Weighted omega-Restricted One Counter Automata

0.15 MB·

by  Manfred Droste

Weighted ω-Restricted One Counter Automata⋆
ManfredDroste1andWernerKuich2
1 Universita¨tLeipzig,Institutfu¨rInformatik,
[email protected]
7 2 TechnischeUniversita¨tWien,Institutfu¨rDiskreteMathematikundGeometrie
1 [email protected]
0
2

n Abstract. LetSbeacompletestar-omegasemiringandΣbeanalphabet.Fora
a weightedω-restrictedonecounterautomatonC withstateset{1,...,n},n≥ 1,
J
we show that there exists a mixed algebraic system over a complete semiring-

1 semimodule pair ((S ≪Σ∗ ≫)n×n,(S ≪Σω ≫)n) such that the behavior
3
kCk of C is a component of a solution of this system. In case the basic semir-
ing is B or N∞ we show that there exists a mixed context-free grammar that
]
L generates kCk. The construction of the mixed context-free grammar from C is
F a generalization of the well known tripleconstruction in case of restricted one
. counter automataandiscallednowtriple-pairconstructionforω-restrictedone
s
counterautomata.
c
[

2 1 Introduction
v
3
RestrictedonecounterpushdownautomataandlanguageswereintroducedbyGreibach
0
[12] and considered in Berstel [1], Chapter VII 4. These restricted one counter push-
7
8 down automata are pushdown automata having just one pushdown symbol accepting
0 byemptytape,andthefamilyofrestrictedonecounterlanguagesisthefamilyoflan-
1. guagesacceptedbythem.
0 LetLbetheLukasiewiczlanguage,i.e.,theformallanguageoverthealphabetΣ =
7 {a,b} generated by the context-free grammar with productions S → aSS,S → b.
1
Thenthefamilyofrestrictedonecounterlanguagesistheprincipalconegeneratedby
:
v L.
i Alltheseresultscanbetransferredtoformalpowerseriesandrestrictedonecounter
X
automataoverthem(seeKuich,Salomaa[15],Example11.5).Restricted onecounter
r
a automatacanalsobeusedtoacceptinfinitewordsanditisthisaspectwegeneralizein
ourpaper.
Weconsiderweightedω-restrictedonecounterautomataandtheirrelationtoalge-
braic systemsoverthe completesemiring-semimodulepair (Sn×n,Sn), where S is a
completestar-omegasemiring.Itturnsoutthatthe wellknowntriple constructionfor
pushdownautomataincaseofunweightedrestrictedonecounterautomatacanbegen-
eralizedtoatriple-pairconstructionforweightedω-restrictedonecounterautomata.
The paper consists of this and three more sections. In Section 2, we refer the
necessary preliminaries. In Section 3, restricted one counter matrices are introduced
⋆Thisworkwaspartiallysupported byDFGGraduiertenkolleg 1763 (QuantLA).Thesecond
authorwaspartiallysupportedbyAustrianScienceFund(FWF):grantno.I1661 N25.
and their properties are studied. The main result is that, for such a matrix M, the
p-entry of the infinite column vector Mω,k is a solution of the linear equation z =
(M (M∗) +M +M )z.InSection4,weightedω-restrictedonecounterau-
p,p2 p,ε p,p p,p2
tomata are introducedas a special case of weighted ω-pushdownautomata.We show
thatforaweightedω-restrictedonecounterautomatonCthereexistsamixedalgebraic
systemsuchthatthebehaviorkCkofC isacomponentofasolutionofthissystem.In
Section5weconsiderthecasethatthecompletestar-omegasemiringSisequaltoBor
N∞.Thenforagivenweightedω-restrictedonecounterautomatonC amixedcontext-
free grammar is constructed that generates kCk. This construction is a generalization
ofthewellknowntripleconstructionincaseofrestrictedonecounterautomataandis
calledtriple-pairconstructionforω-restrictedonecounterautomata.
2 Preliminaries
Fortheconvenienceofthereader,wequotedefinitionsandresultsofE´sik,Kuich[6,7,9]
from E´sik, Kuich [10]. The reader should be familiar with Sections 5.1-5.6 of E´sik,
Kuich[10].
A semiring S is called complete if it is possible to define sums for all families
(a | i ∈ I)ofelementsofS,whereI isanarbitraryindexset,suchthatthefollowing
i
conditionsaresatisfied(seeConway[4],Eilenberg[5],Kuich[14]):
(i) a =0, a =a , a =a +a forj 6=k,
X i X i j X i j k
i∈∅ i∈{j} i∈{j,k}
(ii) X(cid:0)Xai(cid:1)=Xai, if [Ij =I and Ij ∩Ij′ =∅ for j 6=j′,
j∈J i∈Ij i∈I j∈J
(iii) (c·a )=c· a , (a ·c)= a ·c.
X i (cid:0)X i(cid:1) X i (cid:0)X i(cid:1)
i∈I i∈I i∈I i∈I
ThismeansthatasemiringS is completeifit ispossibletodefine“infinitesums”
(i)thatareanextensionofthefinitesums,(ii)thatareassociativeandcommutativeand
(iii)thatsatisfythedistributionlaws.
AsemiringSequippedwithanadditionalunarystaroperation∗ : S →S iscalled
astarsemiring.Incompletesemiringsforeachelementa,thestara∗ ofaisdefinedby
a∗ = aj.
X
j≥0
Hence, each complete semiring is a starsemiring, called a complete starsemiring. A
Conwaysemiring(seeConway[4],Bloom,E´sik[2])isastarsemiringS satisfyingthe
sumstaridentity
(a+b)∗ =a∗(ba∗)∗
andtheproductstaridentity
(ab)∗ =1+a(ba)∗b
for all a,b ∈ S. Observe that by E´sik, Kuich [2], Theorem 1.2.24, each complete
starsemiringisaConwaysemiring.
LetS beastarsemiring.BySn×n wedenotethesemiringofn×n-matricesover
S.
SupposethatSisasemiringandV isacommutativemonoidwrittenadditively.We
callV a(left)S-semimoduleifV isequippedwitha(left)action
S×V → V
(s,v) 7→ sv
subjecttothefollowingrules:
s(s′v)=(ss′)v 1v =v
(s+s′)v =sv+s′v 0v =0
s(v+v′)=sv+sv′ s0=0,
foralls,s′ ∈Sandv,v′ ∈V.WhenVisanS-semimodule,wecall(S,V)asemiring-
semimodulepair.
Suppose that (S,V) is a semiring-semimodule pair such that S is a starsemiring
and S and V are equipped with an omega operation ω : S → V. Then we call
(S,V) a starsemiring-omegasemimodule pair. Following Bloom, E´sik [2], we call a
starsemiring-omegasemimodulepair(S,V)aConwaysemiring-semimodulepairifS
isaConwaysemiringandiftheomegaoperationsatisfiesthesumomegaidentityand
theproductomegaidentity:
(a+b)ω =(a∗b)ω +(a∗b)∗aω
(ab)ω =a(ba)ω,
foralla,b∈S.Itthenfollowsthattheomegafixed-pointequationholds,i.e.
aaω =aω,
foralla∈S.
E´sik, Kuich [8] define a complete semiring-semimodule pair to be a semiring-
semimodulepair(S,V)suchthatSisacompletesemiringandVisacompletemonoid
with
s v = sv
(cid:0)X i(cid:1) X i
i∈I i∈I
s v = s v,
(cid:0)X i(cid:1) X i
i∈I i∈I
foralls ∈S,v ∈V,andforallfamilies(s ) overS and(v ) overV;moreover,
i i∈I i i∈I
itisrequiredthataninfiniteproductoperation
(s ,s ,...) 7→ s
1 2 Y j
j≥1
is given mappinginfinite sequencesoverS to V subject to the followingthree condi-
tions:
s = (s ·····s )
Y i Y ni−1+1 ni
i≥1 i≥1
s · s = s
1 Y i+1 Y i
i≥1 i≥1
s = s ,
Y X ij X Y ij
j≥1ij∈Ij (i1,i2,...)∈I1×I2×...j≥1
whereinthefirstequation0=n ≤n ≤n ≤... andI ,I ,... arearbitraryindex
0 1 2 1 2
sets.Supposethat(S,V)iscomplete.Thenwedefine
s∗ = si
X
i≥0
sω = s,
Y
i≥1
for all s ∈ S. This turns (S,V) into a starsemiring-omegasemimodulepair. By E´sik,
Kuich[8],eachcompletesemiring-semimodulepairisaConwaysemiring-semimodule
pair.Observethat,if(S,V)isacompletesemiring-semimodulepair,then0ω =0.
A star-omega semiring is a semiring S equipped with unary operations∗ and ω :
S → S. A star-omegasemiringS is calledcompleteif (S,S)is a completesemiring
semimodulepair,i.e.,ifSiscompleteandisequippedwithaninfiniteproductoperation
thatsatisfiesthethreeconditionsstatedabove.Forthetheoryofinfinitewordsandfinite
automataacceptinginfinitewordsbytheBu¨chiconditionconsultPerrin,Pin[16].
3 Restricted one counter matrices
Inthissectionweintroducerestrictedonecounter(roc)matricesandstudytheirprop-
erties.OurfirsttheoremanditscorollarygeneralizeTheorem10.5ofKuich,Salomaa
[15]incasethepushdowntransitionmatrixM isarestrictedonecountermatrix.Then
we show that, for a roc matrix M, (Mω) and (Mω,k) , 0 ≤ k ≤ n, introducedbe-
p p
lowsatisfythesamespecificlinearequation.InTheorem1andCorollary1,S denotes
a complete starsemiring; afterwards in this section, S denotes a complete star-omega
semiring.
Arestrictedonecounter(abbreviatedroc)matrix(withcountersymbolp)isamatrix
M in (Sn×n)p∗×p∗, for some n ≥ 1, subject to the following condition:There exist
matricesA,B,C ∈Sn×nsuchthat,forallk ≥1,
Mpk,pk+1 =A, Mpk,pk =C Mpk,pk−1 =B,
andtheseblocksofM aretheonlyoneswhichmaybeunequalto0.
Observethat,fork ≥1,
M = M = A,
pk,pk+1 p,p2
M = M = C,
pk,pk p,p
Mpk,pk−1 = Mp,ε = B,
M = M = 0.
ε,pk ε,ε
AlsonotethatthematrixA(respB,C)inSn×ndescribestheweightoftransitions
whenpushing(resp.,popping,notchanging)anadditionalsymbolpto(resp.,from)the
pushdowncounter.
Theorem1. Let S be a complete starsemiring and M be a roc-matrix. Then, for all
i≥0,
(M∗) = (M∗) (M∗) .
pi+1,ε p,ε pi,ε
Proof. Firstobservethat
(M∗) = (Mm+1) = M M ...M M ,
pi+1,ε X pi+1,ε X X pi+1,pi1 pi1,pi2 pim-1,pim pim,ε
m≥0 m≥0i1,...,im≥1
where,form=0,theproductequalsM .Nowweobtain
pi+1,ε
(M∗) = M ...M M
pi+1,ε X X pi+1,pi1 pim−1,pim pim,ε
m≥0i1,...,im≥1
= M ...M M ·
X (cid:0) X pi+1,pi+j1 pi+jm1−1,pi+jm1 pi+jm1,pi(cid:1)
m1≥0 j1,...,jm1≥1
M ...M M
X (cid:0) X pi,pi1 pim2−1,pim2 pim2,ε(cid:1)
m2≥0 i1,...,im2≥1
= M ...M M (M∗)
X (cid:0) X p,pj1 pjm1−1,pjm1 pjm1,ε(cid:1) pi,ε
m1≥0 j1,...,jm1≥1
=(M∗) (M∗) .
p,ε pi,ε
Here in the second line the pushdown contents pi+j1,...,pi+jm1, m1 ≥ 0 are
alwaysnonempty,i.e.,thetoppisreducedtoεforthefirsttimeinthelastmove.
Corollary1. Foralli≥0, (M∗) = ((M∗) )i.
pi,ε p,ε
Lemma1. Let S be a complete star-omega semiring. Let M ∈ (Sn×n)p∗×p∗ be a
roc-matrix.Then
(Mω) = (Mω) +(M∗) (Mω) .
p2 p p,ε p
Proof. Subsequently in the first equation we split the summation so that in the first
summandthereis nofactor M , while in thesecondsummandthere isat leastone
p2,p
factor M ; since k ,...,k ≥ 2, M is the first such factor. In the second
p2,p 1 m pkm,p
equalityweusethepropertyofM beingaroc-matrix:Mpi,pj =Mpi−1,pj−1 fori ≥2,
j ≥1.Wecompute:
(Mω) = M M ···+
p2 X p2,pi1 pi1,pi2
i1,i2,···≥2
M M ...M M M ...
X X p2,pk1 pk1,pk2 pkm,p X p,pj1 pj1,pj2
m≥0k1,k2,...,km≥2 j1,j2,···≥1
= X Mp,pi1−1Mpi1−1,pi2−1···+
i1,i2,···≥2
X X Mp,pk1−1Mpk1−1,pk2−1...Mpkm−1,ε(Mω)p
m≥0k1,k2,...,km≥2
=(Mω) + (Mm+1) (Mω)
p X p,ε p
m≥0
=(Mω) +(M∗) (Mω) .
p p,ε p
Theorem2. LetS beacompletestar-omegasemiringandletM ∈(Sn×n)p∗×p∗ bea
roc-matrix.Then
(Mω) = (M +M (M∗) +M )(Mω) .
p p,p2 p,p2 p,ε p,p p
Proof. Weobtain,byLemma1
(M +M (M∗) +M )(Mω)
p,p2 p,p2 p,ε p,p p
= M ((Mω) +(M∗) (Mω) )+M (Mω)
p,p2 p p,ε p p,p p
= M (Mω) +M (Mω) = (MMω) = (Mω) .
p,p2 p2 p,p p p p
Corollary2. LetM ∈(Sn×n)p∗×p∗ bearoc-matrix.Then(Mω) isasolutionof
p
z =(M +M (M∗) +M )z.
p,p2 p,p2 p,ε p,p
When we say “G is the graph with adjacencymatrix M ∈ (Sn×n)p∗×p∗” then it
means that G is the graph with adjacency matrix M′ ∈ S(p∗×n)×(p∗×n), where M′
corresponds to M with respect to the canonical isomorphism between (Sn×n)p∗×p∗
andS(p∗×n)(p∗×n).
Let now M be a roc-matrixand 0 ≤ k ≤ n. Then Mω,k is the column vectorin
(Sn)p∗ definedasfollows:Fori≥1and1≤j ≤n,let((Mω,k) ) bethesumofall
pi j
weightsof pathsin the graphwith adjacencymatrix M that have initial vertex(pi,j)
andvisitvertices(pi′,j′),i′ ≥ 1,1 ≤j′ ≤ k,infinitelyoften.ObservethatMω,0 =0
andMω,n =Mω.
LetP ={(j ,j ,...)∈{1,...,n}ω |j ≤kforinfinitelymanyt≥1}.
k 1 2 t
Thenfor1≤j ≤n,weobtain
((Mω,k) ) = (M ) (M ) (M ) ... .
p j X X p,pi1 j,j1 pi1,pi2 j1,j2 pi2,pi3 j2,j3
i1,i2,···≥1(j1,j2,...)∈Pk
ByTheorem5.4.1ofE´sik,Kuich[10],weobtainforafinitematrixA∈Sn×nand
for0≤k ≤n,1≤j ≤n,
(Aω,k) = A A A ... .
j X j,j1 j1,j2 j2,j3
(j1,j2,...)∈Pk
ObservethatagainAω,0 =0andAω,n =Aω.
In the next lemma, we use the following summation identity: Assume that
A ,A ,... arematricesinSn×n.Thenfor0≤k ≤n,1≤j ≤n,andm≥1,
1 2
(A ) (A ) =
X 1 j,j1 2 j1,j2...
(j1,j2,...)∈Pk
(A ) ...(A ) (A ) ... .
X 1 j,j1 m jm−1,jm X m+1 jm,jm+1
1≤j1,...,jm≤n (jm+1,jm+2,...)∈Pk
Lemma2. LetM ∈(Sn×n)Γ∗×Γ∗ bearoc-matrixand0≤k ≤n.Then
(Mω,k) = (Mω,k) +(M∗) (Mω,k) .
p2 p p,ε p
Proof. We use the proof of Lemma 1, i.e., the proof for the case Mω,n = Mω. For
1≤j ≤n,weobtain((Mω,k) ) =
p2 j
X X (Mp,pi1−1)j,j1(Mpi1−1,pi2−1)j1,j2···+
i1,i2,···≥2(j1,j2,...)∈Pk
(cid:16) X X X X (Mp,pk1−1)j,j1...(Mpkm−1,ε)jm,j′(cid:17)·
1≤j′≤nm≥0k1,k2,...,km≥21≤j1,...,jm≤n
(cid:16) X X (Mp,pkm+2)j′,jm+2(Mpkm+2,pkm+3)jm+2,jm+3...(cid:17)=
km+2,km+3,···≥1(jm+2,jm+3,...)∈Pk
((Mω,k)p)j + X ((M∗)p,ε)j,j′((Mω,k)p)j′ =
1≤j′≤n
((Mω,k) ) +((M∗) (Mω,k) ) =
p j p,ε p j
((Mω,k) +(M∗) (Mω,k) ) .
p p,ε p j
Theorem3. LetS beacompletestar-omegasemiringandletM ∈(Sn×n)p∗×p∗ bea
roc-matrix.Then
(Mω,k) = (M +M (M∗) +M )(Mω,k) ,
p p,p2 p,p2 p,ε p,p p
forall0≤k ≤n.
Proof. Weobtain,byLemma2,forall0≤k≤n,
(M +M (M∗) +M )(Mω,k)
p,p2 p,p2 p,ε p,p p
= M ((Mω,k) +(M∗) (Mω,k) )+M (Mω,k)
p,p2 p p,ε p p,p p
= M (Mω,k) +M (Mω,k) = (MMω,k) = (Mω,k) .
p,p2 p2 p,p p p p
Corollary3. Let M ∈ (Sn×n)p∗×p∗ be a roc-matrix. Then, for all 0 ≤ k ≤ n,
(Mω,k) isasolutionof
p
z =(M +M (M∗) +M )z.
p,p2 p,p2 p,ε p,p
4 ω-restricted one counter automata
Inthissection,wedefineω-rocautomataasaspecialcaseofω-pushdownautomata.We
show thatfor an ω-roc automatonC there exists an algebraicsystem overa complete
star-omega semiring such that the behavior kCk of C is a componentof a solution of
thissystem.
Throughoutthissection,S denotesa completestar-omegasemiring.Hence(S,S)
is a complete semiring-semimodule pair. By Proposition 4.1 of E´sik, Kuich [11],
(Sn×n,Sn) is again a complete semiring-semimodulepair (just omit Axiom 4 in the
proof).LetS′ ⊆S with0,1∈S′.AnS′-ω-pushdownautomaton
P =(n,Γ,I,M,P,p ,k)
0
isgivenby
(i) afinitesetofstates{1,...,n},n≥1,
(ii) analphabetΓ ofpushdownsymbols,
(iii) apushdowntransitionmatrixM ∈(S′n×n)Γ∗×Γ∗,
(iv) aninitialstatevectorI ∈S′1×n,
(v) afinalstatevectorP ∈S′n×1,
(vi) aninitialpushdownsymbolp ∈Γ,
0
(vii) asetofrepeatedstates{1,...,k},0≤k ≤n.
The definition of a pushdown(transition) matrix is given in Kuich, Salomaa [15],
Kuich[14]andE´sik,Kuich[10].Clearly,anyroc-matrixisapushdowntransitionma-
trix.
ThebehaviorofP isanelementofS×S andisdefinedby
kPk=(I(M∗) P,I(Mω,k) ).
p0,ε p0
Here I(M∗) P is the behavior of the S′-ω-pushdown automaton P =
p0,ε 1
(n,Γ,I,M,P,p ,0)andI(Mω,k) isthebehavioroftheS′-ω-pushdownautomaton
0 p0
P =(n,Γ,I,M,0,p ,k).ObservethatP isanautomatonwiththeBu¨chiacceptance
2 0 2
condition:ifGisthegraphwithadjacencymatrixM,thenonlypathsthatvisitthere-
peatedstates1,...,kinfinitelyoftencontributetokP k.Furthermore,P containsno
2 1
repeatedstatesandbehaveslikeanordinaryS′-pushdownautomaton.
An S′-ω-roc automaton is an S′-ω-pushdownautomaton with just one pushdown
symbolsuchthatitspushdownmatrixisaroc-matrix.
Remark1. ConsideranS′-ω-pushdownautomatonP withjustonepushdownsymbol.
BytheconstructionintheproofofTheorem13.28ofKuich,Salomaa[15],anS′-ω-roc
automatonC canbeconstructedsuchthatkCk=kPk.
In the sequel, an S′-ω-roc automaton P = (n,{p},I,M,P,p,k) is denoted by
C =(n,I,M,P,k)withbehavior
kCk=(I(M∗) P,I(Mω,k) ).
p,ε p
ThenextdefinitionsandresultsaretakenfromE´sik,Kuich[10,Section5.6]
ForthedefinitionofanS′-algebraicsystem overa quemiringS ×V we referthe
readerto [10], page136,andforthe definitionof quemiringsto [10], page110.Here
we note that a quemiring T is isomorphic to a quemiring S × V determined by the
semiring-semimodulepair(S,V),cf.[10],page110.
Observethattheforthcomingsystem(1)isasystemoverthequemiringSn×n×Sn.
Comparetheforthcomingalgebraicsystem(2)withthealgebraicsystemsoccurringin
the proofsof Theorem14.15ofKuich,Salomaa[15] andTheorem6.4ofKuich[14],
bothinthecaseofaroc-matrix.
Let M be a roc-matrix. Consider the S′n×n-algebraic system over the complete
semiring-semimodulepair(Sn×n,Sn)
y = M yy+M y+M . (1)
p,p2 p,p p,ε
ThenbyTheorem5.6.1ofE´sik,Kuich[10](A,U) ∈ (Sn×n,Sn)isa solutionof(1)
iffAisasolutionoftheS′n×n-algebraicsystemoverSn×n
x=M xx+M x+M (2)
p,p2 p,p p,ε
andU isasolutionoftheSn×n-linearsystemoverSn
z =M z+M Az+M z. (3)
p,p2 p,p2 p,p
Theorem4. LetS beacompletestarsemiringandM bearoc-matrix.Then(M∗)
p,ε
isasolutionoftheS′n×n-algebraicsystem(2).IfS isacontinuousstarsemiring,then
(M∗) istheleastsolutionof (2).
p,ε
Proof. Weobtain,byTheorem1
M (M∗) (M∗) +M (M∗) +M
p,p2 p,ε p,ε p,p p,ε p,ε
=M (M∗) +M (M∗) +M
p,p2 p2,ε p,p p,ε p,ε
=(MM∗) =(M+) =(M∗) .
p,ε p,ε p,ε
This proves the first sentence of our theorem. The second sentence of Theorem 4 is
provedbyTheorem6.4ofKuich[14].
Theorem5. LetS beacompletestar-omegasemiringandM bearoc-matrix.Then
((M∗) ,(Mω,k) ),
p,ε p
isasolutionoftheS′n×n-algebraicsystem(1),foreach 0≤k ≤n.
Proof. Let0≤k ≤n,andconsidertheSn×n-linearsystem
z =(M +M (M∗) +M )z.
p,p2 p,p2 p,ε p,p
ByCorollary3,(Mω,k) isasolutionofthissystem.Hence,byTheorem5.6.1ofE´sik,
p
Kuich[10](seetheremarkabove)andTheorem4,((M∗) ,(Mω,k) )isasolutionof
p,ε p
thesystem(1).
Observethat,ifSisacontinuousstar-omegasemiring,thentheSn×n-linearsystem
in the proof of Theorem 5 is in fact an Alg(S′)n×n-linear system (see Kuich [14, p.
623]).
Theorem6. LetS beacompletestar-omegasemiringandletC = (n,I,M,P,k)be
an S′-ω-roc-automaton.Then (kCk,((M∗) ,(Mω,k) )) is a solution of the S′n×n-
p,ε p
algebraicsystem
y =IyP , y =M yy+M y+M (4)
0 p,p2 p,p p,ε
overthecompletesemiring-semimodulepair(Sn×n,Sn).
Proof. ByTheorem5,((M∗) ,(Mω,k) )isasolutionofthesecondequation.Since
p,ε p
I((M∗) ,(Mω,k) )P =(I(M∗) P,I(Mω,k) )=kCk,
p,ε p p,ε p
(kCk,((M∗) ,(Mω,k) ))isasolutionofthegivenS′n×n-algebraicsystem.
p,ε p
LetnowS beacompletestar-omegasemiringandΣ beanalphabet.ThenbyThe-
orem 5.5.5 of E´sik, Kuich [10], (S ≪ Σ∗ ≫,S ≪Σω ≫) is a complete semiring-
semimodulepair.
Let C = (n,I,M,P,k) be an ShΣ ∪ {ε}i-ω-roc automaton. Consider the al-
gebraic system (4) over the complete semiring-semimodule pair ((S ≪Σ∗≫)n×n,
(S ≪Σω≫)n) and the mixed algebraic system (5) over ((S ≪Σ∗ ≫)n×n,
(S ≪Σω ≫)n)inducedby(4)
x =IxP, x=M xx+M x+M ,
0 p,p2 p,p p,ε
(5)
z =Iz, z =M z+M xz+M z.
0 p,p2 p,p2 p,p
Then,byTheorem6,
(I(M∗) P,(M∗) ,I(Mω,k) ,(Mω,k) ), 0≤k ≤n,
p,ε p,ε p p
is a solution of (5). It is called solution of order k. Hence, we have proved the next
theorem.
Theorem7. LetS beacompletestar-omegasemiringandC = (n,I,M,P,k)bean
ShΣ∪{ε}i-ω-rocautomaton.Then(I(M∗) P,(M∗) ,I(Mω,k) ,(Mω,k) ), 0≤
p,ε p,ε p p
k ≤n,isasolutionofthemixedalgebraicsystem(5). ⊓⊔