8. ANÁLISIS DOFA
8.2 Evaluación de los factores internos
9.5.3 Estrategias de Porter
For any semigroupSwe introduce the families of recognisable, rational and extended rational subsets ofSand establish a hierarchy for finitely generated semigroups.
The family ofrecognisable subsetsof a semigroupSis introduced in Section . and are specified by finite quotients of S. The families ofrational subsets
andextended rational subsetsare introduced in Section . and Section . re- spectively and rely on a specification by formulas. All the named families are of interest in the theory of computation. It isKleene’s Theoremwhich identi- fies the families of recognisable, rational and extended rational subsets of free semigroups.
We then define relations on semigroups that arerecognisable, rational or
polyrational, these will become the point of focus in Chapter and
Subsets of Semigroups
.
The Syntactic Congruence
We associate with every subsetXof a semigroupSa congruence, thesyntactic congruenceofXinS. This congruence is particularly important for recognis- able subsets introduced in Section ..
Definition ..
LetSbe a semigroup and letX⊂Sbe a subset ofS. Then thesyntactic congruence ofXonSis defined by
s≈S,Xtif and only if∀x, z∈S1xsz∈X⇔xtz∈X.
The syntactic congruence is the largest congruence onSsuch that the quotient semigroup can still separateXfrom its complement.
Theorem ..
LetSbe a semigroup and letXbe a subset ofS. IfS−−−−φ→T is a surjective semi- group morphism with X = Fφ−1 for some F ⊂ T then there exists a morphism T−−−−ψ→S/≈S,X with φψ = π≈S,X, or equivalently the following diagram com- mutes.
S T
S/≈S,X φ
π ψ
Proof. To show the existence ofψ and that the diagram commutes, we have to check the hypothesis of Theorem .., that is we show that kerφ⊂kerπ.
t∈skerφ⇒sφ=tφ (.)
⇒∀x, z∈S1xszφ∈F⇔xtzφ∈F (.)
⇒∀x, z∈S1xsz∈X⇔xtz∈X (.)
.. Recognisable Subsets
. Recognisable Subsets
The notion of recognisable subset can be seen as a generalisation of residual finiteness to subsets, and therefore recognisability is a finiteness condition for subsets of a semigroup.
In Section . we will show that the recognisable subsets of a free semi- group are exactly the regular languages. These are precisely the sets of strings that are the behaviour of a finite A-automaton. Therefore one can think of recognisability as an algebraic characterisation of regular languages.
We define the family RecSofrecognisable subsetsof a semigroupSby the following definition of a recognisable subset ofS.
Definition ..
LetSbe a semigroup. A subsetX ⊆Sisrecognisableif there is a semigroup mor- phism S−−−−φ→T whereT is a finite semigroup and such that X = Fφ−1 for some subsetF⊂T.
Note that without loss of generality we can assumeφin the above definition to be surjective. It also follows directly from the above definition that the empty subset∅and the subsetSofSare recognisable.
We will discuss a few important properties of recognisable subsets of a given semigroupS, first of all their behaviour under morphisms. Recognis- ability is preserved under preimages of morphisms, but not necessarily pre- served under morphisms.
Lemma ..
Let S−−−−φ→S′ be a semigroup morphism. IfY is a recognisable subset ofS′ then
Yφ−1is a recognisable subset ofS.
Proof. LetS−−−−φ→S′be a semigroup morphism and letYbe in RecS′. Then there is a semigroup morphismS′−−−−ψ→T, whereT is finite, such that Y = Fψ−1. It follows thatYφ−1=Fψ−1φ−1soφψrecognisesYφ−1.
Subsets of Semigroups
We show that the family of recognisable subsets of a semigroup forms a Boolean algebra.
Lemma ..
LetSbe a semigroup.
• IfX∈RecSthenS\X∈RecS.
• IfX1, X2 ∈RecSthenX1∪X2∈RecS.
• IfX1, X2 ∈RecSthenX1∩X2∈RecS.
It follows thatRecSis a Boolean algebra.
Proof. LetX∈RecS, which by definition means that there is a surjective semi- group morphismS−−−−φ→TwithTfinite andF⊂Tsuch thatX=Fφ−1. Then
(T\F)φ−1=S\Xand thereforeS\X∈RecS.
LetX1, X2∈RecS. There are surjective morphismsS φ1
−−−−→T1andS φ2 −−−−→T2
withT1andT2 finite and setsF1 ⊂T1 andF2 ⊂ T2such thatX1 = F1φ−11and X2=F2φ−21. Define
φ:S −−−−→ T1×T2, s 7→ (sφ1, sφ2),
and
F={(t1, t2)∈T1×T2|t1 ∈F1ort2 ∈F2}.
Consequentlys∈Fφ−1if and only ifs∈F
1φ−1ors∈F2φ−1soφrecognises X1∪X2.
The fact thatX1∩X2∈RecSfollows by applying DeMorgan’s laws.
The following lemma is known as Ogden’s iteration lemma. It was first proven by William Ogden in [Ogd] for context-free languages. The fol- lowing theorem is an adaption of Ogden’s ideas to regular languages, with additional help from the version and proof given in [Ber]. This lemma is a strengthening of the well-known pumping lemma, or iteration lemma, in
.. Recognisable Subsets
automata theory. We prove it in the context of recognisable subsets of the free semigroup on a finite set. Ogden’s iteration lemma is one of the most important tools in our work in later chapters.
Theorem ..
LetAbe a finite alphabet and letXbe a recognisable subset ofA+. Then there exists a natural numbern0 such that for any elements ∈ A+ and any choiceM ⊂ |s|of
marked positions with|M| ≥n0the elementsadmits a factorisations = xuywith x,uandy∈A∗such that the following holds.
• There is at least one and at mostn0marked positions inu.
• xuiy∈Xfor alli∈Nif and only ifxuy∈X.
Proof. LetXbe a recognisable subset ofA+. This means that there is a mor- phismA+−−−−φ→T withT finite, andF ⊂T such thatX= Fφ−1. Letn0 = |T|
and
w=a1a2. . . ak
be an element of A+ and let M = {i1, i2, . . . , in} ⊂ kbe a set of at least n0
marked positions. Define a factorisation
w=w0w1. . . wn0+1 ofwby w0 =a1. . . ai1−1 w1 =ai1 wj =aij−1+1. . . aijfor2≤j≤n0 wn0+1 =ain0+1. . . an
and elementssj ∈T bys0 =w0φandsj+1=sj(wj+1)φfor1≤j≤n0. Since
|T|=n0there exist two indicesj1andj2such thatsj1=sj2, and therefore for
Subsets of Semigroups
it holds that(xy)φ = (xuiy)φ for alli ∈ N>0 and therefore xy ∈ Xif and
only ifxuiy∈X.
For any X ∈ RecS the canonical morphism S−−−−π→S/≈S,X recognises
X and S/≈S,X is minimal in the sense that it is a quotient of any T where
S−−−−φ→T recognisesX. This also shows that a subset ofS is recognisable if and only if its syntactic quotient is finite.
Theorem ..
LetSbe a semigroup and letX∈RecS. ThenS−−−−π→S/≈S,X recognisesXand for any morphismS−−−−φ→Tthat recognisesXthere exists a morphismT−−−−θ→S/≈S,X
such thatπ=φψ, or equivalently the following diagram commutes.
S T
S/≈S,X φ
π θ
Proof. We first show thatS−−−−π→S/≈S,X recognisesX. For this let Fbe the image ofXunder the relation≈S,X. We show thatX = Fπ−1. Ifx∈ Xthen xπ∈Fand thereforex∈Fπ−1. Conversely, lets ∈Fthen there ist∈ swith
t∈X, and by the definition of≈S,Xthis implies thatsπ−1⊂X.
The rest of the proof follows from Theorem ...
.
Rational Subsets
Rationality also is a finiteness condition on subsets of a semigroup in the sense that there is an inductive specification for each set of this class. Inductive definitions are used in formal logic and recursion theory. More specifically, we will define by induction thesyntaxand thesemanticsof expressions. The syntax is the definition ofrational expressionsandextended rational expressions, and the semantics assign to every rational expression a subset of a semigroup. We call the family of subsets thus defined the family ofrational subsets ofS.
.. Rational Subsets
In the previous section, we stated that that recognisability is an algebraic way to specify regular languages. Rational expressions are the formal logic approach to specify regular languages.
Definition ..
Let X be a set. The setRatExpXof rational expressions over X is inductively defined as follows.
• The expressionλis an element ofRatExpX, • any elementx∈Xis an element ofRatExpX,
• ifαis inRatExpX, then the expressionα+is inRatExpX,
• ifαandβare elements ofRatExpX, then the expression(αβ)is an element of
RatExpX,
• ifαandβare elements ofRatExpX, then the expression(α∪β)is an element ofRatExpX.
Given a semigroupSand a mapX−−−−f→S, we assign to each rational expres- sion a subset ofSby inductively defining the map[[]]f. Usually we will choose
X = S andfto be the identity mapιS. IfSis finitely generated, we can do withXbeing a generating set forS.
Definition ..
LetXbe a set,Sbe a semigroup andX−−−−f→Sbe a map. The map[[]]fis inductively defined as follows. • [[λ]]f=∅, • [[x]]f={xf}, • [[α+]]f= [[α]]+f, • [[α∪β]]f= [[α]]f∪[[β]]f, • [[αβ]]f= [[α]]f[[β]]f.
Subsets of Semigroups
We denote the family of all rational subsets ofS with respect toXandfby RatSf. Note thatfimplicitly defines the setX. IfSis generated byX
f
−−−−→S
andY−−−−g→Sis another generating set forS, then RatSf = RatSg. This can
easily be seen in Definition ...
Rational subsets behave dually to recognisable subsets under morphisms: they are preserved under morphism, but not under taking preimages.
Lemma ..
LetS−−−−φ→S′be a semigroup morphism. IfXis a rational subset ofSthenXφis a rational subset ofT.
Proof. This follows by induction over the definition of rational expressions, the map[[]]and from the fact thatφis a map and a morphism.
Given a surjective semigroup morphismS−−−−φ→T and a rational subsetY of
T we can prove the existence of a rational subset ofXofSsuch thatY=Xφ.
Lemma ..
LetS−−−−φ→T be a surjective semigroup morphism. IfY ∈ RatT then there exists
X∈RatSwithY=Xφ.
Proof. Consider the familyR ⊂TˆwithY ∈Rif and only if there isX ∈RatS
withY =Xφ. We show by induction thatR⊃RatT. • ∅ ∈R, because∅φ=∅.
• Sinceφis surjective, for allt ∈T there iss ∈ Switht= sφ, therefore {t}∈R.
• IfYis an element ofR, then there isX∈RatSwithY=Xφ, and therefore
Y+=X+φis an element ofR.
• IfYandY′ are elements ofR, then there areXandX′in RatSwithY = XφandY′ =X′φ, thereforeY∪Y′ = (X∪X′)φis an element ofR.
.. Kleene’s Theorem
• IfYandY′ are elements ofR, then there areXandX′ in RatSwithY= XφandY′ =X′φ, thereforeYY′ = (XX′)φis an element ofR.
We also note that there is a natural definition of rational expressions for monoidsMthat include a rational expressionϵwith[[ϵ]]f = εand a rational expressionα∗with[[α∗]]f= [[α]]∗f.
. Kleene’s Theorem
Kleene’s Theorem, named after its discoverer Stephen Kleene [Kle], identifies rational and recognisable subsets of free semigroups. The technique used to prove that RecA+⊂RatA+has been applied in more general settings and is known today as the Kleene-Floyd-Warshall method.
Theorem ..
LetAbe a finite alphabet. ThenRecA+=RatA+.
A semigroup in which Kleene’s theorem holds is called a Kleene semi- group. A natural task is now to characterise the class of all Kleene semi- groups, which is an open research problem.
Applying Kleene’s theorem and properties of rational and recognisable subsets we can prove the following theorem for finitely generated semigroups, which was proven by McKnight [McK].
Theorem ..
LetSbe a finitely generated semigroup. ThenRecS⊂RatS.
Proof. LetAbe a finite generating set forS. ThenA+−−−−π→Sis surjective and for any recognisable subsetXofSthe preimageXπ−1is a recognisable subset ofA+. By Kleene’s theorem .. the setXπ−1is also a rational subset ofA+
Subsets of Semigroups
For semigroups that are not finitely generated, Theorem .. does not hold. Furthermore, the inclusion RecS⊃RatSdoes not hold in general for finitely generated semigroups. For example in CS(2), which is defined in Section ., the set{ab}+is rational but not recognisable which can be shown by apply- ing an iteration lemma. The following result is known for finitely generated semigroups.
Theorem ..
LetS be a semigroup and let Xbe in RatS. Then there exists a finitely generated subsemigroupT ofSsuch thatX⊂T.
Proof. LetRbe the class of all subsets ofSthat are contained in a finitely gen- erated subsemigroup ofS. We show by induction that RatS⊂R.
• The empty set∅is inR.
• For alls∈S, the singleton set{s}is inR.
• IfX∈RandT ⊂Sa finite subset ofSsuch thatX⊂T+, thenX+⊂T+. • If X ∈ RandY ∈ RandT ⊂ SandU ⊂ Sare finite subsets ofS with
X⊂T+andY⊂U+respectively, thenX∪Y ⊂(T∪U)+
• If X ∈ RandY ∈ RandT ⊂ SandU ⊂ Sare finite subsets ofS with
X⊂T+andY⊂U+respectively, thenXY ⊂(T ∪U)+.
Another consequence of Theorem .. is that in a free semigroup the fam- ily of rational subsets forms a Boolean algebra. This is not true for general semigroups, for consider the monoidM={a}∗×{b, c}∗and the rational sub- setsXandYdefined as follows.
X=[[(a, b)+(ε, c)+]]= { (an, bnck)∈M|n, k > 0 } Y=[[(ε, b)+(a, c)+]]= { (an, bkcn)∈M|n, k > 0 }
.. Extended Rational Subsets
Now
X∩Y={(an, bncn)∈M|n > 0}.
LetM−−−−π→{b, c}be the projection onto the second factor of the direct prod- uct. IfX∩Ywere rational, then(X∩Y)π={bncn∈{b, c}∗ |n∈N}would be rational, which it is not. This can be shown by applying Theorem ...
If we restrict one set to be recognisable, then we can prove a lemma about intersections.
Lemma ..
LetSbe a semigroup. IfX∈RecSandY∈RatSthenX∩Y ∈RatS.
Proof. SinceY ∈RatSby Theorem .. there exists a finitely generated sub- semigroupS′ ofSsuch thatY ∈RatS′. This implies that there is a finite gen- erating setAforS′, that is a surjective morphismA+−−−−π→S′, and by Lemma .. a rational subsetY′ of A+ such thatY′π = Y. Applying Theorem .. yields that Y′ is recognisable. Since by assumption X ∈ RecS, the preim- ageX′ = Xπ−1 is a recognisable subset ofA+. The intersectionX′∩Y′ is a recognisable subset ofA+, and again by Theorem .. rational and therefore
(X′∩Y′)πis a rational subset ofS. Now
( X′∩Y′)π= ( X′∩Yπ−1 ) π=X′π∩Y =X∩Y,
and thereforeX∩Y ∈RatS.
. Extended Rational Subsets
We take the above results as a motivation to define the family ofextended ratio- nal subsetsof a semigroup S. Extended rational subsets are not well-studied in the literature yet. Nothing prevents us from adding intersection and com- plement operations to the basic operations allowed in Definition .. and Definition ... By De Morgan’s laws it would technically suffice to add the
Subsets of Semigroups
complement operation to define the family of extended rational subsets of a given semigroup. We opt to add intersections because we will be interested in intersections in Chapter .
Definition ..
LetXbe a set. The familyERatExpXof extended rational expressionsoverXis inductively defined as follows.
• Forα∈RatExpXthe expressionα∈ERatExpX, • forα∈ERatExpXthe expressionα∈ERatExpX,
• forα, β∈ERatExpXthe expression(α∩β)∈ERatExpX.
The definition of the map [[]]f for extended rational expressions is also straightforward.
Definition ..
LetXbe a set,Sbe a semigroup andX−−−−f→Sbe a map. We define the map[[]]ffor everyα∈ERatExpXinductively as follows.
• [[α]]fforα∈RatExpXis the same as in Definition .. • [[α]]f=S\[[α]]f
• [[α∩β]]f= [[α]]f∩[[β]]f
We call the family of subsets ofSdefined by extended rational expressions
extended rational subsets ofSand denote this family by ERatS.
For extended rational expressionsαwe inductively define some measures of complexity, thedepthd(α),
• d(α) =0ifα∈RatExpX
• d(α) =d(α) +1
.. Extended Rational Subsets
thecomplement complexitycc(α)
• cc(α) =0ifα∈RatExpX
• cc(α) =cc(α) +1
• cc(α∩β) =cc(α) +cc(β), and theintersection complexityic(α)by
• ic(α) =1ifα∈RatExpX
• ic(α) =ic(α)
• ic(α∩β) =ic(α) +ic(β).
For a setX ∈ ERatS, to get well-defined notions of depth, complement complexity and intersection complexity we define the depth d(X), cc(X)and ic(X)ofXto be the minimal d(α), cc(α)and ic(α)among all extended rational expressionsαwith[[α]] =X.