A simple adaptation of our proof for the concrete domain Z shows that the negation-closed structure Q = (Q, <,≡,(≡q)q∈Q) has the property EHD(MSO):
It can easily be shown that for any{<,≡,(≡q)q∈Q}-structureA,A= (A, I) Q if and only if
• (A, E) is acyclic, whereE = Id◦I(<)◦Id and Id is the symmetric transitive closure of the equality relation, i.e. (I(≡)∪I(≡)−1)∗,
• there is no (a, b)∈E+ witha∈I(≡p), b∈I(≡q) and q≤p, and
• there is no (a, b)∈Id with a∈I(≡p),b∈I(≡q), and q6=p.
Since these conditions are easily expressible in MSO, it follows that Q-SAT is decidable.
Thanks to the density of the rational numbers there is no need to use the bounding quantifier to bound the number of elements between any two given rational numbers.
Chapter 6
“Tree-Like” Concrete Domains
We have developed the EHD-method with the initial intention to solve the sat-isfiability problem of CECTL∗ (and CCTL∗) with constraints over the integers.
But the general nature of this method raised the question whether this could be successfully applied to other interesting concrete domains. The complete infinite binary treeT2, i.e. the set of finite words over the alphabet{0,1}, or the infinitely branching infinite treeT∞, equipped with the prefix relation<seemed to be good candidates for domains, and the problem was also mentioned in [19]. Imagine to want to describe all those{<}-structures which allow a homomorphism intoT∞. At first glance it would seem to be enough to avoid cycles and bound the number of predecessors of each element to obtain our purpose, and both these properties can be expressed using boolean combinations of MSO and WMSO+B sentences.
But recall that, aside theEHD-property, a second condition is necessary to apply the EHD-method: negation closure. To obtain this, we need to consider both T∞ and T2 as structures over the extended signature {<,≡,⊥}, where≡ is the equality relation and ⊥ contains all pairs of incomparable nodes. Difficulties arise once we add the “incomparability” relation: It turns out that MSO and WMSO+B are not sufficient to distinguish between those {<,≡,⊥}-structures which allow homomorphism to T∞ and those which do not.
Nevertheless, the study of this case gave rise to two results. Firstly, we prove that the infinitely branching infinite tree T∞ does not enjoy the property EHD(BMW) and, since both T∞ T2 and T2 T∞, this implies that the binary tree does not either.
Theorem 6.1. There is noBool(MSO,WMSO+B)-sentenceψsuch that for every countable structureA (over the signature{<,⊥,≡}) we have: A |=ψif and only if there is a homomorphism from A toT∞.
To obtain this result we develop an Ehrenfeucht-Fra¨ıss´e-game for WMSO+B
and use it to show that the logicBMWcannot distinguish between two structures, one which allows homomorphism intoT∞ and one which does not.
Remark 6.2. We want to remark that the fact that the EHD-method is not ap-plicable to the case of the binary tree does not imply the undecidability of the satisfiability problem for temporal logics with constraints over such structure. In fact there have been some interesting developments in recent work.
In [28], Kartzow and Weidner prove PSPACE-completeness of the satisfiabil-ity problem for CLTL with constraints over trees, enriched with lexicographic ordering, using an automata-theoretic approach.
At the same time Demri and Deters prove decidability of both CLTL and CCTL∗with constraints over the tree, enriched with the ability to compare lengths of longest common prefixes [18]. They also establishPSPACE-completeness in the linear time case.
The result is obtained by a reduction to satisfiability of CLTL and CCTL∗ with constraints over the domain (N,≡, <,(≡a)a∈N), where the linear time case is decidable by [19], and the corresponding result for the branching time setting is established in [11], using the EHD-method.
The idea behind the reduction is to express a prefix constraintx < y(wherex andyare interpreted as words fromN∗) using the equivalent formula clen(x, y) = clen(x, x), which states that the length of the longest common prefix of x and y is the same as the length ofx. One can successively translate these formulas using constraints on the natural numbers by associating to each term clen(x, y) a register variable rx,y and asking thatrx,y =rx,x. In [18] the authors individuate conditions which ensure that a valuation for the register variables rx,y can be compiled into a correct assignment for the string variablesx and y, and are able to express such conditions in CLTL and CCTL∗. In the branching time case, one has to additionally take care of the correct propagation of the values from the register variables in the different branches of the T-Kripke Structures. It is essential here that CCTL∗ has the bounded tree-model property ([25]): From a given formula ϕ we can compute a number d such that ϕ is satisfiable if and only if it admits a d-tree model. Then the issue is solved by creating d copies of each rx,y and regulating their behavior with CCTL∗ formulas (see Lemma 10 from [18]). Since also CECTL∗ has the bounded tree-model property, the same idea should work for this logic. BeingCCTL∗a fragment ofCECTL∗, the formulas from Lemma 10 in [18] would guarantee a correct reduction also in this case.
As a second main result of this chapter, we identify three classes of “tree-like”
structures which do enjoy the EHD-property, namely semi-linear orders, ordinal trees and infinitely branching trees of height h for each fixedh. In the following section we introduce such classes and prove that they are negation-closed and enjoy the property EHD(L) in the sense of Section 4.2.1, where L is eitherMSO
orWMSO, depending on the class. This implies:
Theorem 6.3. ∆-satisfiability (see Def. 4.18) of CECTL∗-formulas is decidable when ∆is one of the following:
(1) the class of all semi-linear orders, (2) the class of all ordinal trees, and
(3) for each h∈N, the class of all order trees of height h.
We would like to remark that for this result (in particular to establish the EHD-property), we do not need to assume that the domain of the considered structures is countable. For example, in the case of semi-linear orders, we can find a WMSO-sentence ϕ such that for all {<,⊥}-structures A, may they be countable or not, A Bfor some B ∈Γ if and only if A |=ϕ. This does not in fact have any effect on the decidability procedure, as the constraint graphs that we obtain through the EHD-method (for which we need to check whether they satisfy ϕ) have countable domain. Nonetheless what we present here is a more general result.
6.1 “Tree-like” Structures
We now introduce the structures we will be dealing with in this chapter.
Definition 6.4. A semi-linear order (in Wolk’s work [43] simply a tree) is a partial order P = (P, <) with the additional property that for all p ∈ P the suborder induced by{p0 ∈P |p0≤p}forms a linear order.
This property is equivalent to the one formulated by Wolk [43]: Given incom-parable elementsp1, p2 ∈P, there is no q ∈P such that p1 < q and p2 < q, i.e., two incomparable elements cannot have a common descendant. Clearly all trees (in the usual sense) satisfy this property, but not vice-versa.
Definition 6.5. We call a semi-linear orderP = (P, <) anordinal forestif for all p∈P the linear suborder induced by{p0∈P |p0 ≤p}is an ordinal.
We call P aforestif if for all p∈P the linear suborder induced by{p0 ∈P | p0 ≤p} is finite.
A forest (ordinal forest) is atree (ordinal tree) if it has a unique minimal element.
A forestF of heighth(forh∈N) is a forest that contains a linear suborder with h+ 1 many elements but no linear suborder with h+ 2 elements. We say that an elementx∈P isat leveliif|{y∈P |y < x}|=i. Thus, every minimal
Given a partial order (P, <), we denote by⊥<theincomparability relation defined byp⊥<q iffp6=q and neitherp < q norq < phold. Given a {<,⊥, ≡}-structureP = (P, <,⊥,≡) such that (P, <) is a semi-linear order (resp., ordinal tree, tree of heighth),≡is the equality relation onP, and⊥=⊥<, then we also say that P is a semi-linear order (resp. ordinal tree, tree of height h).
Proposition 6.6. For any class ∆ of {<,⊥,≡}-structures such that in every A ∈∆
(i) <is interpreted as a strict partial order,
(ii) ⊥is interpreted as the incomparability with respect to<(i.e.,⊥=⊥<), and (iii) ≡is the equality relation,
∆is negation-closed. In particular, the class of all semi-linear orders and all its subclasses are negation-closed.
Proof. For every A ∈∆ the following equalities hold, where Ais the universe of A:
(A2\<) ={(x, y)| A |=y < x∨y≡x∨x⊥y} , (A2\ ⊥) ={(x, y)| A |=x < y∨x≡y∨y < x} , (A2\ ≡) ={(x, y)| A |=x < y∨x⊥y∨y < x} .
Note that for this it is crucial that we add the incomparability relation⊥.
Remark 6.7. Recall the definition of the∼-quotient ˜Aof a structureA(Def.5.6) and the results of Lemma5.8, establishing a connection between theEHD-property of a structure and the EHD-property of its∼-quotient. It is not hard to see that this result applies also in the setting of ∆-satisfiability, where ∆ is the class of semi-linear orders (ordinal trees, trees of fixed height). Therefore, in order to prove Theorem 6.3 it is enough to show that the class of all semi-linear orders (and its subclasses) seen as {<,⊥}-structures have the property EHD(L) for a suitableL.