Logique et Analyse 125-126 (1989), 131-137
A MO DA L EMBEDDING FOR
PARTIAL INFORMA TION SEMANTICS
Juan BARBA ESCR1BA
In [1], a modal embedding o f data semantics into the modal logic S4.1 (S4
plus the McKinsey axiom I 0 A --0 [ I A ) is presented. (The same embedding also appears in 121, with in a useful overview o f the main topics in
partial logics). The existence o f maximal points in data models plays an
important role in the proposed embedding. The author says that it would
be interesting to extend such a reduction to the case in which no maximal
point is postulated, and conjectures that the result should be an embedding
into S4. The aim o f this paper is to show how the conjectured embedding
can be carried out.
1. Partial information Semantics
We assume a formal propositional language PH, (partial information language) whose logical symbols are the unary connectives —
1MUST,
, M and
A Ythe binary
a nones
d A and P a r t i a l information models are
structures D = (I , p , 1 ) , where I i s a partial order on I, and
and a r e relations between elements o f I and propositional variables
which satisfy the following restrictions: for any i I and any propositional
variable p it is not the case that both i p and i p hold simultaneously, and
for any j E l such that i j , i p implies j a n d i l p implies j l p . D is
a data model i f every maximal chain in the partial ordered set (I, ) contains a greatest element j such that fo r every p either j p o r j l p . Such
elements are called maximal elements o f D.
Relations p and l are extended to every wf f by the following clauses:
i f i l A , and i A if f
A B i f a n d iP B , and i A
i
iAf
for all j, i j
,
B i f i forf some ij, il j , j A and j B .
ij A
N I AAY
j,r i j
o An if fl for
y some
o
i, = jI fi A l . B
.
M
j
A
p Y
A
B
.
i
f
f
f
o
132
J
.
BARBA ESCR1BA
i M U S T A iff for all j, luj, there is some k, jdc such that j k and
k A.
MUST A if f there is some j, i 4 such that for every k, jck implies
k A.
Notice that if D is a data model, the clauses for MUST are equivalent to
those formulated in
N411ST A iff for all maximal j, i j
A iff for some maximal j, i j
,MUST
j
, .i A .
A
We have formulated more general clauses because those in [1] make no
sense when D is not a data model.
2. From information models to modal models
Given a model D = (I, Q,
R, V) as follows:
)
we can obtain a modal model M , = (W,
W= i E I
} U { v
(where,
for each i, j El, u
i
ui I' . v)
1:• R is the reflexive and transitive closure of the set
, vi , ea r e
a In yu
t aw ino
The
by:
d divaluation
f )f e V ris defined
e
n ft
:o
V(p, u
o V(p,
b
e
c
ri
vi j
t si s Qo
) =
a Notice
j ne given D and M
)m
1 = that,
d Dp
variable
) i p i f V(Lp, u
i1 f p,
d
i
s
t
i= ,V(
in
f fElo uver
V(111p,
= cr( yE t p= ,
I , iff for every w EW such that u,Rw (or v
)i e =ifi vn tetV
fv) i ih that
r
irecall
p =C (eu I
for
oI awsome
m dthat i 4 if f V(Elp, u,) = V( O p, v ) = 1. The case
iR
r,.: nj such
u
a similar way.
R
I eproven
wi n v in =
)is
e
r
,
i1f fy nfa ,v
iac p ot r n , o pd o s
R
i, i w
r t ) yi p o n a l
V
ii i
f,
f
(Vp pp( )1 1 1
,
,—
i }
w
if .p ,v
PARTI AL I NFORMATI ON SEMANTI CS
1
3
3
3. From modal models to partial information models
The relation R in models M , obtained from partial information models D
is reflexive and transitive. Now we show how a partial information model
D
transitive.
Let M = (W, R, V). Define D
M
M
=
(
I ,
)
c
W
I
a
s
u v i f uRv
fn o l l o w s :
b u i f f V(Elp, u) = 1, and u p i f V( 111
e -Thisip,
definition
thatI for. any u, v I such that t.tv , u p implies
u ) ensures
=
o
v p and u p implies v p
b
,Thiss follows
o
t from
h athet fact that M is a model of the modal logic S4 (because
tR is reflexive and transitive), and so C I A D
M
a
and
,i tmv.
s Then V(Op, u) = 1, and V(1110p, u) = 1. As uu
i-1
V(Elp,
1 1 0 v)
A = i1 and
s v pv . aIn the
l i same
d . way we can prove that u p implies
a
vn
vp
,
u
R
v
,
t
h
u
s
S a pr pt io as e
e
u
l
d
i4. Antranslation
f o from
r mPIL to ML
f
a t i o n
r
m
o
d
By
o M L we understand a standard modal propositional language whose
e
l
. are logical
symbols
m
the
in a l l differences being purely notational. Notice,
1
, same
A appearing
,
a
however,
that in t w o different translations A and A
O
.
n
-T a sentence
f i n roughly
e d speaking,
f
o A ' ris to be interpreted as the
each
hr e e d A,e where,
y
translation
for " A is supported", and A
t r a n s l
jected"
-m a s(seet the
h claim
e on p. 234). However, that double translation can be
a
t i o n
o
simplified
clause for the negation operator (M
t r a n in
s view
l a of
t the
i osamantic
n
p
r
o
p
fidf Ao andr/%/1 o
s
e
d
translation
"Ia A i fAproposed below, T(A) is equivalent to A' (of Ill), while T (
h
e
l equivalent
M
i-is
s to A
r1-A
e
in
[11,
but,
also, it enforces us to include a translation clause for each kind
rmA ) ) e,
iof
instead of a single clause for every formula of the form
.w
B hy iformula,
o negated
c
sh
t h i s
d
For each
PIL-wff A we define a ML-wff T(A) as follows:
b
m
e ea
a t h ks
io
a
d cs
l T(p)
e
=y
lw
l
e
M
A
c-i
a
n
=
a
v
(w
o
i
-h
d
Ii
A
)
t*c
134
J
.
BARBA ESCR1BA
T(
- (
T
-1- A B) = T(A ) A T(B)
T(A
pT )(
1
=T
- (A -B ) = E (T(A )-. T(B ))
T
1 T(
(T(MA
(
1
Y A) = 0 T(A)
T
A
(AA(
-)T(MUS
T A) = L I 0 T(A)
-A4
IT
(
B))
3
M
-=
)
1
InA
)order to prove the adequacy o f this translation we need the following
0
lemma:
Y
M
(=
A
U
T
)S
Lemma
( 1: Let D = (I , u , a n d M = (W, R, V ) be a partial infor=T
mation
v model and a modal model, respectively, and suppose there is a
A
T
A
function
)T
f from W onto I such that for every u, v E W , f (u) = , f (v) =j,
(a)
)A
iu
j
if f uRv
(
A
=
b)
- i i f f V (E p , u) = I
T
)c)
0
(1 i i f f V(E] u ) =
Then,
L
-B fo r every P I L -wf f A,
I1
) A if f V(T(A), u) I
T
B A if f V ( T (
(Notice
() - that, under the hypothesis of the lemma, f (u) = f (v) implies V (0 p,
A )p, , v), and V(111 u ) V ( E v ) . Observe as well that f
u) =- V' (D
)
is onto
I uI, i.e.,
fo r every i E I there is some u E W such that f (u) = i).
=
Proof:
Induction
on the complexity of A. When A is p, the lemma is trivialA
I
.
ly satisfied.
)
When A is either I I I o ronly the
consider
A remaining cases. We shall write u
ments
o f WBsuch ,that f(u,) = i, f ( v ) =
iA
,t -v hi f e f for all j E I, it' i j and j B then j C , if f (induction hypoip thesis
t r o oplus
r o
e p f r e of
s fe, more
n t briefly Eh.) for all u
properties
i V(T(B),
e
i l s us ec uh - t h a t
u , R u
s iiffi V(111(T(B)-.T(C)),
m
p
l
eu,) = I .
. - i) i= f f for some j, i j
S . V(T(
uo
,
1
iji M
j, i j
l -j B
td for some
'
meAapY nB iff
u
R
,
u
i
f
f
V
(0
T(B),
u,)
= I.
s ,j u
)l C
i e s
j, B=
i
f
f
V
,ViI( fT(, f T
( w
. h h ) . ),
V
( i BC
(u e
T
(
B
)
,
r
) ,
=
j e
u
I) u
,
,
i
f= R
r
u o
)
sl i=
m
ao
e
PARTI AL I NFORMATI ON SEMANTI CS
1
3
5
- i M A Y B: this case follows directly fro m the former one, because a
simple inspection to the corresponding semantic clauses shows that
i N I A Y B i f not i M A Y B.
B
if f for all j, iu j, there is some k, j k such that k i f
(Eh.) fo r all u
j1, if f V ( 0 0 T(B), u
- i,i M
u U S T B if f there is some j, iu j, such that for all k, ju k, k B
, (i.h.)
)i i =f there
1 is. some u
u
iR u
k,j u
Now
)i, consider any partial information model D and the corresponding
modal
=
t umodel
R
h e M
r
D 1
each
ieI l
i. Dcan
we
,i e establish
ss u ctheh following lemma:
,f i v its h o a
t
Lemma
in e fm 2:oeGivenr D and M
D
= 1
t fe
E
h e vif f V(T(A),
u
e
,W
f uV
rki An yi cf V T (
.t i (u
,- o n u
, uour attention to a modal model M and the corresponding D
f Let's
(kiI A )turn
As
M
W
i
1
sa a>
,R
u, the
u identity function defined on W satisfies the conditions o f
.i E
=
we can establish a new lemma, simila r to lemma 2:
tslemma
ik) 1. Thus,
sf f IR
s1o ul.
ilLemma
eT
ku•o 3:wgiven M and D
m
s (,cu: A if f VV(T(A),
g u) = 1
u
A
if
f
V
(
T
(
tf, -(h
( 1
h
-t- - - 1
1
3
1
A
)
,
u
)
following
theorem
can be easily proven from lemmas 2 and 3:
u The
e
B
)h
,
=
ci )a
. I : A P1 L -wff A is valid in the class o f all partial information
)Theorem
o
,tI
•
i f T(A ) is valid in the class o f all reflexive and transitive modal
=models
n
u
V
i.e. i f T(A) is Szt-valid.
jmodels,
d
i(
Proof:
Suppose
that A is not valid in a model D. By lemma 2 T(A ) is not
i, )T
tfvalid
=
( in M
D 1
model M in which T(A ) is not valid, and by lemma 3, A is not
i(transitive
B
. .) in D,
vvalid
o
C o•u n v
iM
n
rk•s De be a data semantics model. Then, f o r every i E 1 there is some
s).e Let
j E l such that iL j. Take M
y ),
i=lmaximal
iiI ) =
n
j
b
e
t,f. L e t
a
fT
h
o(m a x i m a l
e
e
l
e
m
e
n
t
rA
h
i
n
)
y
.
piD
136
J
.
BARBA ESCRI BA
Then, as for all p, either j o r j p , v
iis no
d ok eE Issuchnthat
o j tk and j k , there is no w e W such that u
ei xw. Ii f we
s apply
t
i bulldozing
n
modal
methods (see 131) to M
W
,
aRmodel
w D a Mn d
n
d o b contains
.a , maximal
every
w e chain
t a i a'greatest
n
element. On the other hand, i f M
a a modal
smodel such that (W, R) is a partial order and every maximal
D
is
=
chain
h it contains
e
ar greatest
e element, then D
(t W in
m
i
s
a
d
a
t
a
.
this
claim, consider any maximal
chain X in W, and let u be its greatest
m
element.
oR dAs eW l= I. and vRw if f vLw, X is a maximal chain in (I, a n d
,
T
u
o
element. Moreover, fo r any p,
u )
= V (p , u) and
. is its maximal
c, (E hu ) e= V (c u ) k, because u is a reflexive dead end. Thus, if V(p,
V
u)
V = I then i p
. As
a itn isdpointed out in [1], the class o f all partial orders with greatest
o
elements
) t h ef orr maximal chains characterizes the modal lo g ic S4.I (S4 +
w
i s e E A ), so that from lemmas 2 and 3 and the remarks above we
sIII A-s-C>
can
(u establish
i
f the following theorem:
(
cV
—
hTheorem 2 : A P I L -wf f is valid in data semantics i f f T(A) is valid in the
modal
p
, logic S4.I •
t
u
h
)a So we have extended the reduction proposed in I 1 I fo r data semantics to
the
=
t case in which no maximal point is postulated. The result is a reduction
which
embeds partial information semantics into S4, keeping the embedding
I(
of
)W data semantics into S4.1 presented in [ I
li. .
p
,
5.
.R A note on a possible extension o f the above translation to quantified
. languages
)
i Partial information logic could be extended in order to obtain a quantified
predicate
logic. Then, partial information models should be modified in such
s
a
a way that an individual domain is assigned to each point in the model (as
well
p as suitable interpretations for predicates). It seems reasonable to impose
a
a nested domains condition to the resulting models (as we are dealing with
increasing
information stages) and to interpret individual terms as rig id
r
terms.
Then,
it is not difficult to extend the method developed above to
t
obtain
a
translation
from such quantified PIL to a quantified ML with nested
i
domains,
rig
id
terms
and free logic, in a simila r way to the translation
a
presented
in
[41.
O
f
course,
different options concerning the conditions
l
o
r
d
e
r
s
u
PARTI AL I NFORMATI ON SEMANTI CS
1
3
7
imposed on domains or the semantic clauses for atomic sentences and quantifiers would lead to different kinds o f quantified modal logics (see 151 fo r
a classification of quantified modal logics) and, probably, different translations. But that goes beyond the aims and scope of this paper.
Aknowledgements
Thanks must be given to the referee fo r his suggestions, which helped to
improve this paper. I am particularly indebted to Huberto Marraud for his
advice, encouragement and friendship, which lead to this and other works.
Universidad de Valladolid
Departamento de Filosoffa, L ig ica y Filosoffa de la Ciencia
REFERENCES
[1] van Benthem, J.: "Partiality and nonmonotonicity in classical lo g ic" ,
Logique et Analyse, 1986, 114, 225-247.
121 van Benthem, J.: A manual o f intensional logic, 2nd revised edition,
Chicago University Press, Chicago, 1988.
[3] Segerberg, K.: An essay in classical modal logic. Filosofiska Studier,
Uppsala Universitet, Uppsala, 1971.
[4] Barba, J.: Sistemas f o i
- toral Dissertation, Universidad Autônoma de Madrid, 1988.
151
in modal logic", in D. Gabbay and F.
m aGarson,
l e s aJ. f W.
i :n"Quantification
e s
Guenthner
(eds.)
Handbook
of
philosophical
logic, Vol II, p. 249-307,
a
l
a
D.
Reidel
Publishing
Company,
Dordrecht,
1984.
s e m d t z t i
c a
d
e
S i t u a c
i o n e s ,
D
o
c
-