A. Mazurkiewicz, editor, Proc. 5th MFCS, LNCS 45, pages 277–283, Berlin,
1976. Springer–Verlag
AN AXIOMATIC APPROACH TO INFORMATION
STRUCTURES
H.-D. Ehrich
Abteilung Informatik, Universit~t Dortmund
46oo Dortmund 5o / Fed. Rep. of Germany
I. Introduction
Information
structures
are viewed here as formal objects on which certain opera-
tions can take place. Three operations
construction
modelling the integration
Concrete
~6,7J
are considered basic:
selection of parts,
of new objects by vTnaming" given ones, and addition of objects,
or merging of information structures.
examples of the sort of objects we have in mind are Vienna objects
as well as their generalizations
foundations
and modifications
C3,4~.
Axiomatic
of Vienna objects have been given previously by Ollongren ~ 7 ~ and
by the author ~3~- Being broader in perspective,
however,
general axiomatic basis for the type of algebraic
this paper gives a
structures
developed
in ~ 4 ~ .
2. Basic 0perations
Let D be a set, + be a binary operation on D, and O be a special object in D.
The elements of D will be called
Addition:
Ax(A)
(structured)
objects. Here is our first axiom:
D~D-*D
~D,+,O)
is a commutative
monoid.
We adopt the intuition that objects carry information,
and that the addition of
objects forms in some sense the union of their information
contents.
The null
object O carries empty information.
Selection and construction
paths",
are accomplished by means of "names" or "access
represented by the set of words
S*
over a finite alphabet
S
of selec-
tors. The empty word will be denoted by I .
In the sequel, we will maintain the following notational
cD
; s~t,... ~S;
possible.
x,y~... ~ S*. Specifications
Similarly,
versal quantification
what follows,
if quantification
conventions:
a,b,c,...
"ED" etc. will be omitted whenever
is omitted in subsequent
formulas,
for each free variable occuring should be inserted.
index i runs over an index set
I~.
uniIn
278
Selection:
DxS*--*D
Construction:
S*xD--~D
Ax(S1)
al = a
Ax(C1)
la = a
Ax(S2)
a(xy) = (ax)y
Ax(C2)
(xy)a = x(ya)
Ax(S3)
(~--ai)x = ~-(aix)
Ax(C3)
x ( Z a i) = ~ - ( x a i)
Construction
x a may be i n t e r p r e t e d
as g i v i n g object a the name x; addition of
d i f f e r e n t l y n a m e d objects may be v i e w e d as c o m b i n i n g them into one composite
object,
and selection
require
that a d d i t i o n and s e l e c t i o n
ax s e l e c t s that part of a named x. The two last axioms
(resp.
construction)
be independent
in the
sense that they can be a p p l i e d in either order.
Algebraically,
D is a
S*-left(and
are called right t r a n s f o r m a t i o n
An object
e ¢ D
semigroups
is called e l e m e n t a r ~
be the set of all e l e m e n t a r y
e,f,...
r i g h t ) - s e m i m o d u l e [ 2 ] . Right s e m i m o d u l e s
in [ 5 ].
iff, for each x~l, we have
objects.
ex=O. Let E
Its elements will be denoted by letters
, often omitting the s p e c i f i c a t i o n
of D. Let x and y be called independent,
"~E".
x@y
Obviously,
E is a s u b s e m i g r o u p
, iff n e i t h e r is x a p r e f i x of y
nor vice versa.
Interaction
Ax(I1)
(xa)x = a
Ax(I2) x o O y ~ ( x a ) y
Ax(I3)
Va
~x 3b
= 0
:
a = x(ax)
+ b
^ bx=O
The third a x i o m claims that each object a can be s u b d i v i d e d
into two parts:
that part of a named x and the rest b h a v i n g no x-part.
F r o m these axioms,
and x~l implies
following:
we can draw several conclusions.
xa ~ A
since
(i) x a = x b ~
a=b
(ii), E is a submonoid
T h e o r e m 2.1
Proof:
:
~a ~e
, (ii) xO=0
i=1,2,...,n
ai+lS k = O
,
:
a = ~
s(as)
s~S
+
2.2
:
Let
S={Sl,..o,Sn}
. If
a=a I , we
:
ai+lS j =
asj
if
l~k~i<
,
j~n
.
This means that
///
F(a)
:= { e G E
The sets F(a) play an important
and for e l e m e n t a r y
of
e
an+ 1 is e l e m e n t a r y
Definition
a#O
of D.
a i = si(aisi ) + ai+ I = s i ( a s i) + ai+ 1
where
o Obviously,
it is easy to prove the
, and (iii) 0x=0 . As a c o n s e q u e n c e
We apply axiom (I3) repeatedly.
have for
Let A:=D-E
(xa)x=a~O. Furthermore,
I a = ~-- s(as) + e
s~S
}
role in the sequel. Evidently,
objects we have
F(e) = { e } .
F(a)~@
for each a,
279
Theorem
2.3
(i)
:
x~l ~
(ii)
Proof:
F(a)
(i) is evident
e a ¢ F(a)
and
O ~ F(xa)
+ F(b)
from axioms
e b c F(b)o
=
F(a+b)
(II) and
(I3).
In order
to prove
(ii),
let
T h e n we have
a + b = ~
s(as)
s¢S
+ ea
= 7s((a+b)s)
s~S
+
7s(as)
soS
+ eb
+ e a + eb
///
3. E q u a l i t y
It is important
to have
condition
for finite
arbitrary
objects.
Theorem
3.1
Proof:
e ¢ F(a) n F(b) ~
Definition
If
a~b
:
3.2
,
Thus,
'
a
It is evident
to prove
( Vs
a~b
that
~
Theorem
implies
Proof:
These
:
If
Repeated
a
F(ax)n
~ ~
+ e = b
///
.
and it is an easy
, (ii) a x ~ b x
is a c o n g r u e n c e
are finite
, and
relation
{ x !axe0 }
objects,
to
.
and symmetric,
of t h e o r e m
suggest
F(bx)
compatible
A sufficient
extended
a = b
+ e = ~ - - s(bs)
soS
iff the set
b
application
circumstances
of N
finite
and
# ~ ) ~
(i) a + c ~ b + c
closure
a is c a l l e d
3.3
:
w i l l be called
is r e f l e x i v e
a~b
the t r a n s i t i v e
An object
a = 7s(as)
soS
of objects.
will be a x i o m a t i c a l l y
^ F(a) n F ( b )
:~==~ V x
b
for the equality
to be equal
: as = bs
and
that
a criterion
objects
exercise
(iii)
xa~xb
.
on D.
is finite.
we have
:
a~b
~
3.1
a = b
///
the last axiom
in our a x i o m
system
:
Equality
Ax(E)
This
a~b
axiom
~
a = b
is r e l a t e d t o
As a consequence,
Theorem
3.4
:
the c o m p l e t e n e s s
each object
F o r each x, let
a=
Proof:
Let
y~ S*.
Since
of data s p a c e s
can be r e p r e s e n t e d
ex~F(ax).
Z
x¢S*
(~-x~S*
as d e f i n e d
by a s u m of the
in [1 ].
following
form:
T h e n we have
xe
x
xe )y = ~
x
zES*
ze
= e
yz
Y
+ ~
Z~I
ze
= e
yz
Y
+~--sb s ,
s~S
280
where
b s = w ~ S Weysw , we have
eyg F ( ( ~ X e x ) Y ) n F(ay)
.
///
This theorem allows us to represent each object by a formal power series with
coefficients in
~(E),
the power set of E:
a = ~--xE x
xgS *
, where
E x = F(ax) c E
A word x is called peripheral with respect to an object a iff ax is elementary,
i.e.
axy=0
for each
y#l . Since
F(e) = < e } , the coefficients of peripheral
words are singletons.
4- Ortho$ona!it~
The considerations made above show that each model of our axiom system can be
represented by a set
D c ~ ( E ) S* of formal power series, given E and S. The
operations are defined in an obvious way, taking account of theorem 2.3 .
Adopting the notion of orthogonality as introduced in [1 ], we define orthogonal
sets of formal power series as follows. Here, a(x) denotes the coefficient of x,
a(x)~ E •
Definition 4.1 :
A set
D c ~ ( E ) S~
W~DS*3c~D
Wx~S*
is called orthoson&l
:
c(x) =
iff
~(x)(x)
A model of our axiom system which is an orthogonal set will be called an orthogonal model.
Theorem 4.2 :
If D is an orthogonal model, all coefficients are singletons.
Proof:
a
Given
and
x , let
B E D S* be any function such that
~(xy)=O for each y~l. By orthogonality,
a(x)
and
Ic(x)
l = 1
c(xy)= 0
~(x)=a and
there is an object c such that c(x)=
if y~l. Thus, cx is elementary,
and we have la(x) l =
///
As a consequence,
each orthogonal model is represented by a subset of
D ~ E S~ .
Some of these have been considered in E 4 ] . Vienna objects, however, do not
form an crthogonal model: they correspond to the case where each coefficient
of a non-peripheral word is the whole set E. Non-peripheral words will be called
inner words in the sequel.Thus,
in the case of Vienna objects,
not able to carry distinctive information.
In order to include this case, we
generalize the notion of orthogonality slightly.
Definition 4.3 :
A function
Vx ( ( W s
~
: ~(xs)xs
D S*
inner words are
is called admissible
: o ) ~l~(x)(x)l
= I
iff
)
281
Let
~
be the set
Definition
4.4
:
of a d m i s s i b l e
functions
A set
S*
V~c~
Theorem
words
:
3c~o
is called
VxsS*
:
on E
wrt
=
if z=s
otherwise
~1 and ~2 are admissible.
a(x)=b(y).
To complete
e~F(se+e)
by t h e o r e m
shows
a quasi-orthogonal
all singletons,
that
models
power
and
, e~O
,
c2 =
By t h e o r e m
C = ~a(x)
is of the form
C1
relation
~
if z=s
otherwise
, we have
Cl=C 2
and thus
for each e, we have
latter
of all c o e f f i c i e n t s
on E. F r o m t h e o r e m
to
C1
Co
and that among
there
(ii)
of
C
we conclude
is a c o n g r u e n c e
on
on the
(E,+,O)
is
all q u a s i - o r t h o g o n a l
is one c o m p r e h e n s i v e
model
in
is the set of
2.3
to see that each c o n g r u e n c e
model,
members
I aeD^x(S~
corresponding
congruence
This
associating
3.1
that,
{o
C = C O u C 1 , where
is a p a r t i t i o n
a fixed
=
///
the set
as subsets.
series
c I = se + a(x)
eke
.
quasi-orthogonal
implying
of inner
a ( x ) ~ b(x) ~ ~,
z
and
we observe
It is not d i f f i c u l t
by some
all others
2.3
model
and
(E,+,O).
implied
the proof
that
the e q u i v a l e n c e
monoid
b . Let
objects
Io z
Evidently,
theorem
the c o e f f i c i e n t s
by the f u n c t i o n s
~l(Z)
This
model,
a resp.
s c S . Due to q u a s i - o r t h o g o n a l i t y ,
exist
iff
.
Let x,y be inner words
se + b(y)
quasi-orthogonal
c(x) = ~(x)(x)
If D is a q u a s i - o r t h o g o n a l
form a p a r t i t i o n
Proof:
and
4.5
Dc~(E)
in D S* .
model
including
is given by the set of all
with p e r i p h e r a l
words
formal
and members
of
o
C1
with
model
inner words.
implying
It is shown
Structure
subset
the set
fundamental
i.e.
a+b=O
of all finite
equivalence
in [ 4 ]that
of the
zero sums",
The
the same
of V i e n n a
orthog0nal
implies
objects
is an example
of another
relation.
a=b=O.
objects
model,
is isomorphic
provided
This result
that
to a quotient
there
is g e n e r a l i z e d
are "no
in the next
theorem.
Theorem
4.6
:
Let D be a q u a s i - o r t h o g o n a l
is an o r t h o g o n a l
D
Proof:
a(D
cients
model D'
~
and a c o n g r u e n c e
model with no zero sums.
relation
~
on D'
such
T h e n there
that
D'/~
Let D' be the set of power
by the set of all a' where
of inner words
by a r b i t r a r y
that D' is an o r t h o g o n a l
by the q u a s i - o r t h o g o n a l i t y
model.
series
obtained
a' is o b t a i n e d
singletons.
Let ~
from D by r e p l a c i n g
from a by r e p l a c i n g
It is s t r a i g h t f o r w a r d
be the c o n g r u e n c e
of D. We extend
~
relation
to D' as follows:
each
all c o e f f i to prove
on E implied
for a,b ~ D, let
282
a--b
iff
a(x)------b(x) for all inner words x wrt a and b, and
a(x)=b(x)
for
all peripheral words x wrt a or b. In order to show that ~- is a congruence on
D' , we must show that, for each c resp. x, we have
xa_----xb
.
here. That
a+c=---b+c ,
ax~bx
,
and
The last two relationships are very easy to prove; so we drop it
a+c--~b+c
is seen as follows: the sets of inner resp. peripheral
words of a and b are equal. Since there are no zero sums, the set of inner
words of a+c is the union of the respective sets of a and c. The same holds
true for b. Let x be an inner word wrt a+c. Then, if x is inner wrt a, we have
a(x)----b(x); otherwise, we have a(x)=b(x).
(ii) :
In each case, we have by theorem 2.3
(a+c)(x)=a(x)+c(x)--b(x)+c(x)=(b+c)(x)
.
Now let x be a peripheral
word wrt a+c. Then x is peripheral both wrt to a and c, and we have
a(x)+c(x)=b(x)+c(x)=(h+c)(x).
This shows that
Now we consider the quotient structure
a+c~b+c
(a+c)(x)=
.
D'/---- which is, of course, a model of
our axiom system when making the usual conventions about the operations. The
isomorphism of
D'/=
-- and D is established by the 1-1-correspondence between
congruence classes
~/x
[ a ~ on
:
D'
F(a'x)
and objects
=
a'( D
given by
U
F(bx)
b=~a
///
5. Conclusions
The axiom system given here evolved from a specific philosophy of how information is structured in order to be manipulated by computers.
It leads to a gene-
ral framework comprising a great variety of models, from which the Vienna
objects and their derivatives considered so far are only special cases. The
axiom system is consistent in the sense that there is a model for it. The problems of independence and completeness have not been considered. They lie outside the scope of this paper.
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