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An algorithm for the diagnosis of deadlocks
AN ALGORITHM FOR THE DIAGNOSISOF DEADLOCKS* V. S.AROLOVICH and V.M.MIKHELEV (Received
6 January
19721
THIS paper is devoted to the analysis of deadlocks, that is, situations some of the problems being solved by a computer in the multiprogramme cannot
continue
their solution
on account
of interdependencies.
in which mode,
It is proved
that if the dependencies of the problem are represented by a system of equations and their left sides are multiplied, the problems forming a deadlock can be determined from the form of the product obtained. In terms of the matrices constructed possible
from this system of equations, an algorithm is formulated, making to find the minimal sets of problems whose completion by special
of exclusion from the number to be solved removes The discussion is illustrated by examples. In the solution locks may occur; cessed
cannot
other problems
of problems these
on a computer
are situations
be solved or because
the deadlock
from the system.
in the multi-programme
where some of the problems
further because of the absence
of the expectation of the necessary
it means
mode deadbeing
of signals
pro-
from
resources.
In II-31
algorithms were given which for some conditions enable deadlocks to be predicted, and if these conditions are not satisfied they enable the set of problems causing the deadlock to be indicated. In this paper we consider
an algorithm
situations,
where the connections
by Boolean
functions
between
or in an equivalent
for the diagnosis the problems mathematical
of deadlock
are described form.
explicitly
Such an explicit
description enables the solution of the problem to be simplified, requiring a more detailed analysis of the connections, such as a search for the minimal subset of problems which, because of the inclusion of special alarm devices of the operational system or their exclusion, leads to the resolution of the deadlocked situation, and also to an explanation of the deficient resource and the possibility *Zh.
uychisl.
of replacing
Mat. mat. Fiz.,
it.
12. 6, 1543-1553,
213
1972.
V. S. Arolovich
214 We consider
the set of problems
and V.
2 = IT,,
programme mode.
As was shown in Ill,
may be suspended
because
expectation
of explicit
of resources)
M. Mikheleu . . . , Tk! being solved of some of these
(the expectation
of signals)
dependencies
resource
because
R occupied
be continued possibility
of the fact that,
by problems
T,,
only after obtaining of continuing
explicit
and latent,
no direct
effect
T,
firstly,
and T,,
some signal
(the
and secondly,
on whether
where the dependence
and that on T, and T,
of problem T,
it requires
is latent
units of some its solution
from problem T,.
problem T, depends
T,, T, and T, can be continued,
problems
or latent
on other problems.
Example 1. Let Z = {T,, T,, T,, T,, T, 1 and the solution have been suspended
in the multi-
the solution
the solution on problem
only;
can
Then the of problems
T, is both
problem T, exerts
on T,.
The cortege
(Tall
. . . , TaN) of problems of the set Z is called
a
basic
string if for each problem To, 1 ,( l< N, one of the following two conditions is is solved; (2) the completion of some problems satisfied: (1) problem T T a: ., l,(
TaJ (it is considered an (f has emitted
available
conditions
that the completed
1 suppose
Considering
of problem
problem has released
all the problems
only 2 free units;
problems
respectively.
of problems
for the continuation
all its resources
to the other problems).
for 5 units of the resource
resource,
solution
sufficient
all signals
In example T, is waiting of this
O’l
j < 1, creates
are being solved
except
R, and the operative T,,
T, and T,
The deficient
occupy
system
has
2, 1, and 4 units
3 units will be released
T, and T1, or alternatively
T1, where
if the
that of problem T, , is completed.
also the explicit
dependence of T, on T,, we obtain that T, may T, and TB, or T, and T, are completed. Then the basic strings are, for example, (T,), (T,), (T,, T,), (T, , TJ, CT,, T,, T,>, CT,, T,, T,). The strings (T,), (T,, T,), (T,, T,), (T,, T,, T,, T,) etc. are not basic. be continued
if problems
ItiseasytoseethatifC,=(T,r,...,T basic
strings,
also.
This
)andC,=(Tp,...,T
then C,C, = (Tar, . . . , T,
statement
is extended
, Tppi” . . ,
T
PM
)
1 is a basic
in an Obvious way to a greater
PM
>are
string
number of
strings. We introduce to construct
the Boolean
a basic
But if it is not possible
hi = 0.
function
string (Tat,
hi =
. . , , T,
to construct
a bkic
h(Ti) on the set 2.
If it is possible
= Ti) for problem Ti, we put hi = 1. string
for problem Ti we put
An algorithm for the diagnosis We call the function possibility Ti.
h the
expectation
(hi = 1) or impossibility
The recognition
zero expectation It is easy
Its value expresses
function.
(hi = 0) of continuing
of a deadlock
reduces
215
of deadlocks
the solution
the
of problem
to finding a set of problems
with
function. to see that in example
1 all the expectation
functions
are equal
to unity. We introduce
the variable
with each problem Ti whose a Boolean
disjunctive
Boolean
vector
solution
is suspended
a
normal form of the variables p(i) F,(x)=
(1)
where for each fixed
dependence function Fi -
x1,
v
. . . , ~i_~,
Xi+l,
. . . , xk:
A
XVzmn
?
n=*
m, 15 m< p(z], the completion
q(i, m), creates
. . . , xk) and associate
q(i, n2)
m=i
n= 1, ,..,
x = (x,,
sufficient
conditions
of the problems
for the continuation
Tyimn, of problem
Ti. In example
1 for problem T, we have
F,(x) = x,x, We introduce place
the vector
of the corresponding
that hi = 1 implies
h = (h,,
. . . , hk).
components
If we substitute
of the vector
Fi(h) = 1. Indeed, Fj(h) ZmVn
consequently,
‘v’ x,x,.
&,,,,,,=
= 1 means that 1,
for every problem Tyimn we can construct
It is easy to see that
c
.c ‘iml
is a basic
string,
so that hi = 1.
*’
Yim*(i,m)
Therefore,
holds.
(Ti)
the relation
Vi(h)
+
hi) = 1,
‘i(h)
V
hi = 1.
or (2)
its components
x in (1), it is easy to see
the basic
string
Hymn.
in
V. S. Arolovich
216
and V. M. Mikhelev
We now construct the following system of k equations.
If problem Ti is
solved, the i-th equation of the system is of the form Xi = 1. But if the solution of problem T, is suspended, the i-th equation of the system is of the form
We call the set of equation obtained in this way the ~~~~~~~~s~~~ systen. It is easy to see that the vector h is the solution of this system. if problem T, is solved, a basic string (Z’$ exists
for it and hi = l.But if the
solution of problem Ti is suspended, we take the expression substitute
(1) for F, and
it in (21
This reiation can be transformed into the equivalent
Therefore,
Indeed,
form
all the equations of the system are satisfied.
For example 1 the characteristic
system is of the form
We show that the characteristic system permits the set of problems leading to a deadlock to be found without having recourse to basic strings. We first notice that each i-th equation of the characteristic system of the form (3) can be written in the equivalent
form
An algorithm
for the diagnosis
217
of deadlocks
Lemma 1 If (Tal, . . . , T equations
) is a basic *N
of the characteristic
form xa, . . . X,
system
hence
to it.
Let
>. Consider
statement
for the problem of this string
with respect
in the characteristic
T
O[N-*
the product of the left sides
khe statement
system
to N.
For N = 1 problem
the equation
X,
for problem T,. remains T,
of some problems
for the continuation
of problem
T
products
variables
string (T,
valid.
Therefore
of the system
easy to see that the term
N
= 1, the
is of the
of a basic
string,
sufficient
in (1) at least
to only problems
left side by the product xa, . . . X,
, ...,
But if this equation
, 1~ I< N must create
“N’
corresponding
For the aN-th equation
I
TaI is
If it is of the form x,’
form (3), where i = aN, it may be noted that by the definition
contains
are of the
= 1 corresponds
of the lemma be true for the basid
the equation
of the lemma obviously
the completion
of the
.
N
The proof is by induction solved,
string,
conditions
one of the
of the string
(Tal,
. . . , ‘I’,
we now use the form (4) and multiply obtained
N-I
by the inductive
N-I
the
hypothesis.
It is
p(i) s(i,m)
is cancelled
A
A
ml,
7l-i
and only the product xix,
The product of the left sides system,
xvimn J-k,. * * xa_v_,
represented
1
. . .x
aN_1
E x
of all the equations
in disjunctive
remains. OL1’” xa ,PJ of the characteristic
normal form, is called
the
characteristic
product. Theorem 1 If xi occurs We assume exists,
in
in the characteristic
the contrary:
product,
hj = 1. Then a basic
and by Lemma 1, the product of the left sides
characteristic
xa
inversely
xi.
system
We multiply
271. isjunctive
for the problems the remaining
normal form.
of this string
left sides
If then not a single
hi = 0.
string (‘I’,,
, . . . , T, = Ti)
of the equations
of the
are of the form x
...
and represent
their product Prem
term in Prem contains
< as a
).
V. S. Arolovich
218
factor,
3ciwill
contains
not occur
and V. M. Mikhelev
in the characteristic
xi, it is cancelled
product either.
on multiplication
there remain only the term not containing
But if some term
by x,+ . . . Xi, so that in this case
zi, which contradicts
the hypothesis
of the theorem.
Corollary It is possible
to take xi out of the brackets
of the characteristic
equation
if and only if hi = 1. Indeed,
if hi = 1, we have
. ..x aN_l P rem’
P(x)=xixa 1
where P(x)
is the characteristic
the terms,
we have hi = 1, since
Indeed,
the definition
of the system, have P(h)
On the other hand, if xi occurs
product.
hi = 0 implies,
of the characteristic
we have P(x)
= 1;
product implies
in particular,
in all
= 0, but this is impossible.
P(h) since
that if x is a solution
h is a solution,
we also
= 1.
Therefore, characteristic
it is sufficient system,
all the expectation
to multiply
the left sides
of the equations
in order to find from the form of the characteristic
functions
and thereby
determine
which problems
of the product
become
deadlocked.
Example 2.
Let the characteristic
system
be of the form
(52 v X1) (5% v fa v 51) = 1, E, v x2 = 1, x3 = 1, zzv5%vx&=
1,
(ZFui v X5) (2, v 55) = 1. After multiplying
the left sides,
--
-
P(x) =x,x,x,x,x, From this we conclude T,, T,,
we obtain
\/x,x,x,x,xs =X&l
---
(X1X&
VW&,).
that h, = h, = h, = 0, h, = h, = 1, that is, the problems
T,-
We denote
by E the set of problems
whose expectation
function
equals
unity,
An algorithm
2’
= Z\E
for the diagnosis
is the set of problems
We call equations
with zero expectation
the set of problems
S G Z’
of the characteristic
system
of the form Xi = 1 the characteristic corollary
this is equivalent
219
of deadlocks
a resolving
function.
set,
for the problems
product acquires
to all the expectation
if on replacing
the
of the set S by equations
the form X, . . . xk (by the
functions
becoming
equal to
unity). We call a resolving subset
set of problems
of it is a resolving
It is easy to verify resolving
sets,
not minimally
T,!, {T,,
system
a deadlocked if hi = 0, since
equations
hence
2 the sets
(!I’,], IT,!,
lT,I
are minimally
IT,, T,], IT,, T,, T,] are resolving
It is easy functions Conversely,
are suspended
equations
but
of the system
functions
is deadlocked
are equal
it cannot
to construct
to zero.
contain
if there are no equations
and it is impossible
all the expectation
of the form xi = 1 will
to see that a system
of all the problems
h is the solution
of the form Xi = 1.
all the problems string,
T,],
not containing
system.
if and only if the expectation Indeed,
set if no proper
resolving
resolving.
A characteristic be called
that in example
and IT,,
a minimally
set.
of this form,
a single
basic
are zero.
Lemma 2 Let a deadlocked
system
be given and S be a resolving
system
there is an equation
in the left side of which at least
inverse
terms corresponding
only to problems
If S = Z (since 2’
the system
= Z), and the statement
U = Z\ S is not empty.
is deadlocked,
p(i) il(i,jJl) _ /j ( V Xyimn V
Tid
m=l
xi)
It has to be proved that
We assume
Let S c
that this is untrue,
that is,
A Tj”S
n=,
3iarnvn
one factor
in the contains
we have h = 0, consequently
of the lemma is trivial.
/j
Then
of the set S.
We put
P’ E
set.
l’, ,,,,,E S.
x+
2.
Then the set
220
V. S. Arolovich
Then,
P’ in disjunctive
representing
terms
normal form, we obtain
A xv imn(i,m)
TidJ
term is not cancelled,
A
m=i
since
of U, and the non-inverse
contradiction
all the inverse
ones to problems
proves
the equation
element
Yimn
be given,
set
not necessarily
deadlocked.
are of the form (3),
since
between
the characteristic
system
the i-th submatrix.
Each
m-th row of
with the Yimn-th column the
. . . . q(i, m), with the i-th column the element
the remaining
elements
the characteristic
and the
TO each problem Ti in this matrix there corres-
at the intersection
way is called
For example
to problems
of (3) for p(z) = 1, q(i, 1) = 0 (the sum
in the aggregate
contains
,n=l,
x
case
a correspondence
pond p(z) rows, called
correspond
to be equal to zero).
matrix of k columns.
the i-th submatrix
of a resolving
that all its equations
xi = 1 is a particular
We establish following
system
we will assume
of 0 terms is considered
factors
of S.
the lemma.
Now let some characteristic For generality
xj*
Tj”S
On the other hand, from the definition
This
as one of the
p(i) A
This
and V. M. Mikhelev
The matrix constructed
of the row are zeros.
matrix.
2 the characteristic
matrix is of the form
0 0 0 z4 0 Xl 0 .?3 . . . . . . . . . Xl
g2
Zl
x2
0
d
. 0.
. 23.
.
.0 .
.0
(j
‘j;:,‘
0.
.
.
.
.
.
.
.
.
0
0
x3 . 21 0
.
. 0 0
.
.
25
0
0
25
z3
0
55
xi.
All
in this
An algorithm
(the submatrices
are separated
We call the i-th cross obtained
for the diagnosis by dotted lines).
of the characteristic
by the union of the i-th submatrix
We now formulate (1)
an algorithm
all the crosses
are deleted.
221
of deadlocks
matrix the set of elements
and the i-th column.
which generates
a set of problems
in which rows occur not containing
If new rows without inverse
elements
inverse
have appeared,
X c
Z:
elements, then to para-
graph 1). (2)
if the matrix is not exhausted,
(3)
crosses
an arbitrary
row is selected,
otherwise,
end;
of the chosen set X;
with subscripts
row are deleted;
to paragraph
(Explanation:
equal to the subscripts the corresponding
of the inverse
problems
elements
are included
in the
1). after the deletion
of crosses
the submatrices
and columns
are
not renumbered.) We notice
that the deletion
of the i-th equation
value
Xi = 1 in the remaining
may be regarded
of the i-th cross
from the characteristic equations.
as a characteristic
of problem
Ti is carried
number of problems
to be solved:
solution
is a row in the submatrix that is, the corresponding is taken outside hi = 1. as usual. becomes
Hence
deadlocked,
which implies,
to the crosses
(2) is E, and paragraph
deleted
outside
before
(2) and paragraph
of the set Z’ , so that in paragraph in a deadlock
indicates
the completion
in the elimination
Hence
of the corresponding
from the number of problems of those
problems
inverse
l), there elements, Then xi
and by the corollary, and can be solved
(2) the system
obviously
that the set of problems
the first
application
of paragraph
(3) are applied to the system
problems
are deleted.
the deadlock
in particular,
of the
in paragraph
product,
if the
from the
xi on the left side.
(1) to paragraph
problems
their elimination
is deleted
which does not contain
has the factor
from paragraph
in this way
all the dependencies
of the characteristic
problem Ti is situated
On passing
corresponding
equation
the brackets
case
of the
which arises
or it is eliminated
If a cross
of this cross
transformed
for the situation
to completion in either
to the elimination
and the substitution
The system
system
on Ti are cancelled.
other problems
is equivalent
system
(3) the crosses the deletion problems
from the set X.
corresponding
of these by special
to be solved,
for to
crosses means or
which is reflected
222
V. S. Arolovich
It is obvious
from the above that by means
by one more method which problems this by example
and V. M. Mikhelev
2.
In the characteristic
on the completion
of paragraph
of paragraph
have become
deadlocked.
(1) we can determine We will consider
matrix there is the row 00x,00,
(1) the third cross
is deleted,
therefore
after which the
matrix becomes 21
a2
0
0
21
0
54
0
. . . . . . . . 51
0
52
0
. . . . . . . . 0
22
54
%
. . . . . . . .
Since the row 000x, cross
is deleted.
$1
0
0
21
0
0
0
xt
has appeared,
There
remains
paragraph
(1) is executed
again;
the 5-th
the matrix Xl
372
0
0 34 Xl . . . . . . Zl
x2
0'
. . . . . . _ 0
in which there are no longer lock is formed by problems obtained
52
541
rows of the required
form.
Consequently,
the dead-
T,, T, and T,, which is the same as the result
previously.
Theorem 2 The set X generated
by the algorithm
It is easy to see that X c
described
is a resolving
set.
2’ .
We denote by Yl the set of problems corresponding to the crosses deleted at the I-th application of paragraph (1). We replace all the equations for the problems of the set X by equations of the form xi = 1 and show that in the modified system the product of the left sides of the equations for the problems of the set
Z[=
U YjUX j
An algorithm
is equal to
for the diagnosis
of deadlocks
223
A Xi. =1 EZL
The proof is by induction
with respect
are of the form xi = 1 from the definition same form by the construction sides
we obtain
A Xi. TiEZl
form
inverse
After multiplying
In the general
elements
case
Ti E Y,
of Y, that for every problem
there is a row containing corresponding
of the new system.
the required
from the definition
1. For 1 = 1 the equations
to
for Y,
of the set Y,, and for X they are of the the left
it is easy to see
in the i-th submatrix
only at intersections
with columns
to the set
Yj U X = Zl-i.
U kr This
means that in each of the corresponding
factors
of the left side contains
inverse
of the set Z,_, . After representing form (4) and multiplying
equations
of (3) one of the
terms corresponding
the left sides
by the product
A
only to problems
of all these
equations
obtained
xi,
in the
by the inductive
TiC-Zl-1
hypothesis,
we obtain
To complete
the required
the proof of the theorem
fact that if in the operation thenZL
1
A
Ti=Zl
Xi* it is sufficient
of the algorithm
paragraph
to notice
the obvious
(1) is applied
L, times,
=Z.
We continue If in paragraph
the discussion (2) we select
and problem ‘I’, is recorded
of the operation
as an element
Xl * 0
If the row x,0x,
of the set X.
the matrix
x4
of paragraph
the fourth cross
$2
0 . . .
Xl
21
We obtain
2.
is deleted
*
of problem !I’,, and the matrix Xl
on example
cross
54 .
after a twofold application is selected,
the second
0
Xl
which is exhausted
of the algorithm
the row x,xzO or O&4,
22
7
is deleted,
(1). with the recording
V. S. Arolovich
224
arises,
which is also completely
Finally,
on selecting
and V. M. Mikhelev
deleted
on account
the row ,x,0,
we delete
of paragraph the first cross,
(1). record problem
T, in the set X and obtain the matrix 0
X2
*.. 22
which is also exhausted
, II
on the operation
of paragraph
(I).
Therefore, by selecting different rows in paragraph (2) of the algorithm described, different resolving sets can be obtained. In this example we obtained as such sets (T,i, (T,\, {T,!. It can be seen by this example that the algorithm does not give all the resolving sets: thus, the set ET,, T,, T,f, obviously a resolving set, is not obtained for any choice of the rows in paragraph (2). However, those resolving sets which are of greatest practical interest, namely the minimally resolving sets, are completely included in the algorithm. This follows directly from the following theorem.
Theorem 3 If S is a resolving
set, the rows in paragraph
(2) can be so selected
that
x E s. We denote paragraph (3); paragraph
by Xl the set of problems z’, is the set of problems
(2) is applied
S, is a resolving
include in X at the l-th application of corresponding to the system to which
at the I-th time;
sI = S fl 2,‘;
it is easy to see that
set for Z’,.
By Lemma 2, in the characteristic matrix of a deadlocked system there is always a row containing inverse elements corresponding only to problems of the resolving set. Hence if each time we select in paragraph (2) the m-th row of the i-th submatrix
by Lemma 2, then for each Z-th application Since
of para-
graph (3) we will have X1 or Si.
where L, is the number of applications theorem is proved. In particular,
if S is the minimally
of the rows in paragraph
of paragraph
resolving
(2) we can obtain
X = S.
(3), we have X z s.
set, for an appropriate
The
choice
An algorithm
The converse obtained
statement
by this
characteristic
for the diagnosis
is not true,
algorithm
is minimally
that
of deadlocks
225
is, not every
resolving.
set X which
For example,
can be
for the
system
1, 5, v x2= 1, f,Vf,Vx,=l 52 v xi =
the sets
IT’,\ and IT,\
will
be minimally
resolving,
while
if in the characteristic
matrix
on the
application
of paragraph
(2) we choose
the row x,x,x,,
the algorithm
X = IT,, T,I.
gives
The choice of a deadlock
of a particular may be carried
minimally out,
resolving
for example,
set
in the practical
by weighting
resolution
functions
on the set
2. Therefore,
if the algorithm
(2) are selected obtained, must
and each
of them
be performed
serves
in order
as the initial
structed free
in all possible
by starting about
problems
(the
The
vector
The
authors
the manuscript
to remove
about
repeatedly
and the Irows
all the minimally
resolving
which
problems
the deadlock.
for the algorithm. about and also
of the problem
The characteristic
matrix
matrix by some
explicit
resources type
system
particular
dependencies
V, W, A in the notation
for a given
may be con-
of the operative
occupied
of insufficient
if the request
will be operation
This
about
in paragraph sets
any particular
the resources
the resources
problems,
u and the matrices
algorithm,
as a fictitious
with
from information
expected
solution
by a similar
shows
information
for use by problems,
problem,
is applied ways,
between
of [l]).
may be obtained
of resource
is treated
problem. sincerely and made
thank a number
M. F?. Shura-Bura, of extremely
who carefully
useful
read
and important
Translated
through comments.
by J. Berry
REFERENCES 1.
AROLOVICH, V. S. and MIKHELEV, V. M. Some algorithms of the distribution of resources in the multiprogram solution of problems. Zh. uychisl. Mat. mat. Fiz., 12. 4, 1061-1066, 1972.
226
V. S. Arolovich
and V. M. Mikhelev
2.
HABERMANN, A. N. Prevention 373377, 385, 1969.
of system
deadlocks.
3.
SHOSHANI, A. and COFFMAN, E. G. Sequencing taks in multiprocess, Proc. 11-th Annual Symposium resource system to avoid deadlocks. and Automata Theory, 225-233, Oct. 1970.
Commune.
ACM,
12. 7,
multiple on Switching
6 January
19721
THIS paper is devoted to the analysis of deadlocks, that is, situations some of the problems being solved by a computer in the multiprogramme cannot
continue
their solution
on account
of interdependencies.
in which mode,
It is proved
that if the dependencies of the problem are represented by a system of equations and their left sides are multiplied, the problems forming a deadlock can be determined from the form of the product obtained. In terms of the matrices constructed possible
from this system of equations, an algorithm is formulated, making to find the minimal sets of problems whose completion by special
of exclusion from the number to be solved removes The discussion is illustrated by examples. In the solution locks may occur; cessed
cannot
other problems
of problems these
on a computer
are situations
be solved or because
the deadlock
from the system.
in the multi-programme
where some of the problems
further because of the absence
of the expectation of the necessary
it means
mode deadbeing
of signals
pro-
from
resources.
In II-31
algorithms were given which for some conditions enable deadlocks to be predicted, and if these conditions are not satisfied they enable the set of problems causing the deadlock to be indicated. In this paper we consider
an algorithm
situations,
where the connections
by Boolean
functions
between
or in an equivalent
for the diagnosis the problems mathematical
of deadlock
are described form.
explicitly
Such an explicit
description enables the solution of the problem to be simplified, requiring a more detailed analysis of the connections, such as a search for the minimal subset of problems which, because of the inclusion of special alarm devices of the operational system or their exclusion, leads to the resolution of the deadlocked situation, and also to an explanation of the deficient resource and the possibility *Zh.
uychisl.
of replacing
Mat. mat. Fiz.,
it.
12. 6, 1543-1553,
213
1972.
V. S. Arolovich
214 We consider
the set of problems
and V.
2 = IT,,
programme mode.
As was shown in Ill,
may be suspended
because
expectation
of explicit
of resources)
M. Mikheleu . . . , Tk! being solved of some of these
(the expectation
of signals)
dependencies
resource
because
R occupied
be continued possibility
of the fact that,
by problems
T,,
only after obtaining of continuing
explicit
and latent,
no direct
effect
T,
firstly,
and T,,
some signal
(the
and secondly,
on whether
where the dependence
and that on T, and T,
of problem T,
it requires
is latent
units of some its solution
from problem T,.
problem T, depends
T,, T, and T, can be continued,
problems
or latent
on other problems.
Example 1. Let Z = {T,, T,, T,, T,, T, 1 and the solution have been suspended
in the multi-
the solution
the solution on problem
only;
can
Then the of problems
T, is both
problem T, exerts
on T,.
The cortege
(Tall
. . . , TaN) of problems of the set Z is called
a
basic
string if for each problem To, 1 ,( l< N, one of the following two conditions is is solved; (2) the completion of some problems satisfied: (1) problem T T a: ., l,(
TaJ (it is considered an (f has emitted
available
conditions
that the completed
1 suppose
Considering
of problem
problem has released
all the problems
only 2 free units;
problems
respectively.
of problems
for the continuation
all its resources
to the other problems).
for 5 units of the resource
resource,
solution
sufficient
all signals
In example T, is waiting of this
O’l
j < 1, creates
are being solved
except
R, and the operative T,,
T, and T,
The deficient
occupy
system
has
2, 1, and 4 units
3 units will be released
T, and T1, or alternatively
T1, where
if the
that of problem T, , is completed.
also the explicit
dependence of T, on T,, we obtain that T, may T, and TB, or T, and T, are completed. Then the basic strings are, for example, (T,), (T,), (T,, T,), (T, , TJ, CT,, T,, T,>, CT,, T,, T,). The strings (T,), (T,, T,), (T,, T,), (T,, T,, T,, T,) etc. are not basic. be continued
if problems
ItiseasytoseethatifC,=(T,r,...,T basic
strings,
also.
This
)andC,=(Tp,...,T
then C,C, = (Tar, . . . , T,
statement
is extended
, Tppi” . . ,
T
PM
)
1 is a basic
in an Obvious way to a greater
PM
>are
string
number of
strings. We introduce to construct
the Boolean
a basic
But if it is not possible
hi = 0.
function
string (Tat,
hi =
. . , , T,
to construct
a bkic
h(Ti) on the set 2.
If it is possible
= Ti) for problem Ti, we put hi = 1. string
for problem Ti we put
An algorithm for the diagnosis We call the function possibility Ti.
h the
expectation
(hi = 1) or impossibility
The recognition
zero expectation It is easy
Its value expresses
function.
(hi = 0) of continuing
of a deadlock
reduces
215
of deadlocks
the solution
the
of problem
to finding a set of problems
with
function. to see that in example
1 all the expectation
functions
are equal
to unity. We introduce
the variable
with each problem Ti whose a Boolean
disjunctive
Boolean
vector
solution
is suspended
a
normal form of the variables p(i) F,(x)=
(1)
where for each fixed
dependence function Fi -
x1,
v
. . . , ~i_~,
Xi+l,
. . . , xk:
A
XVzmn
?
n=*
m, 15 m< p(z], the completion
q(i, m), creates
. . . , xk) and associate
q(i, n2)
m=i
n= 1, ,..,
x = (x,,
sufficient
conditions
of the problems
for the continuation
Tyimn, of problem
Ti. In example
1 for problem T, we have
F,(x) = x,x, We introduce place
the vector
of the corresponding
that hi = 1 implies
h = (h,,
. . . , hk).
components
If we substitute
of the vector
Fi(h) = 1. Indeed, Fj(h) ZmVn
consequently,
‘v’ x,x,.
&,,,,,,=
= 1 means that 1,
for every problem Tyimn we can construct
It is easy to see that
c
.c ‘iml
is a basic
string,
so that hi = 1.
*’
Yim*(i,m)
Therefore,
holds.
(Ti)
the relation
Vi(h)
+
hi) = 1,
‘i(h)
V
hi = 1.
or (2)
its components
x in (1), it is easy to see
the basic
string
Hymn.
in
V. S. Arolovich
216
and V. M. Mikhelev
We now construct the following system of k equations.
If problem Ti is
solved, the i-th equation of the system is of the form Xi = 1. But if the solution of problem T, is suspended, the i-th equation of the system is of the form
We call the set of equation obtained in this way the ~~~~~~~~s~~~ systen. It is easy to see that the vector h is the solution of this system. if problem T, is solved, a basic string (Z’$ exists
for it and hi = l.But if the
solution of problem Ti is suspended, we take the expression substitute
(1) for F, and
it in (21
This reiation can be transformed into the equivalent
Therefore,
Indeed,
form
all the equations of the system are satisfied.
For example 1 the characteristic
system is of the form
We show that the characteristic system permits the set of problems leading to a deadlock to be found without having recourse to basic strings. We first notice that each i-th equation of the characteristic system of the form (3) can be written in the equivalent
form
An algorithm
for the diagnosis
217
of deadlocks
Lemma 1 If (Tal, . . . , T equations
) is a basic *N
of the characteristic
form xa, . . . X,
system
hence
to it.
Let
>. Consider
statement
for the problem of this string
with respect
in the characteristic
T
O[N-*
the product of the left sides
khe statement
system
to N.
For N = 1 problem
the equation
X,
for problem T,. remains T,
of some problems
for the continuation
of problem
T
products
variables
string (T,
valid.
Therefore
of the system
easy to see that the term
N
= 1, the
is of the
of a basic
string,
sufficient
in (1) at least
to only problems
left side by the product xa, . . . X,
, ...,
But if this equation
, 1~ I< N must create
“N’
corresponding
For the aN-th equation
I
TaI is
If it is of the form x,’
form (3), where i = aN, it may be noted that by the definition
contains
are of the
= 1 corresponds
of the lemma be true for the basid
the equation
of the lemma obviously
the completion
of the
.
N
The proof is by induction solved,
string,
conditions
one of the
of the string
(Tal,
. . . , ‘I’,
we now use the form (4) and multiply obtained
N-I
by the inductive
N-I
the
hypothesis.
It is
p(i) s(i,m)
is cancelled
A
A
ml,
7l-i
and only the product xix,
The product of the left sides system,
xvimn J-k,. * * xa_v_,
represented
1
. . .x
aN_1
E x
of all the equations
in disjunctive
remains. OL1’” xa ,PJ of the characteristic
normal form, is called
the
characteristic
product. Theorem 1 If xi occurs We assume exists,
in
in the characteristic
the contrary:
product,
hj = 1. Then a basic
and by Lemma 1, the product of the left sides
characteristic
xa
inversely
xi.
system
We multiply
271. isjunctive
for the problems the remaining
normal form.
of this string
left sides
If then not a single
hi = 0.
string (‘I’,,
, . . . , T, = Ti)
of the equations
of the
are of the form x
...
and represent
their product Prem
term in Prem contains
< as a
).
V. S. Arolovich
218
factor,
3ciwill
contains
not occur
and V. M. Mikhelev
in the characteristic
xi, it is cancelled
product either.
on multiplication
there remain only the term not containing
But if some term
by x,+ . . . Xi, so that in this case
zi, which contradicts
the hypothesis
of the theorem.
Corollary It is possible
to take xi out of the brackets
of the characteristic
equation
if and only if hi = 1. Indeed,
if hi = 1, we have
. ..x aN_l P rem’
P(x)=xixa 1
where P(x)
is the characteristic
the terms,
we have hi = 1, since
Indeed,
the definition
of the system, have P(h)
On the other hand, if xi occurs
product.
hi = 0 implies,
of the characteristic
we have P(x)
= 1;
product implies
in particular,
in all
= 0, but this is impossible.
P(h) since
that if x is a solution
h is a solution,
we also
= 1.
Therefore, characteristic
it is sufficient system,
all the expectation
to multiply
the left sides
of the equations
in order to find from the form of the characteristic
functions
and thereby
determine
which problems
of the product
become
deadlocked.
Example 2.
Let the characteristic
system
be of the form
(52 v X1) (5% v fa v 51) = 1, E, v x2 = 1, x3 = 1, zzv5%vx&=
1,
(ZFui v X5) (2, v 55) = 1. After multiplying
the left sides,
--
-
P(x) =x,x,x,x,x, From this we conclude T,, T,,
we obtain
\/x,x,x,x,xs =X&l
---
(X1X&
VW&,).
that h, = h, = h, = 0, h, = h, = 1, that is, the problems
T,-
We denote
by E the set of problems
whose expectation
function
equals
unity,
An algorithm
2’
= Z\E
for the diagnosis
is the set of problems
We call equations
with zero expectation
the set of problems
S G Z’
of the characteristic
system
of the form Xi = 1 the characteristic corollary
this is equivalent
219
of deadlocks
a resolving
function.
set,
for the problems
product acquires
to all the expectation
if on replacing
the
of the set S by equations
the form X, . . . xk (by the
functions
becoming
equal to
unity). We call a resolving subset
set of problems
of it is a resolving
It is easy to verify resolving
sets,
not minimally
T,!, {T,,
system
a deadlocked if hi = 0, since
equations
hence
2 the sets
(!I’,], IT,!,
lT,I
are minimally
IT,, T,], IT,, T,, T,] are resolving
It is easy functions Conversely,
are suspended
equations
but
of the system
functions
is deadlocked
are equal
it cannot
to construct
to zero.
contain
if there are no equations
and it is impossible
all the expectation
of the form xi = 1 will
to see that a system
of all the problems
h is the solution
of the form Xi = 1.
all the problems string,
T,],
not containing
system.
if and only if the expectation Indeed,
set if no proper
resolving
resolving.
A characteristic be called
that in example
and IT,,
a minimally
set.
of this form,
a single
basic
are zero.
Lemma 2 Let a deadlocked
system
be given and S be a resolving
system
there is an equation
in the left side of which at least
inverse
terms corresponding
only to problems
If S = Z (since 2’
the system
= Z), and the statement
U = Z\ S is not empty.
is deadlocked,
p(i) il(i,jJl) _ /j ( V Xyimn V
Tid
m=l
xi)
It has to be proved that
We assume
Let S c
that this is untrue,
that is,
A Tj”S
n=,
3iarnvn
one factor
in the contains
we have h = 0, consequently
of the lemma is trivial.
/j
Then
of the set S.
We put
P’ E
set.
l’, ,,,,,E S.
x+
2.
Then the set
220
V. S. Arolovich
Then,
P’ in disjunctive
representing
terms
normal form, we obtain
A xv imn(i,m)
TidJ
term is not cancelled,
A
m=i
since
of U, and the non-inverse
contradiction
all the inverse
ones to problems
proves
the equation
element
Yimn
be given,
set
not necessarily
deadlocked.
are of the form (3),
since
between
the characteristic
system
the i-th submatrix.
Each
m-th row of
with the Yimn-th column the
. . . . q(i, m), with the i-th column the element
the remaining
elements
the characteristic
and the
TO each problem Ti in this matrix there corres-
at the intersection
way is called
For example
to problems
of (3) for p(z) = 1, q(i, 1) = 0 (the sum
in the aggregate
contains
,n=l,
x
case
a correspondence
pond p(z) rows, called
correspond
to be equal to zero).
matrix of k columns.
the i-th submatrix
of a resolving
that all its equations
xi = 1 is a particular
We establish following
system
we will assume
of 0 terms is considered
factors
of S.
the lemma.
Now let some characteristic For generality
xj*
Tj”S
On the other hand, from the definition
This
as one of the
p(i) A
This
and V. M. Mikhelev
The matrix constructed
of the row are zeros.
matrix.
2 the characteristic
matrix is of the form
0 0 0 z4 0 Xl 0 .?3 . . . . . . . . . Xl
g2
Zl
x2
0
d
. 0.
. 23.
.
.0 .
.0
(j
‘j;:,‘
0.
.
.
.
.
.
.
.
.
0
0
x3 . 21 0
.
. 0 0
.
.
25
0
0
25
z3
0
55
xi.
All
in this
An algorithm
(the submatrices
are separated
We call the i-th cross obtained
for the diagnosis by dotted lines).
of the characteristic
by the union of the i-th submatrix
We now formulate (1)
an algorithm
all the crosses
are deleted.
221
of deadlocks
matrix the set of elements
and the i-th column.
which generates
a set of problems
in which rows occur not containing
If new rows without inverse
elements
inverse
have appeared,
X c
Z:
elements, then to para-
graph 1). (2)
if the matrix is not exhausted,
(3)
crosses
an arbitrary
row is selected,
otherwise,
end;
of the chosen set X;
with subscripts
row are deleted;
to paragraph
(Explanation:
equal to the subscripts the corresponding
of the inverse
problems
elements
are included
in the
1). after the deletion
of crosses
the submatrices
and columns
are
not renumbered.) We notice
that the deletion
of the i-th equation
value
Xi = 1 in the remaining
may be regarded
of the i-th cross
from the characteristic equations.
as a characteristic
of problem
Ti is carried
number of problems
to be solved:
solution
is a row in the submatrix that is, the corresponding is taken outside hi = 1. as usual. becomes
Hence
deadlocked,
which implies,
to the crosses
(2) is E, and paragraph
deleted
outside
before
(2) and paragraph
of the set Z’ , so that in paragraph in a deadlock
indicates
the completion
in the elimination
Hence
of the corresponding
from the number of problems of those
problems
inverse
l), there elements, Then xi
and by the corollary, and can be solved
(2) the system
obviously
that the set of problems
the first
application
of paragraph
(3) are applied to the system
problems
are deleted.
the deadlock
in particular,
of the
in paragraph
product,
if the
from the
xi on the left side.
(1) to paragraph
problems
their elimination
is deleted
which does not contain
has the factor
from paragraph
in this way
all the dependencies
of the characteristic
problem Ti is situated
On passing
corresponding
equation
the brackets
case
of the
which arises
or it is eliminated
If a cross
of this cross
transformed
for the situation
to completion in either
to the elimination
and the substitution
The system
system
on Ti are cancelled.
other problems
is equivalent
system
(3) the crosses the deletion problems
from the set X.
corresponding
of these by special
to be solved,
for to
crosses means or
which is reflected
222
V. S. Arolovich
It is obvious
from the above that by means
by one more method which problems this by example
and V. M. Mikhelev
2.
In the characteristic
on the completion
of paragraph
of paragraph
have become
deadlocked.
(1) we can determine We will consider
matrix there is the row 00x,00,
(1) the third cross
is deleted,
therefore
after which the
matrix becomes 21
a2
0
0
21
0
54
0
. . . . . . . . 51
0
52
0
. . . . . . . . 0
22
54
%
. . . . . . . .
Since the row 000x, cross
is deleted.
$1
0
0
21
0
0
0
xt
has appeared,
There
remains
paragraph
(1) is executed
again;
the 5-th
the matrix Xl
372
0
0 34 Xl . . . . . . Zl
x2
0'
. . . . . . _ 0
in which there are no longer lock is formed by problems obtained
52
541
rows of the required
form.
Consequently,
the dead-
T,, T, and T,, which is the same as the result
previously.
Theorem 2 The set X generated
by the algorithm
It is easy to see that X c
described
is a resolving
set.
2’ .
We denote by Yl the set of problems corresponding to the crosses deleted at the I-th application of paragraph (1). We replace all the equations for the problems of the set X by equations of the form xi = 1 and show that in the modified system the product of the left sides of the equations for the problems of the set
Z[=
U YjUX j
An algorithm
is equal to
for the diagnosis
of deadlocks
223
A Xi. =1 EZL
The proof is by induction
with respect
are of the form xi = 1 from the definition same form by the construction sides
we obtain
A Xi. TiEZl
form
inverse
After multiplying
In the general
elements
case
Ti E Y,
of Y, that for every problem
there is a row containing corresponding
of the new system.
the required
from the definition
1. For 1 = 1 the equations
to
for Y,
of the set Y,, and for X they are of the the left
it is easy to see
in the i-th submatrix
only at intersections
with columns
to the set
Yj U X = Zl-i.
U kr This
means that in each of the corresponding
factors
of the left side contains
inverse
of the set Z,_, . After representing form (4) and multiplying
equations
of (3) one of the
terms corresponding
the left sides
by the product
A
only to problems
of all these
equations
obtained
xi,
in the
by the inductive
TiC-Zl-1
hypothesis,
we obtain
To complete
the required
the proof of the theorem
fact that if in the operation thenZL
1
A
Ti=Zl
Xi* it is sufficient
of the algorithm
paragraph
to notice
the obvious
(1) is applied
L, times,
=Z.
We continue If in paragraph
the discussion (2) we select
and problem ‘I’, is recorded
of the operation
as an element
Xl * 0
If the row x,0x,
of the set X.
the matrix
x4
of paragraph
the fourth cross
$2
0 . . .
Xl
21
We obtain
2.
is deleted
*
of problem !I’,, and the matrix Xl
on example
cross
54 .
after a twofold application is selected,
the second
0
Xl
which is exhausted
of the algorithm
the row x,xzO or O&4,
22
7
is deleted,
(1). with the recording
V. S. Arolovich
224
arises,
which is also completely
Finally,
on selecting
and V. M. Mikhelev
deleted
on account
the row ,x,0,
we delete
of paragraph the first cross,
(1). record problem
T, in the set X and obtain the matrix 0
X2
*.. 22
which is also exhausted
, II
on the operation
of paragraph
(I).
Therefore, by selecting different rows in paragraph (2) of the algorithm described, different resolving sets can be obtained. In this example we obtained as such sets (T,i, (T,\, {T,!. It can be seen by this example that the algorithm does not give all the resolving sets: thus, the set ET,, T,, T,f, obviously a resolving set, is not obtained for any choice of the rows in paragraph (2). However, those resolving sets which are of greatest practical interest, namely the minimally resolving sets, are completely included in the algorithm. This follows directly from the following theorem.
Theorem 3 If S is a resolving
set, the rows in paragraph
(2) can be so selected
that
x E s. We denote paragraph (3); paragraph
by Xl the set of problems z’, is the set of problems
(2) is applied
S, is a resolving
include in X at the l-th application of corresponding to the system to which
at the I-th time;
sI = S fl 2,‘;
it is easy to see that
set for Z’,.
By Lemma 2, in the characteristic matrix of a deadlocked system there is always a row containing inverse elements corresponding only to problems of the resolving set. Hence if each time we select in paragraph (2) the m-th row of the i-th submatrix
by Lemma 2, then for each Z-th application Since
of para-
graph (3) we will have X1 or Si.
where L, is the number of applications theorem is proved. In particular,
if S is the minimally
of the rows in paragraph
of paragraph
resolving
(2) we can obtain
X = S.
(3), we have X z s.
set, for an appropriate
The
choice
An algorithm
The converse obtained
statement
by this
characteristic
for the diagnosis
is not true,
algorithm
is minimally
that
of deadlocks
225
is, not every
resolving.
set X which
For example,
can be
for the
system
1, 5, v x2= 1, f,Vf,Vx,=l 52 v xi =
the sets
IT’,\ and IT,\
will
be minimally
resolving,
while
if in the characteristic
matrix
on the
application
of paragraph
(2) we choose
the row x,x,x,,
the algorithm
X = IT,, T,I.
gives
The choice of a deadlock
of a particular may be carried
minimally out,
resolving
for example,
set
in the practical
by weighting
resolution
functions
on the set
2. Therefore,
if the algorithm
(2) are selected obtained, must
and each
of them
be performed
serves
in order
as the initial
structed free
in all possible
by starting about
problems
(the
The
vector
The
authors
the manuscript
to remove
about
repeatedly
and the Irows
all the minimally
resolving
which
problems
the deadlock.
for the algorithm. about and also
of the problem
The characteristic
matrix
matrix by some
explicit
resources type
system
particular
dependencies
V, W, A in the notation
for a given
may be con-
of the operative
occupied
of insufficient
if the request
will be operation
This
about
in paragraph sets
any particular
the resources
the resources
problems,
u and the matrices
algorithm,
as a fictitious
with
from information
expected
solution
by a similar
shows
information
for use by problems,
problem,
is applied ways,
between
of [l]).
may be obtained
of resource
is treated
problem. sincerely and made
thank a number
M. F?. Shura-Bura, of extremely
who carefully
useful
read
and important
Translated
through comments.
by J. Berry
REFERENCES 1.
AROLOVICH, V. S. and MIKHELEV, V. M. Some algorithms of the distribution of resources in the multiprogram solution of problems. Zh. uychisl. Mat. mat. Fiz., 12. 4, 1061-1066, 1972.
226
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and V. M. Mikhelev
2.
HABERMANN, A. N. Prevention 373377, 385, 1969.
of system
deadlocks.
3.
SHOSHANI, A. and COFFMAN, E. G. Sequencing taks in multiprocess, Proc. 11-th Annual Symposium resource system to avoid deadlocks. and Automata Theory, 225-233, Oct. 1970.
Commune.
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12. 7,
multiple on Switching