Above, we proved that the qcWMSO-theory of (ω, <) is decidable. This immediately raises the question whether the result extends to full qcMSO, where quantification over all and not just finite sets is allowed. As qcMSO subsumes MSO+U, it follows from results on MSO+U that this is not the case. Before we say more about the undecidability of full qcMSO, we remark that the topological complexity of qcMSO-definable languages on infinite words is very high, which follows from a result from [57] described below.
6.4.1
The Expressive Power of qcMSO on Infinite Words
In contrast to MSO-definable languages, MSO+U-definable languages exhaust all levels of the Borel hierarchy, and go even further. To formulate the subsequent result on the topological complexity of these languages, we first introduce the projective hierarchy and the notion of topological reductions. The projective hierarchy is the hierarchy obtained from the Borel hierar- chy (see Chapter 2.1.2) by projections. Given a set x of infinite words over an alphabet Σ′× Σ, the projection π(x) of x is the set
π(x) ∶= {w ∈ Σω∣ there exists v ∈ Σ′ω s.th. σ(v, w) ∈ x}. Commonly, we use Σ′= ω.
Definition 6.4.1. The projective hierarchy is inductively defined as follows, for ω levels, where Σ is some alphabet.
Σ11(Σ) ∶= {π(x) ∈ Σω ∣ x ∈ B(ω × Σ)} Π1n(Σ) ∶= {x ∣ xC∈ Σ1n}
Σ1n+1(Σ) ∶= {π(x) ∣ x ∈ Π1n(ω × Σ)}
As for the Borel hierarchy, the levels of the projective hierarchy are contained in higher levels, and the hierarchy properly extends the Borel hierarchy.
To define topological reductions, the notion of a continuous function is necessary: A function f∶ Σω→ Σ′ωis continuous if for every open set x∈ Σ′ω,
the inverse image f−1(x) ∈ Σωis open as well. Given two sets x, y, we say
that a continuous function f reduces x to y if f−1(y) = x. If x is reducible to
some level of the Borel or the projective hierarchy, then so does x. Let Σi α be
a level of these hierarchies, and let x be a set. Then x is hard for Σi
αif every
y∈ Σi
αis reducible to x, and is complete for the level if, additionally, x∈ Σiα.
6.4.1.1 qcMSO Reaches up the Projective Hierarchy
It is known that ω-regular languages are Borel sets of low rank (at most Σ0 3),
and by Büchi’s theorem [16], these are exactly the sets definable in classical MSO. However, this does not necessarily hold anymore when quantitative aspects are introduced. It was shown in [57] that for every level in the pro- jective hierarchy there are languages definable in MSO+U that are hard for the respective level. As MSO+U is subsumed by qcMSO, and in particular the model-checking problem for MSO+U on a structure is reducible to the boundedness problem for qcMSO on that structure, it follows that the topo- logical complexity of qcMSO-definable languages also reaches up the whole projective hierarchy.
Theorem 6.4.2 ([57]). Let n ∈ ω. There exist a finite alphabet Bn and a
formula φn ∈ MSO+U such that the language Ln = {w ∈ Bnω ∣ w ⊧ φn} is
Σ1 n-hard.
Corollary 6.4.3. Let n∈ ω. There exist a finite alphabet Bnand a formula
φn ∈ qcMSO such that the set of words Ln = {w ∈ Bnω ∣ w ⊧ φn = ∞} is
Σ1 n-hard.
Note also that the nondeterministic ω-counter automata considered so far (such as ωB-, ωS- and ωBS-automata) define languages that are at most in Σ0
4, and the alternating variants reach only up to Σ12 [57].
Remark 6.4.4. One should note that the result presented above is of a differ- ent form compared to the decidability result for qcWMSO. For the latter, we did not construct an equivalent automaton model which captures all definable languages, but rather focused on the theory, that is, only on sentences of the logic. In contrast to that, the former speaks about the topological complexity of definable languages, that is, of the boundedness problem for formulas with free variables.
While we did not study this question for qcWMSO, there is work on WMSO+U on infinite words, showing that WMSO+U is equivalent to de- terministic max-automata and that emptiness of these is decidable [5]. As every max-automaton is equivalent to a nondeterministic ωBS-automaton [5], WMSO+U-definable languages are at most in Σ0
6.4.2
The Undecidability of qcMSO
Towards undecidability results for qcMSO, we look at undecidability results for MSO+U, as these results entail the respective results for qcMSO. The first undecidability result for MSO+U investigated MSO+U on the infinite binary tree T2= ({0, 1}∗,≺, L, R). In [7], it was proved that the MSO+U-theory of
T2is undecidable under certain set-theoretical assumptions:
Theorem 6.4.5([7]). Assuming ZF+ (V = L), whether T2⊧ φ or not for a
given sentence φ∈ MSO+U is undecidable.
Only very recently, this result was subsumed by the result that MSO+U is already undecidable on the natural numbers with linear order, which can be proved without any further set-theoretical assumptions.
Theorem 6.4.6([8]). The MSO+U-theory of(ω, <) is undecidable.
The main idea in this proof is a reduction from the acceptance problem of 2-counter machines. In this reduction, accepting runs of 2-counter machines are encoded as MSO+U-definable languages of infinite words. The existence of an accepting run can thus be described by an MSO+U-sentence φ, and such a run exists if φ belongs to the MSO+U-theory of(ω, <).
Corollary 6.4.7. Given a qcMSO-sentence φ, it is undecidable whether (ω, <) ⊧ (φ = ∞). Analogously, whether a qcMSO-sentences evaluates to ∞ on T2 is undecidable.
Accordingly, as full qcMSO is undecidable on the natural numbers and thus also on the infinite binary tree while both the weak variants costWMSO and WMSO+U are decidable on T2 [10, 87], we turned to the weak variant
qcWMSO on the infinite binary tree. In [62], an analogous result to Corol- lary 6.3.9 can be found: given a qcWMSO-sentence, it is decidable whether T2⊧ φ = ∞.