As we observed, the restriction to finitely branching trees essentially affects the horizontal dimension of trees, making it sufficiently coarse to identifyWMSO-expressivity andWFMSO-expressivity. Both these logics turn out to be essentially weaker thanMSOin expressing properties of thevertical dimensionon trees, such as the language
LFppresented in proposition 5.1. Since this is the only ‘relevant’ dimension of the finitely branching case,WFMSO
andWMSOare strictly weaker thanMSOon finitely branching trees, as shown in proposition 5.5.
In this section we observe how this landscape of connections betweenWMSO,WFMSOandMSOis affected by considering the more general setting of trees with arbitrary branching degree.
The first observation is that the relation betweenWFMSOandMSOremains unaltered.
Proposition 5.6. The class of WFMSO-definable tree languages is strictly included in the class of MSO-definable tree languages.
Proof This is an immediate consequence of proposition 5.3, theorem 4.14 and remark 2.32. ◻ The weakness of WFMSO on the vertical dimension is somehow reflected by the semantics of WFMSO
quantifiers: a well-closed set of nodes is limited on the vertical dimension, being included in a well-founded tree. However, it has no limitations of cardinality on the horizontal dimension, meaning thatMSO-quantification and
WFMSO-quantification do not differ much in this respect.
The deeper reason underlying this observation is thatMSOitself is a rather coarse logic, once it comes to specify properties on the horizontal dimension of trees. In fact,MSOcannot even distinguish between trees with finite or infinite branching degree. We indicate this phenomenon as theFinite Branching PropertyofMSO. Its significance is best explained through the automata-theoretic perspective, as shown by the next proposition.
Proposition 5.7(Finite Branching Property). LetAbe an MSO-automaton. If L(A)is non-empty then there is a finitely branching treeTin L(A).
Proof LetA= ⟨A,aI,∆,Ω⟩be anMSO-automaton andTa tree that is accepted byA. By proposition 2.14 we
can assumeAto be non-deterministic. Let f be a winning strategy for∃inG = A(A,T)@(aI,sI), which we can
assume to be full and functional by proposition 2.36 and 2.36. LetTf andπ2f ∶Tf →T be respectively the tree
representation of f and its projection function. As observed in remark 2.35, sincef is full and functional thenπ2f is 1-1 and onto.
The main idea of the proof is that we can pruneTf until we get a finitely branching subtreeT′f ofTf. We can
do it in such a way thatT′f is the tree representation of a winning strategy for∃inG′= A(A,T′)@(aI,sI), withT′
a finitely branching subtree ofTinduced byT′f itself. This implies thatT′is inL(A).
We proceed with the formal part of the proof. We define a treeT′f by induction as follows:
1. The root ofT′f is the root ofTf, i.e. the position(aI,sI).
2. Suppose that(a,s)is a node ofT′f of heighti. By inductive hypothesis the position(a,s)is also a node of
Tf of heighti, and the label of(a,s)inT′f is the label of(a,s)inTf.
Letmf ∶A→ ℘σR(s)be the marking provided by f from position(a,s)andϕ∈SBF+(A)be a disjunct of ∆(a,σC(s)) ∈SLatt(SBF+(A)), such that(σR(s),m) ⊧ϕ. LetSϕ= {t1, . . . ,tk}be the set of nodes inσR(s) witnessing the existential part ofϕ. IfSϕis empty, that is,k=0, then at least we know that there is a node
inT′f, whereb∀is such thattis inm(b∀). Otherwise,Sϕis non-empty and we define the successors of(a,s) inT′f to be the elements of the following set.
{(b,t) ∣t∈m(b)∩Sϕ} (5.4)
Observe that this is a finite subset of{(b,t) ∣t∈m(b)}. It follows that each successor of(a,s)inT′f is also a
successor of(a,s)inTf and we can maintain our inductive hypothesis at heighti+1.
It should be clear by construction thatT′f is a finitely branching subtree ofTf. LetT′be the subtree ofT
obtained by projectingT′f onT, that is, its carrierT′is given asπ f
2[Tf′]. SinceT′f is finitely branching then also
T′is finitely branching. The proof of the main statement is completed by showing the following claim.
CLAIM13. The treeT′is accepted byA.
PROOF OFCLAIM The key observation of the proof is thatT′f is the tree representation of a winning strategyf′for ∃inG′. The strategy f′is defined from each basic position(a,s) ∈A×T′occurring inT′f as a node. The marking
suggested by∃from position(a,s)is uniquely defined by (5.4). The proof of claim is completed by showing that: 1. f′is a full and surviving strategy for∃inG′;
2. f′is a winning strategy for∃inG′.
In order to prove the first fact, recall that by assumption f is full and surviving strategy for ∃in G. By construction f′suggests the same markings of f on all basic positions inG′, meaning that it is full inG′. Given a position(a,s) ∈Tf′, letσ′R(s)be the set of successors ofsinT′andm′the marking suggested by f′from position
(a,s). By definition of f′in terms ofT′f, there is a disjunctϕ∈SBF+(A)of∆(a,σC(s))such that:
a) ifϕonly consists of an universal part, theA-structure(σ′R(s),m′)has a unique element, witnessing the universal part ofϕ;
b) otherwise,ϕhas a non-empty existential part; in this case, by definition(σ′R(s),m′)contains exactly the witnesses for the existential part ofϕ, meaning that it also verifies (vacuously) the universal part ofϕ. In any of the two cases,m′makesϕtrue inσR(′ s), implying that it is a legitimate move for∃. It follows thatf′is a surviving strategy for∃inG′.
In order to prove the second fact, letπbe an infinite f′-conform match ofG′, with basic positions
Bπ∶= (aI,sI),(a1,s1), . . . ,(an,sn), . . . .
By definition of f′, the sequenceBπis just a branch ofT′f. By construction,T′f is a finitely branching subtree of
Tf with the same root. ThereforeBπis also a branch ofTf, meaning that there is an f-conform match ofGwith
the same basic positions. By assumption the strategy f is winning for∃inG. It follows that the minimum parity
occurring alongπis even. Therefore∃wins the matchπ. ∎
The proof of the claim 5.2 completes the proof of the main statement.
◻
The Finite Branching Property qualifiesMSOas a logic which is not very expressive on the horizontal dimension of trees. This aspect is not revealed if we restrict to finitely branching trees. In the same way, the precise nature of
WMSOexpressiveness is disclosed only if we take trees with arbitrary branching degree into account. As we will show in the sequel, it turns out thatWMSOdoes not have the Finite Branching Property, being able to distinguish between trees with finite and infinite branching degree. It follows thatWMSOis essentially stronger thanMSOin expressing properties of the horizontal dimension of trees.
Proposition 5.8. The class of WMSO-definable tree languages and the class of MSO-definable tree languages are incomparable.
Proof By proposition 5.5 there is a tree language which isMSO-definable but notWMSO-definable. For the converse direction, consider the languageLNfbdefined by putting
LetχNfbbe aWMSO-formula defined by putting
χNfb ∶= ∃x¬∃p∀y(xRy→y∈p). (5.6)
Intuitively,χNfbsays that there is a node∥x∥such that no finite set∥p∥can contain all the successors of∥x∥. It is
easy to see that the tree languageLNfbis defined byχNfb.
Suppose by way of contradiction thatLNfbisMSO-definable. By proposition 2.26 there is anMSO-automaton
Asuch thatL(A) = LNfb. Since we are considering trees with arbitrary branching degree, the languageL(A)is not
empty. Then, by proposition 5.7, there is a finitely branching treeT′inL(A). This also means thatT′is inLNfb,
contradicting the definition ofLNfbas in (5.5). Therefore the languageLNfbis notMSO-definable. ◻ Proposition 5.8 does not only state the incomparability ofMSOandWMSO, but also reveals the nature of their relation. The two logics are in some senseorthogonal:MSOis weaker thanWMSOon the horizontal dimension, whileWMSOis weaker thanMSOon the vertical dimension on trees.
In the sequel we bring further our investigation by comparing the expressive power ofWMSOandWFMSO. As a preliminary observation, it is immediate thatWFMSOis weaker thanWMSOon the horizontal dimension, being a fragment ofMSO. The orthogonal question, namely howWFMSOandWMSOrelate on the vertical dimension, requires a finer analysis. What we have seen so far is that bothWMSOandWFMSOare weaker thanMSOon this respect. What we are going to show is thatWFMSOis still stronger thanWMSOon the vertical dimension, implying that the two logics have incomparable expressive power.