系统重用、升级与COTS组件使用的安全性考量
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"这篇学术论文探讨了在系统工程中安全重用系统、升级系统或使用商业现成（COTS）组件的问题。作者A. Wayne Wymore和A. Terry Bahill来自亚利桑那大学的系统和工业工程部门。文章强调了在决定重用、升级或采用COTS组件时，系统之间的I/O等效性和同态映射的重要性。
在文中，作者首先定义了I/O等效性的概念，即两个系统对于初始状态对是I/O等效的，当且仅当它们的最小化形式是同构图像。这意味着尽管状态、输入和输出的名称可能不同，但这两者的本质功能是相同的。然而，这并不意味着这两个系统可以互换使用，因为它们可能不是同态图像。同态图像指的是一个系统是另一个更简单系统的功能等价版本，即使它们的技术实现细节不同。
论文指出，I/O等效的系统并不总是可以互相替换的，因为实现一个系统的方法可能不适用于另一个。这种错误的假设可能导致严重问题。系统工程师在进行功能分析时，需要考虑I/O需求和性能指标，同时也要考虑重用性因素。而系统架构师则需要权衡技术要求和成本，来指定合适的系统设计。
作者强调，确保系统安全重用或升级的关键在于深入理解系统之间的等效性关系以及潜在的同态映射。这一过程涉及到详细的功能和性能评估，以避免因错误的兼容性假设导致的设计失败。此外，使用COTS组件可以降低成本和开发时间，但必须谨慎处理，确保这些组件满足特定系统的所有需求和性能标准。
该论文为系统工程师提供了重要的指导，提醒他们在选择重用、升级或采用COTS组件时，必须全面考虑系统的功能、性能、技术适应性和成本，以及I/O等效性和同态映射的复杂性。这样的方法论有助于确保系统工程项目的成功和安全性。"
Homophones are words that have the same sound, but
differ in spelling, origin and meaning, like night and
knight. They sound alike but look different. This paper
is about systems that have the same input/output behav-
ior, but differ in purpose, states, and structure: They
function alike but look different. Analogously, we say
that systems with the same input/output behavior are
homophonic echoes.
3. THEOREM 1 AND ITS IMPLICATIONS
Theorem 1: I/O equivalence with respect to an initial
state pair is reflexive, symmetric, and transitive.
If {Z1, Z2, Z3} are systems, then Z1 is I/O equivalent
to Z1 with respect to DSZ1 for any DSZ1 ∈ SZ1
(reflexivity),
if Z1 is I/O equivalent to Z2 with respect to the initial
state pair (DSZ1, DSZ2), then Z2 is I/O equivalent
to Z1 with respect to the pair (DSZ2, DSZ1)
(symmetry),
if Z1 is I/O equivalent to Z2 with respect to the pair
(DSZ1, DSZ2), and Z2 is I/O equivalent to Z3
with respect to (DSZ2, DSZ3), then Z1 is I/O
equivalent to Z3 with respect to (DSZ1, DSZ3),
(transitivity).
4. HOMOMORPHISMS
In general, homomorphic images require the definition
of three homomorphic functions: HI, mapping the input
values from one system to the other, HO, mapping the
output values, and HS, mapping the states. However,
in this paper we will simplify the mathematics (with-
out loss of generality) and only consider systems that
have the same input and output values, meaning both
functions HI and HO are the identity over the sets of
inputs and the sets of outputs, respectively. If such
systems are homomorphs, then they are called state-
homomorphs.
4.1. State-Homomorphisms
Definition: The system Z1 is a state-homomorphic
image of the system Z2 with respect to HS if and only
if
1. IZ1 = IZ2,
2. OZ1 = OZ2,
3. HS ∈ FNS (SZ2, ONTO, SZ1), a function map-
ping states of SZ2 to states of SZ1 must exist and
it must satisfy the ONTO property,
4. HS(NZ2(x, p)) = NZ1(HS(x), p) for every p ∈ IZ2
and x ∈ SZ2, the image of the next state of Z2,
started in the state x, is the next state of Z1 with
the initial state HS(x) and the same input, (the
symbol p is used for a particular value of the input
and x is used for a particular value of the state) and
5. RZ1(HS(x)) = RZ2(x) for every x ∈ SZ2, if the
states in RZ1 are mapped according to HS the
result will be equal to RZ2.
Notation: A function specifies a mapping of ele-
ments of set A (the domain) onto elements of set B (the
range). A function must be single-valued, i.e., no two
elements of B are images of the same element of A.
(Terminology: elements of A are mapped onto elements
of B, whereas elements of B are images of elements of
A.) If a function g is 1TO1, denoted g ∈ FNS(A, 1TO1,
B), then no two elements of A are mapped onto the same
element of B. If a function g is ONTO, denoted g ∈
FNS(A, ONTO, B), then each element of B is the image
of at least one element of A. The notation FNS(A, 1TO1,
ONTO, B) refers to all functions relating A to B that are
1TO1 and ONTO.
Corollaries: (1) State-homomorphism implies iden-
tical input sets and identical output sets. HS(STZ2(
f,
x)(t) = STZ1(
f, HS(x))(t) and OTZ2(
f, x)(t) = OTZ1(
f,
HS(x))(t), for every f ∈ ITZ and time t. (2) Systems Z1
and Z2 are state-isomorphic if and only if Z1 and Z2 are
state-homomorphic with respect to HS, and HS is 1TO1.
In other words, isomorphic images have the same num-
ber of states and the state relationship can be thought of
as a mere renaming.
Definition: Z1 is a homophonic echo of Z2 with
respect to HS if and only if HS ∈ FNS(SZ2, ONTO, SZ1)
such that OTZ2(f, x)(t) = OTZ1(f, HS(x)(t) for all ∈
IJS++ and RZ2(x) = I2Z1(HS(x)) for all x ∈ SZ2.
4.2. Homomorphism Examples
Notation: To define a system (Z) we must specify its
set of states (SZ), its set of inputs (IZ), its set of outputs
(OZ), its next-state function (NZ), and its readout func-
tion (RZ). For example, let us define system ZxA.
Specification of example system ZxA: Let ZxA =
(SZxA, IZxA, OZxA, NZxA, RZxA), where
SZxA
=
{1,
2,
3,
4} are th
e
nam
es
of the sys
t
em’s four states,
IZxA
=
{5, 6, 7
}
a
re
t
he
l
egal
v
al
ues
for
the
sy
stem
inputs,
OZxA
=
{8,
9,
10} are the output values produced by the system,
NZxA
=
{
((
1, 5
)
,
2
)
,
((
1, 6
)
,
3
)
,
((
1,
7
)
,
1
)
,
(2, 5), 3), ((2, 6), 3), ((2, 7 ), 1),
((3, 5 ), 2), ((3, 6 ), 4). ((3, 7 ), 1),
((4, 5), 3), ((4, 6), 3), ((4, 7), 1)}.
The next-state function shows the transitions between
states; each entry contains ((a present state, an input
84 WYMORE AND BAHILL
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