Abstract
Recently, we have shown that satisfiability for the temporal logic E C T L ∗ with local constraints over (ℤ, <, =) is decidable using a new technique (Carapelle et al., 2013). This approach reduces the satisfiability problem of E C T L ∗ with constraints over some structure \(\mathcal {A}\) (or class of structures) to the problem whether \(\mathcal {A}\) has a certain model theoretic property that we called EHD (for “existence of homomorphisms is definable”). Here we apply this approach to structures that are tree-like and obtain several results. We show that satisfiability of E C T L ∗ with constraints is decidable over (i) semi-linear orders (i.e., tree-like structures where branches form arbitrary linear orders), (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching trees of height h for each fixed \(h\in \mathbb {N}\). We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fraïssé-games for W M S O + B (weak M S O with the bounding quantifier) and use them to show that the infinite (order) tree does not have the EHD-property. As a consequence, our technique cannot be used to establish whether satisfiability of E C T L ∗ with constraints over the infinite (order) tree is decidable. A preliminary version of this paper has appeared as (Carapelle et al., 2015).
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Notes
A structure \(\mathcal {U}\) is universal for a class Γ if there is a homomorphic embedding of every structure from Γ into \(\mathcal {U}\) and \(\mathcal {U}\) belongs to Γ.
We call (A, <, ⊥ ) a graph to emphasize that here the binary relation symbols < and ⊥ can have arbitrary interpretations and they need not be a partial order and its incomparability relation. We can instead see them as two different kinds of edges in an arbitrary graph.
For the ease of presentation we assume that \(\mathcal {A}\) and \(\mathcal {B}\) are infinite structures.
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Acknowledgements
We thank Manfred Droste for fruitful discussions on universal structures and semi-linear orders.
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This work is supported by the DFG Research Training Group 1763 (QuantLA) and the DFG research project GELO.
Appendix: A universal semi-linear order
Appendix: A universal semi-linear order
In the following we define a semi-linear order which is universal for the class of all countable semi-linear orders. The fact that this order is universal is known to the experts in the field of semi-linear orders. Unfortunately, to our best knowledge there is no proof of this fact in the literature. Hence we provide a proof for this result.
Definition 31
Let \(\mathcal {U}=(U, <, \mathrel {\bot })\) be the countable semi-linear order with:
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\(U = (\mathbb {N}\mathbb {Q})^{*}\) (the set of all finite sequences n 1 q 1⋯n k q k with k ≥ 0, \(n_{1}, \ldots , n_{k} \in \mathbb {N}\) and \(q_{1}, \ldots , q_{k} \in \mathbb {Q}\)),
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< is the strict order induced by n 1 p 1 n 2 p 2⋯n k p k ≤ m 1 q 1 m 2 q 2⋯m l q l iff
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k ≤ l, n i = m i for all 1 ≤ i ≤ k, p i = q i for all 1 ≤ i ≤ k − 1 and p k ≤ q k , and
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⊥ = ⊥<.
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We call \(\mathcal {U}\) the universal countable semi-linear order.
Note that Droste [13] has already studied this and similar orders.
For u = n 1 p 1 n 2 p 2⋯n k p k ∈ U (with k ≥ 1) and \(q\in \mathbb {Q}\), we define
We say that a countable semi-linear order (A, <, ⊥ ) is closed under finite infima if for each finite set \(S \subseteq A\) the linear order {a ∈ A∣a ≤ s for alls ∈ S} has a maximal element, which is denoted by \(\inf (S)\). Let \(E=(a_{i})_{i\in \mathbb {N}}\) be a repetition-free enumeration of A. We say E is closed under infima if for each initial subset \(A_{i}=\{a_{1}, a_{2}, \dots , a_{i}\}\) and each \(S\subseteq A_{i}\) we have \(\inf (S) \in A_{i}\).
Lemma 32
Let \(\mathcal {A} = (A, < ,\mathrel {\bot })\) be a countable semi-linear order. There is a countable semi-linear order \(\mathcal {B}\) that is closed under finite infima and an injective homomorphism from \(\mathcal {A}\) to \(\mathcal {B}\).
Proof
For a nonempty subset \(S\subseteq A\) we set ↓S = {a ∈ A∣∀s ∈ S(a ≤ s)}. Let \(\bar A\) be the set of finite nonempty subsets of A, which is obviously countable. We define an equivalence on \(\bar A\) by setting S ∼ T iff ↓S = ↓T. For all \(S\in \bar A\), [S] denotes its equivalence class. Let B be the set of all equivalence classes. We define an order ⊏ on B by [S] ⊏ [T] if and only if ↓S ⊊ ↓T.
We claim that \(\mathcal {B} = (B, \sqsubset , \mathrel {\bot }_{\sqsubset })\) is a semi-linear order that is closed under finite infima and that the map ϕ given by ϕ(a) ↦ [{a}] is an injective homomorphism from \(\mathcal {A}\) to \(\mathcal {B}\).
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\(\mathcal {B}\) is obviously a partial order. Moreover, note that ↓S is a linear and downwards closed suborder of \(\mathcal {A}\) for every nonempty finite set \(S \subseteq A\). In order to show that \(\mathcal {B}\) is semi-linear, assume that \([S_{1}] \sqsubset [S]\) and \([S_{2}] \sqsubset [S]\), i.e., \({\downarrow } S_{1} \subsetneq {\downarrow } S\) and \({\downarrow } S_{2} \subsetneq {\downarrow } S\). Thus, all elements from ↓S 1 and all elements from ↓S 2 are comparable. Since both sets are downwards closed, this directly implies that either [S 1]=[S 2], \([S_{1}] \sqsubset [S_{2}]\) or \([S_{2}] \sqsubset [S_{1}]\).
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Let us show that \(\mathcal {B}\) is closed under finite infima: Let S, S 1, … , S n be finite nonempty subsets of A and assume that \([S] \sqsubseteq [S_{i}]\) for all 1 ≤ i ≤ n. Thus, \({\downarrow } S \subseteq {\downarrow } S_{i}\). Hence, \({\downarrow } S \subseteq \bigcap _{i=1}^{n} {\downarrow } S_{i} = {\downarrow } \bigcup _{i=1}^{n} S_{i}\). Since \({\downarrow } \bigcup _{i=1}^{n} S_{i} \subseteq {\downarrow } S_{i}\) for all 1 ≤ i ≤ n, \([\bigcup _{i=1}^{n} S_{i}] = \inf (\{ [S_{1}], \ldots , [S_{n}] \})\).
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For a, b ∈ A with a ≠ b we have b ∉ ↓{a} or a ∉ ↓{b}. Thus, ϕ is an injective map from A to B. Moreover, a < b implies \({\downarrow }\{a\} \subsetneq {\downarrow }\{b\}\), i.e., \(\phi (a) \sqsubset \phi (b)\). Similarly, a ⊥ b implies a ∉ ↓{b} and b ∉ ↓{a}, i.e., \(\phi (a) \mathrel {\bot }_{\sqsubset } \phi (b)\).
□
Lemma 33
Let \(\mathcal {A} =(A, < , \mathrel {\bot })\) be a countable semi-linear order that is closed under finite infima. There is a repetition-free enumeration of \(\mathcal {A}\) , which is closed under infima.
Proof
Fix an arbitrary repetition-free enumeration \((a_{i})_{i\in \mathbb {N}}\) of A. Assume that we have constructed a sequence \(b_{1}, b_{2}, \dots , b_{i}\) such that \(B_{j} = \{b_{1}, b_{2}, \dots , b_{j}\}\) is closed under infima for every j ≤ i. Let k ∈ ℕ be minimal with a k ∉ B i . Let \(b^{\prime }_{1} < b^{\prime }_{2} < {\dots } b^{\prime }_{m} < a_{k}\) be the list of all infima of the form \(\inf (S\cup \{a_{k}\})\) for \(S\subseteq B_{i}\) that are not contained in B i . This list is indeed linearly ordered by < since all elements in the list are bounded by a k . Now set \(b_{i+l} = b^{\prime }_{l}\) for all 1 ≤ l ≤ m and set b i + m+1 = a k . The resulting sequence b 1, … , b i + m + 1 contains a k and \(B_{j} = \{b_{1}, b_{2}, \dots , b_{j}\}\) is closed under infima for every j ≤ i + m + 1. This can be easily shown using the fact that \(\inf (X \cup \{ \inf (Y)\}) = \inf (X \cup Y)\) for all sets X and Y.
Repeating this construction leads to an enumeration \((b_{i})_{i\in \mathbb {N}}\) of A with the desired property. □
Lemma 34
Let \(\mathcal {A}=(A, \sqsubset , \mathrel {\bot }_{\sqsubset })\) be a countable semi-linear order. There exists an injective homomorphism from \(\mathcal {A}\) into \(\mathcal {U}\).
Proof
Due to Lemma 32 and 33, we may assume that \(\mathcal {A}\) is closed under finite infima and that \((a_{i})_{i\in \mathbb {N}}\) is a repetition-free enumeration of A which is closed under finite infima. Set \(A_{i}=\{a_{1}, \dots , a_{i}\}\) for i ≥ 1. Inductively, we construct injective homomorphisms ϕ i :A i → U (i ≥ 1) such that
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1.
ϕ i+1 extends ϕ i , and
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2.
for all \(u = n_{1}p_{1}n_{2}p_{2}{\dots } n_{k}p_{k} \in {\mathsf {im}}(\phi _{i})\) and all 1 ≤ j ≤ k we have \(p_{j}\in \frac {1}{2^{i}}\mathbb {Z}\).
Define ϕ 1:A 1 → U by \(\phi _{1}(a_{1}) = 00 \in \mathbb {N}\mathbb {Q}\). Assume that ϕ i has already been constructed. We distinguish two cases.
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1.
If there is some a ∈ A i with \(a_{i+1} \sqsubset a\) let \(u = \inf \{a \in A_{i}\mid a_{i+1} \sqsubset a\}\). Note that \(a_{i+1} \sqsubseteq u\). Since the enumeration is closed under infima, we have u ∈ A i (and thus \(a_{i+1} \sqsubset u\)) and we can define \(\phi _{i+1}(a_{i+1}) = \phi _{i}(u)+ (\frac {-1}{2^{i+1}})\), where we add according to (8). Note that ϕ i+1(a i+1) < ϕ i (u) = ϕ i+1(u). In order to prove that this defines a homomorphism, we distinguish the following cases:
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(a)
If \(a_{i+1} \sqsubset a\) for some a ∈ A i then u ⊑ a. Hence, ϕ i+1(a i+1) < ϕ i+1(u) = ϕ i (u) ≤ ϕ i (a) = ϕ i+1(a) as desired.
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(b)
If \(a \sqsubset a_{i+1}\) for some a ∈ A i , then a ⊏ u. Hence, ϕ i+1(a) = ϕ i (a) < ϕ i (u). Since ϕ i uses only rationals from \(\frac {1}{2^{i}}\mathbb {Z}\), we conclude that \(\phi _{i+1}(a) \leq \phi _{i}(u) + \frac {-1}{2^{i}} < \phi _{i+1}(a_{i+1})\) as desired.
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(c)
If \(a_{i+1} \mathrel {\bot }_{\sqsubset } a\) for some a ∈ A i , then \(a \mathrel {\bot }_{\sqsubset } u\). By induction, ϕ i+1(a) = ϕ i (a) ⊥< ϕ i (u) = ϕ i+1(u). Thus, the assumption ϕ i+1(a) ≤ ϕ i+1(a i+1) leads by transitivity of ≤ to the contradiction ϕ i+1(a) ≤ ϕ i+1(u). Similarly, the assumption ϕ i+1(a) > ϕ i+1(a i+1) yields \(\phi _{i+1}(a) \geq \phi _{i+1}(a_{i+1})+\frac {1}{2^{i+1}} = \phi _{i+1}(u)\). We can conclude that ϕ i+1(a) ⊥< ϕ i+1(a i+1) as desired.
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(a)
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2.
Otherwise, for all j ≤ i we know that \(\inf \{a_{j},a_{i+1}\}\) is strictly below a i+1 and hence belongs to A i (since the enumeration is closed under infima). In particular, the set {a ∈ A i ∣a < a i+1} is not empty. By semi-linearity, \(u=\max \{ a\in A_{i}\mid a< a_{i+1}\}\) is well-defined. Since i m(ϕ i ) is finite, there is some \(n\in \mathbb {N}\) such that φ i (u)n0 is incomparable to all elements from the set φ i (A i ∖{a ∈ A i ∣a ≤ u}). Extending ϕ i by setting ϕ i+1(a i+1) = φ i (u)n0 is easily shown to be a homomorphism.
Finally, the limit of \((\phi _{i})_{i\in \mathbb {N}}\) clearly defines an injective homomorphism from \(\mathcal {A}\) into \(\mathcal {U}\). □
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Carapelle, C., Feng, S., Kartzow, A. et al. Satisfiability of ECTL∗ with Local Tree Constraints. Theory Comput Syst 61, 689–720 (2017). https://doi.org/10.1007/s00224-016-9724-y
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DOI: https://doi.org/10.1007/s00224-016-9724-y