, we can conclu
that the GN-models
are more suitable
for deciding
the sheduling
problems in flexible manufacturing, K K
3. 6. 7:
X
Conclusion
In conclusion, note that
the above constructed GN-models have
the following important advantages in relation to other PN-models. 1) They are considerably more universal compared with
other PN-
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
199
models. A PN-model is constructed for a concrete type of FMSs. The GNmodel discussed here
describes the functioning of the results of
the
work of every element of a complete class of FMSs. 2) As a result of the simulation of more
information about the process
can
the FMS with this be extracted.
characteristic functions defined are constructed
in
GN-model
Note that the
such a way that
they are maximally similar to those for the PN-model. 3) The process
(if it proceeds sufficiently slowly) can be con
trolled on the basis of the constructed model. 4-) The process of the
of the transfer of the workpieces and the choice
processing machines can be optimized on the basis of
the GN-
model. Ibis paper is based on [61-64].
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2-90, Sofia, 1990, 13-14.
208
CHAPTER 3
§ 3. 7: GENERALIZED HET MODELS OF THE ACTIVITY OF NEFTCCHIM PETROCHEMICAL COMBINE IN BOURGAS Stela Dimitrova, Ludmila Dimitrova, Trajana Kolarova, Plamen Fetkov, Krassimir Atanassov and Rumen Christov
Generalized Nets complex
objects
which
(GNs)
are a comfortable
means for designing
are characterized by a large scale of various
parallel processes taking place in the real time. In the general case, it is impossible to show analytical formulas formalizing these proces ses.
For their modelling,
various simulation methods are used. Petri
nets are one of these (new) basic methods. As discussed in
§1,1, the
GNs can also be used as a means for simulating such (real) processes, but (in contrast with the Petri tion condition predicates
nets and
by virtue
of their transi
and characteristic functions) they can also
include in themselves elements of analytical methods for
description.
For example, some characteristic functions can assign to a given token a value that is calculated by analytical functions. "The NEFTOCHIH of the integrated strong
Petrochemical Combine (FCC)
industrial enterprises in
in Bourgas
is one
Bulgaria which exerts
a
influence upon the dynamic development of the State Industrial
Chemical Corporation, the economy of the country,
and the internatio
nal division of labour. " [1] The processes going in PCC can be modelled by GNs and structed
models
can be used
for simulation
combine. The models are investigated following
the con
of the processes in the the ideas
for hierar
chical models and models based on the union and the composition of ot her simpler GN-models introduced in § 1. 1. The modelling starts with the construction of a simplified glo bal
model
of the combine in general.
It has the from of
Fig. 3. 12,
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
209
Fig. 3. 12 The GN used can be, e. g. , a second type intuitionistic fuzzy GN,
i. e.
a ON in which the tokens are replaced by "quantities" (see § 3 in App. 1) which
"flow in the net.
So a token ("quantity") enters place 1 1
and this
action
represents
ceived. This place symbolizes distribution
of
de: "chemical"
the quantity of raw materials (oil) re Petrochemical Sea Port (Terminal) where
raw material through the basic types of units is ma (1 ) and refinery.
The second one has the following
3 two components: a "Catalitic Reforming" (1 ) and an 4
"output of fuels"
(1 ). A part of the production from 1 is directed to 1 and 5 4 6
together
210
CHAPTER 3
with the production from 1 , is going (according to preliminary by set 3 conditions) to 1 9
(for production of plastics, fibres and rubber), or
to 1 (for production 10 A part of the 1 11
of petrochemicals - in "petrochemical units").
"quality" (token) is going from the last place to place
(for production of solvents),
and another
place 1 to the "catalytic cracKing unit" 12
part returns
through
and there takes part
with
newly entered "qualities", in the production of fuels (1 ) and 7 products (1 ). The toKens (qualities) 8 go to place
1 14
other
from places 1 , 1 , 1 , 1 , 1 9 11 7 a 5
and that symbolizes the outcome of the finished pro-
duct from the PCC. By this part of the GN the movement (in general outline) of fuels, petrochemicals and plastics, produced and processed in PCC can be observed. Besides, the GN has a second (not differentiated separately) part
that symbolizes
the entry of information on
the demand
of pe-
trochemicals (1 ), workout of this information (orders) (1 ) and re2 13 port on the orders (1 ) - the effect that they are satisfied or not. 15 The tokens ("quantities") that go into 1 have as initial cha1 racteristics the type of raw material, and during the time of movement in
the GN
they receive the following characteristics:
"the kind and
quantities of processing intermediate fractions" and "the kind and quantities of the finished product" depending on the type (Kind) of places they enter. obtain the
In the end in place
1 these tokens 14
("quantities")
finished characteristic "working expenses" and other
racteristics specified by the model's consumers.
cha-
SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
211
The other type of tokens are going to place 1 with the initial 2 characteristic "information about the demand of goods with sary description, quantities, with
the final
the neces
parameters, etc. and are leaving the GN
characteristic "the order is satisfied" or "the order
is not satisfied" (as well as other data). More detailed processing
models can report on
the moment when the raw material enters the PCC, cesses,
such factors as
the duration of pro
the moments when the products leave the PCC, the duration of
waiting for processing of raw materials or fractions etc. The other tokens can have their own priorities which determine the ways of
the transfer. The transition condition predicates can ha
ve explicite
forms. During simulation of the PPC-processes, different
situations can occur in relation to the initial characteristics of the tokens in places 1 and 1 , and also in relation with the characteris1 2 tics
of the tokens which are in the net at
the initial
time-moment
(they correspond to the available quantities in the PPC). The described GN reflects only the most global connections exi sting
between the different
units of PPC. Mien more details
are re
quired in the positions of the places and the transitions of the above net, new GNs will be
standing.
They will describe
the separate sub-
processes which are included in the most global processes. Another situation is the one when most global one)
corresponds
a new GN
(a subnet
of the
to a part of the places and transitions
of the above GN, but does not cover their content. A GN shown in Fig. 3. 13
is an example,
illustrating this. The
components of this net correspond to places 1 , 1 , 1 and the transi1 4 5 tion Z . 1 Briefly, the sense of the places of the new net are as follows: 1 - Petrochemical Sea Port (Terminal), 1
212
CHAPTER 3
Fig.
3. 13.
SOME APPLICATIONS OF GNs IN ECONCMICS, INDUSTRY AND TRANSPORT
213
1 - production of fuel oil 2 1 , 1 , 1 , 1 , 1 , 1 3 4 5 6 9
,1 - production of other cuts 13 30
1 - production of oil cut 7 1,1 - production of cut C 10 11 4 1 - production of high-octane gasoline 12 1 , 1 ,. . . , 1 16 17 25
- tanks
1 - production of cut C 26 5 1
- production of toluen.
27 The GN-models described here reflect the first steps modelling of
the NEFTOCHIM Petrochemical Combine
proposed text, other
of global
in Bourgas.
In the
aspects and methods for describing the petroche
mical processes are not included. REFERENCE: [1] NEFTOCHIM Economic Combine-Bourgas, 1988.
BulgarReklama Agency,
Sofia,
214
CHAPTER 3
§3.8: A SBCCHD TYPE OF IHTUITIOHISTIC FUZZY GEHERALIZED HET-HCDEL IH THE CHEMICAL INDUSTRY Rumen Christov
and
Stoian Garbov
On the basis of the second type of intuitionistic fuzzy genera lized nets (IPGH2) (see § 3 in App. 1) we shall describe a part of the technological
scheme of the functioning of
a silo-farm with pneumo-
transport in a plasticware company. The raw materials are They are
ferried with
auto-cisterns
provided from 4 places and transferred to 10
and tanks. store-bunkers
(silos) or (if necessary) directly to the production lines. rials must flow continuously to these lines. duction lines from the silos
Raw mate
The resin enters
6 pro
by a distributor. Bach line has an indi
vidual capacity and the material available for a definite time is con sumed. The IFGH2 describing this process has the form of Fig. 3.14. It has 3 transitions and 23 places with: - places 1 1
1 - symbolising the accepting places; 4
- places 1 ,..., 1 - symbolising the silos; 5 14 - place 1
- showing the transportation of the material
to
the pro-
15 duction lines; - place 1 - modeling the distributor; 17 - places 1 1 - symbolising the flow production lines. 18 23 All transitions have equal priority. The places 1 and 1 15 ve
the highest priority;
the other places have equal priority.
transition conditions have the form:
ha-
17 The
SC*E APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
Fig. 3. 14
215
216
CHAPTER 3
1
1 5
1
=
1 7
1
1
8
1 10
9
1 11
1 12
1 13
1 14
15
I 1 1
W
W 2
W 3
W 4
W 5
W 6
W 7
W 8
W 9
W
W 10
11
W 1
W
W 3
W 4
W 5
W 6
W 7
W 8
W 9
W
W 10
11
1 3
I | I I 1
W 10
11
1 4
I 1
W 10
11
W 10
11
1 r
1 6
1 2
1
1
I 16 I
2
W 1
W 2
W 3
W 4
W 5
W 6
W 7
W 8
W 9
W
W 1
W 2
W 3
W 4
W 5
W 6
W 7
W 8
W 9
W
W 2
W 3
W 4
W
W
W 1
5
W 6
W 7
W 8
W 9
where W = "the r e s i n m i s t go t o t h e i - t h s i l o " i - "the resin nust go to
W
(1 i i i
10),
the flow production lines,
because there
11 exists a flow
production line with resin less than the critical
quantity"; 1
1 5
r 2
=
I 1
W
1 I 6 |
W
1 7
I 1
W
1 8
I 1
W
1
I
W
9
16
1 17
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
12
W 13
1 1
I 10 I
W
I 11 I
W
I 12 I
W
1 I 13 I
W
1
W
1 1
I 14 |
SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT
217
where W
; "the resin must go to another silo"; 12
W
= "the resin must go to
a production
line with
resin less than
13 the critical quantity or with a minimal resin quantity"; 1
r
= 3
IS
1 19
1
1 20
1 I W W W 15 I 14 15 16 I 1 I W W W 17 I 14 15 16
1 21 W
1 22 W
17 W
18
W 19
18
W 19
W 17
23
where W
= "the resin mist go to the i-th production line which has
the lo-
i west resin quantity"
and
"the resin is below the critical li
mit" (14 i i i 19). The characteristic functions $ give the infinvm and the suprei mum of the available resin quantity in the place 1 (1 i i £ 23). i The IPGN2-models functioning starts at a certain time-moment according to a certain absolute time-scale o step (t ) being 2 minutes.
and the elementary
T
time-
The features of this net are the discrete values of the predi cates of the transition the tokens are
conditions but the net
is an IFGN2,
"quantities" which flow in the net and
part
because of these
quantities is lost in the process of transportation, i. e. there exists a degree of indetermination. The following problems are solved (addaptively) by the model: a) The optimal schedule of the raw materials ferrying is determined. b) TTie optimal loading of the machines is determined. c) The optimal capacity of the connected elements of the manufacturing is checked.
218
CHAPTER 3
d) The possible waiting of the machines are determined. On the basis of the IFGN2-model, the processes in a large class of productions of the chemical industry can be simulated and led (in real-time).
control
Chapter
4:
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
The first research in the area of medical
diagnosis dates back to 1983.
structed nets are collected. scribed models
by means of
generalized net modelling of In § 4. 1 almost all
The authors plan to realize all program package for
the de
generalized nets.
more general model of diagnostics is introduced in § 4. 2.
219
the con
A
220
CHAPTER 4
H I :
APPLICATIONS OF GENERALIZED NETS IN HEFHROLOGY
Joseph G. Sorsich
and
Krassimir T. Atanassov
In this chapter, an exanple of the application of
GNs in medi
cine is given, Nephrology (see e.g. [1-16]), as a relatively new field in medicine,
has been used to make things clearer
and to demonstrate
the possibilities of GNs. Some other applications in medicine are also available (e.g. [17]). Hie mathematical
foundations of
our research
are essentially
different from other basic mathematical methods applicable in medicine (see e. g. [18-21]). A renal
damage is usually suspected in the course of a routine
examination of a patient with some clinical signs and symptoms ria, renal colic,
edema,
turbid urine),
some disturbances in blood sample or urinalyse are present themia, proteinuria, etc. ). kidney deseases,
(byperazo-
Based upon these common signs relating to
the following GN is constructed. Ibis is not a defi
nite model and it may be a subject of further modifications. which will
be
(dysu-
arterial hypertension or if
New GNs,
the subnets of the general GN described below,
added, and/or the subnets themselves may be modified. to demonstrate that such a subject in biological
may be
Our aim is also
sciences as medicine
is susceptible to comparatively simple and easy to understand mathema tical formulation by GNs. The same may be realized in almost all other areas of medicine. Initially, we shall describe one as subnets other GNs
(general)
describing the process
signs and symptoms in nephrology.
GN
which contains
of diagnosing
Let this GN be E.
different
It has one input
place L . The tokens of this GN will represent the patients of a nephrologic unit. Everywhere we describe the transfer of one token the tokens generated by it,
(patient)
or
but this is valid for every set of tokens
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
221
(patients) transferring in the GN. At the end of this chapter we shall discuss these problems more widely, as well as
the possibilities
for
application of this and similar GNs in different medical units. The token goes to place
L
with
the initial
characteristics
1 "patient with probable Kidney disease", kens which
represent the appropriate
next it splits into investigations of
four to
the patient.
Later these tokens may be united into one token whose characteristics correspond to all the applied investigations, Part of these tokens may leave the GN if some of the signs are missing. The sense of the diffe rent GN's places are as follows: L - abnormal findings (changes) in urine are present, 2 L
- elevated blood urea nitrogen (BON) is present, 3
L - clinical signs and symptoms of renal disease are present, 4L
- arterial hypertension is present, 5
L
- permanent proteinuria is present, 6
L
- hematuria is present, 7
L
- turbid urine is present,
a L
- renal colic is present, 9
L
- dysuria is present, 10
L
- edematous patient is present. (Edemas of renal origin are always 11 related to proteinuria, refer to § 9. 1),
L
- evaluation of arterial hypertension, 12
L
- a specifying the type of renal damage, 13
L
- arterial hypertension of renal (renoparenchimal) 14
origin is sus-
222
CHAPTER 4
pec ted, L
- renovascular hypertension is suspected. 15 The transition conditions are as follows: L 22 rr
= L 11
L
ll ttrue rue
4 4
ttrue rue
true true ;;
11 I I L
L 66
rr
L 33
8 8
V V
= = L
II W W 22 11 11
2
L 77 W W
22
3
where W
= "permanent proteinuria is present", 1
W
= "hematuria is present", 2
W
= "turbid urine is present"; 3 L 99 rr
L
L 10 10
= L II W W 44 11 44 3
11 11
W W 55
W W 6
where W
= "renal colic is present", 4
W
= "dysuria is present", 5
W = "edematous p a t i e n t i s present"; 6 L r
= L 4
L 12
13
l true
true
5 I L
L 12
r
= L 5
where
I 113 3 I
W 7
13
W
fl8
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
223
224
W
CHAPTER 4
= "there is suspicion of renoparenchimal arterial hypertension", 7
W
= "there is suspicion of renovascular arterial hypertension". 8 From the fact that a token
(a unique one or one of many repre
sentations of a given token (patient) that enters GN
E) enters places
L , L , L , . . , , L , L , L , i t follows that this token is directed 3 6 7 42 14 15 to the following subnets: L
- to the GN from § 4. 1. 1, 6
L
- to the GN from § 4. 1. 2, 7
L
- to the GN from § 4. 1. 3, 8
L
- to the GN from § 4. 1. 4, 3
L
- to the GN from § 4. 1. 5, 9
L
- to the GN from § 4. 1. 6, 10
L
- to the GN from § 4. 1, 1, 11
L
- to the GN from § 4. 1. 7, 12
L
- to the GN from § 4. 1. 8, 14
L
- to the GN from § 4. 1. 9. 15 On the basis of this model, we can achieve the following:
- control - optimization -simulation of real processes. Control initial
is realized
state of the patient
the token in L ). 1
as follows:
the physician describes
(this is the initial characteristic
the of
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
225
In the other offices, the token (patient) gets as current characteris tic the
results of the corresponding investigation. Thus on the basis
of the total
(at the moment)
the basis of all existing
information about the patient
(i.e. on
characteristics of the tokens, generated by
the initial token (patient)),
a decision for the subsequent direction
of the tokens is determined (i.e. the subsequent patient's
investiga
tions). When the transfer of more tokens (patients) we can optimize
in the net occurs,
their transfer with the aim of achieving minimum wai
ting-time in queues. Knowing the probable distributions
connected with
the tranfer
of the patients in different offices, on the basis of the total GN-model, the process can be simulated. pital units can be determined
Thus, different parameters of hos
(e. g. workload of offices, average time
for servicing a single patient and of stay of a patient in a hospital
a single office,
unit etc. ).
average time-
This information
can be
used for optimal organization of different medical units. Note, that direct use
of our model is possible
(in this form)
in specialized (nephrological) units because only the processes of the nephrological disease diagnosis are described in it. If we want to ap ply similar models for other medical purposes
(cardiology, neurology,
obstetrics, laboratory investigations and their interpretation, etc. >, we must include in it GN-models of the corresponding case. Some Bulga rian physicians are working on this problem There already exist more expert and/or informative medical sys tems to help physicians in diagnosis and therapeutics. Such expert and informative systems may be included in the framework of the described total GN-model or in some of its parts (subnets).
For example, expert
and/or informative systems for diagnostics can be used frames of the global GN's memory
(see App. 1)
(e. g. , in the
for calculation of the
226
CHAPTER 4
truth-values of some
transition condition predicates and for calcula-
tion of some toKen's characteristics. On the other hand, expert and informative systems for therapeutics may be included in the described GN-model, when the final token's characteristics are to be determined. Below we describe other GN-models of and symptoms
basic nephrological signs
but these models have an independent sense because their
entire inclusion in the GN
E deprives them of
the clearness which is
easyly attainable by the GN-tools. This paper is based on [22-30],
. § 4. 1. i Permanent
»
«
*
proteinwla
Any permanent proteinuria in adults is
a pathological situati-
on. Proteinuria may be detected in different conditions. Usually it is asymptomatic, but the practitioner's first step must always be a careful physical examination of the patient - not only
the urinary tract,
but the cardiovascular system (including arterial blood pressure, eyeground) and other systems as well. The tokens (patients) get into place racteristic
"permanent proteinuria".
1 l
Next they
with the initial chasplit and get in the
sections with initial places 1 , 1 and 1 2 3 4-
so that each gets
propriate
the complete
characteristic:
"the result of
the ap-
urinalysis",
"plasma creatinine's level (or gromerular filtration rate)" and "X-ray examinations of the urinary tract". In the last case, the token leaves this GN and gets in the GN and r
2
have the forms:
from § 4. 1. 2. The transition conditions r 1
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
r 1
= 1 | 1 I
1 2
1 3
true
true
1 5 rr 22
= = 11 II 22 II
The token in place
1
1 2
1 4 true
1
1 7
6
ttrue rue
ttrue rue
227
ttrue rue
8 true true
splits into four tokens which get
into
places 1 , 1 , 1 and 1 according to their characteristics: amount of 5 6 7 8 24-hours proteinuria, presence of glucosuria, microscopic investigati on of urine sediment, urocultures. The transition condition
rr 33
11 99
11
- 11 II W W 33 || 11
W W
10 10 2 2
where W
= "finds out renal insufficiency", 1
W 2
= ~\V . i
The token leaves this GN also and gets into the GN from § 4. 1. 4. 11 11 11 rr = = 11 II W W 44 55 | | 33
11 12 12 W W 4 4
where - "proteinuria more that 3 gr, per 24 hours",
W 3
w - -w . 4
3 11 113 3 rr 55
where
= = 11 II W W 66 II 55
11 14 14 W W
6
228
CHAPTER 4
W = "there is glucosuria" 5 (the token leaves this GN - the patient is directed to diabetologist>,
w =
5
6
r = 1 1 6 7I The token in place places 1 , 1 15 16 results
of
and 1 17
1 7
1 15
1 16
true
true
1 17 true
splits into three tokens which get into
with the corresponding characteristics:
the microscopic examination of
urine sediment
the
for white
blood cells, red blood cells, bacteria The transition conditions r , r 7 8 1 18 r
7
- 1 I W 8 1 7
r
have the forms 13
1 19 W 8
wher W = "urine cultures are positive" 7 (the token leaves
this GN and
the patient is
directed toward the GN
described in § 4. 1. 3), W = -W ; 8 7 1 20 r = 1 8 H
| W I 9
1 21 W
10
where W = "there is a paraproteinuria" 9 (the token leaves the GN - the patient is directed to hematologist), W
10
: nW ; 9
APPLICATIONS OF GENERALIZED NETS IN JffiDICINE
229
230
CHAPTER 4
1 22
1
r = 1 I W 9 12 I 11
W
23 12
where W
= "the proteinuria is combined with hematuria" 11
(the token goes to the GN from § 4. 1. 2); W 12
= nW ; 11 1 24 24 r 10 10
= 1 I W 15 15 I I 13 13
1 25 25 W 14 14
where W
= "leucocyturia is present"; 13
W 14
= nW ; 13 1 26 26 r 11 11
= 1 I W 16 I 15 16 I 15
1 27 27 W 16 16
where W
= "hematuria is present"; 15 = IV
W
16
15
r 12 12
1 28
1 29
= 1 | W 17 17 I I 17 17
W 18 18
where W
= "bacteriuria is present"; 17
W 18
= nW ; 17
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
231
1 30
r
= 13
If the token
1 I 21 I
true
1 I 23 I
true
1 I 14 I I| 1 I 25 I
true true
1 I 27 I
true
1 I 29 I
true
1 I 19 I
true
1 I 10 |
true
gets into place 1 or 1 , it enters the GN from 2428
§ 4. 1. 3 and from place 1 26
to the GN from § 4. I. 2.
Acute nephritic syndrome takes shape if 1 , 26
in places
1 , 1 5 9
the tokens arising from an initial one are gathering
and
(often the
hematuria is macroscopic, the renal insufficiency is transitory, there are edema and arterial hypertension).
The tokens
in places
1 , 1 , 10 14
1 ,1 ,1 ,1 ,1 and 1 transit in place 1 - the patient 19 21 23 25 27 29 30 directed to a nephrologist,
is
where complementary investigations are to
be undertaken (umunological, renal biopsy, etc. ). K K
»
§ 4. i. 2 Hematuria The
presence of blood in urine (hematuria) is a conmon symptom
in nephro-urology. The causal diagnosis sometimes is very complicated.
232
CHAPTER 4
Hematuria may
be isolated
or combined with other signs and symptoms.
Thus a complete examination
of the patient is always indispensable.
The token gets into place 1 with the initial
characteristic -
1 presence of hematuria,
which is confirmed by a microscopic investiga
tion of the urine sediment. The transition condition 1 2 r
= 1 1
1 2
I W
W
1 1 1
2
where W
= "there is a macroscopic (gross) hematuria"; 1
W 2
= nW . 1 The token in place 1 2
splits and gets into places
with the appropriate characteristics:
results of the test
glases and X-ray examinations of the urinary tract,
1 4
and
1 5
with three
because the tran
sition condition is 1 4 r = 1 | true 2 2I Place 1 6 stration
1 5
1 6
true
true .
is included in the framework of this
GN
as an illu-
of one of the methods for constructing a general GN.
presents the relation between the GN from § 4. 1. 1 and
It re
the net presen
ted here. In the framework of the general GN, tokens will transfer he re from place 1 of the GN from § 4. 1. 1 and this corresponds to trans4 ferring from one net to another. 1 9 9 r = 1 | I W 4 4 1 3 4 4 1 3
1 10 10
1
W
W w 4 4
11 11 5 5
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
233
234-
CHAPTER 4
where W
= "the presence of initial hematuria"; 3
W
= "the presence of terminal hematuria"; 4 - "the presence of total hematuria".
W 5
When the token gets into places 1 9 directed to a urologist
r
= 5
(1
and/or
1
the patient is 10
). 19 1 12 12
1
1 5
I true 1
W
1 6
I true 1
W
1
13 13 6
14 14 W 7 W
6
7
1 I true 7 |
W 6
W 7
1 l true 15 I
W 6
W 7
where W
= "there are no contraindications for intravenous urography"; 6
W 7
= ~W . 6
Thus this examination is performed, Transition conditions
r , r , r , r ,r ,r ,r 7 a 9 10 12 14 15
have the forms 1 19 r
= 7
1 | true ; 9 I 1 I true 10 I
and
r 16
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
11 21 21
235
11 20 20
11 II W W 12 II 99 12
W W
rr = - 11 II W W 88 113 3 II 99
W W
11 II W W 114 4 II 99
W W
10 10 10 10
10 10
where W
= "there are
no urographically
detectable changes or
the results
9 are uncertain"; = IV
W
10
9
(in this case the patient is directed to a urologist); 11 222 2
11 23 23
rr = = 11 II W W 99 220 0 II 11 11
W W 12 12
where - "there are no cystoscopic changes
W
or the diagnosis
is uncerta-
11 in";
. -m ;
w 12
11
rr 110 0
11 224 4
11 25 25
11 26 26
= = 11 II W W 222 2 II 113 3
W W 114 4
W W
11 II W W 13 333 3 II 13
W W 1144
W W
15 15 15 15
where W
& IV
= 1W 13
14
15
(the token gets
into place 1 - the patient 24
is left under
lance); W = " t h e r e i s a n e e d f o r complementary X-ray 14 W = "renal biopsy i s 15
needed"
examinations";
surveil-
236
CHAPTER 4
(the token in place 1 usually gets 26 agnosis); 1 29 29 r
= 1 I W 25 I 16 25 I 16
12 12
as characteristic a definite di-
1 30 30
1 31 31
W
W 18 18
17 17
where - "an extended examination of the
W
renal parenchyma is needed";
16 W
= "an examination of the bladder is needed"; 17
W
= "an extended examination of the
vessels is needed" ;
18
r 14 14
1 34 34
1 35 35
1 36 36
= 1 I W 29 19 29 I I 19
W 20 20
W 21 21
where W
= "an arteriography is required"; 19
W
= "an sonography is required"; 20
W
= "a scanning is required"; 21
r 15 15
1 37 37
1 38 38
1 39 39
= 1 I W 30 I 22 30 I 22
W 23 23
W 24 24
where W
= "a retrograde cystoscopy is necessary"; 22
W
= "a retrograde ureteropyelography is necessary"; 23
W
= "an anterograde pyelography is necessary"; 24
r 16
1 40
1 41
= 1 | W 31 I 25
W 26
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
237
where W
= "an investigation of the renal arterial system is indispensable" 25
(aortography, selective renovasography), - "an investigation of the renal venous system is indispensable"
W 26
(cavography, renal phiebography). The section of the GN described above ends by a transition with condition 1 42
1
| 11 I
W
W
1 I 34 I
W
1
I 35 I
W
I 36 I I| 1 | 37 I
W
1
1
r
= 17
1
W
( 39 I
W
1 I 40 I
W
1 41
W
1
I
27
28
27
W 28
27
W 28
27
W 28
W
I 38 |
43
W 27
28
27
W 28 W
27
28 W
27
28 W
27
28
where W
= "urologic cause of hematuria is found" 27
(the patient is directed to a urologist - 1 ); 44 W
= "a nephrologic cause of a hematuria is found". 28
(the patient is directed to nephrologist - 1 ). 45
238
CHAPTER 4
Transition conditions r and r have the forms 9 13 1 27 r = 1 I W 9 23 I 29
1 28 W 30
where W
= "bleeding from the lower urinary tract is discovered" 29
(and the token gets into place 1 ); 44 W
= "bleeding from the upper urinary tract is discovered"; 30 1 32 r
1 33
= 1 I W 13 28 I 31
W 32
where W
= "the bleeding is unilateral", 31
(and the token gets into place 1 ), 44
w
= nw
32
31
(the toKen gets into place 1 and the condition r is verified). 33 10 Transition conditions r 3
and r have the forms 6 1 7 7
r 3
1 8 8
= 1 | W 3 I 33
W 34
where W
= "presence of hematuria combined with proteinuria
that cannot be
33 explained by the presence of blood in urine only"; - "presence of isolated hematuria";
W 34
1 1 1 15 16 r = 1 | W 6 8 I 35
W
1 17 W
36
18 W
37
38
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
239
where - "X-ray examination is needed"
W 35
(and a token's transfer to a 5-th transition follows); W
= "there are definite signs of nephropathy" 36
(the patient is directed to a nephrologist - 1 ); 45
W = "investigations of the concomitant proteinuria i s needed" 37 (the token leaves this GN and gets in a GN from § 4. 1. 1), W
= "investigation of the concomitant arterial hypertension is need38 ed"
(the token leaves this GN and gets into place L
of the basic GN E and 5
from there it goes to some of the GNs from § 4. 1. 7, § 4. 1. 8 or
§ 4. 1.
9). K K
K
§ 4. I. 3 Turbid urine Here some advantages of GNs and their applications
in the ini
tial steps of identification and investigation of a turbid
urine sam
ple will be illustrated Tokens (patients) enter place 1
with the initial characteris-
1 tic
"complains of
voiding turbid urine".
The tokens'
place 1* to place 1" shall be denoted by V -> 1". The transition condition of the GN are 1 12 2 r 1 I true r - 1 I true ; ; 1 1 1 1I I
transfer from
240
CHAPTER 4
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
1
1 2
r
= 1 2
3
I W 2 1 1
W 2
where
W
= "acidification of the urine specimen is performed"; 1
W
= "investigation of urine with test strip"; 2 11 55 rr 33
11 6
= = 11 II W W 33 11 33
W W 4 4
where W
= "the turbidity of urine disapears"; 3
W 4
= IV ; 3 11 77 rr = = 11 4 4
4
11 8
II W W 1 5
W W 6
4 1 5
6
where W = "there a r e no changes i n t h e t e s t 5
strip";
w = -m ; 6
5 1
1 9
1 5
I I
W 7
1 r
= 5
26 W 8
| true 6 1 I 1 I W 7 | 9
false
1
false
| true 8 I
W 10
where W - "the t r a n s f e r 7
1
-> 1 4
i s a c c o m p l i s h e d "; 8
24-1
242
W 8 W
CHAPTER 4
= nW ; 7 = "the transfer 1
9
3
-> 1 is accomplished "; 6
- -\V ;
W
10
9 1 lO r 6
1 11
1 12
= 1 I W W W 9 I 11 12 13
where - "bacteriological investigation of urine is undertaken";
W 11 W
= "microscopic
urine sediment
examination is carried out";
12 - "macroscopic examination of urine is accomplished";
W 13
1 1 13 14 13 14 r = 1 I W W 7 10 14 15 7 10 I I 14 15 where W
= "there is bacteriuria"; 14 - iW
W
15
;
14 1 15
1 16
r = 1 I W S 8 11 I 16
W 17
where - "there is leucocyturia";
W 16 W 17
= iW ; 16 1 17
1 18
r = 1 I W W 9 12 18 19 9 12 I I 18 19 where
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
- "there are urethral filaments in the urine specimen",
W 18
- -W ;
W 19
IS 1 9
1 26
1 r
= 10
I W false 15 I 20 I| 1 I false W 16 I 21
1
I W 17 I 20 17 I 20
W 21 21
where W
= "the transfer 1 20
W
-> 1 11
= nW 21
is accomplished"; 15
; 20 1 9 r
= 11
1 I true 20 I
;
1 I true 21 I
r 12
1 25
1
= 1 I W 24 I 22
W
31 23
where W
= "there is no prostatic hypertrophy"; 22
W
= iW 23
; 22 1 19 1
| true 13 I r = I 13 1 | true 14 I 1 I true 25 I
;
243
244
CHAPTER 4
1 27 r
= 14
1 I true 26 I ; 1 | true IS I 1 22
r 15 15
1 23
= 1 I W W 19 19 I I 2424- 25 25
where W
= "there
are no
X-ray
alternations
in
the kidney
and urinary
24 tract"; W
= nW ; 25 24
r 16 16
1 ] 29
1 30
= 1 I W 23 23 I I 26 26
W 27 27
where W
= "X-ray alternations in the urinary tract are demonstrated" 26 & -W
;
27 W
= "X-ray alternations in the Kidney are present". 27 The transfer 1 -> 1 means that it is indispensable to inves1 2
tigate a fresh urine specimen.
The transfer 1 -> 1 means that the 5 26
turbidity of urine disapears and thus the investigation is discontinu ed. The transfers 1 - > 1 , 1 - > 1 and 1 -> 1 impose an extension 5 9 7 9 8 9 of investigations. The transfers 1 15
-> 1 20
and 1 17
taneously performed, which means that leucocyturia ments are found. The transfers also
-> 1
are simil-
20 and urethral fila
1 -> 1 and 1 -> 1 , which are 16 21 17 21 simultaneously performed, signify the absent leucocyturia, but
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
245
there are urethral filaments. The token in place 1 takes ascharacte24 ristic the result of the investigation of the prostatic gland, place
1 , the characteristics 19
of the results from
and in
the intravenouse
urography. The transfers 1 -> 1 and 1 -> 1 mean the end of the 26 27 18 27 investigation; 1 -> 1 means that the patient is directed to a spe23 29 cialised nepbrologic unit; 1 -> 1 30 32 patient is
and
1 -> 1 means that the 31 32
directed to urological unit; 1 -> 1 means that the pa22 28
tient remains under observation after appropriate therapy.
• § 4. 1.4 Elevated blood urea Estimation of importance
»
nitrogen
the Glomerular Filtration Rate (GFR) is of great
both in the initial evaluation and in subsequent
low-up of patients with renal disease.
the fol
Measurement of blood urea nit
rogen (BUN} is probably the most commonly used, albeit the least accu rate, clinical method for estimating GFR. The serum creatinine concen tration provides a more reliable estimate of GFR. In place 1 we have a token (patient) with the initial charac1 teristic
"elevated BON". Next it splits and gets in places 1 , 1 and 2 3
1 , as the transition condition r has the form 4 1 1 2 - 1
r 1 Each
place
namics
I true 1 I
1 3 true
1 4 true
gets an appropriate characteristics
investigation"
(1 ); "state 2
"results of barody
of bydratation - evaluation
of
24-6
CHAPTER 4
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
247
extracellular volume" (1 ), and "urinalysis" (1 ). 3 4 The transition condition 1 5 5 r 2
1 6 6
= 1 I W 2 1 1
W 2
where W
= "there is a reduction in effective arterial blood volume"; 1
W 2
= IV . 1 The t r a n s i t i o n c o n d i t i o n
r r
1 7
11 88
= 11 II W W 3 33 11 33
W W 44
1 9 W W , 5
where W
= "there are symptoms of hyperhydratation"; 3
W = "normal findings"; 4 W
= "dehydration is present". 5 The transition condition
r 4 4
1 10
1 11
1
= 1 I true 4 4 I I
true
true
1 I true 18 |
true
true
12
From place 1 the token splits and gets into places 1 (deter4 10 mination of urea in the urine), protein), 1 (examination 12
1 11
(evaluation for
the presence of
of urine sediment for broad waxy casts).
The transition conditions
r , 5
r
and 6
r
have the following 7
248
CHAPTER 4
forms: 1 13 13 r
= 1 I W 5 10 I 6
1 14 14 W 7
where W
- "there is an increase of urea excretion in urine"; 6
W
= "there is a decrease of urine urea" ; 7 1 15 15 r = 1 I W 11 I 8 6
1 16 16 W 9
where - "proteinuria is present" (see § 4. 1. 3);
W
a w = nw . 9
8 1 17 17
1 ia ia
r = 1 I W 7 12 10 7 12 I I 10
W 11 11
where W
= "broad casts are present"; lO = lW
W
11
10
and this investigation may be repeated (see § 4. 1. 3). Ihe tokens from
places 1 , 1 and 1 5
9
get into place 1 13
(the-
19
re are signs of functional, reversible renal failure). The transition conditions
r
and r 8
have the following forms; 9
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
249
1 19 1 6 r
I I I | I
= 8
1 9 1
I 13 I
true true true
1 20 r
= 1 9
I W 19 I 12
1 21 W 13
where W
= "BUN is normalized after an appropriate therapy"; 12
.
= IV
W 13
12 The tokens from places 1 , 1 , 1 , 1 ,1 ,1 6 7 8 1415 17
and
1
get 21
into place 1 because organic renal failure is suspected. 22 The transition conditions r
and r lO
have the following forms: 11
1 22 1 5
r
I | I
true
1 I 21 I
true
1 7
true
= 10
1 8 1
l I I I I
true
| 14 I
true
1 | 15 I
true
1
true
| 17 I
250
CHAPTER 4
1 23 r
1 1 2* 24 25
= 1 I true 22 I
11
true
true
Tokens from place 1 mist be submitted to further investigati22 ons: blood (1 ), ultrasonographic (1 ), and radiologic (1 ) evalua23 24 25 tions of kidneys. The transition conditions r
, r , r and r have the follo12 13 14 15
wing forms'.
r 12
1 26 26
1 27 27
= 1 I true 23 I
true
1 1 1 32 33 32 33 r 13 13
= 1 I W 24 I 14 24 I 14
W
1 I W 25 I 14 25 I 14
W
1 34 34
35 35
16 16
W 17 17
16 16
W 17 17
W 15 15 W 15 15
where W
= "the size of both kidneys is normal"; 14 - "presence of bilateral small kidneys";
W 15 W
= "the kidneys are enlarged"; 16
W
= "there is enlargement of the pyelocalicial system"; 17
and
urologist must be consulted because obstructive
pected;
r 14 where W
= "anaemia is present"; 18
1 28
1 29
= 1 I W 26 I 18 Id
W 19
uropathy is sus
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
w
- -m
19
251
;
18 1 30 r 15
1 31
= 1 I W 20 27 I
W 21
where W
= "there is hypocalcemia"; 20
w
- -m ;
21
20 1 36 1 I 28 I r
=
1
true
1 30 I
true
1 I 33 I
true
1 I 34- II 34
true
16
1 37 1 I 29 I r
=
1
1 31 I
true
1 | to I i 32
true
17
Tokens from places 1 , 1 , 26 30 with the
true
1
, and 1 33
get into place 34-
characteristics of chronic Kidney failure.
1 and 1 31 32
the tokens get into place 1 37
1 36
From places 1 , 29
with the characteristic of
acute kidney failure. The above constructed GN describes the a patient presents high level of BUN. K K
It
diagnostic process when
252
CHAPTER 4
§ 4. i. 5 Sena]
colic
The diagnosis of renal
colic is modelled in
the framework of
this GN. The tokens (patients) get into place 1 with the initial chai racteristic "symptoms of renal colic". Then they split up and get into the sections with initial places 1 and 1 and the appropriate charac2 3 teristics - investigation symptomatic
treatment
of
urine specimen
(the subside
and the
of renal colic),
result of
the
because of the
transition condition 1 2 r =1 1 1 1 1 Depending on the values of
1
true
3
true
these characteristics they
continue their
way in the GN. The transition :ondition
r 2
1 4
1 5
= 1 1 2 I
true
true
true .
1 I 31 1
true
true
true
The tokens in places 1 and 1 split 2 31
1
6
and
subsequently
places 1 , 1 and 1 , with the corresponding 4 5 6 the
their
characteristics:
They advance further depending on the values of the transition
conditions r , r , r and r : 3 4 5 6
r -- 1 1 3 41
1 7
1
W
W 2
1
8
1
1 9 W
3
lO W
4
where 1
into
results of macroscopic, test-strip and cytobacteriologic examina-
tions.
W
get
= "the urine is without any microscopic change";
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
253
254-
W
CHAPTER 4
= "there are heiraturia and/or blood coagulums" 2
(the token enters the GN from § 4. 1. 2); W
= "the urine sample is turbid" 3
(the token enters the GN from § 4. 1. 3); W
: "a calculus is emitted"; 4
1 1 11 r = 1 I true 4 10 10 I 1 1 1 12 13
r
= 1 I W 5 5 1 5
W
1 14
W 6
15 W
7
8
where z "there are no changes";
W 5 W
= "there is proteinuria" 6
(the token enters the GN from § 4. 1. 1); - "there is blood in urine"
W 7
(the token enters the GN from § 4. 1. 2); W
= "nitrites are present";
a
1 116 16 1 I true 7 I r = 1 I true 6 12 I 1 I true 5 I The transfer of tokens into place 1
means that a cytobacteri-
16 ological
examination is performed,
of the urine's sediment (1
which is a parallel investigation
) and microbiology (1 ) because 17 18
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
1 17 17 r 7 7
= 1 | true 16 16 I I
255
1 18 18 true
The transition conditions r , r and r have the forms 8 9 10 1
1 19
r = 1 | W 8 17 I 9
1
1 21
20 W
W lO
1 22 W
11
23 W
12
13
where W
= "the urine sediment is normal"; 9
W
= "there is heroaturia" 10
(the token enters the GN from § 4. 1. 2); W
= "there is leucocyturia" 11
(the token enters the GN from § 4. 1. 3); W
= "there is crystaluria"; 12
W
= "there is bacteriuria"; 13 1 24 24r = 1 r I t true rue 9 9 2 23 3I
r r
1 1 25 2 5
1 26
I W 24 I 14
W 15
= 1
10
;
where W
= "there is no bacteriuria"; 14
- -m .
w 15
14 The token's characteristic in place
infection is present" and in place 1 , 26
1 25
is
"appropriate
"no urinary tract therapy must be
256
CHAPTER 4
undertaken". The tokens in
places
1 and 3
1 split and get 22
on parallel
ways. The transition conditions r
, r 11
, r and r have the forms 12 13 14
1 1 1 27 28 r
1 29
30
= 1 | true 11 3 I
true
true
true
1 I true 19 I
true
true
true
r 12 12
1 31
1 32
= 1 I W 27 I 16 27 I 16
W 17 17
where W
= "the transfer 16
W 17
1 -> 1 is previously acconplished"; 3 27
= iW ; 16
r 13
1 33
1 34
= 1 I W 28 I 18
W 19
where W
= "there is no decrease in diuresis"; 18
W 19
= iW 18
(the token enters the GN from § 4-. 1.4);
r 14
1 35
1 36
= 1 I W 29 I 20
W 21
where W
= "there is no abnormal increase in body temperature"; 18
w
= nw . 21
20
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
257
The transition condition
r 15
1 37
1 38
1 39
1 I true 22 I
true
true
= 1 I true 11 I
true
true
1 | true 32 32 I I
true
true
1 I true 30 I
true
true
Therefore, when a token gets into place 1 , 1 , 1 or 1 it 11 22 30 32 splits into three. The transition conditions
r
, r 16 17
the forms:
r 16 16
1 40
1 41
= 1 I W 37 37 I I 22 22
W 23 23
where W
- "the renal function is normal"; 22
w
- -m
23 22 (the token enters the GN from § 4. 1.4);
r 17 17
1 42
1 43
= 1 I W 38 38 I I 24 24
W 25 25
where W
= "the X-ray of the urinary tract is normal"; 24
W = nW ; 25 24
r 18 18 where
1 45
1 46
= 1 I W 39 39 I I 26 26
W 27 27
r 20
have
258
W
CHAPTER 4
= "the metabolic functional tests are normal"; 26 - iV
W
27
;
26 1 44 r 19
- 1 ; I true 43 I 1 47
r 20 When
tokens
= 1 I true 46 I
get into places 1 , 1 , 1 ,1 ,1 ,1 ,1 , 8 9 13 14 20 21 34
1 and 1 , they leave this GN and get to the GN from § 4. 1.4. 36 41 In places 1 , 1 , 1 , 1 and 1 they end their movement in 25 33 35 42 45 the net with the characteristic "normal findings", ces 1 , 1 and 1 end with the characteristic 26 44 47
and tokens in pla "directing towards
appropriate treatment". x it
§ 4. 1. 6
*
Dysui-ia In normal
the ability
micturition cycle,
the continence
phase depends on
of the bladder adapt itself to urine volume
and maintain
pressure in urethra. During micturition, the bladder contraction asso ciated with sphincter relaxation empties the bladder. difficulty or pain associated with voiding.
Dysuria denotes
It may result from a wide
variety of pathological conditions and it is generally due to a cervico-urethro-prostatic obstacle. When this it is not the case one should investigate for a functional obstruction of the striated sphincter and the cervix, or a bladder hypocontractility induced by denervation, in hibition or muscular insufficiency.
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
The tokens (patients) get into place 1 with 1 racteristic "dysuria". Next they split up and get and
1 4
259
the initial cha-
into places
so that in each place they get the characteristics:
1 ,1 2 3 "results
from the case history" (1 ), "general examination" (1 ) or "complemen2 3 tary laboratory investigations" (1 ), because 4 1 2 2 r 11
= 1 I true 11 I I
1 3 3
1 4. 4
true
true
The transition conditions r , r and r have the forms 2 5 4 11 55
rr
= = 22
11 66
i1 iI w W 2 | | 11 2
w W 22
11 33
W W
II W W 11 11 II 11 II ffalse alse 17 II 17
11 II ffalse alse 55 55 II
2 true true true true
where W
= "there is a certain disease that may cause dysuria"; 1
w = -m ; 2
1 1 1 9 9 rr 55
= 11 II W W = 66 11 33
11 11 11 110 0
W w 44
W W 5
where W
= "consultation with a specialist is needed"; 3
W
= "consultation with a neurologist is needed"; 4
260
CHAPTER 4
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
W
261
= "urodynamic investigation is needed"', 5 1 12 r = 1 | true 4 5 I 1 I true 9 I From places 1 and 1 5 9
following characteristic:
the tokens get
into place 1 with the 12
"a specialist's opinion is required"
urologist, gynecologist, etc. ). From there the
(e.g.
token gets back to 1 , 6
passing through place 1 or to 1 according to the condition 17 16 1 16
1 17
r = 1 I W 8 12 I 6
W 7
where W
= "a definite cause of dysuria is found", 6
w = nw . 7
6 The transition condition 1 13 r 9 9
= 1 I W 11 11 I I 8 8
1 1 14 15 W
W 9 9
10 10
where W
= "the urodynamic tests are normal"; 8
W
= "the urodynamic tests are pathologic"; 9
W
= "the urodynamic tests are inconclusive". lO When tokens reach place 1 , 15
they leave
the net with the cha-
racteristic "further medical observation is needed".
262
CHAPTER 4
The transition condition 1 IS 18 r 10
= 1 I lO I
1 19
W
W 12
11
1 I false 13 I
true
where - "a neurological condition
W
that explains
the micturition disor-
11 ders is present";
- -m .
w 12
11 From place 1 (through place 1«
1 ), 35
the token
leaves the net
with the final characteristic "the micturition disorder is symptomatic of
a neurological condition".
Depending on the need
consultation (the transition r
of
psychiatric
has a form: 15
r 15 15
1 20 20
1 21 21
= 1 I W 19 I 13 19 I 13
W 14 14
where W
= "the patient must be consulted by a psychiatrist"; 13
W
= nW 14
13
). The tokens get into place 1 or 1 . From place 1 , the 20 21 20
token gets into place 1 or 1 (the transition r has a form 22 23 10
r lfl 18
1 22
1 23
= 1 | W 20 | 15
W 16
where W
= "the micturition disorder is of functional or anorganic origin"; 15
w
= nw ). 16
15
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
263
The transition condition 1
1 24
r
=
l | 14 I
16
W 17
1 25
W
1
1 27
1 28
29
W
W 20
W 21
22
26 W
18
19
where
W
= "vesical hypercontractility is present"; 17
W
= "a clearcut
pathologic
vesico-urethral dysfunction is manifes-
18 ted"; W
= "vesico-sphincteric dyssynergism is present"; 19
W
r "cervical obstruction is detected"; 20
W
= "there is vesical hypocontractility"; 21
W
= " other findings are present". 22 The transition conition 11 330 0 rr 114 4
= = 11 II W W 16 II 223 16 3
11 31 31
11 32 32
W W W W 25 224 4 25
where W
= "there is a definite diagnosis"; 23
W
= "there are uro-gynecological disorders"; 24
w 25
= nw . 24 Thus, from place 1 the token gets into places 1 , or 16 31
or 1 depending on 30
the characteristics it has acquired
(there is or is not uro-gynecological disorder, or another
1 , 32
in place 1 »2 disease is
present). When a token splits and engenders tokens in places
1 and 2
1, 3
264
CHAPTER 4
and when at a given moment these newly engendered tokens get into pla ces 1
, 1
31
and 1
21
(i. e. there is an accumulation of data for a gi24
ven patient from different channels), these three tokens fuse into one in place
1
with the characteristic
"micturition disorders of uro-
33 gynecological origin", because of the transition condition 1 33 1 I 21 I r = I 19 1 I 31 I 1 I 24 I
true true true
The transition condition 1 34
-=
r 21
1 I 32 I
true
1 I 23 | I 1 I 24 I
true
1 I 25 I
true
true
Therefore, when tokens engendered by an initial one simultaneo usly get into places 1 , 1 , 1 32 in place 1 34
with
23
and 1 25
,
they fuse into one
token
26
the characteristic "micturition disorders of
the
isolated neurogenic bladder". From 1 the token leaves the net with the characteristic "non29 dysuric disease". The transition condition
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
265
1 35 1 I 30 I
true
1 I 10 18 I|
true
1 I 22 I r = | 25 1 I 33 I
true
1
Therefore,
true
I 34 I
true
1 I 27 I
true
1 I 28 I
true
1 I 54 I
true
from each of the places
and 1 , the token gets into 1 54 35
1 ,1 ,1 ,1 ,1 ,1 ,1 30 18 22 33 34 27 28
which symbolizes
the performance of
an appropriate treatment. The transition condition 1 7 r 3
- 1 l 4 I
true
1 8 true
The token from 1 splits into two new tokens which get into 1 (inves4 7 tigation for hyperazotemia) and 1
(urinalysis).
8 The transition condition
r 6
= 1 I 7 I
1 136 36 W 26
1 1 37 37 W 27
where W
= "elevated blood urea nitrogen is present"; 26
266
w
CHAPTER 4
= nw 27
26 From 1 the token gets into place 1 or 1 according 7 36 37
dition r 7
to con-
(hyperazotemia is present or is not). In the first case the
token leaves the net and gets into the GN from § 4. 1. 4. From 1 , the token splits into three which 0 to places: 1 ("two glasses" test), 38 ons, including for b. Koch) and 1 40
1 39
(bacteriologic investigati-
(micturition flow), because
1 38
1 39
1 40
r = - 1 I true 7 7 I
true
true
The transition conditions r
, r 11
r 11 11
get correspondingly
and r 12
= 1 | 38 38 I I
have the form 13
1 41
1 42
W 28 28
W 29 29
where W
= "turbid urine is found"; 28
= -m
w 29
28
1 1 43 r 12
= 1 I 39 I
W 30
1 1 44 W 31
where W
= "a positive bacteriologic finding is present"; 30
w 31
= nw 30
APPLICATIONS OF GENERALIZED NETS IN HEDICINE
1 45 45 r 13 13
s 1 | 40 I 40 I
W 32 32
267
1 46 46 W 33 33
where W
= "there is an increased urine flow"; 32 = iV
W 33
32 The increase of urine flow means polyuria, so the token in pla
ce 1 leaves the net with this characteristic. 45 From places 1 , 1 , 1 and 1 the tokens get into place 1 37 42 44 46 48 (combined excretory and mictional urography) because 1 48 1 | true 37 I r = 1 I true 20 42 I 1 I true 44 I 1 I true 46 I The transition conditions r , r , r and r have the form 23 24 25 22 1 49 49 r = 1 I W 22 48 22 48 I I 34 34
1 50 50
1 51 51
W
W 36 36
35 35
where W
= "the X-ray picture reveals distinct pathological finding", 34 - IV
W
35 W
,
34 = "the X-ray report is in conclusive".
36
268
CHAFFER 4
r
= 1 I 23 51 I
1 52
1
W
W 37
53 38
where W
= "suprapubic cysto-urethrography is performed"; 3?
W
= "retrograde urethrography is performed" ; 38
r 24
1 54
1
1 I 49 I
W
W 40
= - 1 I 50 I
W
1 I 52 I
W
1 I 53 I
W
39
W 39
40
39
W 40
39
W 40
where W
= "there is a definitive diagnosis"; 39 - iW
W 40
. 39
Finally, 1 47 r
= 17
1 I 41 I
true
1 I 43 I
true
K
It
It *
55
APPLICATIONS OF GENERALIZED NETS IN IVEDICINE
§ 4. 1. 7 Arterial
269
hypertension
The purpose of the diagnostic examination of a patient
is with
hypertension to determine both the severity of the disease and whether the hypertension is primary (essential) or secondary to another disor der. The results of this procedure
will determine the need for treat
ment as well as the type of treatment. The systemic arterial hypertension is usually astolic pressure
defined as a di-
(DBP) of 90 nnHg or higher recorded on
minations of a subject with normal salt intake
several exa
( 1 0 + 3 g/day) regard
less of age and sex. A token in place 1 means a patient 1
with arterial hypertension
(as initial characteristic). The transition conditions of the GN have the form 1 2
r 1
- 1
I W 1 1 1
1
1
1
3
4
W 2
W 3
1 5
W 4
6
W 5
where
W
= "recording the DBP under appropriate conditions"; 1
W
= "fundoscopic grading of retinal changes
(according Keith-Wagner-
2 Barker classification)"; W
= "each of left ventricular hypertrophy"; 3
W = "investigation for renal damage"; 4 r
- "evidence
of target organ damage or
presence
5 risk factor other than hypertension";
r 2 where
1 7
1 8
1
= 1 I W 2 1 6
W 7
W 8
9
of cardiovascular
2T0 270
CHAPTER 4
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
W
= "the DBP is within the limits 90-1O4 imHg"; 6
w
= "the DBP is within the limits 105-114 mnHg"; 7 = "the DBP is i 115 imHg";
W ft
1 10 10 r 3
= 1 I W 3 1 9
1 11 11
1
W
W w 11
12 12
10
where W
= "no retinal changes or such of KWB group 1"; 9
W
= "retinal changes KWB group 2"; lO
W
= "retinal changes KWB groups 3 and 4"; 11 1 13
1 1 14 15
r = 1 I W W 4| 12 13 4
W 14
where
- -m
W 12 W
& nW ;
13
14
= "paterns of left ventricular 'strain'"; 13
W
= "left ventricular hypertrophy is present"; 14 1 16 16 r 5
1 17 17
= 1 I W W 15 16 5 I
where W
= "no renal damage"; 15
W 16
= nW ; 15 1 1 1ft 18 1 19 9 r = 1 I W r 6 6 7 6 6 II 117
W 1ft 1ft
271
272
CHAPTER 4
where W
= "no evidence of other target organ damage
or other cardiovascu-
17 lar risk factors"; W
= nW 18
; 17
- 1
r 7
1 20
1
I W 10 19 10 I
W
21
20
where W
= "there is a token, related to the present token, at least in one 20 of the following places: 1 , 1 , 1 , 1 ,..., 1 , 1 "j 8 9 13 14 17 19
= -m ■,
w 19
20
r
= 1 8
1 22
1
I W 13 I 21
W
23 22
where W
= "there is a token, related to the present token, at least in one 22 of the following places: 1 , 1 , 1 8 9 = iV
W 21
,1 11
,1 ,1 "; 12 17 19
; 22
r
= 1 9
1 22
1
I W 16 23 16 I 23
W
25 24
where W
= "there is a token, related to the present token, at least in one 24 of the following places: 1 , 1 , 1 , 1 , 1 , 1 ■'; 8 9 13 14 15 19 = iV
W 23
; 24 1 26 r
= 1 I W 17 I 10 25
1 27 W 26
APPLICATIONS OF GENERALIZED NETS IN MDICTNE
273
where W
26
= "there is a token, related to the present token, at least in one
of the following places: 1 , i , i , i "; 9 12 16 19 W = iH ; 25 26 1 1 28 29 r
11
= 1 | W 27 19 1
W 28
where W
28
= "there is a token, related to the present token, at least in one
of the following places: 1 , l , 1 , 1 "; 9 12 15 17 W = nW . 27 28 The remaining transition conditions' predicates
(of the 12-th,
13-th and 14-th transitions) are only "true". K K
§ 4. i. 8 Arterial
hypertension
«
of renal
The tokens get into place 1 1
origin
with initial characteristics for-
med by three groups of data: - group X: "signs and symptoms
directing to renal origin of the arte-
rial hypertension" - group Y: "signs and symptoms directing to renovascular origin of the arterial hypertension" - group Z: primary
"there are no signs, symptoms or conditions, excluding the participation of kidneys in the
origin of the hypertensive
state". Hie transition condition
r
1 2
= 1 1 true 1 11
1
3
true
274-
(
CHAPTER 4
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
275
Thus the token splits into two, which get into place 1 (urine is sent 2 for analysis)
and place 1 3
(blood sanple is taken
for examination).
The token in place 1 splits into three new tokens that simultaneously 2 get into places 1 (biochemical examination of urine), 1 (urinary se4 5 diment examination) and sition condition r
1 6
(bacteriologic examination) because tran-
has the form 2 1 4
1 5
r = 1 I true 2 2 I i 2 2
1 6
true
true
The transition conditions r and r have similar form 4 5 1 10 r = 1 I true 4 4 I 1 1
12 12
r
= 1 I true 5 5I
when the token gets into places 1
true ;
1 1 1
1
13 13
true
14 14 true
, . . . , 1
10 taneously for proteinuria,
1 11
,
if means testing simul-
14
glucosuria,
hematuria,
leucocyturia
and
bacteriuria. The transition conditions r , r , r 6 lO 11
r = 1 I 6 6 1
1 15
1 16
W
W 1
where W
= "there is positive urine culture", 1
2
r
have the form 14
276
W
CHAPTER 4
= nW ; i
2
1 ) 24 24 r 10
= 1 I W 3 10 I
1 25 25 W 4
where W
= "there is protenuria"; 3
W
-- nW ; 3
4
1 26 r
= 1 I W 11 10 I 5 11 10 I 5
1 27 W 6 6
where - "diabetes mellitus is suspected";
W 5 W 6
= lW ; 5 1 28 28 r 12 12
= 1 I W 12 I 7 12 I 7
1 29 29 W 8 S
where W
= "there is hematuria"; 7
w = nw ; a
7 1 l 30 30 r 13
= 1 | W 13 I 9
l 1 31 31 W 10
where W
= "there are bacteria in urine sediment"; 9
W 10
= nW ; 9
APPLICATIONS OF GENERALIZED NETS IN MBICTNE
1 32 r
= 1 | W 14 1414 I 11
277
1 33 W 12
where W
= "there is bacteriuria"; 11
W 12
= iW ; 11 The transition condition
r
= 17
1 36
1
1 I 25 I 25 I
W 13 13
W 14 14
1 I 27 I | 1 | 29 I
W 13
W 14
W
1 I 31 31 I I
W
1 I 33 I 33 I
W
1 16 16 | |
W
37
W 13
14
13 13
W 14 14
13 13
W 14 14
13 13
W 14 14
where W
= "glomerular nephropathy is suspected"; 13
w
= nw . 14
13 The token in place 1 splits into three tokens which get into 3
places 1 (serum creatinine by urea determination), 1 (serum potassi7 8 urn), and 1 (blood sugar), because the transition condition r has the 9 3 form: 1 7 r 3
= 1 I true 3I
1 8 true
1 9 true
278
CHAPTER 4
The transition conditions r , r , r , r and r have the form 7 ft 9 10 17
r 7
1 17
1
= 1 I W 7 I 15
W
18 16
where W
= "there
is increase in serum creatinine level";
15
w
= nw 16
15 1 19 19 r = 1 I W ft 17 ft ft ft I I 17
1 20 20
1 21 21
W
W 19 19
18 18
where W
= "hyperpotassemia is present"; 17
W
= "nomaopotassemia is present"; 18 - "hypopotassemia is present";
W 19
r 9 9
1 22
1
= 1 | W 9 20 9 I I 20
W
23 21 21
where W
= "there is hyperglycemia"; 20
W 21
= iW ; 20 1 34r
= 1 | W 18 I 22 10
1 35 W 23
where W
= "there is a low glomerular filtration rate"; 22
w 23
= iw 22
;
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
279
1 39
Al1 tokens place
1 39
1 I 37 I
true
r - 1 | 17 35 I
true
1 I 21 I
true
1 I 23 I
true
pertaining to a specific patient 'when getting
into
fuse into one token with characteristics incorporating all
the previously
acquired characteristics by the tokens
the investigations carried out up to now. Place 1
resulting from
symbolizes "a test
39 for iodine sensitivity is performed". The transition conditions r 19
r 19
and r have the form 20 1 1 1 1 40 41
= 1 I W 2439 I 24
W 25
where W
= "the test for sensitivity is negative"; 24-
W 25
= nW ; 24
r 20 20
1 42
1 43
= 1 I W 40 40 I I 26 26
W 27 27
1 | false 41 I
true
where W
= "intravenous urography is performed"; 26
W
= "other 27
instrumental
investigations
of kidneys
are
necessary
280
CHAPTER 4
(isotopic, sonography, etc. )". The place
1
corresponds
to the following actions: isotopic
43 investigations (renography, scintigraphy,
gamma-camera >,
sonography,
computer tomography, and the place 1 to excretory urography. 42 The transition conditions r , r and r have the form 21 22 23 1 1 44 45 44 45 r = 1 I W 21 42 21 42 I I 28 28
W 29 29
where W 28
= nW ; 29 - "there are urographic data for renovascular hypertension";
W 29
1 46 46
1 47 47
r = 1 | W 22 43 22 43 I I 30 30
W 31 31
where W 30 W
= nW ; 31 = "there are signs of renovascular hypertension";
32 1 48 48
1 49 49
r = 1 I W 44 I 32 23
W 33
1 I W 46 I 32 46 I 32
W 33 33
where W 32 W
= nW ; 33 = "the initial
characteristics
of the token
has positive
siqns
33 from group Y". When tokens get into place
1 24
there is
a call
to
GN
from
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
§ 4. 1. 3, in place 1 28
281
- to GNs from § 4. 1. 2 and § 4. 1. 5, in places 1 15
and 1 to GNs from § 4. 1. 3 and § 4. 1. 5; in places 1 , 1 30 17 19 GN from § 4. 1.4; in place 1 - to GN from § 4. 1. 3; 32
and 1 34
in places 1 47
to and
1 - to GN for § 4. 1. 9. 49 The tokens in places 1 , 1 and 20 22
1 26
leave the GN because of
the indications for other diseases, and in places 1 and 1 they re38 48 ceive the characteristic: appropriate treatent and medical observation of the patient.
* X
§ 4. i. 9 Renovascular The tokens
*
hypertension (patients) get into place 1 from a GN from § 4. 1. 8 1
with the initial characteristics
"signs and symptoms directing to re-
novascular origin of the arterial hypertension", The transition condition r
has the form: 1
1 2 r
- 1 1 true 1 11
A token from place places
1
1 1
1
3
true
1 4 true
5 true
splits into four tokens
1 ,..., 1 . These places signify: 2 5
that get
into
1 - check-up for endocrine 2
ne disorder (primary hyperaldosteronism, pheochromocytama, hypercorticism etc. ), 1 - performing an iodine sensitivity test, 3 gation
of
plasma renin activity, 1
tests. The transition condition
1
4
- investi-
- performance of pharmacological
282
CHAPTER 4
1 6 r 2
1 7
= 1 I W 2 | 1
W 2
where W
= "an endocrine
disease as
a probable cause
of the
hypertensive
1 state is found", W 2
-- -W . 1 Place 1 is the final one for the net. 6 The transition conditions r , r , r , r , r 3 4 5 8 9
and
r 6
have the
forms
1 3
1 8«
1 9
II 11 rr = II 3 11 II 10 II 10
W W 33
W W 4
W W 3
W W 4
1 II 17 I 17 I
W W 3
W W 4
1 II 19 I 19 I
W W 3
W W 4
where W
= "pharmacological test gives negative result and it is possible to 3 perform a vasography";
w
= w 4
; 3
- "high
W
level of plasma renin activity is found";
5 W 6
= nW
; 5
1
1 12
r
= 1 5
where
I W 5 | 7
13 W 8
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
W
283
= "angiotensin test is performed"; 7
W
= "test with saralasin is performed";
a
1 1 17 17 r d 8
= 1 | W w 12 12 I I 9 9
1 1 18 18 W w 10 10
where - "the angiotensin test is positive";
W 9 W
10
= nW ; 9 1 19 19 r - 1 I W 13 I 11 9
1 20 20 W 12
where W
= "the saralasin test is positive"; 11
W 12
= nw ; 11 1 14 r = 1 I 6 7I
true
1 | 8I
true
The tokens from places 1 and 1 reach directly place 1 (per7 8 14 forroance of
vasography - abdominal aortography and/or selective renal
arteriography). The transition conditions r , r , r and r have the form 71 10 11 12 1U 11 1-
r 10 where
1 21 21
1 22 22
= 1 I W w 14 I 15
w 16
284
W
CHAPTER 4
= "there are signs of renal arterial stenosis"; 15
W 16
= nW ; 15
r 11 11
1 23
1 24
= 1 I W 21 I 17 21 I 17
W 18 18
1 | true 15 I 15 I
false
where W
= "there is
evidence that
the stenosis is
functionally signifi-
17 cant", = iW ;
W
18
17 1 25 r
= 1 | true 12 23 I
Places 1 , 1 25 26 tests to
26
1
1 27
1 28
29
true
true
true
true
1 indicate the performance of appropriate 29
determine whether
1 (Howard's test), 25
1
the stenosis has functional significance:
1 (Rapoport or Stanley test), 1 (separate he26 27
modynamic investigation of Kidneys), 1 (determination of 28 nin activity in renal
plasma re
veins separately), 1 (other investigations of 29 separate Kidney functions).
r 13
1 30
1 31
= 1 | W I 25 | 19
W 20
where W
= "the test of Howard is positive", 19
w 20
= -m ; 19
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
r 14 14
1 32
1 33
= 1 | W 26 26 I I 21 21
W 22 22
285
where W
= "the test of Rapoport or Stamey is positive"; 21
W 22
= nW ; 21 1 34 r
= 1 I W 15 27 I 23
1 35 W 24
where W
= "unilateral decrease of ERBF is found during
separate hemodyna-
23 mic isotopic investigation of kidneys"; w
- -m ■, 24 23 1 36 r 16 16
= - 1 I W 28 2ft I I 25 25
1 37 W 26 26
where w
= "augmented plasma renin activity is determined unilaterally"; 25
W 26
= nW ; 25
r 17
1 38
1 39
= 1 I W 29 I 27
W 28
where W
= "other functional 27 on"; 28
27
tests for
determining unilateral renal lesi-
286
CHAPTER 4
1 40
1 41
1 I 30 I 30 I
W 29 29
W 30 30
1 I 32 I r = I 18 1 I 34 I
W 29
W 30
W
W 29
30
1 I 36 36 I I
W 29 29
W 30 30
1 I 38 I 38 I
W 29 29
W 30 30
where - "a biopsy of the contralateral kidney is necessary";
W 29 W 30
= nW . 29 Place 1 signifies the performance of renal biopsy. 40
r 19 19
1 42
1 43
= 1 I W 40 40 I ! 31 31
W 32 32
where W
= "evidence for lesion of the contralateral kidney is present"; 31
W 32
= nW . 31 1 44 r 20
= 1 | true 43 I 1 I true 41 I
Place 1 signifies aa check-up for contraindications to surgi44 cal intervention. The transition condition
r 21
1 45
1 46
= 1 I W 44 I 33
W 34
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
287
288
CHAPTER 4
where W
= "there are contraindications"; 33
W 34-
= nW 33
(a surgical investigation (1 ) is possible). 45 From
places
1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 11 16 18 20 22 24 31 33 35 37
1 , 1 and 1 the tokens get into place 1 . Here, they acquire the 39 42 46 47 characteristic
"appropriate treatment and medical observation
of the
patient".
REFERENCES'. [1] F. Balag, G. Petranyi and F. Renyi-Vamos,
Nephrologia,
Medicina
Acute renal failure,
Saunders
Konyvkiado, Budapest, 1980 (in Hung. ). [2] B. Brenner and J. Lazarus (Eds. ), Co. , Philadelphia, 1983. [3] B. Brenner and F. Rector (Eds. ), The kidney, Vol. 1 and 2,
Saun
ders Co. , Philadelphia, 1986. [4] N. Bricker and M. Kirschenbaum (Eds. ), The kidney - diagnosis and menagement, Wiley Medical Publ. , New York, 1984. [5] J. A. Halsted and C. H
Halsted (Eds. ) The laboratory in clinical
medicine - interpretation and application,
Saunders Co. , Phila
delphia, 1981. [6] J. Hamburger, J. Crosnier and J. -P. Grunfeld (Eds. >, Nephrologie, Vol. 1 and 2, Flanmarion, Paris, 1979 (in French). [7] J. Jones, J. Briggs and T
Hargreare, Diagnosis and menagement of
renal and urinary diseases, Blackwell, Oxford, 1982. [8] S. Klahr and S. Massry, Contemporary Nephrology, Plenum Publ. Co. New York, 1981. [9] H
Kremling, W. Lutzeyer and R. Heintz,
Gynakologische
Urologie
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
289
und Nephrologie, Urban Schwarzenberg, Munchen, 1982 (in German). [10] M
Legran and J. -M Sue,
Abrege de nephrologie,
Masson,
Paris,
1981 (in French). [11] G. Majdrakov and . Popov (Eds. ) Gidney diseases, Sofia,
Medicina
i Fizkultura, 1980 (in Bulg. ). [12] F. Marsh (Ed. ) Postgraduate nephrology,
W. Heinemann Medical Bo
oks Ltd. , London, 1985. [13] S. Massry and R. GlassocK, Textbook of nephrology,
Vol. 1 and 2,
Williams and Wilkins, Baltimore, 1983. [14] E. Patev, Acute renal failure, Sofia, Medicina i Fizkultura, 1981 (in Bulg. ). [15] D. Vasilev, Vasorenal hypertension, Sofia, Medicina i Fizkultura, 1989 (in Bulg. ). [16] D. Vasilev and G. Stefanov
(Eds. ),
Clinical nephrology,
Sofia,
Medicina i Fizkultura, 1990 (in Bulg. ). [17] J. Sorsich and K. Atanassov.
Application of generalized
medicine (Acute Attack of Gouty Arthritis),
nets in
Proc. of Third Symp.
Int. Ing. Biomed. , Madrid, Oct. 1987, 643-64-5. [18] R. Bellman, Mathematical methods in medicine.
World Sci. Publ. ,
1983. [19] S. Kiforenko, Methods of mathematical biology, Kiev,
Vischa sko-
la, 1984 (in Russian). [20] J. Piemann, Modeling hospital information systems with Petri nets Methods Inf. Medicine, Vol. 27, No. 1, 1988, 17-22. [21] Mathematical methods
in medicine
and biology,
Proc. of
UNC of
Academy of Sciences of USSR, Sverdlovsk, 1986 (in Russian). [22] J. Sorsich,
An example of application of generalized nets in me
dicine. Proc of II Int Symp. "Automation
and Scientific
Instru
mentation", Varna, May 1983, 387-389. [23] J. Sorsich and K. Atanassov, medicine
Application of generalized nets
in
(Diagnostics of arterial hypertension of renal origin).
290
CHAPTER 4
Proc. of III Int. School
"Automation and Scientific Instrumenta
tion", Varna, 1984, 233-236. [24] J, Sorsich and K. Atanassov,
Application of generalized
nets in
medicine (Renal colic), Lect. Notes in Medical Inform 24, 1984, 352-355. [25] J. Sorsich and K. Atanassov, medicine
(Renovascular
Application of generalized nets
in
hypertension). Proc. of Third Int. Symp.
"Automation and Sci. Instrumentation", Varna, 1985, 167-169[26] J. Sorsich and K. Atanassov,
Application of generalized nets
in
medicine (Permanent Proteinuria), Proc. of National Sci. Session "Automation of Biotechnology Processes",
Sofia, Oct. 1985,
138-
141 (in Bulgarian). [27] J. Sorsich and K. Atanassov,
Application of generalized nets
medicine (Haematuria), Proc. of III International Symp.
in
"Automa
tion and Scientific Instrumentation", Varna, Oct. 1985, 163-166. [28] J. Sorsich and K. Atanassov,
Application of generalized nets
in
medicine (Dysuria), Proc. of III Symp. Int. Ing. Biomed. , Madrid, Oct. 1987, 639-642. [29] J. Sorsich, Application of the generalized nets in medicine (Ele vated blood used nitrogen),
First Sci. Session of
the Math. Fo
und. of AI Seminar, Sofia, Oct. 10, 1989, Preprint IM-MFAIS-7-89, Sofia, 1989, 57-59. [30] J. Sorsich,
Application of the generalized nets in medicine (Ar
terial hypertension), Ninetieth Session of
the Nat. Seminar
Informatics of the Union of Bulgarian Mathematicians
and
of
Fourth
Sci. Session of the Math. Found. AI Seminar, Sofia, Nov. 5, 1990, Preprint IM-MFAIS-5-90, Sofia, 1990, 37-39.
APPLICATIONS OF GENERALIZED NETS IN MEDICINE
291
§4.2: MODELLING OF DIAGNOSTIC AND THERAPEUTIC PROCESSES IN MEDICINE BY GENERALIZED NETS Borjana Jordanova
and
In this article we describe
Joseph Sorsich
a general outline of
the approach
to a patient who needs medical advice. This net is more general than the ones lacks the details of the others, that most patients pass through
described in § 4. 1.
It
but it gives some idea of the stages in a medical department.
So the nets
described in § 4.1 are its concrete applications. In the
net presented it is possible to
see more
clearly the
ways of introducing waiting and performing times of medical examinati ons
and carrying out the
time may
appropriate investigations,
duration of the whole cycle of treatment.
as well as the
These time-parameters
be obtained by a characteristic function of
the net as presented
below. The GN models
of
the diagnostic process
and
the appropriate
treatment of other (non-nephrological) diseases may also be settled in this scheme. The token
(patient)
enters place
1
with the characteristic
1 "need to consult a physician"
(with some complaints or for prophylac
tic needs). The GN below described is a reduced GN from the class A ,A ,A ,A ,n ,n ,c,6 ,6 , n ,G ,b 3 4 6 7 A L 1 2 K K
I
(everywhere the transition type is "v", ties are
"", 3 where 'servernode'
is a node in the system where the request
is to be serviced. $ -> "", 6 where 'trans_id' is a transaction identifier system;
'TIME' is the current time,
assigned by the
'6' is the period during
which the transaction is active, 'result' signifies whether the
GENERALIZED NETS AND COMPUTER SCIENCE
operation has or has not been successful: result € ( true, false ).
Fig. 5. 2. The t r a n s i t i o n c o n d i t i o n s are defined as f o l l o w s : 1 2 r
w*iere W ='f 1
E Fr";
W = "f € Fo"; 2
1
=
1
I W 1 1 1 I 1 I W 2 I 3
1 3 W 2 true
303
30*
W
CHAPTER 5
= oc a = "c(l , X ) < r(pr2 x )" 3 3 0 0
where
c(l, X)
being a function giving the number of tokens
in place
'1' with characteristic 'X';
rr -2
1 i 4 4 W W 44
1i
5 5 W W 5
1 II 33 || II 1 II ffalse alse 4 II
true true
where
4
a a = "x - pr x "; cu 10
5
= IW ; 4
W W
rr 3
::
1 6 6
11 77
1
1 II W W 55 || 66 II 11 II W W 6 1 9 6 1 9
W W 77
W W 8
W W 10 10
W W 11 11
88
where W
= "f e Fr"; 6
W 7
a a = "f e Fo" St "pr x i Tr" & "pr x = true" 3 cu 4 cu where 'Tr' being the maximum time allowed by
the system
which a transaction may be active, a a W > = W = = "f "f ee Fo" Fo" & & "pr "pr xx > Tr" Tr" vv "pr "pr xx = ff aa ll ss ee " " ;; 8 3 cu 4 cu 8 3 cu 4 cu = a a w = "c(l , x ) < r ( p r x )"; 9 6 0 2 0 = a a a a W = " c ( l , x ) = r ( p r X )" & "pr x " & "pr x
6 0
20
3cu
a > Tr" & "pr x
4cu
= false";
during
GENERALIZED NETS AND CCWUTER SCIENCE
r
l1 I 7| I 1 I fl II
= 4
1 99 wW 12
1
W
W
305
10 10 Ww 13
12
13
1 I false 9 I
true
where - 1W
W
;
12
13 a a W = "x = pr x ". 13 1 10 The presented GN model of a DFS does not describe all sible situations in ally during are made and
the pos
a full-scale system, which is rarely needed. Usu
the time when
a DFS is designed,
some restrictions are imposed.
particular assumptions This is done to prevent
the possibility of eventual indefinite states in the system, to avoid deadlocks and other undesired ten achieved after
effects.
a reasonable
A predictful behaviour is of
compromise between the complexity of
a model and the clarity of its operation. In this particular case some potential events are ignored, such as conmunication failures,
invalid client requests or
These situations are easily solvable,
access rights.
but the model would be unreaso
nably complicated and will not be adequate for its present purpose. Further research work in the field of DFS design will probably bring forth
other problems, and methods for
thor's belief is that a precise GN model of
their solution.
The au
any distributed file sys
tem could suggest helpful ideas about its design and development.
306
CHAPTER 5
§ 5. 2: GENERALIZED NET MODELS OF DATA TRANSFER IN INDUSTRIAL LOCAL AREA NETWORKS Alexander Savov
§ 5.2.1
Introduction This
paper by means of the Generalized Nets (GNs) presents the
results of a simulation study carried out by an Network (ILAN).
Industrial Local Area
The results are very close to the designed specifica
tions, The ILAN is composed of three layers: physical, data
link, and
application [1]. The data link layer
is built on a polling mechanism
because of its deterministic behavior.
All stations are polled either
centrally
by a supervisor or decentrally by a token
[£]. There are two types of stations:
passing protocol
initiators and responders. Ini
tiators provide the following data transfer services [3]: (a) SDA - send data with acknowledgement, (b) SDH - send data with no acknowledgement, (c) RDR - request data with reply. The initiators connected to the bus cooperate by using the sha red channel. The right to use the channel is named a token. A token is passed
from a station to a station in a cyclic way,
thus defining a
logical ring. The services SDA and RDR
are a couple of frames
sent via the
medium: the frame - request - passes from the initiator to the responder, the frame - answer - passes from the responder to the initiator. The service SDN has only one frame request passed from the ini tiator to the responder. The stations
included in the ILAN
are Programmable
Devices
(PDs) - components of manufacturing systems such as programmable logic controllers, robots, process controllers, personal hers.
computers, and ot
Their base functions are calculating the control
programme and
updating inputs and outputs. The control programme works cyclicly. The
GENERALIZED NETS AND COMPUTER SCIENCE
cycle is called
307
a Treatment Cycle (TC). Tie PD takes inputs -when the
TC starts. This is done in order to avoid hazards due to value changes during a cycle.
For the same reason data exchange is impossible
when
the TC runs. Two models of ILAN exist: (a) model 1 - only initiators are included in a logical ring, (b) model 2 - all stations are included in a logical ring. Model 1 behaviour 1. The initiator receives
the toKen and sends
the frame - request to
the responder, 2. The receipt of this frame causes
the responder to wait
the
TC to
end, to compose and transmit the frame- answer. 3. the receipt of this frame causes the initiator to pass the token to another initiator. Model 2 behaviour 1. The initiator receives the token,
sends the frame - request to the
responder, and passes the token to the next initiator. 2. The receipt
of this frame causes the responder to wait
for the TC
to end, to compose the frame-answer, and to wait for the token. 3. Tne responder receives the token and transmits the frame - answer.
§ 5. 2. 2 Generalized
net model
By means of GNs we shall describe
a GN-model
(see Fig. 5. 3)
of data transfer in ILAN. N is the number of stations included in the ILAN. I is the number of initiators included in the ILAN. Lmin is the minimum length of the data field
that can be tras-
ferred by one service. Lmax
is the maximm length
trasferred by one service.
of the field
of data
that can be
308
CHAPTER 5
Fig. 5. 3 CI1 is a TC in the initiator when it composes a frame-request. Cli is a TC in the initiator when it processes a frame-answer. CR
is a TC in the responder when it processes a
frame-request
and proposes a frame-answer. The GN-model consists of two parts. tions of
the ILAN and the second
The first models the func
(transition r , place 1 and token 5 9
GET'ERALIZED NETS AND COMTJTER SCIENCE
309
a) - the determined protocol on the data link level. The tokens p , . . . ,p 1
in the GN describe
the transfer and pro-
N
cessing of the data in the media and the station. The tokens in the first part of the GN have the following tial characteristics:
ini
<SA, DA, L, typo, where
(a) SA is the address of the station initiator, the source of the ser vice; (b) DA is the address of destination station - responder, the receiver of the service; (c) L is the length of the
data field
that is transferred during the
service; (d) type is the type of service (SDA, RDR, SON [2]). These tokens pass consequently through the places 1 , 1 , 1 , 1 , 1 , 1 , 1 6 3 6 4 6 5 7
1 , 1 , 1, 8 1 2
and 1 . 8
One service is modelled by passing a token through the net. The places model the following: 1 - setting up of the frame-request in the initiator; 1 1 - frame-request's buffer in the initiator; 2 1 - transmission of the frame-request and frame-answer in the media; 6 1 - processing 3
of the
frame-request and setup of the frame-answer's
buffer in the responder; 1 - frame-answer's buffer in the responder; 4 1 - no physical meaning; 5 17 - processing of the frame-answer in the initiator. The time intervals,
when the tokens are in
the different pla
ces, depict the processing of the data in the stations and the time of transmission in the media
These time intervals are a function of the
310
CHAPTER 5
length of the data-field and depend mostly on U ) the speed
of the microprocessors
and
the hardware in
the data-
transfer section (DMA or other), (b) the data transmiting speed in the media, (c) the frame organization and the data representation. The following equations are used for estimation of the duration Pf a tokens' stay a certain place: (a) for 1 : Tl - the time for setting up the frame-request 1 Tl = Kl+L. tl+RCIl (for the services SDA, SDN), Tl = Kl+R. CI1 (for the service RDR), where Kl - a constant, tl - a time constant depending on the speed of the hardware, R - a random number in the interval [0, 1], "k. T . ((Log k) - N + l)/(2. (N-l) )", where T 5 c 2 c r a t i o n computation time; 2
$
6
$
-> "k. (T
+ T )/(2x(N-l) t C
-> "k. T ";
7
t 2 -> " ( n - l ) ".
$ 9
);
is thetoasicope-
GENERALIZED BETS AND COMPUTER SCIENCE
319
§ 5. 4: IMPLEMENTATION OF GENERALIZED NETS FOR SIMULATION CF HQDE-TO-NQDE CQMMUNICATIOU IN A REGULAR STRUCTURE Stefan Stefanov
The case
of node-to-node
commmcation in
a massively linked
network of nodes is discussed. A structure
which
we shall
call
a massively linked
regular
graph (MLRG) is shown in Fig. 5. 5. Each node must have 8 bidirectional links. In case the hardware has only
6 or 1 bidirectional
cluster
like the ones shown in
links each node may Fig. 5.6.
be replaced
In that case
by a
we call
the
structure a massively linked regular cluster graph (MLRCG). Bearing in mind that a) the nodes are very close to one another in space, and, b) the main difficulties
in conmini cation
are
the conflicts between
the messages trying to pass through the same link at the same time, the distance (or the message price) between two nodes N
and N i
be defined as the number of links which the message in order to reach S
may j
passes through
starting from N . j i
The distance (message price) between N i
and N is J
P(N , N ) = max (lv(H ) - v(N ) I, lh(N ) - h(H ) I) , i j i j i j where h(N ) i
v(H ) i
is the relative vertical position of H
is the relative horizontal position.
in the graph and i
Note that this result
is
optimal (even better than could be expected). Let D = IE, W, H, S, NE, HW, SE, SW] be the set of the possible "world" directions. Let us substitute v(N ) - v(N ) = v i j
320
CHAPTER 5
h(N ) - h(N ) = h. i j Then P(v, h) = max(|v|,
|h| ).
The primary direction calculation mechanism (PDCM)
is given in
the table below, where /'"-", if x < 0 sgn(x) -)
"0", if x : 0 , I " + ", if x > 0
d(v, h) € D, "NONE" means that no direction is calculable. ( v , hh) ) I 1 1I sgn(v) sgn(v) I ssgn(h) g n ( h ) I dd(v, 11 0 0
1 00
1 None
II
I 1
00
I
--
IW W
||
1I
00
I
++
I EE
I1
1I
-
I
00
I NN
II
1I
-
I
--
I NW
II
1I
-
I
++
E I NNE
||
1I
++
I
00
I SS
II
1I
++
I
-
I SW SW
I|
1I
++
I
++
E I SSE
II
The secondary direction calculation mechanism type of A (SDCMAI is defined recursively: applicable if h / 0 and v ^ 0 and v / h (if (|v| > Ihl) then d(v, 0) d'A(v, h) -1 \ if (Iv| < |h|( then d(0, h) The secondary direction calculation mechanism type of B (SDGMB) is given by the table below which is based on the PDCM The "nearest possible direction" to d(v, h), where D, is defined as follows:
d'B(v, h) €
GENERALIZED NETS AND COMPUTER SCIENCE
321
I1 d(v, d ( v , h) h ) I d'B(v, d'B(V, h) h ) 1I
1 I
NE
I NNor orEE
I 1
E
I NE NEor orSE SE 1I
1 I
SE
I E or S
II
1 I
I 1
S
I SE or SW 1 I
1 I
SW
I S or W
I 1
W
I SW or NW I
1 I
NW
I WWor orNN
I 1
N
I NW NWor orNE NE 1I
I
1I
The tertiary direction calculation mechanism (TDCM) determines d"(v, h) € D - {d(v, h)I, i.e. , d"(v, h) is any member of D different from d(v, h). In the case of a MLRCG
each node is a cluster and the in-clus
ter directions are calculated similarly (using tables). Note that if the SDCMA is chosen, instead of the FDCM, the mes sage' s price
P does not increase, i. e. an optimal routing is achieved
if the FDCM or the SDCMA, is used. If the SDCMB is chosen instead of the FDCM then P
is increased
If the TDCM is chosen instead of the PDCM then
P
is increased
Tertiary directions are chosen in emergency cases
(hardware or
by 1.
by more than 1.
software malfunctions). A problem that Let
is solved by the simulation is the following.
C be the set of the vertices of the graph,
Cs c C be
the
set of message senders, Cd c C be the set of message destinations, nl - |Cs|, n2 = |Cd|. The time W ; W(C, nl, n2) during which all messages wait to re ach their destinations is a topic of importance.
322
CHAPTER 5
The generalized net (GN) representing the case is shown in Fig. 5. 7. The model
consists of 7 transitions
further minimized,
but this is not done to
and 11 places.
It can be
preserve clarity
and the
closeness to physics of the process. Tokens from the initial place L
pass through the trivial trani
sition
T
where they receive their initial location coordinates
cc €
1 Cs, destination coordinates
cd € Cd,
with priority equal to the time
when they enter the transition, Wc and Wg being zeroes and all the ot her elements of the characteristic vector blanks. Transition T
stays inactive until all nl tokens gather in 2
Passing through the trivial
T
a token receives
L, 1
its next partial de-
2 stination en e c,
calculated
using the PDCM Then it continues thro
ugh T and goes either to L (1 £ i £ K, and K is the number of no4 7, i des in the graph) if it is empty, or to L if otherwise. From L 8 7, i tokens go to
L
and from there either to L 9
if cc = cd,
or to
o
L
to 2
continue moving in the net. A token from
L
goes either to 8
tial destination, or, Passing
through
T
L
to retry to reach
its par-
5
if it has tried for more than a new partial destination
X
times, to L , 6
en € C
is calculated
3 using the SDCM. The simulation stops when all tokens gather
in L . o
is the maximum retry count. o * The time-parameters of the net are: T = o , t = 1 , t =oo. The transition conditions are
x
GENERALIZED NETS AND COMPUTER SCIENCE
323
L 1 r
= L
1
I
W
I I I
where W = "TIME > nl"; 1 L 3 r 2
= L I true 1 I L l true 2 I L 4
r 3
- L | true 6 I
L 7,1 r 4
L 7,2
...
L 7,N
L fl
- L I W 3 1 2
W 2
...
W 2
W
L I W 4 -| 1 22
W 2
...
W W 2 2 3
L I W 5 1 2
W 2
...
W 2
3
W
where
W = "C(L "c(L 2
, TIME) = 0" & "Cn = : cC(l (l 7, i
)", 7, i
where "i" i s the ni«l>er m«i4>er of the current place; W
= nW ;
3
2
L 5 r : L I W 5 8 I| 4 4 where W
4
= "W
C
W - iv ; 5 4
< X",
L 6 W 5
3
324
CHAPTER 5
Fig.
5. 7.
GENERALIZED NETS AND COMPUTER SCIENCE
325
L 7, 1 L r r = = 6 6
| true 7, 7, 1 1I I l I L | L | true true 7, 2 7, 2 I I
L
| true 7, N I L O
r =L I W 7 9 1 6
L 2 W 7
where W 6
; "C = C "; c d - nW .
W 7
6 Every toKen has only one (its current) characteristic "", where (a) cc are the current coordinates; (b) cd are the destination coordinates; (c) en are the coordinates of the next partial destination; (d) Wg is the global wait time (retry count); (e) Wc is a scratch (current retry count). The characteristic functions related to the places are $
-> "", 1
where (a) cc is the initial location (the message source); (b) cd is the destination; (c) en is undefined; (d) Wc is 0; (e) Dg is 0;
326
$
CHAPTER 5
-> ""; 3
where
"en" is the next partial destination calculated using the PDCM,
all the rest remaining unchanged; $ = ""; 4 where
"en"
is the next partial destination calculated using the SDCM
(A or B), all the rest remaining unchanged; $ = ""; 5 where (a) Wc = Wc + l; "*";
2 $ -> "*"; 7, i $ -> "»"; 8 $ -> "<Wg, TIME - nl>", O (this is the final token's characteristic;
TIME
is the current model
time). All the places and arcs have infinite capacities. All the tran sitions and places have equal priorities. All transition types are "v".
GENERALIZED HETS AND COMPUTER SCIENCE
327
§ 5. 5: GENERALIZED HET MODELLING OF DISK SUBSYSTEMS Bistra B. NiKolova
Disk Drives (DDs) and the subsystems one of
the most important components
built on their bases, are
of the computer.
They are used
for expansion of the RAH of the computer. The productivity of the com puter depends on them An example of a structure scheme shown
in
Fig. 5.8.
It consists of
and H numbers of DDs. The whole
of a disk subsystem
an input/output path
(DS) is (i/o path)
software and hardware, necessary for
the data exchange between the RAH and DDs is represented as a separate functional unit, placed between the RAH and DDs [1]. This unit is mar ked as an i/o path. part in
If a certain
the i/o path of
part of
the hardware,
which takes
the concrete subsystem, is occupied by a DD,
the other DD cannot exchange data. Data exchange is realized by direct memory access.
Fig. 5. 8. There are some kinds of architectures
in which in order to in
crease the parallelity in the functioning of the DS, new i/o paths are added. They are
called "subsystems with a dynamic distribution of the
control between a certain number of i/o paths". They consist of H num bers of DDs and connected DDs and
H
numbers of i/o paths,
to every i/o path.
2 i H i H [4]. Every DD is
The control of the i/o operations
with
their parts can be done with the help of as i/o path. The re
quests to an i/o path form a queue [2,5].
328
CHAPTER 5
Usually, the i/o path is used for but also added, for
of control information.
two buses
are formed
of data,
If a separate path for commands
for data and commands.
positioning at every moment of
the data bus
exchange, not only
the DD
is
The opportunity
is created,
nevertheless
and the i/o path are busy with the data exchange from/to
a DD. The result of an i/o operation is given in Fig. 5. 9, where A - the request goes into the DS (in the queue to the DDs) B - the i/o operation is started C - the positioning of the DDs is completed and the request goes
into
the queue of the i/o path D - the data exchange begins E - the request service ends and the request leaves the DS T - the time during which the request is in the queue to the DDs DQ T - position time DS T - the time, during which the request is in the queue to i/o path CQ T - data exchange time CS
Fig. 5. 9. On the other hand, an i/o operation in the DS consists of positioning, searching and exchanging data. The purpose of the article is to make a model of some DS archi-
GEI>JERALI2ED NETS AND COMPUTER SCIENCE
329
tecture in terms of Generalized Nets (GNs). Some specifications are made: 1) all DDs are equal and
the input requests
are equally
distributed
between them [2, 5]; 2) the fulfilment of the i/o
operations includes the steps from Fig.
5.9 [2,5]; 3) the queues to the DDs and the i/o path are realized by means of the FIFO [2, 3]; 4) every D has its own queue of requests,
while the queue to
the i/o
path is conmon; 5) DD positions the head over the wanted cylinder by itself; 6) the i/o operation starts from a free DD and i/o path; otherwise the request
moves either to the queue of the DD or to the queue of the
i/o path, according to the architecture of the DS; 7) the i/o path
takes
part in
the search for the block on the track
and in the exchange of information between the DDs and the i/o path [33. The requests in the DS are the tokens in the given model. token a goes in the net with an initial characteristic:
Each
"" (= x ), where NDS, NCYLN, NB, LDBL are natural numbers 0 and NDS is the number of the DD, 1 i NDS i MaxNDS, where MaxNDS is the ma ximum number of the DD, NCYLN is
the
number
of
the cylinder,
MaxNCYLN is the maximum NB is
the number
0 £ NCYLN ", if a goes into b for the first time K 2 $
-> ■
2
a "<TIME, TIME - pr x >", otherwise V. 1 cu
1
GENERALIZED NETS AND COMPUTER SCIENCE
333
a where x is the current characteristic of token a; cu a $ -> "<TIME, TIME - pr x >"; 3 1 cu a a $ -> "h (pr x , pr x )", where the function h determines the ti4 1 2 cu 2 cu-1 1 me for waiting in place b ; 4 a a $ -> "h (pr x , pr x )", where the function h determines the time 5 2 20 4 0 2 for positioning; a a $ -> "h (pr x , pr x )", 7 3 20 4 0
where the function h determines the time 3
for searching for the blocK; a a $ -> "h (pr x , pr x )", 8 4 30 4 0
where the function h determines the time 4
for exchange between the DD and the RAM. Second, we shal1 discuss a DS which consists of an independent command path, an i/o path, and M DDs, as it is shown in Fig. 5. 11. The independent command path controls the search and exchange of in formation. Each toKen a with the above initial characteristic enters the net. We accept that in practice the independent command path is always free. The queue to the DDs and the i/o path, as above, are pro cessed by means of a FIFO-discipline. The GN modelling this architecture is characterized by compo nents, which are similar to the above ones. The transition condition r is 0 b b ... b 0, 1 0,2 0, M r =b I W 0 0 | 1 where
W
... 1
W 1
334
CHAPTER 5
GENERALIZED NETS AND COMPUTER SCIENCE
W
335
a = "the number of place is equal to pr x ". i 1 0 The transition conditions r
are i, i
r
= b 1, i b
b b
1, 1, i i | true 0, i |
| true 2, i I
for 1 i i i VL The transition conditions r
r , r ,. . . , r 2, 1 2, 2 2, M
are similar to
with exactly the same notation. 1 The transition conditions r
and r 3
have the forms: 4
b" 3 b r
= 3
| true 2, 1 I I b | true 2,2 I
b
I true 2, M I
b 4, 1 r = b" I W 4 3 1 1
b
. . . 4, 2
W
b 4, M
. . . 1
W 1
Noting that the capacity of place b" is 1. 3 The transition conditions r
r , r ,..., r are similar to 5,1 5,2 5, M
with exactly the same notation. 5 The transition conditions r
and r 6
have the forms: 7
336
CHAPTER 5
b 6 b r
= 6
I I true true 5, 11 II 5, I I bb II true true 5, 22 II 5,
b
I I true true 5,M 5,M II
b' b' 7
II true true II
b b 77 rr ss bb II W W 77 66 11 55
b' b' 77 W W 6 6
where W
= "c(b , TIME) = 0"; 5
W
7
= nW ; 6 5 The transition condition r
is the same as the one in the first
8 GN. The characteristic functions in both GNs coincide exactly in nota tion. Finally,
we shall discuss a DS which consists of DDs in number
M and i/o paths in number N, as it is shown on Fig. 5. 12.
The discip
line to the DD and the i/o path, as above, is FIFO. The differences between the second and the third net are in the forms of transitions conditions r , r and r . They have the forms: 0 3 7 b b .. .. .. b b b b 0, 0, 0, 0, 11 0, 22 0, M M r = b I W WJ ... W 00 0 1 11 1 1 bb"" "" II 33 1
W 11
1
W 1
...
W 1
GENERALIZED NETS AND COMPUTER SCIENCE
337
338
CHAPTER 5
b"' 3
bb"" "" 3
I 2, 1 I I b I 2, 22 II 2,
W
W 8
b
W
b r
= 3
7 W
W 77
I 2,M I
7
88
W 8
where W 7
= "c(b"', TIME) : 0"; 3
w = -m ; 8
7 b
b 7, 1
r
= b 7
I W 6 1 9
. . . 7, 2
W 9
b 7, N
b" 7
. . . W 9
W 10
where W
= "This i/o path is free"; 9
W
= "There is no free i/o path". 10 The transition conditions
r , r 8, 1 8, 2
r
are similar to 8, N
r with exactly the same notation. 8 The function $ is similar to the one from the second GN. The theory of the GNs makes it possible to create a cannon for mal description of DS. The concept GN allows
all phases of an
dependency between them when i/o requests be shown.
i/o operation
in the DS
and the
are processed to
It also allows all specialities of the separate architectu
res to be shown.
GENERALIZED NETS AND COMPUTER SCIENCE
339
REFERENCES: [1] J. Major, Processor, i/o path and DASD configuration capacity, IBM System Journal, Vol. 20, No.
1981. 63-85.
[2] I. Donev, A research over the productivity of disk subsystems, [3] D. Brown, R. EiQsen and C. Thorn, Channel and direct access device architecture, IBM System Journal, Vol. 1 No. 3, 1972. 186-199. [4-] R. Wilson,
Designers rescue
superminicomputers from
i/o bottle
neck, Computer Design, Vol. 26, No. 18, 1987, 61-71. [5] I. Donev,
A research over
the productivity
of disk
subsystems.
Proc. of Symp. "Electronic and computing technics and technology", May 1987, Sofia, (in Bulgarian). [6] I. Donev, L. Michov and N. Sinyagina,
A research over the organi
zation of highly-productive disk subsystems with stratification of information. Automation Control, Systems, Sofia (in Bulgarian).
Computing Technic
and Automated
340
CHAPTER 5
§ 5. 6: SOFTWARE TOOLS FOR GENERALIZED HETS Rumen Christov
§ 5.6.1
and
Stojan Hibov
Introduction During the last ten years concurrent processes
ve been developed.
Too many big
researches ha
and complex systems do not allow
an
analytic study. Petri Hets (PNs) and their extensions and modificati ons are a powerful theoretical tool for the study of the properties of complex concurrent systems. Too many software tools for implementation of the PNs and some of
their modifications
work of scientists easier.
are produced to make
the
These tools contribute to further develop
ment and extension of the practical applications
of PNs and their mo
difications [1], Each extention of
the FHs is
designed to solve problems in a
determined area. Howadays a trend to generalization and comparison mathematical
models
But there is no
of processes
from different areas
of
is observed.
compatibility between the FHs' extentions.
It
makes
the comparison of the processes' models very difficult. Generalized Hets
(GHs) were defined in 1982.
They are a po
werful mathematical tool for describing and studing the properties
of
complex concurrent systems. These nets are a continuation of the trend to make tools for modelling of parallel processes. GHs are an extension of the PNs. It is proved, that all exten sions
and modifications of -the PNs are particular cases of
fact proves
that each process
GHs. This
which may be described and studied
by
FHs or with their extentions may be described and studied by GHs. This means
that GNs
are a universal
language for describing and studying
the processes from different scientific and practical areas. This fact proves the neccesity
of software tools for
the im
plementation of GHs. They are developed by the group, working over GNs and the tools are named STGNs. The STGHs include:
GENERALIZED NETS AND CCMTJTER SCIENCE
34-1
(a) presentation of GN-based hierarchical models of processes; (b) computer tool support, with which it is possible to create and si mulate hierarchical GN; (c) tools for analysing results of a GN-based model simulation.
§ 5. 6. 2 GTfs - a short
description
GNs are a complex
mathematical object.
Many of
their compo
nents are presented in the other FNs extentions. However, the GNs are not an aggregate
of the components of other types of PNs.
All compo
nents of GNs have a specific content. GNs have a static and a dynamic structure, temporal components and a global memory. Static structure The
static structure includes the net transitions, net places
and arcs which connect the places with the transitions. The information necessary for each transition description is: (a) a transition
identifier,
which is
a unique name in
the net for
them; (b) a transition name (only for user applications); (c) priority towards other transitions in the GN; (d) temporal components of the transition: time of the next activation and duration of this active status; (e) type of the transition; (f) indexed matrix expressing the transition conditions; (g) a list of the input places for the transitions; (h) a list of the output places for the transitions. The transition type, condition matrix and the functions which assign temporal transition
components are included in the dynamic net
structure. The transition type contains the necessary conditions
for the
342
CHAPTER 5
transition activation.
In the GNs' theory
the transition type
has a
specific structure, but in software tools it may be an arbitrary logi cal expression, which may depend on arbitrary events in the net at the same or a certain previous step of the model simulation. The information necessary for each place description is (a) place identifier, which is a unique name in the net for than; (b) place name (only for user applications); (c) priority towards other places in the GN; (d) place capacity - maximum number of tokens which may be in the pla ce in one step from the net simulation; (e) tokens which
are in
the place
before
the GN-roodel
simulation
starts. A place may be (a) a GN input, if it is
a member only of
the input places' list for
any net transition; (b) a GN output, if it is a member only of the output places' list for any net transition; (c) internal to the net, if it is a member of for any net transition and is a member of
the input places'
list
the output places' list
for another (eventually the same) net transition. From
each input place for each transition in the GN,
each of its outputs are defined. For each arc a predicate from
A capacity
arcs
to
is assigned to each arc.
the transition condition matrix corres
ponds. Dynamic structure Components of the dynamic structure are tokens, types of tran sitions,
conditions of
the transitions and temporal functions of the
transitions. The information necessary for each token description is (a) a token identifier, which is a unique name in the net for them;
GENERALIZED NETS AND COffUTER SCIENCE
34-3
(b) a token name (only for user applications); (c) priority towards other tokens in the GN; (d) the time when the token enters the GN; (e) a list of the net input
places,
through which
a token may enter
the GN. At the moment
when the net starts to simulate, the tokens may
be in any place in the net or may be waiting to enter the net.
In the
beginning, the tokens may have a list of any initial characteristics. The token's way
through
the net depends
When the tokens are going in the net,
on
these characteristics.
to the lists of characteristics
new characteristics are appended The characteristics are the greatest difference between GNs and other kinds of PNs. dividuality
The tokens have an in
during the whole time of net simulation.
This fact means
that many of the properties of the concurrent processes may be descri bed
by characteristics.
helps models
the user
It simplifies
the static net structure and
to understand the model of the process.
are more compact
The GN-based
than models based on other kinds of PNs.
It
allows more detailed models of the studied processes to be made and to get more complex results. Token characteristics are useful but their software implementa tion is very difficult.
They may not be formalized in their full the
oretical definition. Software implementation is the implementation
of
a class of GNs. However, this class is enough to describe and to study real
and theoretical processes
and to use these software
tools for
practical applications. The characteristics nents of
have two aspects.
First,
they are compo
the net PSI-function. It is a definition of the way for cal
culating token characteristic values. Second, they are the list of va lues, which a certain token gains when it comes into
a certain place.
The following types of characteristic values are supported: - a whole number;
344
CHAPTER 5
- a real number; - logical values TRUE and FALSE; - a string; - a list, containing an arbitrary combination
of
values of
the above types; - a matrix of whole or real numbers. Conditions of transitions The conditions of
net transitions
are
an indexed matrix
of
predicates (see App. 2). The matrix elements are indexed by transition input and output place
identifiers.
Each predicate of these matrices
corresponds to one of the transition arcs. The values of
the predica
tes are calculated at each step of the model simulation for each token in the corresponding input place. Predicates may sions which return the values TRUE or FALSE. ons between all net attributes.
be arbitrary expres
They may include relati
A very important limitation
is
that
the predicates cannot depend on further events in the net. Type of transition The type of transition is a logical expression. It assigns the necessary set
of
tokens at the transition input
places for
regular
work of the GN. If the type of transition returns TRUE, then the tran sition has been activated. Global memory The global memory includes; (a) initial token characteristic values; (b) characteristic functions,
which act on
each token at
the places
corresponding to them; (c) the number of
the characteristic
token is in the net.
values which
are kept when the
GENERALIZED NETS AND COMPUTER SCIENCE
Wien the token is at a net's characteristic
values.
net. It allows the
output place,
345
it owns a list
of
The list corresponds to the token path in the
process to be studied by the history of its compo-
nents represented by tokens in the GN model. Hierarchical models based on GNs GNs allow one to create hierarchical models of the studied pro cesses. It is useful for their complex study. of subnets.
A GN may contain
a set
In the primary GN these subnets are represented by places
or transitions. If the subnet has only one input and one output, it is represented by a place. outputs,
If the subnet has
several inputs and several
it is represented by a transition.
If a token comes
into a
certain subnet input, then the subnet simulation starts. The simulati on of the primary net and its
subnets is synchronized by time towards
an absolute time scale. The subnets can use information for the events in the primary net and other subnets tion.
As a result the tokens gain
during the time of their simula
the result of the subnet action as
a characteristic. As a further development be
other PNs extentions,
it is provided
that
the subnets may
as Predicative/Transition Nets
or Coloured
Petri Nets for example or Neural Networks. This fact means: - models, based on different PNs'
extentions may be compa
red by their actions and results during the time of their simulation; - models, based model;
already built, may be included as parts of
- if a Neural Networks
based model
based model, then each Neural Network
is a subnet of the GNs
may be taught and it may define
the action of the primary net,
§ 5. 6. 3. Tools for , 1 2 where a) LJ
and
L"
are finite, non-empty sets of places (the transition's
input and output places respectively); J
1.1 they are L'= (1 , 1'
1 b) t
2
for the transition in Fig.
1'J and L" = II", ]",...,]"].
1
m
2
n
is the current time-moment of the processes's firing; 1
c) t is the current value of the duration of its activity; 2 d) r is the transition's condition determining the
349
tokens which shall
350
APPENDIX 1
Fig. 1. 1. transfer from the
transition's inputs to
its outputs;
it has the
form of an index matrix (see App. 2): 1" . . . 1 1'
r
-
1
V i : 1' m
(i, j)
1" . . . 1" j n
I | I I I I I I I
denotes the element
r i,j (r
- predicates) i. j i,
( l i i i m l i j i n ) which corresponds to the
i-th input and
j-th output places; these elements are predicates and when the truth value of the
(i, j)-th
element is true,
the token from
place can be transferred to j-th output place; otherwise, possible;
i-th input it is not
REMARKS ON THE GENERALIZED NETS
351
e) M is an index matrix of the capacities of transition's arcs: I"
1
1' M
I' m f) D
.
.
1" . . .
j
1"
n
1 1 1
: 1' i
=
.
is an object having
1 m 1 i,j ; I 1 (m i 0 - natural lumbers) 1 i, j 1 | |1 U U 1 i j ( n| a form similar to a Boolean expression.
In
it the variables are all symbols which mark the transition's inputs nemes, and the Bollean operations "A" and
"v" determine the
fol-
lowing conditions: Ml
i
.1 1
,...,1 ) - every place 1 , 1 i i i i 2 u i 2
1 i
must
contain
u
at least one token, v(l
i
, 1 , . .., 1 i i 2 u
1
) - in all places 1 , 1 i i 1 2
1 i u
tain at least one token, where (1 , 1 , . . . , 1 i i i 1 2 u
must contain
) c L'.
The ordered four-tuple O
K
E = > A L 1 2 K K is called a Generalized Net (GN), if: a) A is a set of transitions; b) n A
is a function giving the
priorities of the transitions,
i. e. ,
TI : A -> N, where N = 10, 1, 2, . . . ) U too); A c) TI L
is a function
L -> N,
giving the
priorities of the places,
where L = pr A U pr A, and 1 2
i. e. , TT : L
pr X is the i-th projection of i
the n-diroensional set, where n € N, n i i and 1 N; e) f is a function which calculates the truth values of the predicates of the transition's conditions
(for the GN described here, let the
function f have the value "false" or "true", i. e. , a value from the set
(0, 1).
In § 3. 1,
we shall
describe other
types of nets in
which this function will have a value in the interval [0, 1] or in the set [0, i ] x [0, 1 ] ); f) © is a function giving the 1
next time-moment
when a given transi-
* © (t) = t', where t, t' € [T, T + t ] 1
tion can be activated, i. e. ,
and t £ t'. The value of this function is
calculated at the moment
when the transition ends its functioning.
Below we shall discuss a
special representation of this function; g) © 2
is a function giving the
duration of the activity
of a
given
K
transition, i.e., © (t> = V , where t € [T, T + t ] and t' £ 0. The 2 value of this function is calculated at the moment when the transi tion starts its functioning.
It has
another representation as the
above one; h) K is the set of the GN's tokens. In some cases, it is convenient to consider this set in the form K = U I 1€Q where K
K , 1
is the set of tokens which enter the net from place 1, and 1
I Q i) TT K
is the set of all net's input places; is a function giving the
priorities of the tokens,
i. e. ,
TI : K
K -> N; j) 9 K
is a function giving the time-moment when a given token can en-
REMARKS CM THE GENERALIZED NETS
353
ter in the net, i. e. , 6 (a) = t, where K a 6 K,
*
t € [T, T + t ] ; K) T
is the time-moment when the GN starts functioning.
This moment
is determined about a fixed (global) time-scale; o 1) t is an elementary time-step, related to the fixed scale;
(global) time-
M
m) t
is a duration of the functioning of the net;
n) X
is the set of all initial characteristics
which can receive the
tokens when they enter the net; o) $
is a
characteristic function which gives new characteristic to
every token when it makes a given transition.
a transfer from input to output place of
As some of
the above functions
this function
can be represented in another form, shown below. p) b
is a function giving the maximum number of characteristics which
can receive a given token, i. e. , b: K -> N.
If for a certain token
a, b(a) = 1, the token will enter the net with an initial characte ristic (as a zero-characteristic). After this, it will receive only the characteristic of the previous. When b(a) = a>, the token a will receive all possible characteristics. When b(cc) = k < co, except its zero-characteristic, the token a will keep the last
k
as its cha
racteristics (previous characteristics will be "forgotten"). Hence, in general, every token a has b(a) + 1 characteristics. We must note that this definition similarly to
the ordinary ON
definition is not fully formalized, because if we fully formalize transition
conditions and the characteristic functions
the
of the GNs, a
smaller class of GNs will be obtained. It is convenient
to assume
that the functions
f, 6 , © i 2
and
354
APPENDIX 1
$ have the following forms, also: IAI f = U f , where f calculates the truth-values of i-1 i i the i-th
GN transition
the predicates of
(as we shall see in
§3.1,
they
cannot have values "true" and "false"; they can have fuzzy or intuitionistic fuzzy transition
conditions
(see can
App. 2)
be
values;
calculated
different
by
different
IAI i i 6 = U 9 , where 9 calculates the next time-moment of the 1 i= l 1 1
activati-
ways);
on of the i-th GN transition; IAI i i Q - U 0 , where 9 2 i=l 2 2
calculates the
duration of the active state
of
the i-th GN transition; ILI $ = U $ , where $ calculates the i= l i i
tokens'
characteristics,
which
they will receive in the i-th GN place. A given GN may not have some of the components. Then the places of such components will be marked by "*". The GNs with such form gene rate a special class of GNs, which we shall describe in § 2.1, The static part of a given GN is determined by the set pr
{i
i i
A, where for a given n-dimensional set X (n £ 2) i, 2, 6, 7 K pr X - n pr X i ,i i j=l i 1 2 k j
in, l i j i k , j
of
the elements of
i i- i JJ J"
for j' t j"),
a GN is determined by input and output
i.e. the static part
transition places,
dex matrix of the arcs and by the type of the transition. cal character of the net comes from the GN's tokens and ons conditions
(pr A), 5
by in
The dynami the transiti
the temporal character comes from
the compo-
REMARKS ON THE GENERALIZED NETS o * nents T, t , t and from the elements of the set pr components $, X and
b
355
A. 3,4
Finally, the
play the part of the GN's memory.
The different functions are also related to these four GN com ponents: functions TT , n , c to the static structure; f, IT to the dyA L K namical elements; 9 , Q and 6 to the temporal components. 1 2 K One very important restriction for
the GNs
is the
following.
The transition condition predicates cannot be related to future events for the GN. X K
K
§ 2. Reduced GNs For two subsets £' and E"
of E - the class
of all GNs
define (see also definitions of the relations n, n , s, * E' t- £" iff the functioning and
x
in
let us § 5:
*
the results of the work of every ele
ment of E" can be described by some element of E'; E' i- E" iff "HE* H E"), E' H E" iff <E' i- E") &
T
T
, 2 K
where d - [ f , t"] and T s t' i t" < T + t . When
the tokens
in the
input places of Z are already enough to satisfy the transition type
D
and TIME € a, the transition Z will be fired. Let the GN E be a net which has at least one transition of the form described above. For this the function © is extended to function 1
362
APPENDIX 1
9 , which to every time-interval d = [V, t"] 1 1 1
juxtaposes a new timeK
interval [t\ t"j 2 2
for which (T i)
t" i t' i t" i T + t , 1 2 2
i.e. ,
the
next time-interval for firing of the transition Z will come not earli er than the moment when this transition ends its former firing. GNs with a complex structure The transition forms of GNs are graphically similar to those of the E-nets. For the Petri nets the graphical structure is not the same as for GNs.
It is proved
that if a constructed GN which
than one arc entering each of its places
allows more
and more than one arc coming
out of each of its places, then this extension of the GNs will also be conservative and there it is called "a GN with a complex structure". GNs with global memory In the modelling
of real processes it
is appropriate
to keep
data (while the GN is functioning) or to determine the values of ferent parameters related to these processes.
dif
Thus, the definition of
the concept GN can be extended with the addition of a new component B, which we shall call "global memory". The component
B can be seen as a
list of such functions that will change their values in the process of GN functioning.
Let a GN with component B be called "a GN with global
memory" (GNGM). The formal definition of this object is: *
E
O K
= , where all components are as in § 1,1 and the component B is as descri bed above.
REMARKS O N THE GENERALIZED NETS
363
GNs with optimization components Some GN's models correspond to real processes
whose functions
do not stop and which determine standard (given before) the process's continuation. components
solutions for
F o r these G N s it is necessary to add such
that secure their non-stop functioning.
For this aim,
we
shall construct a new extension of the ordinary GNs - GNs with additi onal clocks (GNACs). Itiey have transitions with the form: Z - < L \ L", t , t , r, t , r', M, Q>, 1 2 3 where the components conmon with the ordinary transitions are the same and a) r' is a (0, 1)-index matrix with the form 1" . . . 1
1" . . . j
1" n
1' I 1 1 I I : 1' i
r' =
I r' I i, j I I (r € (0, 11) I i. j I I (1 S I $ m, 1 < J ,j i n)
: 1' m where L' = IV,
1 b) t
1 J ) and L" = 11", 1"
1'
2
is the maximal
1
m
,
1"),
2
n
time-duration for a check
of the truth
of the
3 predicates of the transition condition
r. In the case that time is
increased the checking of the predicates of
r shall be reduced and
their respective values from the matrix r' will be assigned. Obviously, t 3 the predicate's
s t , but if 2
truth can
t 3
> t 2
the process of checking of
be stopped before
the time necessary
for
364
APPENDIX 1
this tume-moment. If A is the set of the GNAC's transitions the GNAC has the form o E
=
t
3 does not receive new value.
+ t , 1
the compo-
2
3 b) G
is a function which to the predicates of r Juxtaposes (0, 1)-va
lues by an advance determined way.
Its calculations are made inxne-
diatelly after each checking of the values of the function © . 3 c) #'
is a function which
tokens which
determines the new characteristics
enter the transition's
output places
of the
after applying
the specifics of the new nets mechamism,
» x
§ 4-. GNs and other
it
objects
In this chapter,
it is proved that the functioning of
the re
sults of the work of the different types of modifications of the Petri nets and of the finite automata and the Turing machines can be sented by GNs.
A GN which makes the same for each
repre
ordinary Petri net
REMARKS ON THE GENERALIZED NETS
365
is constructed. •
§ 5. Algebraic
aspect
of the theory
it
of OVs
For the two t r a n s i t i o n s Z and Z we s h a l l 1 2 Z = Z i f f ( v i : 1 i i £ 7 ) ( p r Z = pr Z ); 1 2 i 1 i 2 Z c Z 1 2
iff
define
(Vi: 1 £ i £ 2)<pr Z c pr Z ) & i 1 i 2 (Vi: 3 £ i £ 4 ) ( p r Z = pr Z ) & i 1 i 2 (Vi: 5 £ i £ 6>(pr Z c i l l
pr Z ) & (pr Z c l 2 i l 2
pr Z ), 12
where c
is a relation of inclusion over index matrices and if 1 A = (X , L , la I], B = [ K , L , 1 1 i, j 2 2 A C B iff 1
c
(K 1
lb I], then i,j
c K ) & (L C L ) & (Vi€K >(Vj€L ) ( a 2 1 2 1 l i ,
= b ) j i , j
i s a r e l a t i o n of i n c l u s i o n over Boolean e x p r e s s i o n s and, f o r two of 2 these expressions a and b a c
b iff it is getting the expression
a after removing
the argu-
2 merits of b, which are not arguments of a and the logical operations, associated to them Over the transitions Z 1),
i i i i i i i = < L , L , t , t , r , M , D > i 1 2 1 2
we shall define three operations.
It is necessary for
(i=l,
the follo
wing to be valid: if place 1 € pr Z n pr Z and 1 € pr Z , then 1 € 1 i 2 i s 3-i pr Z f o r l £ i £ 2 , l £ s £ 2 . These o p e r a t i o n s are 3-s 3-i
366
APPENDIX 1
a) a union
(the necessary conditions
for this operation are
(j = 1,2), and (as if 1 e pr Z , this is s i pr
not possible),
1 2 t - t j J that
1€
Z for 1 i x i 2, 1 an i n t e r s e c t i o n (with the above c o n d i t i o n s ) : 1
Z HZ 1 2
=.
c) a composition (with the above c o n d i t i o n and w i t h t h e c o n d i t i o n 1 2 L nL 1 1
= L 2
Z
=, where D results from D after removing all its arguments, whose iden1
tifications are elements of the set L 1 It is possible that L Z
o o n Z = Z , where Z 1 2
2 n L 1 1
2 n L. 2 1
- f> and
L
1 2 n L = . In this case 2 2
is the empty transition
(for
some other coro-
ponents as M, r, Q will be degenerated). "Hie operations described below are unique in the Petri net the ory.
Tliey can be transfered on practically
nets
(obviously with some modifications
the corresponding
nets).
all other types
related to
These operations are
of Petri
the structure of
very useful
for con
structing GN models of real processes. Before introducing the different GN's operations, we shall for-
REMARKS ON THE GENERALIZED NETS
367
mil ate same conditions which the arguments of these operations (diffe rent GNs) must satisfy. We shall assume that
if a place
takes part
simultaneously in
two nets, then in both locations the place has (a) equal capacities, (b) the same characteristic functions (c) equal possibility to accept tokens net
from K,
(which is a model of some process)
ments of the set
K' c K
could pass
i.e. if
in the first
the tokens which are ele
through the examined place,
while in the second net tokens which are elements of the set
K" c
K could pass through this place, then K' = K", (d) equal values of the priority functions, (e> one and the same numeration/notation. Similarly,
we shall expect that if two places are respectively
an input one and an output one for two transitions in both nets, then (a) the capacities of the connecting arcs are equal, (b) the time for passing from the input to the output place
should be
one and the same, (c) they should be in the same consequence, i. e. the place which is an input
(output) one in the first of the transitions
should be the
same in the other transition too. Let E
and E 1
be two GNs and let for i = 1, 2: 2
i i i i E = , ))>, i i 1 2 1 2
max (T iixi2 i <X 1
U X , $ U $ , b U b >>, 2 1 2 1 2
where A
U A 1
= 2
2 U ( Z / ( Z € A ) & (V Z' i=l i 2 U fZ/(3Z' i=l
€ A
) (Z D Z'
o = Z )) U
3-i
o € A )(3Z" € A ) (Z' D Z" ?! Z > & (Z = Z' U Z " > ) . i 3-i
A composition of the above nets will be called the object: (E , if T + t < T 1 1 2 2 1 E o E = 1 1 2 \ x E , i f T / ( 1 " € L")
M, D > / 1 "
and 1 € 2
E l " l | D
& ,
K
K
t , t , r, 1 2
o * ,
M, D> € A ) ] > ,
369
71 , TI , c, A L
f,
9 , 9 >, 1 2
>.
I t can be e a s i l y seen, t h a t the s e t s f, d) t
- a real number which corresponds to the length of a time-inter2
val in the above mentioned time-scale, e) r
- an index matrix (see App. 1) having the form 1" . . . 1" . . . 1" 1 j n 1' 1 r
=
1' i : 1J m
(i, j)-th
element of
I I I I l I I I l
r i,j (r
- predirates) i, J
(i i i i m, 1 S. j S n)
which correspond to the
output places, these elements being predicates, f) M - an index matrix having the form
i-th
injwt and j-th
372
APPENDIX 1
1" .. .. .. 1 1' 1'
M = M
1" .. .. .. 1" jj n
II 1 II
1' 1' i i : : 1' rm m
I I ID m l l i,j i,j II I (m i O I 0 -- natural natural lumbers) numbers) II i,i, jj I iI (l£i£m,l£j*n) I ( l £ i £ m , l £ j * n )
g) Q is an object having a form similar to an Boolean expression. variables are exactly the symbols which marK the names; the Boolean operations
"A" and
"v"
Its
Z's input places'
determine the analogo
us conditions from § 1 and 1)A(1
, ] ,...,1 ) - every one of the places 1 , 1 i i i i i 1 2 u 1 2 contain at least one token,
2) v(l
,1 i 1
,..., 1 ) - all the places 1 , 1 ,. . . ,1 i i i i i 2 u 1 2 u
at least one token, where
1 i
must u
must contain
(1 , 1 ,...,1 ) c L'. i i i 1 2 u
The object described above will be called a transition. The inductive definition of the concept GN is as follows: (1) The object which has the form of a transition
is called a GN,
if
to it are added a) TT - a function giving a natural number, being the priority of the A transition, b) n - a function giving the priorities of the transition's places, L c) c - a function giving the capacities of the transition's places, d) f - a function
which calculates the truth values of the predicates
of the index matrix r; e) ©
- a function giving 1
the next tune-moment
when the
given tran-
REMARKS ON THE GENERALIZED NETS
373
* sition can be activated, i.e. © (t) - t', where t, t' e [T, T+t ] 1 and t i t ' . The value of this function is calculated
at
the mo
of
a given
ment when the transition ends its functioning; f) © 2
a function
giving the duration of
the activity
transition, i. e. , 6 (t) = t\ where t e [T, T + t ) and 2 The value of this function is calculated
at the moment
t' z 0.
when the
transition starts its functioning; g) K - a set of tokens; h) ii - a function giving the priorities of the tokens (IT : K -> N); K K i) © - a function giving the time-moment when a given token can enter K the net, i. e. © (a) = t, where K a e K, K
t 6 [T, T + t ]; j) T - a time-moment
when the GN starts functioning.
This moment is
determined on a fixed time-scale and the first value of t = T; 1 o k) t - an elementary time-step related to the fixed time-scale; K
1) t
- a duration of the functioning of the net;
m) X - a set of all initial characteristics which can
receive the to
kens when they enter the net; n) § - a characteristic
function
which gives
new characteristic
to
every token when it makes a transfer from input to output place
of
a given transition; o> b - a function
giving the maximal number of characteristics
which
can receive a given token (i. e. b: K -> N) transfering in the net. (2) If E, E
and E 1
a) E
U E 1
are GNs, then 2
is a GN, 2
APPENDIX I
374 b) E o E is a GN, 1
2
c) Ex is a GN. THEOREM 5, 3, t: The two definitions of the concept "GN" are equivalent. K
» J 6. Topological
»
aspect of the theory of GNs
The graphical structure of a given GN is a two-coloured graph. Thus the study of such a structure and checking of the properties of a GN following from these structures are important for the theory of the GNs.
Some operators which assign to every GN one- or two-coloured
graph and some other operators, which juxtapose to every GN one natu ral
number corresponding to the complexity of the net with respect to
a certain criterion, are defined, The class £ is ordered by relations, generated from the above two types of operators, and some topological properties of £ are studied. x K
j 7. Logical
*
aspect of the theory of GNs
Logical operators which are similar to the modal operators "ne cessity"
and "possibility"
are defined and some of their basic pro
perties are given. * *
x
§ 8. Operator aspect of the theory of GNs The operator aspect has an important place the GNs.
in the theory of
Six types of operators are defined in its framework.
Every
operator assigns to a given GN a new GN, which has some preliminaryly, introduced properties. Particular types of operators are: global (one
REMARKS ON THE GENERALIZED NETS
which
transforms globally
transforms
some
some transition
375
GN's components),
components
local
of a given GN),
(one which
hierarchical
(one which substitutes a place or a transition for some part of
a gi
ven GN, or makes reversal), reducing (one which removes some of the GN components), extending (one which transforms a given GN to the corres ponding extending GN)
and dynamical (one which determines
rent strategies for the tokens' transfer
in the net).
the diffe
The basic task
of this aspect of the theory of the GNs is a research on the connecti ons and influence between the separate operators.
* § 9. Other extensions of the 0Vs By analogy with the universal Turing machine, versal GN (UGN) as a GN E
the object Uni-
for which the following is valid: K
(V E € £)(E c is intriduced.
E ),
It is noted that at the moment of writing of [K] (NOV.
1990) , the question for the existence of an UGN is open. Let Q be a some fixed time-scale. For T' e Si we define 2 = I E / E E J SprprE tprprEST'!. T' 1 3 3 3 K
We shall call the GN E
for which the following is valid, K
(V E € £
)(E c
T'
E ),
it
a T'-UGN about the time-scale Q. Obviously, the global time-components of O
E
=
. X
K
THEOREM 9. 1. 2: There exists a UGN for the class £ . M
Let E be some GN. Let K be a set of tokens of E, and X be a set of initial
characteristics for these tokens of K.
The object
"Self-
Modifying GN" (SMSN) is constructed. x Let K
be a set of tokens which will be called
The defining of this set is possible in the
"control
framework
tokens".
of the GN the-
x ory.
The tokens of K
can be interpreted as GN-tokens
with initial
K
characteristics
(elements of the set X ) which contain the identifier
"control token"
(this is also possible).
frame of a given GN places
(in which
characteristic
E
is that
there will be
The second addition in the
the characteristic functions of transferred control tokens)
some
give as
identifiers and parameters of some types of operators,
defined over the GN.
This is also not in contradiction with the defi
nition of GNs. Finally (really, the new): when a certain control token gets as characteristic the identifier of some operator with parameters former
or current characteristics of
this or
another control token,
then at this time-moment over the GN E the corresponding operator will be realized. Obviously, not-every operator can be applied to the net
in the
REMARKS ON THE GENERALIZED NETS
377
time-interval when it functions. Thus, we can define the SMGN as a GN, for which there exists a mechanism p for realization of the over it.
This mechanism functions in the following way:
moment when a certain control token of a given ristic the identifier
of a certain
operators
at the tiroe-
GN gets as a characte
(added) operator and the
values
of its parameters, the mechanism p realizes this operator. The basic aims in [») are: 1. to show the operators which can be realized by p; 2. to show the relations between GNs and SMGNs.
and £
If £ is the class of all SMGNs, then obviously J c J SMQN SMQN H £, because every GN is a SMGN with an empty set of con-
SMGN trol tokens. THEOREM 9. 2. 1: For every SMGN, there exists a GN which represents it. The SMGNs are the basic aim of the operator aspect of
the the
ory of GNs. x
* § 10. Methodological
aspect
of
»
the theory
of GNs
The b a s i c ways for c o n s t r u c t i n g o f a GN of a given r e a l process and same ways for u s i n g and modifying of already c o n s t r u c t e d GN-models are d i s c u s s e d . X *
§ 11. Open
X
problems
There are many unsolved
or unforroulated problems in the theory
of GNs. Here we shall introduce some of these problems which are rela ted to the text of the book. 5. To research the different classes of reduced GNs without two or mo re components, and to show the relations between these classes.
378
APPENDIX 1
7. To prove that the constructed
reduced GN in § 2. 2 from the book on
GNs (from § 1. 1.2) has the least possible number
of different GN's
components, or to give a counter example of this. 8. What other conservative extensions of the GNs can be defined? 9. To construct GNs which are universal for the classes (sets) of the different
Petri net modifications,
by analogy
with the
GNs from
§ 4. 3. 12. To construct an algorithm which for every set of transition phical structures
(obviously they must satisfy
some
gra
conditions)
and for every given GN, construct a GN for which: - all its transitions have
the forms, which are elements
of the
given set, - both nets have equal tokens and as a result of their
with equal
initial characteristics
functioning equal tokens receive equal
final characteristics. 19. To prove or disprove the existence of a universal GN for £. 20. To construct other extensions of the GNs and to prove
or disprove
their conservativness. 21. Chapter 10 is a superficial attempt to give the basics of
the m e
thodology of work with GNs. To continue and enrich the research in this area. 22. A list of the basic applications to this moment (Nov. 1990) is gi ven in Chapter 0. To extend the area of the GN's applications.
Appendix 2: GENERALIZED INDEX MATRICES
A definition of "generalized index matrix" (GIB) or "index mat rix" is given in [1-3]. Here we snail introduce only the results which are related to this basic text. Let bers.
I be a fixed index set and H be the set of all real num
A GIH with generalized index sets K and L
(K, L c I) will be
called the object: 1 1
k
1
k 2
k m
1
1 2
|a i kK , 1l
1
1
a k ,l
1
. . .
1 n
. . .
2
a k , 1
I n
la a . . . I k ,1 k ,1 I 2 1 2 2 I I . .
a k ,1 2 n
I a a I k ,1 k ,1 | m 1 m 2
a k ,1 m n
< =
[X, L,
.
fa ]]) k ,l i j
),
where X = Ik , k , . . . , k 1, L = 11 , 1 , . . . , 1 !, for l ! i ( n an
1
2
1 2
m
for 1 i j i n: a
n
€ K. Let H be the set of all GIHs with index k , 1 X, L i j sets X. and L, and let H = U H . K, Lcl
K, L
379
APPENDIX 2:
380
In [31, for the GIMs A = [K, L,
(a
k ,1 i j
1], B = [P, Q, (b
a r e defined ordinary matrix o p e r a t i o n s "addition" on"
t ,v u w
/a / k ,1
"miltiplicati-
! ] . where
, if t
- k
u
k
1
b
v "i
t , u w
a
, if t
p ,q r s
t + b
K ,1 i j
p ,q r s
v0
t ,v u w
-
a
k,l i j
, if t
. b
P,q r s
t ,v u w
u
= k
€L-Qor
j
€ K - P and v = 1 e L w j
i
€ P - K and v
r
= p
i
r
6 K DP
w
€Q-Lor = q
and
e Q
s v
= 1 w
= q j s
na
= k i
u
The b a s i c p r o p e r t i e s of
= p r
e K D P and v
=1 w
= q € j s
L n Q; fc
t
u
v
!],
where
w
if t
1 b 1 p ,q r s E a .b 1 =p 6LTP k , 1 p ,q j r i j r s
,°
=1
1], where
, for t
(v. t ,V u w
= p
u
w
= p € P and v = q r w s
e L
A . B = [K U (P-L), Q U (L-P),
c
u
€ K and v
i
, otherwise
A x B = [K n P, L n Q, fc
c
!)
and a l s o the f o l l o w i n g o p e r a t i o n s
A + B = [K U P, L U Q, fc
c
and
p ,q r s
if t
if t
u
u
u
= k
= p
i
r
€ K and v
w
=1
e P - L and v
w
j
€ L - P
= q
s
e Q
= K € K and v = q € Q i w s
otherwise the s e t M w i t h t h e above o p e r a t i o n s are
GENERALIZED INDEX MATRICES
381
studied in [3]. There some GIMs measures are introduced (for GIMs de terminants do not exist). Ihe given mathematical
apparatus may be
with elements from the sets (0, li,
applied to
the
GIMs
[0, 1), or from the class of all
predicates, etc. In the first two cases, the operations " + " and ". " in R will be substituted by
"max"
and
"min" respectively,
and in the
third case - by the operations "v" and "*". REFERENCES: [1] Atanassov K.
Conditions in
Generalized
nets,
Proc. of the XIII
Spring Conf. of the Union of Bulg. Math. , Sunny Beach, April 1964, 219-226. [2] Atanassov K. Dynamical elements in the Generalized nets,
AMSE Re
view Vol. 1 (1985), No. 4, 1-9. [3] Atanassov K. Generalized index matrices, Comptes rendus de demie Bulgare des Sciences, Vol.+0, 1987, No. 11, 15-18.
l'Aca-
Appendix 3 : INTUITIONISTIC FUZZY SETS
Following [1,2] we shall give short remarks on
the intuitioni-
stic fuzzy sets (IFSs) and other intuitionistic fuzzy objects
related
to them. Let a set E be fixed. An IFS A«
in E is an object having
the
form a A
: |/X€E),
A where
the functions
A
y (x) : E -> [0, 1] A
fine the degree of membership
and
t (x) : E -> [0, 1] deA
and the degree of non-membership of the
element x€E to the set A, which is a subset of
E,
respectively,
and
for every x€E 0 i \i <x] + T (x) i 1. A A Obviously, every ordinary fuzzy set has the form |<x, v (x), 1-p <x)>/x€E]. A A 11
For simplicity, below we shall write A instead of A . For every two IFSs A and B, the following relations tions are valid: A c B iff (VX€E) <JJ (x) /xeE); A A A D B = (<x, minlp (X), p (x>), max(T (x), A B A
t
(x))>/xeE);
A U B = (<x, max(p (x), p ( x ) ) , min(T (x), A B A
t (x))>/x€E); B
B
A + B = (<x, p (x)+p ( x ) - p (x). p (x), T (X). T (x)>/x€E); A B A B A B A . B = [<x, p (x). p (x), A B
T (x)+T ( X ) - T (x). T (x(>/xeE); A B A B
A - B = A fl B. The concept IFS is the basis for the definition of the concepts Intuitionistic Fuzzy Propositional Calculus
(IFPC)
and such extensi
ons as IF predicative calculus, IF modal logic, etc. (see [3-5]). To each proposition (in the classical meaning) its truth value: truth
denoted by 1, or falsum by 0.
one can
assign
In the
case of
fuzzy logics, this truth value is a real number in the interval and can
be called "truth degree" of a particular proposition.
case of
IFPC is
added one more value - "falsum
[0, 1] In the
degree" - which will
be in the interval [0, 1] as well. Thus one assigns to
the proposition
p two real numbers p(p) and T(p); moreover the following constraint is valid: p(p> + T(P) i
i.
Let this be done by an evaluation function V defined so that V(P) = . Hence the function V: S — >
[0, 1] x [0, 1] gives the truth and
falsuni degrees from the class of all propositions and it
can be defi
ned so that it assigns to the logical truth T: V(T> = , and to the logical falsum F: V(F) = . Ibe negation np of the proposition p will be defined through
384-
APPENDIX 3:
V H P ) = .
When T(P) = 1 - p, for np we get V(np) = , p(p)>, which coincides with the result, e. g. , from [6]. Depending on the way of defining, of the operation
"3"
can be
obtained by different variants of IFPC. One of them is as follows: V(p & q) = <min(p(p), p(q)>, max,
V(p * q> = <max(p(p), p(q)), min(T(p),
T,
V(p 3 q) = <max(T(p), p(q)), min(p(p>,
r(q))>.
A given propositional form A propositional form; if
(c. f.
[7):
each proposition is a
A is a propositional form, then iA is a propo
sitional form; if A and B are propositional forms, A D B are propositional forms)
then A & B, A * B,
will be called a tautology,
iff
V(A) = , for all valuation functions V,
and an intuitionistic fuzzy tautology,
iff "if V(A) = , then a i b". There exist some forms of an intuitionistic fuzzy
Modus Ponens
(see [3,5]). The definition for the quantifiers is as follows: V(VxA) = <min p(A>, max T ( A ) > x x and V(3xA) = <max p(A), min T(A)>. x x For the proposition p for which V(p> = , in [4] are defined the following operators: V(Qp) = ,
INTUITIONISTIC FUZZY SETS
385
V(Op) = , vdiich are analogous to the nodal operators "necessity" and "possibili ty". REFERENCES: [1] K. Atanassov,
Intuitionistic fuzzy sets,
Fuzzy Sets and Systems,
Vol. 20 (1986), No. 1, 87-96. [2] K. Atanassov,
More on intuitionistic fuzzy sets.
Fuzzy
Sets and
Systems, 33, 1989, No. 1, 37-46. [3] K. Atanassov,
Two variants
of intuitonistic fuzzy
propositional
calculus. Preprint IM-JFAIS-5-88, Sofia, 1988. [4] K. Atanassov,
Two variants of
intuitionistic fuzzy
modal logic.
Preprint IM-MFAIS-3-89, Sofia, 1989. [5] K. Atanassov, G. Gargov, Intuitionistic fuzzy logic.
Compt. rend.
Acad. bulg. Sci. , Tome 43, N. 3, 1990, 9-12. [6] A
Kaufmann,
Introduction a la theorie des sous-ensembles
flous,
Paris, Mas son, 1977. [7] Mendelson E. , Introduction to mathematical logic, D. Van Most rand, 1964.
Princeton,
NJ:
LIST QF AUTHORS
Alexander Georgiev - Institute for Microsystems, Sofia Alexander Savov -
Institute for Microsystems, Sofia
Antoaneta Eirova - High Economical Institute, Sofia Antonia Dimitrova - Institute for Microsystems, Sofia Bistra Nikolova - Institute for Microsystems Borjana Jordanova - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Evgeni Tzolov - Institute for Microsystems, Sofia Ewgeni Dimitrov - High Transport School, Sofia Ilya Eazalarsky - Institute for Microsystems, Sofia Ivan Hristozov - Institute for Microsystems, Sofia Joseph Sorsich - 2-nd City Hospital, Sofia Katja Stefanova - Institute for Standardization and Sertification, Sofia Erassimir Atanassov - Institute for Microsystems, Sofia Lilija Atanassova - 105 Al. Dimitrov School, Sofia Ljubomir HadJyisKy - Technical University, Sofia Luomila Dimitrova - Institute of Chemical Engineering, Burgas Maria Stefanova-Pavlova - Institute for Microsystems, Sofia Martin Tetev - Faculty of Mathematics, El. Ochridsky Univ. , Sofia HiKo Pehlivanov - Institute for Microsystems, Sofia Favlin Gyurov - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Peter Georgiev - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Flamen Fetkov - HEFTCCHIM Petrochemical Combine, Bourgas Radosvet Todotov - Centre for Scientific Information of Bulg. Acad. of Sciences, Sofia Kossen Petrov - Institute for Microsystems, Sofia Rumen Christov - Institute for Microsystems, Sofia Sergej Bedev - Technical University, Sofia
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367
Stanislav Pishmanov - Faculty of Mathematics, Kl. Ochridsky Univ. , Sofia Stefan Stefanov - Institute for Microsystems, Sofia Stela Dimitrova - NEFTOCHIM Petrochemical Combine, Bourgas Stojan Mihov - Faculty of Mathematics, Kl. Ochridsky Uhiv. , Sofia Stoian Garbov - Central Laboratory of Automation, Sofia Trajana Kolarova - NEFTOCHIM Petrochemical Combine, Bourgas