Download 1. Introduction 2. Minimal grammar logics VAR set of

G R A M M A R LOGI CS
Luis FARINAS DEL CERRO & Mart t i PENTTONEN
1. Introduction
In this paper we present a simple method to define modal logics f or
f ormal grammars. Given a f ormal grammar, we associate wit h each rule
an axiom of a modal logic. By this construction, testing whether a word
is generated by a f ormal grammar is equivalent wit h proving a theorem
in t he logic. Firs t constructions produce mult imodal logics possessing
several agents [Ko], but the technique can be applied to construct also
more classical logics wit h only one modal operator.
Our approach is suggested by a met hod used in logic programming,
where the analysis or generation of a sentence is transformed to theorem
proving [Co, PW]. Other related work is done by Wolper [Wo], who defines
an extension of the linear temporal logic of programs to include regular
expressions.
2. Min ima l grammar logics
We begin wit h logics that are rather art if ic ial but are closely related
wit h f ormal grammars.
The expressions of the minimal grammar logic are constructed of the
f ollowing symboles :
VAR s e t o f propositional variables,
MO D m o d a l alphabet, interpreted as a set of agents,
modal operator constructors, and
c onditional connective,
where VAR and MO D are nonempty, dis joint sets. The set FOR of the
f ormulas of the minimal grammar logic is defined as f ollows :
VAR F O R ,
if A, B E FO R then A - * B E FOR,
if A E FO R and a E M O D then [ a] A E FOR.
124
L
.
FAR I F4 AS D E L C ER R O a n d M . PEN T T O N EN
The constructions [a], where a E MOD, are called modal operators,
and can be considered as agents. Furthermore, for arbitrary n 0 , a, E
MOD, [a,a,...a
nword.
] A
i stype 0 grammar is a quadruple G—(V,P,S), where V and T are disjoint
finite
S E V, and P is a finite set o f pairs (uy), u E
s h o
r t alphabets,
h
(V
U dV ( V U T
a n
)f * another
to
word y, x y , if x and y can be decomposed to x= x'u c" ,
o
,r xy vx ", Ewhere (u,v) E P. The relation derives, x 4
y=
transitive
closure o f i s
'([ y , Vai s closure
t h o fe T hr ee non-reflexive,
f l e x i vtransitive
e ,
U
denoted
yT T h e language generated by G is the set L (G) =
]
[ ba
)21wI * S w , w E
.] .It is. well
. known (see [Sa]) that the membership problem of type 0 grammars,
i.e.
the question whether a given word is generated by a given gramA
[ a
mar,
i
s
undecidable.
We use th is result t o construct undecidable
w
o
n
propositional
modal
logics.
r] , d
xa Given a type 0 grammar G = (V,T,P,S), we associate to it a minimal gramo
v L , that has the ax iom schemes
nmar logic
e
r [ v ] A for all (uy) E P, where A does not contain a n d the
dAl. [u ]A
V
w
inference rules
U
e
A B
C
T
cRI. transitivity
d
e
a
rl
i
R2. conditional necessitation
vl
e
[a]A [ a ] B
si
and rules, A, B, and C range over formulas, and "a" ranges
d
i
t In axioms
over
the
modal
alphabet.
ra
e
Under
the
epistemic
interpretation, the axiom A l defines an in cm
t
terdependence
between
th e knowledges o f agents. F o r example
lo
y
d{al[b]A [ c ] A can be read as follows : " I f a knows that b knows A, then
c knows a".
a
The notions of a proof and a theorem are defined as usual. A proof
l
of a formula A from a set S of formulas is a finite sequence of formulas,
each of which is either an axiom or a formula in S or is obtained by a
rule of inference from earlier elements of the sequence. This is denoted
by S H A . A formula is a theorem, 1— S, i f it has a proof from the
empty set.
G R AM M AR L OGI C S
1
2
5
Theorem 2.1. I t is undecidable whether a formula o f a minimal grammar logic is a theorem.
Proof. Cons ider a type 0 grammar G=(V,T,P,S). Because the membership problem of type 0 grammars is undecidable, it is sufficient to prove
that i - [ L I N [ v ] q i f f u = ,
1 - We shall show first that for any theorems of the f orm [u]q -+ [v]q there
v , a derivation
w h e r e
is
u ' v. The axioms of IL, correspond t o derivations of
q
s t hat [ L I
length
1. Asi s ume
]a[v]q
q -0
[ w
Iwlq] by
q RI . By induc t ion hypothesis, u ' v and v ' w, and
r s o p o u s w i . t I f i [ alI t t iq [ a
ip
consequently
o
l v e d
p
] [ nr ao[u]ci
theorem
a
r o pi rmauo v e d
fvduction
r
* I vi l sq
hypothesis
0
.
l y eI .
[b
b For
T
h Lthe
e inverse, consider a derivation u v x w v y w , where (x,y) E
]P.
R Bs yqsinduc
2e t ion hypothesis, [ u ] q [ v ] [ x ] [ w] q i s a t heorem. A s
a
Irf t i r ov n o i m
I x l[ wlq [ y ] [ w] c i is an axiom, by repeated application of R2 we see that
q
t
h
e
a
[v][x][4:1 [ v ] [ x ] [ w] q is a theorem. Now N A -0 I v ily llwlq is proved by
a
n
u
an application o f RI.
d
a
v
f
o
3.
Grammar
logics and Thue logics
l
l
o
w
s We shall now prove a similar theorem for more standard propositional
modal
d
i logics, which include the classical proposional logic as a sublogic.
r call
We
e these logics grammar logics. We study also a special case, where
c rules
the
t o f the grammar are symmetric. These logics are called Thue
logics.
l
y
f The alphabet of a grammar logic contains the propositional constant
r
false
i i n addit ion t o the symbols o f section 2. a n d propos it ional
o
variables are formulas. I f A and B are f ormulas and a E MO D, then
m
A B , and [a]A are formulas. The connectives ( n e g a t i o n ) , v (disjuncttion), A (conjunction), and 4-0 (biconditional) are introduced as usual :
h
AvB=
A A B = -1(A -0 e
1
B
)
,
a
n d
A
(A B ) A (13 i
+
4
1
3
<
a
>
A
=
0 A ) .
n
ah] e- a type zero grammar G=(V,T,P,S), and denote M 0 D = V UT.
T1 [Consider
I
A
.
m grammar
o
d
The
logic L , is axiomatized as f ollows . I t has t he ax iom
schemes
a l
o
p
e
r
a
t
o
r
<
a
>
i
s
d
e
f
i
126
L
.
FARINAS DEL CERRO and M. PENTTONEN
A l A -.(B -.A), (A ( B C ) ) . ( ( A((A - ) +
AB , )
*A2.( [a](A
A B ) ( [ a ] A [ a ] B ) for all a E MOD,
C
A3. [u][w]q
)
) [ v ],[ w ] q for (u,v) E P, w E MOD*, and q E VAR,
and the inference rules
RI. Modus Ponens
R2. Necessitation
A, A -.B
A
lalA
The proof and the theorem are defined as in section 2. We also have
the undecidability theorem, but now the proof is less trivial.
Theorem 3.1. I t is undecidable whether a formula o f a grammar logic
is a theorem.
Proof. Let G=(V,T,P,S) be a type 0 grammar and consider the grammar
logic 1,, associated with it. We shall show that S w if f H
[w]q, where q is a propositional variable.
G
[ SAssume
]q
-first+ that S I f the length o f the derivation is 0, then
w=S and H [S]A [ w I A follows from Al by RI. Otherwise assume that
S •
w
H [S]q
'
[ w' ] [ u ] [ w" ] q . Because [u][w")q [ v ] [ w " ] q is an axiom, by
u
repeated
application of R2, A2, and R1, we see that H [w '] [u][w "]q -0
[w
w "l[v][w "
]w We shall now prove that if H
Q
' G
by
defining the concept of G-truth - i f a formula is G-true then it corT
v [ hS ]eq to a- derivation.
responds
4
We shall show that all theorems are G-true,
a
s the
e] axioms
w [s w
because
are
G-true
and the inference rules preserve G-truth.
q
r" By
tt ithe
o
of LG, we can assume that the formulas do not
h axiomatization
e
n
n
=S
contain
other connective than T h e context c(p,A) of an occurrence
pH
w of
w a variable
. in a formula A is defined as follows:
[. W S
e
c(p,p)= empty,
]B d q
o (or c(p,C)) if A=B C and p occurs in B (resp. in C),
[y c(p,A)=c(p,B)
tc(p,A)=ac(p,B)
h
ifi A
w
i s
-[ a ] B .
]n Intuitively, c(p,A) is the sequence of modalities governing p in A. A
q
d
n
u
o
c
w
t
fi
o
ln
lh
o
y
GRAMMAR LOGICS
1
2
7
subformula Mp N q , where M and N are modal words and p and q are
variables, is called a pair, and occurrences p and q are paired. I f an occurrence of a variable is not paired, it is single. A pair Mp ---, Nq is strict
if p=q and there is no single occurrence r of the same variable such that
c(r,A)=c(q,A) or c(r,A)=c(q,A). The 0
- t r u At his defined
mula
o f recursively:
a
s u b f o r m u l a
_l_ is 0
i
n
variables
not occurring in a strict pair are G-false,
a
f pair
a l s Mp —*Nq is G-true if f it is strict and c(p,A) G
a
f
o
r
,e ( q , A
.enon-pair
a
subformula
),
B C is G-true iff B is G-false or C is G-true,
[a113, containing a connective, is G-true if f B is G-true.
We say that a strict pair gets its G-value by derivation, other subformulas
get their values by definition or by computation (two last lines). A formula A is G-sure iff for all modal words u, [u lA is G-true in NIA.
We shall now prove by induction on the length o f the proof that all
theorems of L
G Aal. rAe ( 1 3 --, A), ( A ---. (B C ) ) —. ( ( A -- B ) ( A C ) ) ,
G - I ) --+ 1 ) ---, A. All of these are proved in the same way, we choose
((A
s usecond
the
r e .as an example. The first A is not within a pair because the
right hand side is not of the form Mp with a modal word M and a variable
p. Hence A B and A —• C are not strict pairs. I f (B —• C) is not a strict
pair, then all of A, B, and C get their values uniformly and the whole
axiom gets the value 0
- t ar ustrict
is
e pair,
b ythe latter B and C are 0
becomes
-af a l s eG-true,
a nanddhence the
( axiom
A is 0
sureness,
-B
o( r dthat
e the
r addition of a modal word in front
pt rr u oe ).pweoIhave
sn itot establish
A
t-i othenoaxiom
of
does
p
not
r
violate
o
the
v
0
e
-a l 0
change
the
0
-tC
c t r ou t hhm. ) pI enu d e e d ,
-a
t htthe i-whole
G
truth
does not depend on the 0
t t rauof
o axiom
n
m
-o
r fu ot[a](A
h d—*B)
o af --, ([a]A
lt h
—. [alB).
e
I f A and B do not form a strict pair,
. t A2.
the
axiom
is
0
o
m
sp
t
r
r
i
e
c
t
f
i
p
x
a
i
r
s .
I
-e
t
r
u
e
a
strict
pair,
also
ialA
l
a
l
B
is,
and
their
0
cf
a
n
derivation.
Hence
A2
is
G-true.
0
b
y
-s( t r tu t rh i d e p e n d s
-o
ut r[u][w]q
sThe axiom
a
cBsA3.
ne n e s--,s [v][w]q.
t i h
e
is a strict pair which corresponds
sscp
e
e
n
to
a
l
derivation
s
s
of
one
step,
and
consequently
is 0
a
a
m
e
—
vious,
s.ls 0 arbitrary context can be added.
i, t cr ur aebecause
-a
op n ev sAssume
. that
p
-a
rub or ePonens.
se
i A and
s A B are G-sure. As A is G-sure,
,C sModus
o
b
) s bi t i o
u n a l
ct
a l
ca u l
a
t i
s
o
n .
i
It
fw
A
a
128
L
.
F A R I 4 A S D EL C ER R O a n d M . PEN T T O N EN
it gets its value by a computation and therefore it is G-sure in A B
Therefore
B must be G-sure.
9 t o o also
.
Necessitation preserves 0
sureness
- s u rrequires
e n e s G-truth
s,
in all contexts.
b
e
c
a
u
s
e
We have proved that all theorems are G-sure. Hence 1— G
implies
h [G-sureness
[t S ] qthe
we ] q o f [S]q [ w ] q , i.e. the existence of a derivation
S dw , ewhich
f iwasnto be
i proved.
t i o n
The special form of the axiom A3 does not look nice in comparison
o
f
with the other axioms. It remains open whether A3 can be replaced by
t
h
the scheme [u]A —.,[v]A. However, if the rules of the grammar are syme
metric, i.e. (u.v) E P implies (v,u) E P, then we can prove the undecidability
G such a scheme. Indeed, if the rules of the grammar are symmetric,
with
we- can have axioms of the form [u]A4--.[v]A. We have come to what we
call Thue logics.
A Thue system is a pair T= (V,P), where V is a finite alphabet and P
is a finite set of pairs of words over V, i.e. P V
gruence
. xV
C V
.. x V
=
.(i)
. xD
a y ,es f i n e
t
h
e
f(ii)
c o x =loxl' uoxn" , y=- x 'vx", where (u y) E P or (v,u) E P,
w
(ii) sthere
: is a word z such that x z and z y .
X
The theorem 3.1 holds fo r Thue systems with the modification A 3 '
T
[u]Ai-.[v]A for all (uy) e P.
Y
The concepts o f T-truth and T-sureness are defined as 0
i- t r u t fh
n a dstrict pairs
G Mp- —
sureness, but anow
fformula A if f c ( p , A )
.Nq a n d
M p + +- N inqthe senseaof section
valid
r 3.eThis is true because for any paired variables
-p
,
c
(
q
,
A
)
.
I
t
T a n d- q ti n A
r , uc(p,A)m(p,A)
u
e
vL
ii c ( sp , A )nm ( p , A )
stronger
T
f ,i Ac ) m
i e
as . ut cf ( theorem.
A
q
(q , A )
vin cTheorems
(t q ,f A ) m2.1 and 3.1 do not speak o f a particular undecidable pro(at q , A )n. omodal
positional
d logic, but tell about the impossibility o f a decision
algorithm
common
H
oc o hw
n e e vl eto yall grammar logics. Indeed, there is an undecidable
logic.
ric , We
k use
f an undecidability result o f algebra.
w
t It washe
proved by Cejtin [Ce] that the word problem " x y ? " is
undecidable
pa
r t e for fthe Thue system T=([a,b,c,d,e},P), where P consists o f
the
pairs
eA
r
3
t'
oi
ss
t
a
tT
e
as
o
m
e
w
h
a
t
G R AM M AR L OGI C S
1
2
9
ac=ca, ad = da, bc =c b, b d = db, abac = abace, eca=ae, edb b e .
Using this result, we get
Theorem 3.2 There is an undecidable propositional modal logic.
Proof. The construction is analogous to the one of the previous theorem,
now wit h Cejtin's system T. Consider the logic wit h the following axioms
and rules o f inference:
A l . A -) (B A ) , ( A ( B -) C)) ( ( A -* B) ( A -) C)),
((A -) I )
A ,
A2. [ a] (A B ) ( L a i A -> 1a1B)
A3. [ u] A(-)[ v ] A f or all (u,v) E P,
RI. Mo d u s Ponens,
R2. Necessitation.
The undec idabilit y n o w f o l l o ws f r o m t h e as s ert ion U
T 1[u]p4-.1v1p f or an arbitrary propositional variable p. We omit the proV
i f it is analogous to the proof of Theorem 3.1 and we have already
of because
sketched the change in the proof of A3.
4. Soundness and completeness o f Thue logics
Unt il now we have concentrated in the undecidability of some proposit ional modal logics. We shall now prove that the Thue logics are "wellbehaving" logics in the sense that they are complete in terms o f Kripke
style semantics.
In the following, we define the semantics of the Thue logics. To define
the meaning of formulas, we shall fix the set of states or worlds, and the
relations between the words corresponding t o the modal operators.
Formally, a model is a t riple
M - (W, [ R a E MO D) , m),
a
where W is a nonempty set of states, f or each a in MOD, R
a i on
tion
s Wx
a W (to
r be
e characterized
l a later), and m is the meaning function
mapping each variable a to a subset of W, consisting of those states where
this variable is true.
130
L
.
F A R I N A S D E L C ER R O a n d M . PEN T T O N EN
Given a model M= ( W, [ R
I
a
E in a state w, denote M, w sat A (resp. M, w unsat A),
A a
is (un)satisfiable
if f M
oneOo f D
the }f ollowing
,
conditions holds :
m
)
,
(i) M , w unsat I ,
w
e
(ii)s M , w sat
a p (p yE V A R) i f w E m( A ) else M, w unsat p,
(iii)t M, w sat
B if f M, w unsat A or M, w sat B,
h A a
(iv)t M, w sat [ a] A if f f or all w ' such that w R
a a
The set ex t , , (A)= [w I w E M, M, w sat A l is called the extension of A
w f M o, w r
'm
s
a
t
in
A is t rue in a model M, i f ext,,(A)=W. A f ormula A
B uM. A. lf ormula
a
is valid, A , if f A is true in all models. A f ormula A is a logical consequence of a set S of formulas, S A , if f for every model M and f or every
w i n M, i f M, w sat B f or all B E S t hen M, w sat A . A f ormula A is
satisfiable if f f or some w in W and f or some model M of A, M, w sat
A. A set S of formulas is satisfiable in a model M and a state w, M, w
sat S, if f M,w sat A for all A in S. Finally, a set S of formulas is satisfiable
if f M, w sat S f or some M and w.
We shall show that i f the relations in the models are restricted in a
suitable way, the axiomatization of Thue logics is sound and complete.
Consider any model M = (W, [ R
word
a e,alet R, be
E the identity relation on W, and f or any a E M O D and
OO DD, let
} R
,
wM
E M
relations.
)
.
a m
F= T=(RoMO D, Pr ) be a Thue system. We say t hat a model M =(W,
„ Let
a
• h
R e
[ Rt
e,
m
p
t
„a
speak of T-satisfiability, T-validity etc. and use the notation , for them.
y
w
h
e
r
I
e
a
Theorem 4.1. (Soundness) I f 1— „A then r ,A.
•
E
r
e
f
e
M
Proof. One can easily check that axioms A l t o A3 are T-valid, and RI
rO s
and
R2 preserve the T-validity. For example, the T-validity [u]A4-*[v]A is
t
D
seen
as follows. I f M,w does not satisfy [v]A, there is a state w ' such that
}o
(w,w
' ) E hR , a n d M , w ' d o e s n o t s at is f y A . N o w b y R , Ç R„,
,t
(w,w
')
E R
e
m
uc wa satisfies
M,
n
o d m [ ulik 4)
,hp The
Ne A
. s
result, the completeness, states that all valid formulas are
onconverse
i
ctheorems.
e
is
t
i We o
prove it by the method of canonical models, f ollowing the
M
,
guidelines
o f [ HC] .
an
w
o
T
df
o
e
s
m
n
o
o
d
t
e
s
a
l
t
i
,
s
f
i
y
GRAMMAR LOGICS
1
3
1
In order to simplify our reasoning, we use the classical connectives ,
v, etc. such as they were defined in section 3.
A set S of formulas is T-consistent, if there is no finite subset [B,,...,B
of uS such that 1— T
( 3l
T-consistent,
i f it is T-consistent and f or any f ormula A, either A E S
1 AA . .E. S.
A The f ollowing lemmas are quite obvious :
or
B „ )
(Lemma 4.1. [HC1 Let S be a max imal T-consistent set of formulas. Then
f—
o r any 1f ormulas A and B,
A
)
.
(a) exactly one o f A and —
A
1 AA v Bi EsS i f fi AnE S or B E S,
(b)
s
e
S
(c) i f ,1—T A then A E S,
t
(d)
S i f T A a n d A E S then B E S.
o
Lemma
4.2. ( HC] Any T-consistent set S can be extended to a max imal
f
T-consistent
f
o
r set.
For
any
set
m
u
l S of formulas and u E MO D' , denote
a
s
Su = [ A I [ u] A E SI.
i
s
m
a
The canonical
T-model (W, [ R
xa I
ia
E
(1)
W is athe set of all max imal T-consistent sets of formulas,
m
M O D
) ,
(2)
w,w ' E W, (w,w ') E R
lm f o r any
)
u if of rf any wvariable
a
w A ) = [w I A E
(3)
A, m(
i
s
d Wee shall
f now
i prove
n
e
that the canonical model, indeed, is a model :
d
a
s
Lemma 4.3. The canonical T-model M=( W, [ R
fT-model.
o
l
l
o
a I
a
E
M
O
M
w
s
m
)
i
s
:
a
Proof. We must show that if (u,v) E P, then R„ = R
ysufficient t o prove that R
.order
s. ythe
m inclusion
m e t r Ry ,
y ç B yt o Rshow
iyAç s t sR u
Therefore,
f or m
anye Mi A E w,swe should prove A E w I f [ u] A E w, then
(
w
u
by
[ v , ] Aw and Lemma 4.1.(d), [ v ] A E w. A E w ' n o w f ollows
f,' rom wv
w ) we
E The
m
u completeness
s
t
theorem is largely based on the f ollowing property
sR
h
o
of the canonical wmodel :
it
t„
h
a
e
(.
w
,.
w
'w
)
c
E
w
R
.
I
n
132
L
.
F A R I N A S D E L C ER R O a n d M . PEN T T O N EN
Lemma 4.4. Let M=(W, [R
a Then
T.
I a for any
E formula A and any world w EW, M,w sat A if f A E w.
M O D } ,
m
Proof.
The
) Lemma holds f or propositional variables, by the def init ion
of
M.
We
b
e shall prove that if it holds f or formulas A and B, then it will
hold
f
or
t
h A , eA v B, and [a]A, too.
A . By induction, M, w sat A if f A E w. Hence,
c Consider
o
n theof ormula
n
M,
i wcsat —
a1 A lif f not M, w sat A if f A e vv. But now, by Lemma 4.I.(a),
A
m e w if fo—1 A E dw.
e By thel def init ion o f sat, M, w sat A v B if f M, w sat A or M, w sat B.
By
o induc t ion hypothesis, M, w sat A or M, w sat B i f f A E w or B E
fBy Lemma 4.1.(b), this equivalent t o A v B E w.
Consider now [ a] A. " i f " . I f [ a] A E w t hen by the def init ion o f R
A E
a w' f or all w' satisfying (w,w ') E R
a .these w' , L, w' sat A , whic h implies M, w sat [a]A.
all
. "only
B y if". iI f n[a]A
d ue w,
c then
t i oby nLemma 4.I.(a), [ a ] A E w. First we shall
h
y
p
o
t
h
e
s
i s ,
show t hat wa U A l is T-consistent.
Otherwise there would exist f orfmulas B,,...,B„
o
r
E w
a s u cand
h by Al, also H
necessity
ttions
h[ aofa] Al
T
( tBand
, H
propositional calculus, H
T
A
.
.
.
A
B
T [ because
now,
a ] B , taIB,
A . E
. .w,A by
[ aLemma
] B 4.1.(d) also [ a] A Ew would hold,
(contradicting
n
a B ,
—
)A
] w'
' T-consistent.
a be] a max
A imal
. T-consistent set containing it. By
1
[ a. ][ A . a [.Let
is
B
A
.
B
u hypothesis,
t
e
induction
M, w' satisfies —
a
B
w Ainit ion
1
def
a yon f dR
d o e s
f)a
u
t
h
a Wenoare dnow
n
tr
ready to prove the completeness theorem :
A
e
r
h ,t w i s f y
,st ( aw
.e
a
p T . -p
l
'A
)
Theorem
4.2. (Completeness) I f T
iB
c
a
c o n R
s
B
E
A
e yn
H
T
yi s t th e
tProof.
a
h
e A is T-valid, it is true in all T-models, by Lemma 4.3
A
.
I
f
a
f
ormula
tn c y
.especially in the canonical T-model. Assume that A were not a theorem.
h
o
H
e An is T-consistent. Let w be a max imal T-consistent set containHence,
e
fc
e
,
rw A . By
ing
u Lemma 4.4, M, w satisfies —) A, and consequently cannot
M
l.
satisfy
A, econtradicting our assumption.
,
o
H
e
w
fRemark
n
c4.1. The logics we have introduced are mult imodal. The above
d
e
,
proofs
remain
valid i f we replace every a, in M O D =
b y
an afo
w
firmative
modalit y 0
e
1
sa
[U1 1 a n d
n
[
u
s e
o
tA
h
e
L
ea m
m
sl
o
ni
t
cs o f r r e
sy p o n d
e
n c e ,
[
w
h
a
GRAMMAR LOGICS
1
3
3
provides every af f irmat iv e modalit y 0 wit h an accessibility relat ion R'
characterizing 0 (see Lemmon [Le], pp. 62-63).
5. Conc lus ion
In this note we have presented a method to define modal logics that
simulate the behaviour of grammars. As a corollary we get a simple proof
for the undecidability of a propositional modal logic. Of course, if more
restricted classes of grammars (eg. regular grammars) are considered, these
logics may be decidable. This methodology is related with the logic grammars (see [Co] and [PW1) that have become popular in logic programming. However, grammars based on classical logic need predicate calculus,
while in our case propositional calculus was sufficient t o get f ull c omputational power. St ill f or the easiness of expression and practical efficiency the int roduc t ion o f predicates is useful.
We have not considered here the mechanical proof procedures that are
very import ant in programming. I t will be the matter of future research.
Acknowledgements. We are grateful to David Makinson for valuable comments, both on general structure of the representation and on details such
as Remark 4.1, and t o Andrés Raggio, wh o revealed an error i n t he
undecidability proof of an earlier version of this paper. This work was
funded by the ALPES project of ESPRIT, and by the Academy of Finland.
Université Paul Sabatier &
University o f Joensuu
L
.
M
FARISiAS DEL CERRO &
A
RTTI PENTTONEN
REFERENCES
[Ce] G.S. Cejtin: An associative calculus with an unsolvable problem of equivalence (In
Russian). Doklady Akademii Nauk SSSR 107, 370-371 (1956).
[Co] A . Colmerauer : Metamorphosis grammars. In L. Bloc (ed): Natural language Communication with Computer, 133-189. Springer Verlag 1979.
[HC] G.E. Hughes, M.J. Creswell: A Companion to Modal Logic. Methuen 1984.
[Ko] K . Konolige: A deduction model of belief and its logic. SRI International, Technical
Note 326, 1984, Menlo Park, California, USA.
134
L
.
FARISIAS DEL CERRO and M. PENTTONEN
[Le] E.J. Lemmon : An Introduction to modal logic. American Philosophical Quaterly, Oxford 1977.
[PW] F.L.M. Pereira, D.H.D. Warren : Definite clause grammars for language analysis. Artificial Intelligence 13, 231-278 (1980).
[Ra] MO . Rabin : Decidability of second order theories and automata on infinite trees. Transactions o f American Mathematical society 1969, 1-35.
[Sa] A Salomaa: Formal Languages. Academic Press 1973.
[Wo] P. Wolper : Temporal logic could be more expressive. Information and Control 56,
72-99 (1983).