UNIVErlSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. C.
AFOSR Report No.
ON THE DEFINITION OF A CERTAIN CLASS OF AUTOMATA
by
M. P. Schutzenberger
January, 1961
Contract No. AF 49(638)-213
•
In this note we verify that a large
part of Kleene's Theory of regular
~vents still applies to a class of automata with an infinite set of states.
Qualified requestors may obtain copies of this r~port from the
ASTIA Document Service Center, Arlington Hall Station, Arlington
12, Virginia. Department of Defense contractors must be established for ASTIA services, or have their "need-to-know" certified by the cognizant military agency of their project or contract.
Institute of Statistics
Mimeograph Series No. 273
ON THZ DEFINITION OF A CERTAIN CLASS OF AUTOMATA
M. P. Schutzenberger
I.
1. Introduction.
This note is concerned with the definiton of
a class of one way one tape automata,tOt, that includes that of finite
automata. (1) In a schematic manner an automaton from the class;Acconsists of an
~
way,
~
tape, finite automaton(2)controlling a very
elementary computational process in which enter a finite number of finite
but not bounded integral parameters.
Three main restrictions are
imposed:
i)
There is no feed back from the parameters to the finite
state machine and only addition and subtraction are used;
ii)
For each input letter only a bounded number of additions
or subtractions are performed;
iii)
Drastic
conditions are imposed on the rules by which, from
the computed values of the parameters it is decided to accept or not an
input word.
As expected, the definitions insure that a good part of Kleene's
theory remains valid for the class
it .
In order to make the paper self contained, some concepts of the theory
of finite automata are recalled in 1.2; the definitions are discussed
at heuristic level in section I.3, I.4, I.5; a formal definition is given
in 1.6 and in I.7 this definition is reduced to a simple standard form.
The second part is devoted to the discussion of examples and counterexamples in support of the definitions.
In the last part Kleene's
theory is applied to the class J4:and an intuitive interpretation of the
set of words accepted by an automaton of this class is given.
1. This research was supported by the United states Air Force through the
Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No.AF 49(638)-213. Reproduction in whole or in part is permitted for any purpose of the United States Government.
2
1.2 Finite Automata
He recall the definition of a finite, one wa;)' , one tape
liutomaton (0:, F).
t 1 and Z = 1er } be t'-10 fin! te
Let X = x
sets; X is usually
called the input alpha'Lel:. and E is the set of the state of the
automaton.
To each x g X is associated a rnappint;, of
",hich \le denote simpl;)T by
LTX.
into i tseli'
L
An initial state 0"1 and a subset
z:'
have been distinguished in Z.
If presented with an input word f = x.
~l
x.
~,.,
c
",x
i
in the letters
n
xi ~ X, we compute successively the states O'lxi ,ulx
x. (=(erlxi )x. ),
i 1 ~2
1
1 ~2
Jlx. xi xi ••• , erlxi xi ",xi ' .•• ,erlf = Jlx. xi ",xi ",xi
~l 2 3
12m
~l 2
m
n
and we say that Q accepts the word f (in sjmbols f ~ Fa) if the
state vlf belongs to the distinguished subset E'
E.
The definition given here is in fact that of a right automaton
",hich reads the input word f from left to right and a perfectly
s~!unetric
definition could be given for a left automaton.
the family
However,
R of those subsets of words which can be considered as
o
the set of the 'Words accepted by some finite automaton ,haS t.he·,
special property that to··eacbrig)it finite automaton (Q,F)l.there
t
corresponds a left finite automaton ex'for which Fc(l = ~
The main
property of H is contained in Kleene's Theorem which can be presented
o
in the following manner:
1. ~o qont~ins
a.lJ". the fini;te
&J.l~ c;f
J;
2. If F' ,F"g{if6'r. tl;1eIlcF"~l.,1I, F' F"", Fi<~FtI(F ,rf=f'f" wi1;Jlf'Gf.': and
fIlW"] )and F,*=~n(With :F,n= F,n-~'land-F,l~F') all belong to~.
(3).
111,
then the complement F-F' ~f F I also belongs to (A,o .
3·
If F\
4.
(}Lo is the smallest family of subsets of F that satisfies (1),(2).
3
In this note we propose to verify that tha left-right symmetry and
the statements 1) and 2) in Kleene's theorem are still valid for a somE;·
,'\
what larger class f4cin which E is replaced by a finite dimensional spac8
V with integral coordinates and where the operations are restricted to
N
ordinary arithmetic and, even
1.3
more to the addition of integers.
Finite dimensionality and elementary nature of the operations.
We shall constantly assume that all the information
m letters of any input word f
abo~the
first
= xi
xi
xi ..• x. that is available
12m
~n
for further computation is summarized by a vector v(f ) belonging to V
m
N,
the N-dimensional vector space
~ii'h~in:teg1'a:l'·'~bO:Pdina:te~f'.'whereN ~sl.a :1'1nite
fixed constant characteristic of the automaton.
Thus, by definition, we are given independently of the input word
f:
1)
A vector v(e)€ VN from which we start the computationsj
2)
For each letter x.
~l
€ X and each number j
= 1,2, ... ,N
a
function ¢.(Vjx) which expresses the value of the j-th coordinate
J
and of the N coordinates
v,(f 1) of v(f 1) as a function of x = x
J m+
m+
im+l
of v = v(f )(with v(f ) = v(e». This we summarize by saying that for
m
0
each x € X we are given a mapping ¢x: VN ~ VN·
3)
we declare
A decision rule by which from the knowledge of v(f)(= v(f »
n
f
to be accepted or not.
In themselves, these assumptions do not imply any restriction whatsoever upon the set of words FQ; accepted by our automaton.
This is shown by the display of a simple computational rule such
that for any two words f, f'
€
F, v(f) = v(f') only if f = f'; or, in
4
bctt~r
terms by the display of a simple isomorphic representation of
the free monoid
F.
This, of course, is quite well known and we take
an example for which a proof if readily available. (4)
Example 1.3.1.
Let us suppose, first, that the input alphabet
contains two letters x and y only.
for any word
f
Let v(e) = (1,1) and inductively
with v(f) =(V (f),v (f))., let
l
2
v(f·x) = (v l (f), v l (f) + V2 (f)); v(fy) = (vl (f)+v (:t;,J1''''2(t,,»);(as usual "fz"
2
denotes the words made up of "f" followed by the letter z. It can be
shown that if f
r
VI of the space V
2
~
f' then v(f)
v(f').
and by deciding that
Thus, by selecting any subset
f
is" accepted" when v(f)
€
VI,
we could obtain any F'e: F as the set of word accepted by our automaton.
The process easily generalises to any finite alphabet x ,x2 ' •.• x n .
l
With Xl we associate the function v(f)
~
v(fx) described above; with
any x.(i = 2,3, ••• n-l)we associate the function V(f)~(fyi-lx) and
1
finally with x
n
we associate the function v(f) ~ v(fyn).
Since any
word in the letters x and y can be factorized in a unique manner as a
n'
product of the words x, y
x(l
~
n'
~
n-2), and y
n-l
we again obtain
a one to one correspondence between the set of words in the Xi and a
subset of V and this proves our contention.
2
Thus, restrictions have to be imposed either on the operations
involved in computing v(fx) from v(f) and x, and/or on the
that is on
the final operations by which
f
decision rule
is accepted or not.
In order to formulate the restrictions we introduce the following
definition:
5
Definition 1:
A mapping ¢:V ~ V is said to be elementary if
N
N
every coordinate of ¢(v) can be obtained from those of
v € V by a
N
finite number of addition, subtraction, multiplication and reduction modulo
p (cfbelow).
This mapping is said to be very elementary if it satisfies
the supplementary condition that only multiplications with a left factor
bounded by an absolute constant are allowed.
It is positive if it maps
into itself the subset v~os of all vectors with non-negative coordinates
and if only addition, multiplication by a non-negative number and "reduction modulo p" are allowed.
In this definition, by "reduction modulo p" we mean the usual mapping
y
p
Z (the
of
set of all integers) onto the interval
(0,
p-l) defined by
=-
Y n
n (mod p).
p -
The reader may observe that in the above example the algorithm was
a very elementary and positive one, since only addition and multiplication
by
° or by
1 were .used.
Definition 2.
A rule for deciding from v(f) if f is accepted or not
will be said elementary if it is given by the union VI of a finite number
of linear subspaces of V together with the prescription that f is accepted
N
if and only if v(f) ¢ VI.
It will be said positive if VI belongs to the
subspace v~os of the vector with non negative coordinates.
It is frankly admitted that no argument is given for deciding to say
that f is accepted when v(f) ¢ VI rather than when v(f)
€
VI.
However these
two possible defiIlt10nsantail somewhat different structures as it is shown
by the example I.7.4 below.
6
1.4
Definition of the
class~
Even with the extremely restrictive conditions that both
¢:
VN~
V and the decision rule are elementary, I am not able to
N
verify the not trivial parts of Kleene's Theorem.
Definition 3.
Thus, we propose:
~
An automaton a will be said to belong to the class~
X the mapping ¢x: V ~ V is very elementary and i f the
N
N
decision rule is elementary.
if for all x
E
Definition 4.
An automaton a
e~will
be said to belong to the
sUbclass~
if the initial vector v(e) has non negative coordinates, if
'0
for all x ex,
¢·x is
very elementary and positive and if the decision
rule is elementary and positive.
If one accepts to consider that "addition" and "reduction mod p"
are operations taking one unit of time irrespective of the value of
the operands one can think
Of~S
of the class of those automata which
use only elementary operations and which work" in real time."
in each computation (v(f); x)
~
Indeed,
v(fx) only a bounded number of units
of times is needed since because only multiplications with a bounded
factor appears, these can be replaced by bounded cascades of additions.
If we had also imposed the condition that the length of any vector
v(f) is bounded it is intuitive that we would have reduced ourselves to
the class of finite automata.
It can be shown that because of our re-··
strictive definition of the decision rule it turns out
lent to the class of the finite automata.
that~
is equiva-
(c.f. 111.4.3 below).
1.5 Preliminary reduction.
We want to bring any elementary mapping ¢x:VN ~ V into a simpler
N
form by getting rid of the "reduction modulo p" operations.
ThiS, of
7
course, is done at the cost of increasing the dimensionality.
By hypothesis, there exists only a finite number of integers p which
are used for performing reductions in all the computations
¢x (v(f»
and
we denote their product by q.
Let now)for any f g F) V'(f) be the direct sum of v(f) and of a N-v8ctor
Vi
(f) whose coordinates are those of v(f) reduced modulo q.
Thus, by definition v"(fx) is obtained by computing first
v(fx)
= ¢x(v(f))
modulo q.
and, then, by deducing Y'(fx) from v(fx) by reduction
However, as we shall show, the same result can be obtained in the
following different way:
i)
we obtain Y(fx) as a polynomial function of v(f) with coeficients
depending upon v'(f) and x.
ii)
we obtain Y'(fx) as a function of x and of
y
l
(f) only.
Since "for';: 'any of' ~Ft~oo;r.di"'nate.S o'f ytfl)a:re bounded by. q,; the ma,pplng..
described in ii) is just a mapping from a finite set to itself and does
not bother us any more since it can be realized by a conventional finite
state automaton.
Now to the
proof~
By hypothesis the computation of any coordinate of Vi (fx) involves only
iterations of the following operations:
Computation of the sum or product of two partial results a and b, reduction
modulo p of some partial result c.
We use induction and we assume that
every partial result is either a polynomial in the coordinates of v(f)
with coefficients depending upon x and the coordinates of v'(f) or a
function of x and of the coordinates of v'(f) which consequently is
bounded in absolute value.
8
Thus, trivially, this is true of a+b and ab if it is true of a and
of b.
'
o
If c is a pOlynomial in the coordinates of v(f), c(mod p) is a
polynomial in the values (mod p) of the coordinates of v(f) and consequently, since p is a divisor of q, c (mod p) is a function of the coordinates
of vl(f); finally if c is a function of vl(f) the same is true of c(mod p).
We now apply our requirement that only mUltiplications by factors less
than K are allowed.
Admittedly we do this by giving to the definitions
their most restrictive interpretation and consequently we offer the present
derivation as being a heuristic introduction to the formalized definition
1 1 given below.
Nonetheless it may be argued that the instruction to carry out a
mUltiplication by a factor can only be under the control of x and Vi (f)
because we only allow arithmetic operations.
Accordingly, if a coordinate
v.(f) happens for some f e F to exceed K in absolute value, then it must
1
appear in anyone of the polynomials only under at least one of the two
conditions.
i)
ii)
as a linear term (with a coefficient depending upon (x, vl(f), i»
in a monomial of degree two or more in the other coordinates of v(f)
but with the provision that this monomial has a numerical coefficient
k(x, vl(f),i) which is zero for all the fwhich are such that
IV.(f)1 > K.
1
A liberal interpretation of the definitions would only lead to the
colclusion that when a product vi(f)vi,(f) appears in some polynomial the
construction of the algorithm is such that)for all f) Vi(f) and Vi(f)
are not both r > K.
However, if we admit that the ,sequential time order
in which the operations have to be performed is part of the algorithm,
9
this interpretation would result in taking, at least provisionally, unbounded numbers as multipliers.
Now, reverting to our above assumptions i and ii, we show that we can
entirely eliminate the case ii.
Indeed let p be a prime larger than K
and replace every coordinate Vi(f) by the pair made up of vi(f) itself
and of the value vi
,p
(f) o,f v. (f) reduced module p.
].
Also in the com-
putation of every monomial of degree> 2, replace vi(f) by vi (f).
,p
Thus, the (possibly unbounded) Vi(f) appears only linearily in the
polynomials and, because of the condition ii, the final result remains
unchanged.
values v.
]., p
As in the previous section, it is easily verified that the
(fx)'s are in fact independent of the possibly unbounded
coordinates of v(f) and thus we are led to believe that the following
definition (1') is equivalent to the initial definition (1).
I.6
Formal definition.
Definition 1'.
An
automaton a ~:A: is the structure given by
=1. xl;
1)
a finite alphabet X
2)
a finite set E
3)
for each x S X a mapping E
4)
for each x
~
=i(]'}
;
~
E;
X and (]' SEan integral N x N matrix
~(x,(]');
E; v(e) ~ V ; E'C E;
N
V'C V where V'is the union of a finite number of linear subspaces of VN ..
N
5)
(]'l~
the following distinguished objects:
For any input word f
= xi
xi
1
cessively:
xi ' the automaton computes sue2
n
10
~lx. ; VeX! ) = vee) ~(x. , ~l)
11
1
1
1
(~lx. )x. ; v(x
11
12
i 1 Xi 2 ) = v(e) ~(xi1 ; ~l) ~(xi 2 ; ~lx.1 )
1
or, for short, v(f)
V(f) ¢ V' •
Definition 1":
= v(e)~f.
Then,f is accepted by a if ~lf S ~' and
The subclass/{ 0 C
fr is
characterised by the following
restrictions:
~(x,~)
1)
all the matrices
2)
v(e) has not negative coordinates;
3)
the decision rule is
.~ .
·,t
':
have not negative entries;
positive~
, . ~;
, ,1 ' .. 1
i~(""'l"'~:
(";
•
.
"
If
~
reduces to a single state,
~f
can be simply written as the product
~(Xil)~(Xi2)... ~(Xim) and we shall refer to ~ as to an homomorphism F~~.
Such an a
sA with ~ = ~'= {~l\ will be said in reduced form.
We now verify the following statement.
I.6.1
Every automaton a
Proof :
S~iS
equivalent to one in reduced form.
'Let M be the number of states
-
~.
J
of L
To each X S X we associate the (M x N) x (M x N) matrix ~x defined
as follows:
(~X)ij,i'j'= 0 for 1 < j, j' < N if ~ix
I
~i"
(1 ~ i,i' ~ M).
(~x)., i' .,= (~(X'~i)' j' for l~'j, j' < N if ~.x ='~i' (1 ~ 1; i' ~ M).
1J,
J
J.
-
1
-
11
It is easily verified that for any f = x. x •.. X, if we write
1
1
1 i2
m
we also have
~Xi
tn
Cilf) J.J,
., i
I
.,
J
= 0 if
0"1' f
~ O"i I
(~f)i' iljl= ~(xi jO",)~(xi jO".x )~(x jO"x x. ) •.. Il(Xi ;O"lx x ..•. X.
)
i 1 J.
i 1 J.
J,
1 1
2 1 i1
i3
In
lm_l
2
2
if (J'i f = (J'i'
Thus, taking an initial M x N-vector vee) with (v(e))lj= (v(e))j and
(v(e)),.= 0 for i ~ 1, v(f) = vee) ~f satisfies the following relations:
lJ
2)
the vector v(f) which for each 1 ~ j ~ M has coordinates
~ (v(f)) .. = (v(f)). (with I' = i:O"iS ~I
iSI'
J.J
J
distinguished subspace.
) does not belong to VI, the
By adding eventually a dummy coordinate with value 1 for all f we
can always suppose that the sUbspace V' is defined by homogeneous linear
equations ,i.e., that v ¢ VI only if not all the vectors product vw
zero where ~k~ is a fixed finite set of vectors.
k
are
It is trivial that
these conditions reduce to homogeneous linear conditions on
~f
since
vee) is a fixed vector.
Example r.6.l:
Let ~ = {(J'} be a finite automaton with N states.
each x S X we associate the N x N matrix
~x
with entries
Then for each f = Xi Xi ... x. , the matrix
1 2
J.m
~f
= ~xi
~xii'=
1
~x,
J. 2
1 if
••• ~xi
m
To
12
also has entries
= rr i ,
cording to rrif
~
which satisfy the condition
~fii'=
1 or 0 ac-
Thus i t I' = {i:rris ~I} the linear function
or not.
is 1 or 0 according to whether f is accepted or not.
~f
i¢I'
~fi,i'
Ii
Remark:
As a trite consequence of I.6.1 we may observe that for the class
A
there is no difference between "left" and "right" automata in the sense
of Ll.
I.7 We now find it convenient to introduce a few more words and
notations j by an homomorphism
~ :F~
zw
we mean a mapping from F to
the ring of the integral N x N matrices that issuch that
for any f, f' S F.
arbitrary
~x
S
~
~,
~ff'= ~f~f'
This condition is clearly satisfied when we take an
for each x S X and define
~xi
xi ••• xi
as
12m
By a projection
~
=~
~:~~
Z we mean any fixed linear function
p .. ,m " of the entries m , of an element of Z-..
i~
ii
iN
1 < i,i'< N ~~
P is the matrix from
~ =
~
l<i<N
ZW
whose transpose has entries
Clearly i t
Pii' we have
(mp)i i= Tr(mp) . (the "Trace·." of mp).
'
We very explicitly state the following well known result:
L 7.1
Given ~:F~ ~ and ~' :F~ ~ we can find ~":F~ ~,,(N"= NN' )
such that for any
~:~~
Z,
~'
:ZW' ~
having the property that, identically,
of
~~f
and
~'~'f.
Z there exists a ~":~,,~ Z
~"~'f
is a given bilinear function
13
Proof.
~'f
Let
~f
be the kroneckerian product of
matrix from ~f' with entries:
=
~lIff'
Since any fixed bilinear function of
bilinear function of the entries of
function of the ent:r;:1es of
~"f
take N" = N+N' and for
~lIfJ
~f
n~f
and
and
n'~'f
is, the
Then, by
identically.
n'~'f
and
1s a fixed
it is a fixed linear
~'f,
and the statement is proved.
n~f
if only a linear function of
~'f,that
~"fii' ,jjl= (~fij)(~'fi' jl)'
~"f ~"f'
induction, one verifies that
and
Of course
is wanted, it is enough to
the direct sum of
~f
and
~'f.
Let ID = ~ Fa1
(F: the set of the words accepted by a), then
V\. 1 JasJc a
r. 7.2.
Fa,Fal S CR.. implies Fa U Fas ([land Fan Fas
CR:,
Let us consider firstly Fa-= if: n~f,lo } where ~ is an homomorphism
Proof:
By taking
F--;>~.
~2 :F---> Z~J
the kroneckerian square of
2
n2 :ZN2--;> Z such that, for all )n2~2f = (n~f).
l
Fa = f: n~f
~
0 } ={ f: n2~2f
~ 0 ~;
kroneckerian square ~2 of ~'.
~,
we can find
Consequently
if Fa' : { f::If 1 ~'f
~
0 } we
t~ke
also the
Then, if ~ is the direct sum of ~2 and ~2'
·,'.I~.~ we can choose ;r:Z~iN'2--;> Z such that identically
n
~
f =
(n~f)2+ (n'~'f)2j
thus Fa"={ f:'ir ilf
~
O} = FaU Fa"
of definition 1 1 ) every general a has the form F~U F
u
F
ak
=f:nk~f ~
.~
where
o}, the result is proved for the operation·U.
have the side advantage that we know how to reduce any F
a
in which a single n appears.
Fall Fa l since, if
~
and
~'J
~II
We also
to the simpler form
This instantly de1ivers the result for
is the kroneckerian product of the corresponding
we just take the projection
ty nil ~II f =
Since (because
(n~f)( n' ~ If) •
i':~,~
Z which gives the identi-
14
We still go a step further in the reduction process and we prove:
1.7.3
To any pair (n,~)(~:F~ ~; n:~~
~:F~ ~2+2 such that identically n~f
Proof.
z) there corresponds one
= ~fl,2+N2'
For every f g F let us construct ~f as the N'x N' matrix
(N t = 2+~) with the following entries:
(i)
(ii)
~fN' . = ~f. 1 for 1 < j ::: N' •
,J
J,
"iIf1,j+kN+l
for each (j,k)(l ::: j ::: N; 1 ::: k ::: N) is equal to the
(k,j) entry of
p(~
f) where p is the N x N matrix cor-
responding to n.
( ii)'
f
l.1 j+kN+l, N'
for each (j,k)(l ::: j ::: N; 1 ::: k ::: N) is equal to the
(k,j) entry of
(iii)
(~
f)p.
The restriction of ~ f to the set of indices i,j(l < i,j < ~+ 2) is
the direct sum of N matrices identical to
The verification that
P:
~f.
is an homomorphism is straight-forward and the
result is proved because of (iv).
II Examples and counter-examples.
We want to display first an example which shows that some automata
a g;Atcan accepts sets which cannot be accepted by any finite state devices,
at least when the alphabet has two letters or more.
We write F' Sd\(£Ro)for
denoting that there exist some a g~ (some finite state automaton) such
that F' :: F .
a
Example 11.1.
Let!: = ~ cri~ i=1,2,3,4,5' X
=i x'Y1
'
and the mappings
15
(JIY
= (J3Y = (J4Y = (J5 Y = (J5;
(J2 Y = (J3'
Let also ~«J,x) be the following 2 x 2 matrices:
~«Jl;X) = ~«J2;x) = (~
~«J2Y)
1
= (0
~«J3;x)=~«J4;x)=(~
i);
= (0o
0
1); ~«J,x)
-i);
0
0) in any other case.
initial state (Jl with the initial vector v(e) = (1,0),
l+n
1+n
l
2
it is clear from the diagram that (Jlf = (J4 if and only if f = x
Yx
and that, then, v(f) = (1,n -n ); in any other case the second coordinates
l 2
of v(f)is O.
Thus, this algorithm can be made to accept all the input
words except those which have the form f = x
l+n
Yx
l+n
fact(5)that this cannot be done by any finite automaton.
It is a well known
Thus
O(F
~.
The same example also shown that, because of our very restrictive
definition of an accepted word, it may be that F SeA and F - F ¢ rh
Ct
Indeed, let F , = F-Fa = {n+l
x
Y xn+17J n = 0,1,2, ••..
FaS~ and we want to prove that F ' ¢ d{ .
contradiction that
F~ ~,.
~a
F'={ f:(~'f)l,N,F
Under this hypothesis F
a
I
Ct
VL.
j
We have just seen that
Let us assume for the sake of
o}where
= F-F
-
~I
is an homomorphism
={ f:(~f)l,N,F
o}.
Now, since
is a Nix N' matrix it satisfies an equation of degree at most N' and
consequently there exist for each fixed pair (i,j) an homogenous linear
relationship independent of m S
sequences of N' matrices
~,
2
m~x, m~x
between the (i,j) entries of any
N'
, •.. ,m~x
.
But by hypothesis, for
every n we have
( , n+l 2)
( , n+l n)
0
d ( ,n+l)
.1 0
( ~ ,n+l)
X yx 1,N'= ~ x
yx 1,N'="'= ~ x
yx 1,N= an
~ x
1,N,r;
16
thus, N' has to be larger than n+l and, since n is unbounded, N' cannot
have any finite value.
Example 11.2
F
~ ~,
¢(t)
=1
Let X have a single letter x,
~
be any homomorphism
and consider the usual integral power series
co
+ ~ tn(~ xn)l N in the ordinary variate t.
n=l
'
According to our last reduction, the set F
a
by an automaton a
of the words accepted
sJt with associated homomorphism~, is
of those integers n for which (~ xn)l,N f O.
just the set
But,as a function of t,
¢(t) is the Taylor series of an ordinary integral rational function whose
denominator is a divisor of det(l - t
~
x).
Thus, according' to Sk6lem's
theorem (6), there exist natural numbers n,n' ,d ,d , •.• d'k(With
l 2
m
0< d. < n') which are such that when m > n the coefficient of t in
-
~-
¢(t) is zero if and only if m is congruent mod n' to one of the d.'s.
~
Consequently, the complement F - F
a
theorem, F
of F
a
(and, because of Kleene's
itself) is accepted by some finite automaton.
a
X has a single letter our class
Thus, when
does not differ from that of the
finite automata and there are qUite simple sets (as that of the words of
the form f
=
n2
x
where n runs over all integers) which cannot be
accepted or not accepted by some a S
Example 11.3
A
We want to show that the familYCR
closed under set multiplication.
- Fa
1 s,A
a
is not
More explicitely we verify that there exist
homolllQI:Ph16malJ..':'F~zwmd
. ~':~""'~tsuc'm~!that
be factorised in at least one manner as fll
~lfll,N'. 0
=iF
the set F1L . of' all f"'whioh 'can
= ffl
with
~fl,N=
0 and
cannot be written as F" = tf:~lIfl,NII= O}for any finite Nil.
thus, for the
class~there exists a lack of symmetry even at the set
theoretic level between the accepted and not accepted types of sets.
\
1 0
Let X = t x'YJ; ~e = ~'e = (0 1); ~x = ~'x
~Iy
=
(~-i); Fctlf:~fl,2=
Fa (Fa ,)
Thus
0 \;
= (01
1
1);
~y
-1)
1;
= ( 01
F~,= {f:~lf12= o}.
is the set of the words f such that If I
x
-
If I
y
=0
(If Ix- 21fl y = 0) where If I z denotes the number of times the letter z
appears in f.
By definition F -
Fa
and F -
Fa I
both belong to "\
m and
we want to verify that this is not true of F - F" where F" = F F "
Let
aa
W(f) = \f' :ff' SF"] for any f S F.
1)
For any f" S Fa' W(f)C W(t"f), indeed, ffl S F" means that ffl
with flS Fa and f 2S Fa"
trivially, F F C F .
aa
2)
We have
If f,f"S
Thus f"ff' = (f"f l )f 2 with f"2 f l S Fa' since,
a
Fa- te") then W(f) ~ W(f"f); indeed,let f"fS Fa' that is
If"fl = If"fl = k, say.
x
y
k
The word f"fx k satisfies If"fxkl = 2k;/f"fx , =k.
x
Y
Thus this- element belongs. ;to;· F' ,and since
'.
a
k
we have x S W(f"f).
= If31y
F',
a .C
F" because e S
Fa f'I•Fa ,
'
k
-
Now it is impossible that f x = f f with flS Fa
l 2
would imply f
that is k
= f l2
since f, flS
= flf3
with If
k
3
X
Ix=
If 'x+k = 21f31y'
3
Fa entails f 3S Fa and If 3 'x= If 3 1 .
y
Let us suppose now that F"= tf:~"fl,N"= 0} with ~:F~Zrr"
For any
given f the set W(f) of the vectors consisting of the N~th row of the
matrices ~"f' (f' S W(f»is a linear space of dimensions at most N".
we have seen how to build at least on infinite strictly increasing
Since
18
W-sequence W(f) C W(fllf) C . . . C W(f"mf )
-
- -- - --~
... ....",
--- --
•.. we also have 8.A'1\infinite
---
~---_._,.
strictly increasing ~~po~n~ti.sequence and N'I cannot be finite.
Remark:
In the next section it will be proved that for any
formal power series
~ f(~f)l
fSF
~
fSF
maton.
the
N is in a sense a rational (non-commutative)
'
function of the input letters x.
of the sum
~:F~ ~,
This, of course, is true in particular
P where F I is the set of words accepted by a finite auto,i
Of course also
f is not in general a rational function when
~
fSF
a
F is the set of words accepted by an arbitrary a S~.
(~)
a very special subclass it is easily verified that
However for
a
~
f is an algebraic
f¢F '
a
but not necessarily rational function of the
XIS
and that apart from
this special subclass the same sum can be a transcendant function.
This
again points to the asymmetry mentioned above between "accepted" and
"not accepted" sets of words.
III
KLEENES I S THEOREM
III.1 Although this part could be written without explicitly using the
notion of the ring
Aof
the formal integral power series in the non-
commutative variates x S X, it seems more natural to do so and we recall
without proofs a few definitions and results on A.
These are very
special and shallow cases of theorems used by many authors, in the study
Two especially valuable references are
(7)
and
A is the ring of all formal infinite sums a
=~
f(a,f)
of other problems.
Definition 1:
fSF
with integral coefficients (a,f).
are defined respectively by:
The addition and mUltiplication
(8).
19
a + a l = ~ f«a,f) + (a'f»; aa l = ~ f (
fSF
fSF
where, as always in this section,
~
(a, f 1 ) ( a I ,f" ) )
f'f"= f
means a summation over all
~
f'f"= f
factorizations f = f'f" of f.
It may be easier to visualize any element of
A as a generating function in which every_word fhas a (positive or
negative) integral coefficient (a,f).
Thus, in particular to each sub-
set FIeF there correspondsthe formal sum (its characteristic function)
=
~ f
fgF'
=1
~ f ~,(f) with X-,(f)
'F
'F
fSF
or 0 according to fgF ' or not.
The mUltiplication is simply the ordinary multiplication of series
using infinite distributivity, that is aa l can also be formally expressed
as
~
fgF
f(a,f)a ' =
~
fgF
a f"(a ' ,f) _
~
ff' (a,f)(a I ,f ' ).
f,f'SF
It may not be unnecessary to stress that this product is
Hadamard product
~f(a,f)(a' ,f)
B2!
the
to which we were led by the construction of
the kroneckerian product of matrices in 1.7.
We shall always denote the empty word by e and byA*.the subset, of' a.ll
a g A in which (a,e), the coefficient of e, is zero.
The elements of
A* are usually called quasi regular and we denote by a* the mapping
A~A* defined by a - (a,e)e.
If and only if a is quasi regular
(i.e. a = a*), it has a quasi inverse a
0 0 0
aa + a
-
=a
a + a
= a.
a g A has an inverse a
O
=
~ an which satisfies
n>l
In a perfectly equivalent manner an element
-1
(a
-1
a = a
-1
a = e) if and only if it belongs to
the group GCA of the elements a l which are the sum of e and of the quasi
regular element a l *; then a'
-1
=e
0
+ (-a'*) .
We shall find it more
20
convenient to deal with
t~e
quasi notions because if a has non-
negative coefficients the same is true of (a*) o but generally not of
(e + a*) -1.
All the above operations are legitimate because A* is a continuous
topological algebra where the distance between a and a' (a
F a')
is
the supremum of the inverse of the length of those f for which
F (a' ,f).
(a,f)
III.2
R~A is the subset of all formal power series which
Definition 2:
have the form r: f(l-lf)l N for some homomorphism I-l:F~~ (N < co).
fSF
'
Proposition III.2:
R is the smallest subring of A such that its inter-
section with G is a group and whicb:.contains everyi.x
~
x.
The proof is simple but we find it clearer to break it into four
independent statements:
III.2.1
Proof:
R is a submodule.
if a
= r:
fSF
f(l-lf)l N' a'= r: f(I-l'f)l N' (for short, if a
'
fSF
'
is produced by I-l and a' by I-l') we take the direct sum 1-l":F~~+N' of
I-l and I-l' and we apply I.7 for reducing to the desired form.
III.2.2
Proof:
R is a subring.
We have to prove that if a is produced by I-l and a' by I-l' we
can construct some
~'
which produces aa'.
It will be simpler to prove
the result under the additional assumption that a, a' S A* and to observe
that the general case follows from II.2.1 because
aa'= a*a'* + (a,e)a'* + a*(a' ,e) + (a,e)(a' ,e)e.
I-le and I-l'e are the identity matrices of
~
and
We can also assume that
~,respectivelY.
these preliminaries we proceed to the actual construction.
After
21
For each x
~
X we define j.LII X g
~+N'
as the matrix
(6
X
(~~~u)
where by (j.Lx)u we mean the NxN ' matrix in which all columns are zero
except for the first one which is equal to the n-th one of j.Lx.
~+N"
after taking j.Lll e as the identity matrix of
an homomorphism j.L1I
:F-->~+N'
Then,
we extend j.L1I to
in the usual manner.
Because of our assumptions the following relations are surely true
if f = e or x:
""f
I'f
when 1 < i <_ N,' IllIf
L: IIfi
II Ifll
when
.... l,i= .... l,i
.... 1,N+l flfll=f.... 1,N....
l,i
1 < i < N' .
for fx.
Let us verify now that if they hold for f they ,also hold
Indeed we have for 1 < i < N:
j.L1I fx
= L:
j.L1I f
j.L1I x, . = L: j.L If
j.Lx. = j.Lfx .
l ,J.
l,i l:::j::;,N+N' l,j
J,J. l:::j::;,N
l,j J,i
and for i = N + i I < N+N ' :
j.LllfXl,N+il= L:
j.Lf
l
l~j~'
j(j.LXU). i l + L:
j.Lllfl N+,j.L'X j
,J
l~j~'
J,
The first sum is just j.LfXl,N when i = 1 and zero otherwise.
'1'
,J.
By the
induction hypothesis the second sum is
~
~
~
~
j.Lf' 1
f'ftl=f l~j~
-~ 'fll 1·1J. I Xj
,N
,J
,
When i ~ 1 this can also be written as
i'=
'"
~
1
j.Lf' 1 N....1I fll xl i"
f'f"=f'
,
L: j.Lf'l Nj.L' g"l i since
gl gll=fx'
, '
j.L' e1 ,i'= 0; on the contrary when i = 1 we have
j.LllfXl,N+l= j.LfXl,N+
L:
glgll=fx
gil ~ e
j.Lgi
1,N
j.LI g"
L:
1,1 g'g"=fg
above relations are true for all cases.
j.Lgi
1,N
IJ.' gil
1,1
Since they imply that
and the
22
t f(l-l"f)l N N'= t f
t (I-lf')l N(I-l'f")l N'= aa l the result is proved.
fSF
,+
fSF f' f" =f
'
,
III.2.3 R contains the quasi inverse of each of its quasi regular
elements.
Proof:
As above we assume that I-le is the identity matrix and we
define iie as I-le.
a matrix (I-lx)u S
For each x S X, we take I-lx equal to the sum of I-lX and of
~
which has all columns zero except for the first one
which is equal to the n-th column of I-lX.
I-lf 1, i = I-lf 1, i + f' f~ =fiif ' 1,N I-lflll,1.
-f
I-l xl ·=
,~
0:"
~
Ilf
~ xl
1<i<N
ii
.~x.,
'~
~,
i+
For f = e or x we have
As in the lastproof above:
0:"
0:"
iif'
1 .,
~
~
~
1 N "f"
~
f' f" =f 1<i ' < N '
,~
~x.
II
~,
i"
Thus»
if i = 1) iifX ,; = IJ.fx
+ I-lfx
+ t iif'
I-lfll x
+ t iif' _J.lfll x.. •
l ...
1, i
1,N f' fll =f
1,N
1,1 ft fll =f
1,M
1.,N
Thus
the~initial
relation is valid in all cases.
Let us now compute
In
particular for i
= N we
have a
N= a
+ aNa, that is e = (e +
since a was assumed to be quasi regular, a
= (e_a)-l_
N
e =a
aN) (e-a)
O
and,
•
III.2.4 Reciprocally, any element a = t f(l-lf)l N' of R, can be obtained
fSF
'
from the generators x S X by a finite number of ring operations and
formation of the quasi inverse (of quasi regular elements).
Proof:
ality.
As usual we can assume that a g
A*
without loss of gener-
Let us introduce as a tool the ring ~ of the formal sums
23
~
fm
fSF
AN'
with coefficients mfS
f
~
and consider the element s ;
~
Xllx of
xSX
If we extend in a natural fashion to ~ the notions introduced
for A, s can be considered as a quasi regular element of ~ and the
o
r
r
infinite sum s ; U ; ~ sn does exist since each entry Ui,j of u is the
n>l
~ f~fi j from
element
fSF
A.
'
Thus, the problem reduces to that of showing that any entry u, j
~,
can be obtained from the generators by the above listed operations.
This is trivial if N
= 1,
because, then,
the result is already proved for N - 1.
AN= Aand
we assume that
Let us consider t S
~-l
obtained from s by replacing by zero all the entries sl,i and si,l
(l~i~);
by the induction hypothesis t
o
=v
exists and all entries
vi ... (2 < i, j < N) satisfy the desired conditions.
,J
-
-
We define u S "L_
-~
by the following expression for its entries:
u li = sl'+ ~
sl' vJ'~; u is ; siS+ ~ v' j sJ's
J
...
j:;:29f ~
~ ~j~
If i, j ~ 1
u ij ; v ij + uiS(e + u )
ll
Ulj
.
By the induction hypothesis, all the u .. 's can be obtained from
~J
generators by the specified operations and we verify that u ; sO
by showing that us
= u-s.
We have
24
(US)ll= ulls ll + ~ ul·s· l = ulls ll + ~ Uljs. l + U
~ U S
2:9::N J J
~j~
J
11 2:;:j::N lj j 1
= ulls ll +
(e + Ull )(
~ sl,s'l+
~
sl"v"j S., )
2:;:j~ J J
2:;:j,j'::N J J
JS
= ulls ll +
(e + ull)(e - sll- (e + ull)-l) = u ll - sll'
If 2 <: i < N
(US)li= ullS li + (e+u ll ) ~ Ul'S ji
. 2:;:j~ J
= ullS li + (e+u1l )( ~ SljS"+
~
Sl"v , .S .. );
2<j<N
J~ 2<j j'<N J j J J~
--
-'
-
Because of the induction hypothesis, this is equal to
u
= 11Le+(e+u ll )(sll+ ~ 6 1 ,S'l+ ~
sl"v" .s'l)_7- sil
.
2:9:::N J J 2:;:j, j , :;:N J J J J
= uilLe+(e+ull)(e - (e+u ll )-1)_7- sil= uil(e+u ll ) - sil= u il - sis'
Finally, if i, j
~
1
(US)'j:::; U. si'+ ~ uij,s.,.= UilSl'+UilullSi'+v, .-S,.+ ~ u'l(e+ull)uljlS<l'
~
~s J 2<j'<N
J J
J
J ~J 1J 2:;:j':::N ~
JJ
25
Proposition III.2 can be interpreted as meaning that R is the ring of
the rational function with integral coefficients in the (not commutative) variates x.
As a further justification of this terminology
we verify the following property that has some application in problems
involving probabilistic considerations.
fin
Let A ~ A be the subring of the (not commutative) polynomials i.e.,
the subring of the a S
Awhich
have only finitely many coefficients (a,f)
different from O.
Let also
~
be the canonical homomorphism which sends -fin
A
onto
the ring of the ordinary polynomials with integral coefficients in the
variates AX = x; A extends to
Ain
a natural fashion and we prove
For each r g R, Ar is a power series, converging in some domain
III.3
around zero and representing there a rational function of the variates
X = AX.
We consider the matrix E x~x = s
Proof:
xSX
and the ordinary polynomial det
x = AX.
(I -
s) in the commutative variates
For small enough €/det(I - s) has its value arbitrarily close
to 1 when all the
IiI
are less than E.
Under this condition the matrix
I + E sn= (I - s)-l= (det(I - s»-l Adj(I - s) exists and its (l,n)
n>l
entry, that is Ar, is a rational function of the ordinary variates x.
Example III.2:
the matrix s =
Let us use the notation of example I.6.1 and consider
E~x S~.
Since for any f S F and rriS E, the state
rrif is unique, the elements (so)liS
Aobtained
by the construction of
26
III.2.4 have only coefficients 0 or 1 and the same is true of any
sum E
Hall
(so)l"
Thus, by this construction we can exhibit a strictly
J.
algebraic form of the sum E f(~f)li=
iQI'
istic function of the set F
E f which is the charactera
f~F
accepted by the automaton.
a
reg
Definition 3: Let 'R
be the smallest subset (in 'fact, the smallest
semi-ring) of
x ~ 'R
(i)
Awhich
reg
if a, a'
(ii)
satisfies the following conditions:
reg
for any x g X and e ~ 'R
;
~
-reg
-reg
R
then a + a' and aa' alS0 belong to R ;
if a g 'Rreg , then a*o~ 'Rreg •
(iii)
Proposition III.3.1
-reg
a S R
is that a = E
A necessary and sufficient condition that
N where
f~fl
f~F
~:F~
ZNpos
and where
ZNpos
denotes
'
the subset (in fact, the semi-ring) or the integral NxN matrices with
non-negative entries.
Proof:
It is enough to revert to I, III.2.2 and III.2.3 and to
observe that if a, a' are produced by homomorphisms into ~os the same
o
is true of a + a', aa' and a ; also trivially
~x belong to ~os.
~
and all the matrices
The construction performed in III.2.4 does not use
subtraction either, and consequently E
f~fl,N~
-reg
R .
R is the smallest submodule of -A that contains -reg
R
and any
reg
r S 'R can be written under the form r = r' - r ll with r' ,r"~ R .
III.3·2
Proof:
Since every r
~
R can be obtained from the
generat~rs
by a finite number of additions, subtractions, multiplications and
formation of inverses it is enough to prove that if the result is
true for r , r2~
l
R it is still true for r
3
= r l + r 2;
r 4= r - r 2 ;
l
x
27
l
r 2= r 21- r"2 where r l'
r 2l r"1 and r"2 belonD's
to
c
Rreg . We have:
r = (r'+ r')-(~'+ ~,). r = (r 1+ r")-(~'+ r 1). r = (r1r 1+ ~'r")-(r'~'+ ~'r')
312
12' 4
1 2
12' 5
1 2 1 2
1 2 1 2
-reg
where again all the elements between brackets belong to R
since
r*= r'*- r"* and that r S
111
then, r* S -reg
R
since r*=
Rreg implies (e - r*)-l_ e
(e _ s)-l_ e.
Now, for any a, b g R with a = a*, b
= b*
we have
e - a+b = (e - a) (e +(e-a) -~)
= (e - a)(e+(e-a)-~)(e -(e-a)-~)(e _(e_a)-~)-l
• (e - a)(e _«e_a)-~)2)(e - (e_a)-~)-l.
From this we get the
identity
(e-a+b)-l= (e-(e-a)-~)(e_(e_a)-~(e_a)-~)-l(e_a)-l
= Lre-(e-a)-~(e-a)-~)-1(e-a)-:7
- Lte-a)-~(e-(e-a)-~(e-a)-~)-.1(e-a~-:7.
Thus, taking, a = ri* and b = ri*, we can display (e - ri*+ ri*)-l as
reg
the difference of two elements from R
and the result is proved.
III.4 We now revert to the proof of Kleene' s theorem for
III.4.10
If F
Proof:
a
,F IS
a
6\,
then F F IS
aa
Let Fa = {f:lJ.flN~
1
(R = { Fa tfs A'
OZ.
o} ; Fa ,= if:IJ.'fl,N'~ 0 }.
We can
assume that for all f,lJ.fl,N and lJ.'f lN , are both negative (cf I.7).
Then,
28
if r = I: f(~f)lN and r' = I: flJ.f "
lN
l'SF
l'SF
we have (rr',f) =
I: (lJ.f')l N(IJ.'f''), N
1" 1''' =F
'
...., '
that is, (rr' ,f) F 0 if and only if there exist at least one factorization l' = 1"1''' for which
Fcla,={f:(rrl,f) F
OJ;
IJ.f'l,NF 0 and lJ.'fi,N,F
o.
Thus
since we know by III.2.2 how to construct
~':F~~+N' such that (zz' ,f) = ~'fl,N+N' the result is proved.
III.4.2
If FaS
~:
above.
OZ then F*a S rn
.
V'\..
Let
F
a={f:(z,fL= IJ.fl,NF O}With (z,f) ~
We can write
~= { e)
U (F0:" {e
assume that Fa dOBS not contain e.
0 for all 1', as
1)* and consequently we
can
Then,as in 111.4.3 it is easily
OJ and the result follows from 11I.2.3.
checked that ~ = { 1': (rO ,f) F
Let us recall that according to the definition
~'
of 1.6, and the re-
ductions to simpler form carried out in the same section an automaton
a belongs tofto
if
and only if the associated homomorphism IJ.:F-'>~
is a ~pping into ~os.
reg
Thus R = tflJ.f
lN
where IJ. corresponds to some
lt.
a S IJHt/ o .
111.4.3
A necessary and sufficient condition that FaSID
(the family
\.1\0
of the sets of words accepted by some finite automaton) is that
Fa=
t
l' : (r ,f) F 0
~:
1 for some r
reg
S 'R
.
The condition is necessary beacuse as we have seen in
example 11I.2, the sum r = I:
l'SF
f(;~~felOngS to 'Rre~1'
a
. . . . -~---. "----------"
For proving the sufficiency we start with any
consider the mapping
~
IJ.:F~
ZNpos
and we
which sends 0 onto 0 and every positive element
29
onto
1,
1
where:: Q and
= Q = Q + Q and 1 .1
=
are boolean elements, i.e., Q Q =
.1 = 1
+
Q = Q + 1 = .1 + 1j t3 is an homomorphism
of semi-ring and it can be naturally extended to
ZNpos
f3m when m g ~os as .the matrix whose entries are f3(m
I
Trivially, for any m,m g
ZNpos
2. 1 .=1 Q
= f3mf3m
we have f3mm'
ij
) g {Q,~}
and
pos
f3ZN
.
has at
1 fW is a finite monoid-
2
most 2N < co distinc t elements.
I
by defining
Thus, {f3lJ.f
1is the inverse image by the homo-
01
M and Fa ={f:lJ.fNf
= \f:f3lJ.f l ,Nf Q
morphism t3IJ. :F~ M of a subset of M.
It is enough now to verify the slightly more general statement:
A necessary and sufficient condition that F'g
6\0
-1
is that F ' = ¢ ¢ F'
where ¢ is an homomorphism of F into a finite monoid G.
The necessity is trivial since the mapping ¢ defined by ¢f
if and only if crf
= crt'
= ¢f '
for all states cr of the automaton is an homomor-
phism of F into a finite monoid and since F' is union of ¢-classes.
In order to prove that the condition is sufficient let E
a set in one to one correspondence with G
cr x
g
= CT
= ¢F
=
t
crg} be
and for each xgX define
¢ j the initial state is s¢ and E' = (cr l
g x
e
l g J g€¢F ' =G
.
Because
¢ is an homomorphism, cr f g E' is equivalent to ¢e¢f = ¢fg¢F ' and the
e
-1
result is proved since by hypothesis F'= ¢ ¢ F. In consequence, we have
shown that(R"o ={Fa
1~ J+.o •
Another simple characterization of
III.4.4
0<..0
is the following one:
A neeessary and sufficient condition that Fag~ is that
30
Fctif:(Z,f)
Proof:
I:
°~ where r S R is such that (r,f) is bounded flDr all f.
This is just a more formalized version of the'heuristic
reasonings of I.5.
= l~fINI
If l(r,f)1
< K for all f we take Yp the
homomorphism that sends every integer upon its residue mod p where
p
~
K.
t
Then, as above, Yp~F is finite and Fct f
:~f 1,NI:
O}..:= {f: Yp~f lNl:
01
is the inverse image of a subset of it.
III.5 An intuitive interpretation of
Let us consider a finite automaton with set of states E
We assume that an initial state
~,y:~ ..
and
~
= fy I .
input alphabet Y
for",a,;)jJJt,·'j'
=
nave:.
0-
associated a word f
.
b<;~n
•.•
o' t,.
.
and a Bink\mo.. s:W::};l;~tp.~11l'~\:)Y' ;;:
.....
....
·'tl1st1LguislJedr.
I:
To each state
in a certain output alphabet
Qo-'
Given an input word g
when ~I:~'
= Yi
1
~o
"
~.r
= Qo- ~e;
is ~ required that Q~
~l
I: ~0
X ={ x}
~
is
; it
.
y .•
1.2
y.
J.
.• ,
we compute successively the
m
If
~i =~ig·
m
there exists a maximal index m'< m such that
~i
m'
I:
~
~
i
= ~0
m
and we define
0
We call this process a (right) transduction Q;
in fact it is a "right coset IDaFPing."
Given a transduction, Q, we can count for each word fSF how many
distinct words g are such that Qg
= f and
~lg
I:
~o'
This number which,
of course, can be zero is denoted by (t,f) and we have:
III.5.1 The infinite sum
t
= E f ( t;f) belongs to -pos
R
f~F
31
Proof:
Let N be the number of states in E distinct from
(f
o
and consider for each y S Y the NXN matrix vy with entries vYi'=
J
if (flY
= (fj
and
=0
otherwise; also let m = E vy.
ySY
Aand
n
above in III.2. 4 we can define its quasi inverse m0 = Em.
n>l
1
e for all (Ji~
(f
0
Q(f.
J
The matrix m belongs
to the ring ~ of the NxN matrices whose entries belong to
Q(f.~
.,\'.
as
Because
no word fSF of length less than n appears in
n'
0
m when n'> n and consequently each entry (m )i,j is a sum
a ij = E f(aij,f) with finite (but unbounded) coefficients (aij,f).
a
Again,
. is equal to the sum of all words Qg where g satisfies the equation
l ,J
Thus, t
=
E f(t,f)
=
fSF
(f.
1
.E Qg
gSG
=E
g/:cr"0
alj
We revert to the proof
j'
of III.2.4 and we verify by performing the same constructions that
any entry a
belongs to Rreg •
ij
III.5.2 To any FaS ~, there corresponds at least one pair (Q,Q') of
1fand only if (t,f)~(t',f':)'
a
~: By definition there exists some r S R which is such that
According to III.3.2, there exist r', r 1l S -pos
R
f: (r,f)~O
transductions such that fSF
a=
F
which are such that r
= r'·
r 1l and, consequently, that Fcttf:(r' ,f)~(rn,f)}
Thus, it is enough to show that to any r S RPos there corresponds at
least one transduction Q such that r
this is trivially true if r
=0
or r
= t.
=x
As usual, we verify that
S X and we perform a construction
corresponding to each of. the,threer·operations.
32
i)
Addition
=
Let r
E f(t,f)j r'
fgF
=
respectively to transductions
a~
F and 9'
Q)
:G'~
F.
We can assume
and Y' (for 9') are disjoint and
= 15.
Now let 'I' = y
we define
9:G~
alphabetsY~or
that the two input
also that E (\ E'
E f(t' ,f) where t and t' correspond
fgF
V
1
Y' and E" = E u E' v a~ ~where a~ is a new in!tial state j
y to be O"lY if Y g Y and to be a1y if y g y'.
the other transitions are kept and
original transductions.
Q(]'
We also define
and 9 1 a' are defined as in the
The verification that this process gives r + r'
is trivial.
i1)
Multiplication.
We use the same hypothesis as above and we take 'I' = y
E" = E VEl.
u
Y' and
For each y g Y we define:
ay = O"y when
= a'0
when
0"
g E,
(j
g E' •
For each y' g y' we define"
ay' = a' yl
for each agE - { a 0
a y'= a'
o
0
for
1
1)
agE'
o
1
a'y'= aryl for each a'g E'.
Trivially again this gives rr' when the initial state is alg E.
iii)
Quasi inverse
We take two copies E1 and E" of E and two copies Y' andY' of- Y.
33
All the transitions of the form
{!:~ y' )~!:'
or (!:ll ,!:yll)~ !:"
U'~nthe
same as in (!:,Y) but we define
(j' y"
= (j"1 y"
and (j" y' = (j' y' for any (j' g !:' , (j" g !:", y' g y' and y" g. '1' ,
1
The initial state is (jig !:l'
With this process (t",f) is equal to the
sum!: f(t",f) = r o ,
m>l
Example III, 5
t
Let x = xl ,x2 ~ and consider the mapping F--.;> Z4 defined by
1"1'00
1000
100 0
0 000
~X = (0 01 0);
~x2= (0 0 1 1) ,
1
O· 0 0 0
0 0 1 0
induction shows that
(~f)ll
is just the number of ways of factorizing
f as a product of xl ,x2 ' and xlxl and, similarly,for (~f)33 with respect
to x ,x and x x '
l 2
2 2
any l~i, j~3; Q(jl= ~; Q(j2= X2 j Q(j3= xlx '
l
Then (t,f) is precisely
equal to (~f)ll' and a similar construction gives (t' ,f)
Thus F = f:(t,f) ~ (t' ,f)
ex
and, for instance,
= (~f)33'
REFERENCES
(1)
Automata Studies, Princeton, 1956.
(2)
M. Rabin and D. Scott, IBM Res. Journal 2, 1959, p.114.
(3)
cf
(4)
W. J. Harrington, American Math. Monthly, vol. 58, 1951, p. 693-696.
(5)
C. c. E1got, Decisions Problems of Finite Automata Design,
Univ. Michigan, 2722, 2794, 2755, 6 T, (June 1959).
(6)
T. Sko1em, Comptes Rendus du 8-eme Congres des Mathematiciens
Scandinaves. stockholm 1934, p. 163-168.
(7)
M. Lazard, Anna1es Sci. Ecole Norma1e Sup (3) 72, 1955, p 299-400.
(8)
K. T. Chen, R. H. Fox and R. C. Lyndon; Ann. Math. 68, 1958,
P 81-95.
-
s. C. K1eene's paper in ref. (1).
I
M. P. schutzenberger, SeminaireDUbre11-Pisot Institut H. Poincare
(Paris Dec. 1959).