(1.1) It is assumed that (1.2)
Lab. y. Schee,psbouwkunde
Technische
HogeschoGl
Deift
[1038] 13ULLETIN DE L'ACADEMIE POLONASE DE.S SCIENCESO Strie des sciences techniques
Volume XXXII, No. 12 - 1975
The Structure of Safety Regu1atons
by
J. S. PAWLOWSKI
Presented by J. WIECKO WSKI on September 25, 1975
Summary. The term "safety regulations" used in this paper denotes the regulations restricting the
parameters of mechanical devices (such as ships, for exampk) in order to assure that their operations
will be safe. The author attempts to present the structure of the regulations, used more or less
intuitively in their composition, in the terms of the fundamental notions of set theory. Such a
pre-sentation visualizes several assumptions made tacitly in the intuitive approach. One of those
assumptions, which seems to be disputable in some cases at least, and is fundamental for the whole structure, is discussed more closely at the end of the paper.
1. Introduction and fundamentals. The term "safety regulations" used in the
title denotes the regulations restricting the parameters of mechanical devices (such
as ships, for example) in order to assure that their operations will be safe. The author holds that all those regulations are of the same nature and their internal
structures are the same. Hence one can write "the regulation", instead "the
regula-tions" when referring to their structure, and that convention is adopted in what follows. The structure of the regulation presented in this note is generally used more or less intuitively. The notions of set theory employed in the presentation make possible the visualization of several assumptions tacitly involved in the
intuitive approach.
Let X, Y, Z be pairwise disjoint, nonempty sets and Z0 be a proper subset of Z. Moreover, let f be a mapping f: XxYZ and X0 be defined as:
X0={xeX: A(f(x,y)eZ0)}.
yEY
V (x eX0).
The sets and the mapping mentioned above are regarded as representing,
respecti-vely: X - the space of the devices subjected to the regulation, Y - the space of the external conditions, under which the devices are intended to operate, Z - the space of the physical states of the devices likely to be experienced in operation,
Z0 - the physical states considered to be safe, f the cause-effect relation between
97[1033J
ARCHES
V A R J A (COJIIPUTER SCIENCE) possible that icture of he des, it shows 8(1-952 GDAÑSIC 1972. MesatoTca npa-.1 MexaHwiecKJ{x .cTanneT co6ott cxeMTbi,npuMe-98 J. S. Pawowski
[10341
the members of X x Y and the members of Z, X0 - the devices which are safe
under the conditions Y. The assumption (1.2) states the existence of a dvice being safe under the conditions Y. The notions mentioned above, even if not explicitly expressed in the formula of the regulation, are fundamental to it. As an example
ships may constitute X, weather conditions (waves and winds), occurring on different
areas of seas and oceans, may form Y, and then all possible performances of the ships sailing in the conditions constitute Z.
2. The criterion of safety. Besides the assumption (1.2) the existence of map pings k1 : X_Rk, k2: Y-+R",
k3: Z-R"
is postulated with R denoting the set of real numbers. An order relation in R' is introduced in a natural way:(2.1)
A [(xy)
A (tr'(x) ir'(y))],x,yR" 1=1 q
where 7t (y) is the projection of y n R" on the i-1h factor.
Itis assumed that:
(2.2) V
A (x<c=-k1(x)c:Z0),
CER" XEk3(Z)
c need not be unique but for the construction of the regulation only one element of R", for which the proposition of (2.2) is valid, must be chosen. That element
may be called "the criterion of safety". The proper choice of the criterion of safety is not intended to be discussed here and "cj' will be used to denote the criterion or safety in the sequel.
Some further remarks are introduced to explain the properties of the criterion. of safety. Let the mappings defined so far be exemplified in the form of a diagram.
shown below. In order to simplify notation X x Y is denoted by W, RkxR" by
R+m and k1xk, by g.
(2.5)
A (h(x,v)
e),
p6k(Y)
[1035]
for xE k1 (X). Howi
and sufficient that:
(2.6) u, (see, e.g. [1]). In tF of safety can be fo (2.7) A yEk2(
where the expressic second internal bra the condition
ueg
the probabilistic corbility of the event)
and for
xekj(X).
3. The structur shown which arises follows the structure
t ions fundamental t
their compatibility ' is not always obvio part of the paper.
Let I be a natur Therefore an equiv (3.1 W Z with pr (x) denotin over, let
R' be or
g k3 4- ... I (3.2) Rk+m R"[t can easily be verified that:
(2.3) A
j (A
A ((k3 of)(u) c) (k 'xk1(X) yk(Y) UEg'(x.y)
later on the antecedent of the implication in the brackets will be called "the condition
ofsafety", for any x E k1 (X).
If/i: R"-. R" exists which closes
the diagram in such a way that it becomescommutative, that is:
(2.4) A ((Ii og)(u) =(k3 of) (u)),
ueW
(3.4)
then the condition of safety can be written in an equivalent form:
If (2.6) is valid, fo
(3.3) B
In the opposite cas be replaced by (2. cases. For further below*) in R1. Hei
by "br", 2Z (br) is
*) Ati equivalent
L1035 The Structure of Safety Regulations
99
for x E k, (X). However, it should he recalled that for the existence of/i it is necessary
and sufficient that
(2.6)
A [(g(u)=g(:)) =.((k3of)(u)=(k1of)(:))],
(see, e.g. [fl). in the case when (2.6) is not true another probabilisticconditions of safety can be formulated as follows:
(2.7) A [Prob {(k3 of) (u) c u e ' (x, )} ii],
where the expression on the left-hand side of the inequality sign outside the
second internal brackets denotes the probability of the event (k3 of) (u) ( e, under
the condition u e g1 (x, ï). where xc k, (X) and ;e <0, 1>. With
the use of
the probabilistic condition of safety it can be said, if it is satisfied. that the
rroba-hility of the event f(u) e Z is at least 7/ for any randomly chosen u e kI'(x) x Y
and for x e k1 (X).
3. The structure of the regulation. Up to no'w the general situation has been shown which arises when one attempts to construct the safety regulation. In what follows the structure of the regulation generally used will be presented. The
assump-tions fundamental to it are important despite of their mathematical simplicity because
their compatibility with the physical nature of the objects subjected to the regulation is not always obvious. That situation is illustrated by Iwo examples set in the final part of the paper.
Let ¡ be a natural number and ¡<k, then R' can be considered as R' x R'.
Therefore an equivalence relation p can be introduced in Rk as follows:
(3.1) A [(xp.v) (pr2 (x) =pr2 (Y))]
x, y Rk
with pr (x) denoting the projection of xc R' x R''on the i-th factor, i2.
More-over, let R' be ordered by a relation analogous to (2.1) and
(3.2) C={y e R" : ypx}.
If (2.6) is valid, for every x e k1 (X) the set B is defined as: (3.3)
B={prj(y) E R': (j' e CX)A A (h(y, z)
c)}.z E k2 (Y)
In the opposite case the second component of the conjuction in the brackets should
be replaced by (2.7). In the sequel no distinction will be made between the two
cases. For further consideration it is assumed that B is nonempty and bounded below*) in R'. Hence there exists the greatest lower bound of B in R' denoted by "br", z' (bE) is simply the greatest lower bound of the set:
(3.4)
D,={e R :V (n'Ú')=)},
y E 15,
*) An equivalent assumption would be that Bz is bounded abovewith the resulting change of greatest lower bounds to the least upper bounds and vice versa in the sequel.
[1034J which are safe
a device being flot explicitly As an example ing on different rmances of the tence of map-noting the set ray: ly one element That element tenon of safety the criterion of Df the criterion n of a diagram
W, RcxR by
"the condition hat it becomeslOo J. S. Pawlowki [10361
i<I. From the definition it follows that:
(3.5)
A (yb)
yEB
and
(3.6)
A ((y> b)
V (z y)).'ER'
Let for a
Pc:kj(X), Pø,
the set B, be defined as:(3.7)
B9={b: xeP}.
The assumption is made that B9 is bounded above in R'. Therefore there exists
the least upper bound ofB9 in R' denoted by "b9". From now on every element y e R' will be called safe for an xE k1 (X), if and only if ye B,.A sufficient condition
for b9 to be safe for every XEP is:
(3.8)
A A
(y b y E Br).XEP YER
In the case when V
(P=C) there is:
XEk1(X)
(3.9)
b9=b,
and then
(3.10) b E B
is the sufficient and necessary condition for b9to be safe for every X EP.
The regulation of safety assigns b9 to a set P as the restriction from below for the regulated parameters y E R' (z' (y) is simply the i-th regulated parameter of an object of the regulation). Therefore the validity od (3.8) constitutes an assumption
which is fundamental to the structure of the regulation. In the author's opinion
cases in which (3.8) is not satisfied are possible in practice. An illustration of such a situation is given below as an example.
Some additional remarks can be made concerning the assumptions that B,,
and B9 are bounded. Excluding the necessity of restricting the regulated parameters
from above and below at the same time, and taking into account that every
para-meter restricted from above becomes restricted from below after reversing its sign, the restriction of the parameters can be reduced to the restriction from below (cf. the footnote on thepage 99 [1035]). On the other hand, if some parameters need to be restricted from both sides, restriction from above can be treated separately from that from below which leads to another regulation of analogous structure.
Subsequently the case of restriction from below is sufficiently general. The
admis-sion that B is unbounded below leads to the concluadmis-sion that there exists a natural number i ¡ and a sequence (y,,), Y'1 EB, such as that:
(3.11)
!imir'(y,,)=c'o.
n-.
In practice it means that the i-th parameter need not be restricted at all by the regu-lation. Therefore the dimenison of R' is reduced by one after eliminating the i-th
.
-parameter. Such a for practical reason
ments apply to the
above there exists
as that:
(3.12)
In practice it mean made, there is alw
parameter should (X,,) having the pro
4. Examples. L
deck of the vessel
Moreover, let the w
of the frequency o safety. A safe freeb The heights of the
If the frequency of the height of the de stations, then the c to establish the rest
required by the reg
a station depends n of the deck at the ot For example, let r
-4{0,l},
xek,(X)
(4.1) X where d is a real (4.2)and d,, i=l, ..., 1,
(4.3) and therefore: (4.4) for any y E R'.all by the
regu-nating the i-th
11037] The Struct'ure of Safeti Regulations 101
parameter. Such a process of elimination shows that B should be bounded below for practical reasons or otherwise there is no need for the regulation. Similar
argu-ments apply to the assumption that B, is bounded above. 1f B, is not bounded
above there exists a sequence (x), x,Ek1(X), and a natural number isI such
as that:
(3.12) um
' (b)=cx
n- r
in practice it means that no matter how large the value of the z-th parameter is
made, there is always an object of the regulation, for which the value of the
parameter should be larger in order to ensure its safety. In fact the existence of
(xv) having the property (3.12) seems most unlikely in practice.
4. Examples. Let the freeboard of a vessel be considered as the height of the deck of the vessel above still waterplane specified at / stations along the vessel.
Moreover, let the wetness of the deck be restricfed, then the largest permitted values
of the frequency of wetness at the respective stations constitute the criterion of safety. A safe freeboard is the freeboard, for which the condition of safety is satisfied. The heights of the deck at the stations 1=1, ..., ¡ constitute an element r e R'. If the frequency of the wetness at a station is a continuous decreasing function of the height of the deck at the station and is independent of the heights at the other
stations, then the condition (3.8) is satisfied, and one can look for b,, or b, in order to establish the restriction of freeboard height, for a vessel or a group of vessels, required by the regulation. On the other hand, if the frequency of the wetness at a station depends not only on the height of the deck at the station, but on the heights
of the deck at the other stations as well, then the condition (3.8) need not be satisfied. For example, let ir'(y)=, for y e R', and let the family of- functions (o,,:
R'--
{O, i }, x ek1 (X) be defined as follows:if
d)A(y
b,,),if
(yb,,),
¡=1
where d is a real number such that:
(4.2) A ((xpy)..(d,,=d,,)),
x,y
k,
(X)and d, 1=1,...,!, are nonnegative real numbers. Moreover, let: (4.1) (4.3)
((o,,U')=1)(y
E B,,), and therefore: (4.4) (ç,,(y)=0) (y B,,), for any Y E R'. 11036] re there exists every element :ient condition C EP.rom below for parameter of an assumption ithor's opinion Iration of such
ptions that B
.ted parameters hat every para-'ersing its sign,rom below (cf.
eters need to be
eparately from
:ructure.
rai. The admis-exists a natural
(4.5)
¡=1
it is easy to see that P (b)=O, and consequently b B. It is also possible that for Pck1(X):
(4.6)
v(
xP 1=1
(2r' (be) d.
and then (3.8) becomes not true.
The last example shows that such situations to which the structure of
the-regulation presented in this paper is not aDplicable are possible. Besides, it shows
that the understanding of the structure may be essential in practice.
SHIP RESEARCH INSTITUTE. TECHNICAL UNIVERSITY. MAJAK0wsKIEG0 11/12. 80-952 GDAÑSK (INSTYTUT OKRÇTOWY, POLITECHNIKA GDAÑSKA)
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[1] K. Maurin. Analiza, Czes 1, PWN, Warszawa, 1971.
[21 K. Kuratowski, Wstp do leoni mnogoci i topologii, PWN, Warszawa, 1972.
51. C. HaBoaKlr, CrpyzTypa IIPaBHJI óe33naduocnl
Coepaiisie. B npeLic-raa:HHotÍ pa6oie nOEa npaaMnaMi. 6e3onacHocTll notpaayeaaioca ripa-eiuia Hajiaraloatife, rio coc6paeHHsM 6e3onacHocTH, orpaHwleH}IR Ha napasiepsr Mexaimlecsax
KOHCTPYKUISVI, TOKHX Kab Harip. caMosleml .iis6o Kopa6Jlif. H3CTOHUIIa1 pa6ora npecTaBiDleT CO6Ot 11O[IbITKY nocTpoeHsssl, c }idnOsls300aHlleM tTOH2TI1ìI 113 TeOPHuí MHO3CCCTB, O6u1e1Î cxeMlO,
ripilMe-H1IMOk KO aceri BHLIM npaaLl 6e3onacilocTil.
Summary. The term
parameters of mechani
will be safe. The auth
intuitively ini their co sentation visualizes se assumptions, which se structure, is discussed
1. Introduction title denotes the r
as ships, for exam'
author holds that
structures are the s
tions" when referr
follows. The struc
more or less intui
make possible the intuitive approach. Let X, Y, Z be of Z. Moreover, le (1.1) It is assumed tha (1.2)
The sets and the
vely: Xthe spa
tite external condi space of the phys Z0 the physical 102 J. S. Pawlowski [10381 8 nuLLETrN DE LA POLONAISE DES Série de sciences
Volume XXIII, No. Then, if for some x e k1 (X):
