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Computer Science Technical Reports
Department of Computer Science
1971
An Axiom System For the Weak Monadic Second
Order Theory of Two Successors
Dirk Siefkes
Report Number:
71-056
Siefkes, Dirk, "An Axiom System For the Weak Monadic Second Order Theory of Two Successors" (1971). Computer Science Technical
Reports. Paper 478.
http://docs.lib.purdue.edu/cstech/478
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AN AXIOM SYSTEM
FOR THE WEAK MONADIC SECOND ORDER THEORY
OF TWO SUCCESSORS
D i rk S i efkes
July ,
CSD
1971
TR-56
ABSTRACT
A comp l e t e ax i om sys t em for the weak monad i c second order
t heory of two successor func t i ons , W 2 S , is presen t ed .
The
ax i om sys t em cons i s t s , rough l y , of the genera l i zed Peano
ax i oms and of an induct i le
def i n i t i on of the f i n i t e
se t s .
For the proof , me t hods of J . R . Buch i and J . Doner are used
to ob t a i n a new dec i s i on procedure for W 2 S , whose proofs are
eas i l y forma l i zed .
D i fferen t f i n i t eness ax i oms are d i scussed .
-1-
0.
In t roduc t i on .
Le t W2S be the weaK monad i c second order
t heory of two successor func t i ons , i . e . the t heory of the
fu l l b i nary tree wh i ch a l l ows quan t i f i ca t i on over bo t h
e l emen t s and f i n i t e subse t s of the t ree .
Doner 13] , and
Independen t l y t hough somewha t l a t er Tha t cher-Wr i gh t [7] ,
have shown t ha t W2S Is dec i dab l e .
W 2 S is t hus t r i v i a l l y
ax i oma t i zab l e by i ts true sen t ences .
It is no t just for
aes t he t i ca l reasons , however , t ha t this paper presen t s a
"neat" ax i om sys t em for W2S .
When work i ng w i t h monad i c
second order t heor i es one ac t ua l l y worKs w i t h fragmen t s of
se t t heory .
Thus there is no abso l u t e frame of monad i c
second order l og i c , and i t is ques t i onab l e whe t her there is
such a frame even for on l y the dec i dab l e monad i c second
order t heor i es .
Therefore when prov i ng the dec i dab i l i t y of
a monad i c second order t heory , one shou l d spec i fy -what par t
of se t t heory one needs for the proof .
The s i t ua t i on is
l ess uncer t a i n in case of a weak monad i c second order t heory .
S t i l l there are d i fferen t def i n i t i ons of i nf i n i t y , some of
wh i ch are equ i va l en t on l y by the ax i om of cho i ce .
So one
shou l d find ou t wh i ch def i n i t i on(s) one is us i ng .
In case of a dec i dab l e t heory , to revea l the very
con t en t of the t heory one has to forma l i ze the dec i s i on
procedure , and under the way to co l l ec t al l pr i nc i p l es one
needs for dec i d i ng .
The resu l t for W2S is a comp l e t e ax i om
sys t em . wh i ch cons i s t s of t hree par t s :
( i ) Ax i oms for the
e l emen t ary l og i c of this l anguage , (i i) Ax i oms for the two
successor func t i ons , wh i ch are genera l i za t i ons of the Peano
ax i oms for one successor .
(i i i) Ax i oms charac t er i z i ng the
-de-
f i n i t e sets as an Induc t i ve s t ruc t ure genera t ed from the
emp t y se t by the opera t i on of ad j o i n i ng an e l emen t .
In
o t her w o r d s , one ge t s exac t l y the f i n i t e sets by repea t ed l y
add i ng s i ng l e e l emen t s , s t ar t i ng from the emp t y se t .
Th i s
def i n i t i on , however , is i nduc t i ve and no t exp l i c i t , since
"repea t ed l y" means "f i n i t e l y of t en" .
As usua l w i t h monad i c second order theories. , Doner ' s
dec i s i on procedure for W2S i nvo l ves f i n i t e au t oma t a- - t ree
au t oma t a in this case .
It is h i s d i scovery t ha t one has
to have t ree au t oma t a worK i ng backwards , from the branches
to the roo t , in order to make de t erm i n i s t i c au t oma t a usefu l .
The au t oma t a no t i ons are eas i l y forma l i zed for our comp l e t eness proof .
Doner ' s ma i n proof t oo l , however , tree i nduc t i on ,
canno t be expressed in the l anguage of W2S ; so we canno t
forma l i ze h i s proof .
Instead we g i ve a new dec i s i on
procedure , wh i ch uses Doner ' s backward au t oma t a , bu t resemb l es
in s t ruc t ure the dec i s i on procedures for the weaK and the
s t rong monad i c second order t heory of one successor of
Buch i [l] and [2] , as presen t ed in [6 j .
I w i sh to express m y t hanKs to J . R . Buch i who grea t l y
i nf l uenced this worK and i ts presen t a t i on ; some of h i s ideas
can be. found in this i n t roduc t i on .
I a l so w i sh to t hanK
J . Doner for many he l pfu l d i scuss i ons on the sub j ec t .
And I thank G . H . Mu l l er who or i g i na t ed and s t i mu l a t ed my
i n t eres t in monad i c second order t heor i es .
-131.
The b i nary t ree and the sys t em W 2 S .
se t of al l f i n i t e sequences of
O's
Le t
and
T2
l ' s.
be the
T2
can
bes t be p i c t ured as the fu l l b i nary t ree , where the roo t
represen t s the emp t y sequence
two successors ,
xO
and
t, u, . . . ,z,
equa l i t y sign = , two
Xz ,
Tg
cons i s t s of i nd i v i dua l
se t var i ab l es
unary
and an i nd i v i dua l cons t an t
form
and each e l emen t
x
has
xl .
The l anguage to descr i be
var i ab l es
e,
x = y , and X = Y .
U,V, . . . ,Z,
the
func t i on symbo l s s^ and
e.
s^ ,
Pr i me formu l ae are of the
Arb i t rary formu l a
are bu i l t
up from pr i me formu l ae us i ng sen t en t i a l connec t i ves and
quan t i f i ers for bo t h types of var i ab l es .
The i n t erpre t a t i on
of the formu l ae is sugges t ed by the no t a t i on :
i nd i v i dua l
var i ab l es range over the e l emen t s of
T 2 , se t var i ab l es
range over f i n i t e subse t s of
T g , so
Xz means "z is an
e l emen t of
e
X " , the cons t an t
denotes the roo t of the
t ree , and the two func t i on symbo l s are used for the two
successor func t i ons .
iff i t is true in
T^
We s t i pu l a t e t ha t a sen t ence is true
under this i n t erpre t a t i on , and ca l l
the resu l t i ng sys t em ( = i n t erpre t ed t heory) W 2 S , Weak
monad i c
second order t heory of 2 Successors .
c
Th i s name is
adap t ed from Rab i n ' s S2S for t he correspond i ng s t rong
monad i c second order t heory of [ 4 J . — F o r the res t of the
paper "set" w i l l norma l l y mean "fini te subse t of T 2 " .
In formu l ae of W 2 S we w i l l use the fo l l ow i ng
no t a t i on :
We w i l l wr i t e xO and x l instead of
s-j^x) respec t i ve l y , espec i a l l y
and s ^ e ) .
0
and
A l so we w i l l of t en wr i t e
1
SQ( ) and
for
z 6 X
sQ(e)
instead of
X z , and we w i l l use free l y the usua l se t - t heore t i ca l
no t a t i on
e.g.
z £ X U {yj s t ands for
[Xz v z=y j .
S i m i l ar l y , sets and func t i ons of e l emen t s or subse t s of
T2
w i l l be def i ned by comprehens i on ; i t should be kep t
in m i nd t ha t thus defined terms are used on l y as abbrev i a t i ons
for express i ons of the forma l l anguage . --We w i l l use Greek
cap i t a l l e t t ers to deno t e formu l ae of W 2 S .
The symbo l
w i l l deno t e l i t era l equa l i t y of formu l ae in def i n i ng
abbrev i a t i ons .
=
2.
The ax i om sys t em .
The purpose of this paper is to show
t ha t the fo l l ow i ng three sets of ax i oms t oge t her cons t i t u t e
a comp l e t e ax i om system (for der i vab i l i t y) for W 2 S ,
i.e.
exac t l y the t rue sen t ences of W 2 S are der i vab l e from this
ax i om sys t em .
Par t A :
An arb i t rary ax i om sys t em for the e l emen t ary l og i c of W2S ,
regarded for the momen t as a t wo-sor t ed e l emen t ary t heory .
Here we add fur t her two equa l i t y ax i oms :
(LEIBNIZ EQUALITY)
(VZ)[Zx
(EXTENSIONALITY)
- Zy]
(V z )[Xz -
Yz]
*
x = y
«
X = Y
Par t B :
Genera l i zed Peano ax i oms for the two successor func t i ons :
(OEl)
xO ^ e
(0E2)
>
xl £ e J
(SE1)
xO = yO
(SE2)
xl = yl
(SE3)
xO £ y l
(IE)
$(e) A (V 2 )[ § (z)
(the roo t has no predecessors)
x = y
-
x = y
>
(branches do no t merge)
j
-
$(z0) A $(z l )]
-
(Vz)S(z)
( i nduc t i on schema for e l emen t s of the t ree)
Par t C :
Ax i oms for f i n i t e subse t s of the t ree :
(OS)
(3X)
X = 0
(SS)
(*X)(Vy)(SZ)
(existence of the emp t y se t )
Z = X U [y}
(the un i on of a se t w i t h a s i ng l e t on is a se t )
(IS)
4(0) A (VZ)(Vx)[*( $
-
i(Z U- t x})]
-
(VZ)s(Z)
( i nduc t i on schema for subse t s of the t ree)
Remarks :
The ax i om sys t em of par t A cons i s t s of ax i oms for
a t wo-sor t ed f i rs t order pred i ca t e ca l cu l us res t r i c t ed to
our l anguage .
An examp l e of such an ax i om sys t em may be
found on p . 4 / 5 of t he au t hor ' s [6 j , if one (i) changes the
subs t i t u t i on ru l e (SP) for pred i ca t e var i ab l es into a rule
for chang i ng free se t var i ab l es , and (i i) adds the usua l
equa l i t y ax i oms wh i ch make
=
a congruence re l a t i on .
(SP)
mus t be de l e t ed , s i nce i t is equ i va l en t to t he fu l l
comprehens i on pr i nc i p l e , and t hus is fa l se in W 2 S .
We
w i l l i ns t ead der i ve the comprehens i on pr i nc i p l e res t r i c t ed
to f i n i t e se t s .
There seems to be no f i n i t eness def i n i t i on
in W2S wh i ch wou l d a l l ow us to rep l ace par t C by this res t r i c t ed
comprehens i on pr i nc i p l e and the ax i om t ha t a l l sets are
fini te.
(See the d i scuss i on fo l l ow i ng propos i t i on 3 . 6 ,
P . 16)
If we had no t i nc l uded equa l i t y as a pr i m i t i ve no t i on ,
we could def i ne i t as usua l :
x = y
(VZ)[Zx
Zy]
X = Y
) (Vz)[Xz
Yz]
These two equ i va l ences are der i vab l e from par t A , and have
to be used to ge t the forma l coun t erpar t s for the ax i oms of
par t C .
Ma i n t heorem :
Exac t l y the true sen t ences of W2S are der i vab l e
from the above ax i oms .
It is easy to see that a l l the ax i oms are true in W 2 S .
I t rema i ns to show t ha t the ax i oms are comp l e t e , i . e . that
a l l true sen t ences are der i vab l e .
To prove this we w i l l
-7(i) descr i be a dec i s i on procedure by wh i ch any sen t ence of
W2S is t ransformed into an equ i va l en t t ru t h va l ue , and (i i)
show a t the same t i me t ha t the equ i va l ences in t he s i ng l e
steps of the procedure are der i vab l e .
comp l e t e in an effec t i ve sense :
The ax i oms are thus
For any true sen t ence of
W 2 S , we can f i nd effec t i ve l y a der i va t i on .
3.
Bas i c proper t i es o f t he t ree .
We w i l l t ry to ge t some
i ns i gh t i n t o t he bas i c s t ruc t ure of
T2,
va l uab l e l a t er in t he dec i s i on p r o c e d u r e .
wh i ch w i l l prove
For t he res t of
the p a p e r , in a l l l e m m a t a , propos i t i ons and t heorems ,
the reader shou l d add t he phrase "The fo l l ow i ng i s der i vab l e
from the ax i oms" .
N o r m a l l y , h o w e v e r , t he proofs w i l l be
g i ven in a ha l f-forma l w a y , on l y i nd i ca t i ng how a der i va t i on
cou l d be bu i l t u p .
We s t ar t by der i v i ng t wo vers i ons o f t he comprehens i on
pr i nc i p l e for f i n i t e s e t s , s t a t i ng t ha t def i nab l e par t s
of se t s are se t s .
Propos i t i on 3. .1:
(COMPfin)
(3W)(Vx)[4(x)
-
Wx]
(COMP*in)
)(Vx)[Ux
i Wx
-
(3U)(Vx)LUx
$(x)J
$(x) J
Proof :
I t is easy to der i ve t he equ i va l ence of t he t wo
forms .
We w i l l der i ve ( C O M P * i n ) :
We w i l l show
(VW) } (W)
schema (IS) .
i|i(0) means ac t ua l l y
Le t
w i t h t he he l p of the se t i nduc t i on
(VY)C(Vx) I Yx
-
¥(Y)L
Now by pred i ca t e l og i c we have
(Vx) "1 Y x A (Vx)- i Ux
(V x )[Ux t —> Yx A 4(x)] ,
-
and thus
(Vx)iYx A
)(Vx)->Ux
-
(3U)(Vx)[Ux f - j Yx A $(x)]-
Ax i om (OS) y i e l ds
(Vx) "lYx
-
i.e.
*(#) .
U [z}) is correc t l y expressed as
(VY){
(Vu) t Yu
u = z v Wz]
-
*(Y) j
-9Then
xjf(W)
-
wh i ch y i e l ds
U {z})
(VW) t (W)
is derived by s i m i l ar s t eps ,
by
(IS) .
•
I t is mos t i mpor t an t for our purpose t ha t the naturalpar t i a l order of
be
def i nab l e in W 2 S .
Def i n i t i ons :
1)
Trans(U)
U
s
(V z )[Uzo v Uz l
x ^ y
= d f (VU) [.Trans (U) a U y
3)
x < y
=
4)
x ~ y
=df> x ^ y
^
Propos i t i on 3 . 2 i
T
UzJ
is t rans i t i ve , i . e . c l osed under predecessor .
2)
of
-
2
Y
A
X
-
UxJ
^ y
v
x and y are comparab l e .
- and < are the na t ura l par t i a l order i ngs
i nduced by the successor
(a)
A
(b)
Z
x ^ z
(c)
A
(e)
y
(f)
y A y ^ x
(g)
-
y
= y
e
(h)
(i)
(J)
y
xc
" » * y v xl s y
xl
z - y y z.
The proof cons i s t s of a l ong cha i n of l emma t a , wh i ch
we w i l l no t g i ve here .
The i n t eres t ed reader should no t e
t ha t (e) is bes t der i ved before (d) and ( f ) . — I t should be
remarked that Propos i t i on 3 . 2 is a consequence of the ax i oms
of par t s A and B a l one , t oge t her w i t h
(COMPfin) .
The fo l l ow i ng def i n i t i ons concern subse t s of
Tg .
The
-10-
t erm i no l ogy is par t i a l l y t aken from Doner [3] and Rab i n C4J .
S i nce in W2S we dea l on l y w i t h f i n i t e se t s , however , the
def i n i t i ons of "fron t i er" and "fron t i ered tree" are d i fferen t
from Rab i n ' s defini t ion for S2S , a l t hough the no t i ons are
the same ; the no t i on of "path" is weaker t han in S2S .
Doner ,
on the o t her h a n d , ca l l s "fron t i er" wha t we ca l l "border" ;
so no t every fron t i er in Doner ' s sense is a fron t i er in our
sense .
Def i n i t i ons : '
1
Tx =df
Cy ; x ^ y l :
the t ree w i t h roo t x .
Px =df
ty; y - x i :
the pa t h up to x .
3
C1(U) = d f {x ; (3y £ x)UyJ :
4
Br(tJ) = d f tx €
5
B r ( U ) = d f tx % C1(U); (Vy < x ) y ^ C l (U)) :
xo %
U;
the t rans i t i ve c l osure of
v
C1(U)
xl %
C1(U)J:
+
border of
+
U
7
U" -
= d f C1(U) U B r ( U ) :
f
U - Br(U) :
Fr(U)
the border of
U.
the ou t er
U.
+
6
U.
the ou t er c l osure of
the i n t er i or of
U.
U.
= d f (*z)(*y e U ) y ~ z A (Vy ,
z
€ U)Lz ~ y
-
z = y ]
U is a fron t i er .
9
FrTr(U)
F i n(U)
Pa t h(U)
U
= d f Trans(U)
s
Fr(Br(U)) :
(^W)CFr(W)
=df U ^ 0
u ^ C1(W)J :
Trans(U)
U is a fron t i ered
U is f i n i t e .
(V z )[Uzo
- nUz l J :
is a pa t h .
I t shou l d be remarked once more t ha t def i n i t i ons 1 - 1 1
are mere abbrev i a t i ons , e . g .
x ^ y ; bu t
Tx
y £ T A„ means forma l l y just
is no t a se t of our m o d e l . — W e w i l l use
no t a t i on l i ke U € Trans instead of Trans(U) .
t ree .
-10(a)-
An examp l e m i gh t I l l us t ra t e t hese def i n i t i ons .
e
Here the fo l l ow i ng sets occur :
•
U = [10 , 11}
o,»
C1(U) = {e , l , 10 , l l j
•
Br(U) = {e , 10 , l l }
•
B r ( U ) - [0 , 100 , 101 , 110 , 111}
+
The reader w i l l more eas i l y unders t and the proofs of this
paper if he draws s i m i l ar p i c t ures .
-11Up to now we Know on l y t ha t the emp t y se t ex i s t s ; we
w i l l show now t ha t there are a l o t more se t s .
Propos i t i on 3 . 3 :
Ex i s t ence of se t s :
P
• X
= Br(W)
U = v n W
U = V U W
= C1(W)
+
= Br (W)
(S)
Proof :
We ge t (a) by comb i n i ng (OS) and (SS) .
Us i ng the
formu l a
*(x)
s
d f
U = tx]j
we prove (b) by (IS) , using (a) and (SS) .
(i) are i ns t ances of ( C O M P f i n ) .
(c) , (d) , and
Se t i nduc t i on (IS) on the
formu l a
*(w)
= j f (VV)(^U)
gives (e) w i t h the he l p of (SS) .
U = V u w
(f) fo l l ows from (b) and
(e) by (IS) , if w e no t e t ha t
C1(U U txJ) = C1(U) U P x .
We canno t ye t prove (g); a proof fo l l ows from Propasi t ion 3 . 4(a)
t oge t her w i t h (a) , (e) , and ( C O M P f i n ) .
d i rec t consequence of
(h) f i na l l y is a
(f) , (g) , and (e) .
•
The nsxt proposi t ion prov i des two o t her i nduc t i on schema t a
for se t s :
-12-
We w i l l show (VU) t (U)
t rans i t i ve
U,
by se t i nduc t i on (IS) .
S i nce for
U = C 1 ( U ) , we ge t t he wan t ed conc l us i on
(<U e T r ans ) * ( U ) .
C1(0) = 0,therefore •((*) h o l d s .
«(U) .
We w i l l show (Vx)f(U U [x i )
S t ar t w i t h
Case 1 .
U
be g i ven such t ha t
by i nduc t i on (IE) .
x = e:
U = 0.
fo l l ows from
Case 2 .
Le t
Then
U U lei = t e} = C l ({ e} ) , thus
U te
4( t e}) .
U ^ 0.
Then
e £ C 1 ( U ) , t herefore
C1(U U {e}) = CI
wh i ch t oge t her w i t h ^r(U) i mp l i es *(U U [e i ) .
Now l e t
V(U U (x j )
be proven j we have to show
¥(U U t xo3)
^(u U {.xl}) .
S i nce
Trans (CI (U U t x})) A
x
e C1(U U t x}) A 4(C1(U u { x j ) ) ,
we ge t from t he hypo t hes i s of the propos i t i on
Bu t
C1(U U £ X J ) U t xo)
wh i ch i mp l i es
^(U U t xo} ) .
=
C1(U u {xo J) ,
Ana l ogous l y we ge t
Y(U U t x l i ) .
-13-
(a
(b
L y t p A y ^ x
-
y o e p v y i e ^ j
(c
(d
(e
(f
(g
(h
(i
(*
(1
(m
(n
Proof : (a) , (b) , and (c) fo l l ow eas i l y from Propos i t i on 3 . 2 .
j
(d) fo l l ows d i rec t l y from t he def i n i t i ons .
(e) 3 (g) , and (h) are
j
easy.To prove
W = d f , U fl (T
Then
(f) l e t x £ U be g i v e n .
B r ( W ) ^ Br(U) n (T
Le t
- [x j ) . |
- UJ).
Case 1 :
W = P.
Then
Case 2 :
W £ i>.
I t i s easy to prove
I
x t Br(U) .
V ^ <t> -
by
B r ( V ) ji 0 .
se t i nduc t i on t ha t
-14So l e t y
Br(W) .
Then
B r ( U ) , and x < y .
R r (i): x £ C1(U) i mp l i es t ha t
+
+
txo fc C1(U) v xo € B r ( U ) J A [x l t C1(U) V x l € B r ( U ) J
and thus
On the o t her hand ,
x 6 Br(U )
xo % U
+
+
V xl % U .
+
Therefore , x € B r ( U ) i mp l i es
+
x e u ,
+
thus x £ B r ( U ) .
+
Then
x £ U .
i mp l i es by (e) and by def i n i t i on
x % C1(U) , and t r i v i a l l y ,
+
Converse l y , l e t
x £ Br (U) .
It rema i ns to show
xo % U
+
+
v xl % U .
+
Assume
xo t U :
Case 1 :
xo 6 C1(U) .
+
Then
x £ C1(U) , thus x £ B r ( U ) ,
con t rad i c t i on .
Case 2 :
Thus
+
xo £ B r ( U ) .
Then x € C1(U) , aga i n con t rad i c t i on .
+
xo % U .
Tb prove (J) , use
t rans i t i ve se t s :
Propos i t i on 3 . 4(a) , i nduc t i on for
+
+
B r ( p ) = EeJ , B r ( t e J ) = t o , l } ; t hus the
i nduc t i on beg i nn i ng is easy .
let
x t U , suppose
Now l e t
+
Fr(Br (U))
U
be t rans i t i ve ,
as i nduc t i on hypo t hes i s .
We have to show
+
+
F r ( B r ( U U t xOJ)) A F r ( B r ( U U t x l j ))
Case 1 :
xo € U .
Then
U U t xo} = U .
Case 2 :
xo £ U .
Then
x € Br(U) A xo € B r ( U ) .
+
+
Thus
+
B r ( U U t xo}) = ( B r ( U ) - t xo}) U {xoo , xo l } .
Le t
If
+
y , z e B r ( U u Ixoi), y ^ z :
+
y , z 6 B r ( U ) , t hen y •/ z
To show y ^ z .
by i nduc t i on hypo t hes i s .
y = xoo , z = x o l , or converse l y , then y ^ z .
If , s a y ,
If
-15+
y £ B r ( U ) - Cxo3 ,
= x o o , then y / x o , thus
Th i s
proves the second c l ause in the def i n i t i on of a fron t i er .
To ge t the f i rs t c l ause , l e t
t ha t
+
z € Br (U) A y ~
y
be g i ven , l e t
z
be such
(by i nduc t i on hypo t hes i s) .
z
We
have to show
+
(3z € B r ( U U t xo}))
If
z = x o , t hen
y ~ z.
xoo ~ y v xo l ~ y .
+
z £ B r ( U U t xo)) .
If z ^ x o , t hen
Thus we have shown
+
F r ( B r ( U U {xo})) .
The proof for x l is ana l ogous .
Thus we have proved (j) for
t rans i t i ve se t s wh i ch by (d) is enough .
from (e) , ( i ) , and ( j ) .
+
(U ) ' = U
+
(k) fo l l ows
To prove (1):
+
- Br(U ) = U
by (i) and def i n i t i on ,
+
+
- B r ( U ) = C1(U)
(m) is a d i rec t consequence of
(1) , s i nce
Trans(U) ^
For (n) l e t F r T r ( U ) .
from (f) .
So l e t
U = C1(U) .
The t rans i t i v i t y of
IT
fo l l ows eas i l y
+
x e B r ( U ~ ) , thus x % U"
Since Fr(Br(U)) , t here is
(Vy < x ) y £ U~ .
z € Br(U) such that
x ~ z.
If
z < x were t rue , t hen z € U ~ , wh i ch con t rad i c t s z € Br(U) .
Thus x ^ z , and t herefore by the t rans i t i v i t y of
-
S i nce x £ U , we have x 6 Br(U) .
U, x E U.
Thus
+
B r ( U ~ ) c Br(U) .
The converse i nc l us i on is proved s i m i l ar l y .
converse l y Trans(U")
the t rans i t i v i t y of
from ( j ) .
+
Br(U) = B r ( U ~ ) .
U , whereas
Now l e t
Th i s i mp l i es eas i l y
Fr(Br(U)) fo l l ows d i rec t l y
-16-
For (o) aga i n l e t F r T r ( U ) :
(lQ
by (n) .
+
+
= C1(U") U B r ( U " ) = U" U B r ( U ) = U
The converse d i rec t i on fo l l ows from (k) .
Propos i t i on g » 6 :
•
F i n i t eness pr i nc i p l es (two o t hers are to
be found in Propos i t i on 3 . 5
( j )and(k) .
(a)
(VU € Pa t h)(3x)
U = Px : every p a t h has a m a x i m u m .
(b)
(vu) F i n (U) : every se t Is f i n i t e .
(c)
(VU € F r ) [ U = [e} v (ax)LUxo
U x l ] ] : every fron t i er
has a m a x i m u m .
Proof :
(a) and (b) are eas i l y proven by Propos i t i on 3 . 4 ( a ) ,
i nduc t i on for t rans i t i ve s e t s , on t he formu l ae
Pa t h(U)
-
(3x)
U = Px
and F i n(U) , respec t i ve l y .
(c) i s t r i v i a l w i t h he l p of i nduc t i on for fron t i ered
t r e e s , Propos i t i on 3 . 4 ( b ) .
u
I t is easy to see t ha t a l l f i ve f i n i t eness pr i nc i p l es
of Propos i t i on 3 . 6 wou l d be fa l se in the s t rong monad i c
second order t heory of two successors , Rab i n ' s S 2 S , i . e .
t hey don ' t ho l d in t he t ree w i t h arb i t rary subse t s .
Bu t
some of t hem are t rue in i n t ermed i a t e s t ruc t ures , wh i ch
have some i nf i n i t e s e t .
I ndeed , ca l l a pa t h
U
i nf i n i t e i ff
( V x t U) t Uxo v u x l ] ,
ca l l a Be t
(Vu
Al l
W
t h i n i ff
(U)
-
(ay 6 U ) W n T y = 0 ]
so
.g.
comb ,
-17i . e . the se t
n
{O l ; n <
.
N o w , in the s t ruc t ure wh i ch adm i t s a l l t h i n sets bu t no
o t her o n e s , Propos i t i on 3«5( j ) and (k) and Propos i t i on 3 . 6
(a) and (b) are t rue , bu t Propos i t i on 3 . 6(c) is fa l se .
This
shows t ha t the fo l l ow i ng se l f-sugges t i ng ax i om sys t em is
no t comp l e t e :
Ax i oms par t s A and B , t oge t her w i t h (
and the f i n i t eness ax i om Propos i t i on 3 . 6(b) .
C 0 M P
fin)
(In the case
of one successor func t i on , the ana l ogous ax i om sys t em is
comp l e t e , s i nce there the f i n i t e se t s are un i que l y def i ned
as the bound se t s .
See L6 j , p p . 117 ff•)
The thin se t s
are no t n i ce anyway , s i nce the c l osure of a thin se t need
no t be t h i n , and t hus Propos i t i on 3 . 3(f) and (h) are fa l se .
Bu t if we en l arge the s t ruc t ure under cons i dera t i on by
t ak i ng the c l osure of se t s . Propos i t i on 3 . 6(a) becomes
fa l se , t oo , whereas t he t hree o t her pr i nc i p l es rema i n
true . —
I t m i gh t be t ha t the above ax i om sys t em becomes comp l e t e
if we add (a) and (c) as ax i oms (or j us t (c)?) . We did n o t ,
however , i nves t i ga t e t h i s ques t i on .
The s t ronger t heory S2S is more powerfu l here .
In
S2S , a pa t h is wha t we ca l l ed above an inf ini te pa t h (see
Rab i n t 4 i ) .
And a fron t i er is then def i ned as a se t wh i ch
mee t s every pa t h in exac t l y one po i n t .
W i t h this concep t
of a fron t i er , the def i n i t i on of a f i n i t e se t as a se t bounded
by a fron t i er works proper l y .
Bu t in W 2 S , where i nf i n i t e
pa t hs are no t ava i l ab l e , "fini te" does no t mean "bounded" ,
bu t "bu i l t up po i n t by po i n t " .
As an app l i ca t i on of the f i n i t eness pr i nc i p l es
we
-18prove now two l emma t a wh i ch we w i l l need l a t er in the
dec i s i on procedure :
Lemma 3. .7:
(3x)$(x) «—f (ttW)LWe A ( V x 6 W)L-iWxo A -Jtfxl
Proof : - :
Le t 0(x) be t rue .
-
4(x)J]
I t is easy to see t ha t
W = P x sa t i sf i es the l e m m a .
- :
Le t
W
have the s t a t ed proper t i es ,
+
impl ies t ha t
e 9- B r ( W ) .
e € W
Therefore , by Propos i t i on 3 . 5( j )
and Propos i t i on 3 . 6(c) , t here is an
+
x
such t ha t
+
xo £ B r ( W ) A x l £ B r ( W ) .
Thus fc(x) ho l ds .
•
D i fferen t forms of the i nduc t i on pr i nc i p l e for e l emen t s
are easy consequences of Propos i t i on 3 - 2 , e . g . the m i n i mum
pr i nc i p l e , s t a t i ng t ha t every se t has a m i n i ma l e l emen t
(i t can have more t han o n e , of course) , or t he max i mum
pr i nc i p l e for subse t s of a pa t h .
The fo l l ow i ng i nduc t i on
pr i nc i p l e is d i fferen t , and w i l l be usefu l for the hand l i ng
of tree au t oma t a .
I t Is ano t her form of the max i mum
pr i nc i p l e of Propos i t i on 3 . 6(c) :
Lemma 3.-8:
Downward i nduc t i on :
+
Trans(U) A ( v x £ B r ( U ) ) * ( x ) A ( V x € U)[4(xo) a $ ( x i )
Proof :
Le t
U, 4
-
+
(Vx € U ) i ( x )
sa t i sfy the hypo t hes i s of the l emma .
By ( C O M P f l n ) , t here is a se t
W
such t ha t
(Vx)[Wxf—> Ux A "I4(x)j .
Assume W ^ 0 .
t ha t
By Propos i t i on 3 . 6(c) t here ex i s t s
,
,
xo € Br (W) A x l 6 Br (W) ,
x
such
-19and thus
x 6 W , xo £ W , x l fL W .
Since tf 5 U , and U is t rans i t i ve , by Propos i t i on 3 . 5(h) we
have
+
+
B r ( W ) £ B r ( U ) U U, .
t herefore
+
xo , xl € Br (U) U U
If
+
xo € B r ( U ) , t hen by hypo t hes i s
*(xo) ho l ds .
If
xo j? Br (U) , t hen xo € U , and t herefore again 4(xo) ho l ds
(by def i n i t i on of
W,
since
xo % W ) .
Ana l ogous l y one
ge t s «(x l ) , and t herefore by hypo t hes i s 4(x) .
i mp l i es t ha t x £ W , con t rad i c t i on .
•
Bu t this
-20Tree au t oma t a , recurs i on and norma l forms .
— — .
•
• — — —
.
the set of a l l n- t up l es of t ru t h va l ues
T,F.
Le t £
n
be
Our def i n i t i on
of tree au t oma t a is abou t the same as Doner ' s [3J , bu t we
can use bo t h , the se t of s t a t es and the i npu t a l phabe t ,
chosen among the
Rab i n
Our t erm i no l ogy is par t i a l l y that of
U L
Def i n i t i ons ;
1)
A
^ n - t r e e is a func t i on from a f i n i t e t rans i t i ve subse t
of
2)
T2
Ij .
In t o
A de t erm i n i s t i c t ree au t oma t on over the a l phabe t
is a quadrup l e
=
se t of s t a t es ,
—
.
'
j:
and
x ^
K
^
2J: U
-*
is the
EC
is the t rans i t i on func t i on ,
Z^- t ree
+
1)
is the i n i t i a l s t a t e ,
—
-
X:U
The run of
as i npu t is the
def i ned by
+
A
(VZ fc B r ( U ) ) Zt = s Q
We wr i t e Z «= rn(j*,X).
3)
where
is the se t of f i na l s t a t es .
sd over the
tree
>
®
£ 2*
s
o
x ^
K
Is
(Vt € U ) Z t = J ( X t , Z t o , Z t l ) .
4 accep t s X Iff Ze £ K .
S i m i l ar l y , a nonde t erm i n i s t i c _tree au t oma t on over
<si ~ <
1, L, K> ,
where
i n i t i a l s t a t es , and
L £
re l a t i on .
&
any
A run of
Z^.-tree
Z: U
I 5
is the se t of
i s
x
over the 2-/n-tree
-
is
t h e
t r a n s i t i o n
X: U
2-/n
is
sa t i sfy i ng
+
(Vt € B r ( U ) ) Z t € I A (Vt € U)(X t , Z t , Z t o , Z t l ) € L .
We wr i t e
run
4 )1
Z
Z t Rn(jrf,X).
of
A se t of
&
over
accep t s
such that
X
iff there ex i s t s a
Ze fc K .
Ltn - t rees is au t oma t on def i nab l e iff there is a
tree au t oma t on over
the se t .
X
&
wh i ch accep t s exac t l y the trees of
-21Thus , tree au t oma t a are genera l i zed in the na t ura l
way from the case of one successor .
one s t r i k i ng d i fference :
There i s , however ,
Tree au t oma t a run down the t ree ,
i . e . t hey s t ar t read i ng the i npu t tree at i ts border and
end up a t the roo t .
(For this reason t hey have to be O-sh i f t
au t oma t a , i . e . the s t a t e at "t ime"
t
depends on the i npu t
at the same " t i me" , whereas in the 1 - s h i f t au t oma t a of the
l i near case the s t a t e a t t ime
the prev i ous t i me . )
t
depends on the i npu t a t
The reason Is t ha t upward de t erm i n i s t i c
tree au t oma t a are ra t her w e a k , s i nce a t any po i n t they
carry the same i nforma t i on to both successors .
It was
for t h i s reason t ha t Doner i nven t ed downward au t oma t a in
[33 .
(As a ma t t er of fac t , nonde t erm i n i s t i c t ree au t oma t a
do no t prefer a d i rec t i on ; we t h i nk of them as runn i ng
downwards j us t for ana l ogy . )
Tree au t oma t a share w i t h ord i nary au t oma t a the fo l l ow i ng
fac t s , wh i ch we sha l l use :
To any nonde t erm i n i s t i c au t oma t on
there is an equ i va l en t de t erm i n i s t i c one ; the au t oma t on
def i nab l e sets form a Boo l ean a l gebra ; the emp t i ness prob l em
is so l vab l e .
For more i nforma t i on abou t tree au t oma t a
see Doner [3-1, Tha t cher-Wr i gh t L?] , and Rab i n [4J .
As remarked in the i n t roduc t i on , the dec i s i on procedure
for W2S presen t ed here w i l l fo l l ow c l ose l y the dec i s i on
procedure for the Sequen t i a l Ca l cu l us SC of Buch i [2] , as
d i scussed by the au t hor in
[6 j .
The presen t a t i on here w i l l
be se l f-con t a i ned , bu t we w i l l refer to C6J for proofs and
for exp l ana t i on of t he me t hods used .
-22We i den t i fy
of
I^- t rees w i t h n- t up l es of f i n i t e subsets
T 2 (monad i c pred i ca t es res t r i c t ed to a common f i n i t e
t rans i t i ve se t ) , i n a manner ana l ogous to Buch i L l J .
Thus ,
we can represen t in the l anguage of W2S the cond i t i ons
spec i fy i ng a t ree au t oma t on by propos i t i ona l formu l ae
i nvo l v i ng se t var i ab l es .
We use X , Y , 2 , some t i mes w i t h
the upper i ndex n , for n- t up l es of se t var i ab l es , i . e . for
iL^-trees.
The l e t t ers U , V , W w i l l be used as before for
se t var i ab l es ,
F
or
F
n
s , s^ w i l l deno t e t up l es of t ru t h va l ues ,
the t up l e cons i s t i ng of
F ' s o n l y . — I n this way
we use formu l ae of the fo l l ow i ng t hree norma l forms as tree
au t oma t a in W2S (for de t a i l s see
Def i n i t i on ;
6 , p p . 25 f f . and
ff . ) :
Au t oma t a norma l forms are the fo l l ow i ng :
+
(az).(Vt € B r ( U ) ) Z t = s Q
A (Vt € U ) Z t = J[X t , Z t o , Z t l ] A K t Ze]
2P:
U)
2Jn :
+
(3Z) . (V t € B r ( U ) )
iLzt] A
A (Vt E U ) L[Xt , Zt , Zto . , Zt l j A K l ze j
( ^ Y ) (az) . KCZe] A (V t ) L l X t , Y t , Z t , Z t o , Z t l J
H e r e , I , K , L are propos i t i ona l formu l ae i nvo l v i ng at mos t
the i nd i ca t ed pr i me formu l ae ;
formu l ae .
J
is a t up l e of propos i t i ona l
(AY) is a s t r i ng of n-1 a l t erna t i ng blocks of
se t quan t i f i ers where the l as t one is un i versa l .
o
o
Obv i ous l y , a ^ - f o r m u l a or a £ -formu l a is true for
some
X
and some t rans i t i ve
U
if and on l y if the correspond i ng
de t erm i n i s t i c , respec t i ve l y nonde t erm i n i s t i c , t ree au t oma t on
accep t s the tree xfu (the func t i on X res t r i c t ed to U ) .
U)
A la' - formu l a corresponds to a nonde t erm i n i s t i c au t oma t on
-23-
o
Ij
Z « rn(4 , xfu)
p
o
xfu)
uu
U)
Tj
<u
A
(
V T
) L J ^ L
J Z ^ . Z
-24-
(3W)
([We
K^(e)
A
LgtFjFjFsF]]
A
A
I ^ L F J F , V
[-IWE
A
K
2
( e )
A
v
( t 6 U) { [ W t W t o j A [wt <—, WtlQ A
{ [Wt A ^ ( t ) ] V [iwt A L 2 ( t ) j j j .
Put t ing these two equivalences together and using the
defini t ion of Trans , one sees that the disjunct ion of two
UJ
^ - f o r m u l a e can be wr i t t en in
Theorem 4 . 2 :
UJ
1^-form .
•
Any formu l a no t contain ing free individual
UJ
var i ab l es
is equ i va l en t to a
^ - f o r m u l a for a su i t ab l e n .
The proof is the same as for the correspond i ng theorem
I . l . d . l of [63 , pp . 20-23 .
We have to use Lemma 3 . 7 to
e l i m i na t e con j unc t i ons of ex i s t en t i a l individual quant if i ca t i ons , and the above l emma to reduce the number of
d i s j unc t i ons In the d i s j unc t i ve norma l form .
To be able to swi tch back and forth between determinist ic and nondetermi n i st i c au t oma t a , we have to prove
o
that for a
run .
I^-formu l a to any Input there exists a unique
This is i mp l i c i t In the no t a t i on , so in reading the
nex t propos i t i on the reader should recal l that we prove
der i vab i l i t y , no t t ru t h .
Proposi t i on 4 . 3 *
(a)
Trans(U) a ^
- (Vx € U j L z ^
(b)
Trans(U)
-
o
Downward recursi on :
= rn(xfu) A
For any I^-formu l a
= rn(4 , x[u)
$
-
z2x j
(az) Z = rn{i }
xfu)
The proof is ana l ogous to the proof of Lemma t a I . l . b . l + 2 ,
p p . 10-11 of [63; i t uses downward i nduc t i on , l emma 3 . 8 ,
and set i nduc t i on .
Propos i t i on 4 . 3 can be eas i l y general ized
to more general forms of recurs i ons ; we w i l l , however , need on l y
-25this form .
A l so we do no t s t a t e the correspond i ng
propos i t i on
upward recurs i on .
o n
I t is by downward recurs i on , t oge t her w i t h downward
i nduc t i on and i nduc t i on on fron t i ered t rees , Propos i t i on 3 . 4 ,
t ha t we avo i d Doner ' s t ree Induc t i on and t ree recurs i on
p . 409) .
([3] ,
Doner ' s pr i nc i p l es are no t express i b l e in t he
l anguage of W 2 S .
sub t ree of
I n d e e d , even t he no t i on
beg i nn i ng a t
i"fw,
" t he
w" (Doner , I . e . ) , wou l d make
W2S undec i dab l e , s i nce i t a l l ows one to def i ne conca t ena t i on .
Theorem 4 . 4 :
o
formu l a .
To any
D -formu l a t here is an equ i va l en t
The proof is essen t i a l l y the same as in the l i near
c a s e , c f . t heorem I . 2 . C . 2 on p .
o
of the
of 16 j .
S i nce t he run
Z^-formu l a is cons t ruc t ed by downward recurs i on ,
Propos i t i on 4 . 3 , the equ i va l ence has to be proved by downward i nduc t i on , l emma 3 . 8 .
JD
Coro l l ary 4 . ^ :
Proof :
Z^
is c l osed under Boo l ean opera t i ons .
As in t he l i near c a s e , us i ng t heorem 4 . 4 for nega t i on .
See e . g .
uu
£
p.
To der i ve from Coro l l ary 4 . 5 our ma i n t heorem , t ha t
w
is c l osed under n e g a t i o n , we have to use the fac t t ha t
Is dec i dab l e .
Th i s fo l l ows d i rec t l y from the fo l l ow i ng
cons t ruc t i on , wh i ch is due to RabU) i n L53 P « 3 0 :
L e t $ be a sen t ence i n
con t a i n i ng K se t quan t i f i ers ,
i
B
df (
Def i ne se t s R , c X
a Z
* ) KlZel
A
(
V t
as fo l l ows :
) L£z t , Z t o , Z t l ]
-26-
1
RQ
R
=
D F
=
R
i+1 df
S i nce
i
U
^ h: '
R1 5 R . +1
= Rm
Proof :
ex
6
*
S
o'
€
1
R
i
s , t
i £ m.
Le t
R =df ^
Wr i t e
2
holdsj
such t ha t
R1<
*L(x) shor t for the r i gh t side of t he l emma .
We w i l l show by (me t ama t hema t i ca l ) i nduc t i on on
-
for i , l e t s £
•
E i t her
i nduc t i on hypo t hes i s .
t ha t
hypo t hes i s
t
5O_
s £ R ^ , t hen we can use the
Le t
be g i ven .
By t he i nduc t i on
(xo) and i|r (x l )
bi1
are t rue .
Le t
we def i ne a run
Us i ng Propos i t i on 3-3
for
Z
by
(C0MP
s = T( t rue) , the formu l a def i n i ng
& C1(Zq)
For
s Q , s^ € R ^ such
x
be the respec t i ve runs .
and
So l e t i t be proven
Or e l se t here are
L l s , s0 , s^] ho l ds .
i:
(Vx) * g ( x )
Th i s is t r i v i a l for i=0 by ax i om (OS) .
U C1(Z1)
A
[[xo
S t A
ZQt] V
).
F L N
Z
Z
O
and Z ,
^
(e) and (d) ,
For
k = 1
wou l d be
[xl
S t A
^ t j
v
s = F(fa l se) , t he c l ause t = x wou l d be dropped .
arb i t rary
k
one has to use
componen t s of
~
k
p,
s € R <—y (az) lZx « s A (Vt ' € T )L[Z t , Z t o , Z t l ]]
s 6 R±
t
o
Rq
' LCs , sojSlJ
for a l l 1 , t here k e n m ^
m
for a l l
Lemma 4 . 6:
2
iff L C F J J , ^ ] ho l ds ; o t herw i se
LFK)
U
on
*(x)
U
0 1
k
A
t (az
sfcEK
and
We w i l l prove by downward
U)(Zx = s
-
t ha t (Vx £ U )ty(x) .
x € U
i mp l i es t ha t there is a
(Vt 6 T ) L[Z t , Z t o , Z t l J .
be t rans i t i ve .
+
formu l ae to def i ne the
^s(x)
such t ha t
(3Z c U) . Zx = s
So l e t
For
Z.
By Propos i t i on 3-3 (e) ,
t rans i t i ve se t
k
t=x}
S
(vt
i nduc t i on
T )L l Z t , Z t o , z t i ]
€ r]
The l emma w i l l fo l l ow .
-
-27+
For
x £ Br (U) ,
x £ U,
$(x)
s €
let
is eas i l y seen to be t rue .
Z
Le t
be such that
Z c u a Zx = s A (Vt € T x ) L[Z t , Z t o , Z t l ]
Le t
S q = Zxo ,
s 1 = Zx l .
Z f U A zx j *= Sj
Thus ,
s o , s-^ G R
L[s ,
S
Q
S-J^j,
,
A
Then for j = 0 , 1 ,
(Vt 6 T x J ) L l Z t , Z t o , Z t l ] .
by the induct ion hypo t hes i s .
we have
s G R.
Since
ui
Since the set R is compu t ab l e , we ge t :
UJ
Theorem 4 . 7 :
is dec i dab l e . In fac t , for any sentence
UJ
?
In
e i t her
we can effec t i ve l y cons t ruc t a der i va t i on of
4
or
^ 4.
No t e that l emma 4 . 6 impl ies that a l l sub t rees
Tx
are
" isomorphic re l a t i ve to input free au t oma t a" , i . e .
(1)
(SZ)tZx = s A (Vt £ T ) L[Zt , Zto , Zt l jJ
<
y
? (az)tzy = s A (Vt G T ) L[Z t , Z t o , Z t l J} .
v
By re l a t i v i z i ng the comp l e t eness proof for W2S , the " i somorph i sm
re l a t i ve to W2S-sen t ences" is also der i vab l e , i . e .
(2)
4(x)
for any formu l a
(Tx)
4(x)
i(y)
(Ty
^
con t a i n i ng x as the only free var i ab l e
(T )
and no t con t a i n i ng the cons t an t
e.
re l a t i v i za t i on of a l l quan t i f i ers in
(Here
$
to
x
' is the
Tx . )
No t e that
(2) canno t be extended to formu l ae con t a i n i ng o t her free
var i ab l es , since w i t h i n W2S we canno t map e l emen t s or subsets
of
T
into the correspond i ng•e l emen t s or subse t s of
T .
-28The usua l proof for t he dec i dab i l i t y of
^
fac t t ha t , if an au t oma t on adm i t s a run a t a l l
adm i t s a "shor t" one
(see e . g . Doner
uses the
t hen i t
p . 413) .
To
forma l i ze t h i s proof one needs (1) to cu t down a g i ven run
wh i ch is too l ong .
A l so one needs a s t ronger vers i on of
l emma 4 . 6 , wh i ch is more cumbersome to der i ve .
The
recurs i ve charac t er of t he cons t ruc t i on of Rab i n is be t t er
sui ted for our i nduc t i ve proofs .
We need a fur t her l emma :
Lemma 4 . 8 :
Fr(U)
-
VX 6 U)(3Z)[K[Zx]
(az)(Vx € U) KtZx]
A
(Vt fi
(vt E T ) L t Z t , Z t o , Z t l j }
u") L[Z t , Z t o , Z t l J :
If one can s t ar t a g i ven au t oma t on on every po i n t of a
fron t i er , then t here is a s i ng l e run from wh i ch one can ge t
a l l the separa t e runs by res t r i c t i ons .
Proof :
I t i s easy to prove by se t i nduc t i on and Propos i t i on 3 . 3(e
(Vx , y €U)Lx ~ y
A
(Vt
E T
X
)
-
x = y ] A ( V x € U)(3Z)[KCZx] a
L[Z t , Z t o , Z t l J j
-
(3Z)(Vx €
U)[K[ZX]
A
A (Vt 6 T ) L[Z t , Z t o , Z t l ]} .
Th i s d i rec t l y i mp l i es t he l e m m a . " •
u>
Theorem 4 . 9 :
is c l osed under Boo l ean opera t i ons .
U)
Proof :
Con j unc t i on is easy .
(az) . KLze]
So l e t
$(X) €
be t he formu l a
(Vt) L[X t , Z t , Z t o , Z t l ] .
By res t r i c t i ng t he cons i dera t i on to
of l emma 4 . 1 , we see t ha t
$(x)
C1(X)
as in the proof
is equ i va l en t to
-29(1)
(VU € Trans ){, (Vt £ U)TX t
-
(3Z) t K[Ze] A
A (Vt 6 U ) L[X t , Z t , Z t o , Z t l ] A (Vt % U ) L[p , Z t , Z t o , Z t l ]}}
Us i ng the formu l a
R
LlF,Zt ,Zto,Zt l]
we def i ne the se t
as in l emma 4 . 6 , and cons t ruc t a propos i t i ona l formu l a
s.t.
I t s]
? s € R.
(2)
(VU e Trans)((V t £ U) i X t
A (vt e
-
(3Z){KlZeJ a
Llxt ,zt ,zto,zti] A (vt e Br
Indeed , (1)
-*
For
(1)
(2)
Then (1) is equ i va l en t to
(2)
is i mmed i a t e from l emma 4 . 6 .
use l emma t a 4 . 6 and 4 . 8 t oge t her w i t h
Propos i t i on 3 . 3(e) .
2-P-formula
^(X , U) .
t-P-formula
i | i2(X,U)
Therefore T $(X)
iLzt]}}.
The second ha l f of (2) is a
Thus by coro l l ary 4 . 5 there is a
equ i va l en t to
"^(XjU) .
Is equ i va l en t to
(3U € Trans)( (Vt 9- U ) "T X t A . * 2 ( X , U ) } ,
ui
wh i ch is eas i l y t ransformed into
•
I
-30-
REFERENCES
J . R . BuchI :
au t oma t a .
J . R . Buch i :
Weak second order ar i t hme t i c and f i n i t e
Z . Ma t h . L o g . Grund l . Ma t h 6
On a dec i s i on me t hod in res t r i c t ed second
order ar i t hme t i c .
In "Logic M e t h . Ph i l . Sc . , Proc .
I960 S t anford In t . Congr . " .
J . Doner :
66-92 .
S t anford
1-11 .
Tree accep t ors and some of t he i r app l i ca t i ons .
J . Comp . Sys t . Sc . 4 (1970) , 406-451 .
M . O . Rab i n :
Dec i dab i l i t y of second-order t heor i es and
au t oma t a on i nf i n i t e t rees .
M . O . Rab i n :
au t oma t a .
D . S i efkes :
Transac t i ons AMS 141
1
Weak l y def i nab l e re l a t i ons and spec i a l
Techn i ca l repor t N o . 3 2 , Jerusa l em , June , 1969 .
1
Dec i dab l e t heor i es I ; -Buch i s monad i c
second order successor ar i t hme t i c .
L e c t . No t es M a t h .
V o l . 1 2 0 , Ber l i n-He i de l berg-New York 1970 .
J . W . Tha t cher , J . B . Wr i gh t .
Genera l i zed f i n i t e au t oma t a
t heory w i t h an app l i ca t i on to a dec i s i on prob l em of
second-order l og i c . Ma t h . Sys t . Theory 2
57-81 .