Bisimulation Invariant Monadic-Second Order Logic in the Finite

We consider bisimulation-invariant monadic second-order logic over various classes of finite transition systems. We present several combinatorial characterisations of when the expressive power of this fragment coincides with that of the modal mu-calculus. Using these characterisations we prove for some simple classes of transition systems that this is indeed the case. In particular, we show that, over the class of all finite transition systems with Cantor-Bendixson rank at most k, bisimulation-invariant MSO coincides with L_mu.


Introduction
A characterisation of the bisimulation-invariant fragment of a given classical logic relates this logic to a suitable modal logic. In this way, one obtains a correspondence between a family of classical logics and a family of modal logics. Such characterisation results therefore help with ordering the zoo of logics introduced (on both sides) over the years and with distinguishing between natural and artificial instances of such logics. The study of bisimulation-invariant fragments of classical logics was initiated by a result of van Benthem [2] who proved that the bisimulation-invariant fragment of first-order logic coincides with standard modal logic. Inspired by this work, several other characterisations have been obtained, the most prominent among them being a characterisation of bisimulation-invariant monadic second-order logic by Janin and Walukiewicz [10]. The table below summarises the results known so far.
bisimulation-invariant fragment modal logic reference first-order logic modal logic [2] monadic second-order logic modal µ-calculus [10] monadic path logic CTL * [12,13] weak monadic second-order logic continuous µ-calculus [4] weak chain logic PDL [4] There are also similar characterisations for various variants of bisimulation like guarded bisimulation [1,7] or bisimulation for inquisitive modal logic [5]. Researchers in finite model theory started to investigate to which extent these correspondences also hold when only considering finite structures, that is, whether every formula of a given classical logic that is bisimulation-invariant over the class of all finite transition systems is equivalent, over that class, to the corresponding modal logic. For first-order logic, a corresponding characterisation does indeed hold. Its proof by Rosen [15] uses tools from finite model theory and is very different to the proof by van Benthem. The above mentioned result by Janin and Walukiewicz on bisimulation-invariant monadic second-order logic has so far defied all attempts at a similar transfer to the realm of finite structures. The main reason is that the original proof is based on automata-theoretic techniques and an essential ingredient is a reduction to trees, via the unravelling operation. As this operation produces infinite trees, we cannot use it for formulae that are only bisimulation-invariant over finite transition systems.
In this paper we start a fresh attempt at a finitary version of the result of Janin and Walukiewicz. Instead of automata-theoretic techniques we employ the composition method. For certain classes of very simple, finite transition systems we characterise the bisimulationinvariant fragments of monadic second-order logic over these classes. We hope that some day our techniques can be extended to the general case of all finite structures, but currently there are still a few technical obstacles to overcome.
We start in Section 2 by recalling the needed material on bisimulation and by listing all known results on bisimulation-invariant monadic second-order logic. We also collect some low-hanging fruit by proving two new results concerning (i) finite classes and (ii) the class of all finite trees. Finally, we lay the groundwork for the more involved proofs to follow by characterising bisimulation-invariance in terms of a combinatorial property called the unravelling property. In Section 3, we collect some tools from logic we will need. The emphasis in on so-called composition lemmas. Nothing in this section is new.
Finally we start in Section 4 in earnest by developing the technical machinery our proofs are based on. Sections 5 and 6 contain our first two applications: characterisations of bisimulation-invariant monadic second-order logic over (i) the class of lassos and (ii) certain classes of what we call hierarchical lassos. The former is already known and simply serves as an example of our techniques and to fix our notation for the second result, which is new.
Before presenting our last characterisation result, we develop in Section 7 some additional technical tools that allow us to reduce one characterisation result to another. This is then applied in Section 8 to the most complex of our results. We characterise bisimulation-invariant monadic second-order logic over the class of all transition systems of a given Cantor-Bendixson rank.

Bisimulation-invariance
We consider two logics in this paper: (i) monadic second-order logic (MSO), which is the extension of first-order logic by set variables and set quantifiers, and (ii) the modal µ-calculus (L µ ), which is the fixed-point extension of modal logic. A detailed introduction can be found, e.g., in [8]. Concerning the µ-calculus and bisimulation, we also refer to the survey [17]. Transition systems are directed graphs where the edges are labelled by elements of a given set A and vertices by elements of some set I. Formally, we consider a transition system as a structure of the form S = S, (E a ) a∈A , (P i ) i∈I , s 0 where the E a ⊆ S × S are (disjoint) binary edge relations, the P i ⊆ S are (disjoint) unary predicates, and s 0 is the initial state.
We write S, s to denote the transition system obtained from S by declaring s to be the initial state. A central notion in modal logic is bisimilarity since modal logics cannot distinguish between bisimilar systems. Definition 2.1. Let S and T be transition systems. (a) A bisimulation between S and T is a binary relation Z ⊆ S × T such that all pairs s, t ∈ Z satisfy the following conditions.
(b) Let s 0 and t 0 be the initial states of, respectively, S and T. We say that S and T are bisimilar if there exists a bisimulation Z between S and T with s 0 , t 0 ∈ Z. We denote this fact by S ∼ T.
(c) We denote by U(S) the unravelling of a transition system S.
The next two observations show that the unravelling operation is closely related to bisimilarity. In fact, having the same unravelling can be seen as a poor man's version of bisimilarity.
Lemma 2.2. Let S and T be transition systems.
Proof. For (a), note that graph of the canonical homomorphism U(S) → S forms a bisimulation. (b) follows by (a) since U(S) ∼ S ∼ T ∼ U(T) .
As already mentioned modal logics cannot distinguish between bisimilar systems. They are bisimulation-invariant in the sense of the following definition. (b) We say that, over the class C, bisimulation-invariant MSO coincides with L µ if, for every MSO-formula ϕ that is bisimulation-invariant over the class C, there exists an L µ -formula ψ such that A straightforward induction over the structure of formulae shows that every L µ -formula is bisimulation-invariant over all transition systems. Hence, bisimulation-invariance is a necessary condition for an MSO-formula to be equivalent to an L µ -formula.
The following characterisations of bisimulation-invariant MSO have been obtained so far. We start with the result of Janin and Walukiewicz. Theorem 2.4 (Janin, Walukiewicz [10]). Over the class of all transition systems, bisimulation-invariant MSO coincides with L µ .
The main step in this theorem's proof consists in proving the following variant, which implies the case of all structures by a simple reduction.

xx:4
Bisimulation Invariant MSO in the Finite Theorem 2.5 (Janin, Walukiewicz). Over the class of all trees, bisimulation-invariant MSO coincides with L µ .
There have already been two attempts at a finitary version. The first one is by Hirsch who considered the class of all regular trees, i.e., unravellings of finite transition systems. The proof is based on the fact that a formula is bisimulation-invariant over all trees if, and only if, it is bisimulation-invariant over regular trees. Theorem 2.6 (Hirsch [9]). Over the class of all regular trees, bisimulation-invariant MSO coincides with L µ .
The second result is by Dawar and Janin who considered the class of finite lassos, i.e., finite paths leading to a cycle. We will present a proof in Section 5 below. Theorem 2.7 (Dawar, Janin [6]). Over the class of all lassos, bisimulation-invariant MSO coincides with L µ .
In this paper, we will extend this last result to larger classes. We start with two easy observations. The first one is nearly trivial. Proof. As any two non-bisimilar, finite transition systems can be distinguished by an L µformula (in fact, even by a formula of modal logic, see e.g. [17]), we can pick, for every pair of non-bisimilar transition systems S, T ∈ C, an L µ -formula satisfied by S, but not by T. Let Θ be the resulting set of formulae. The Θ-theory of a transition system S ∈ C is By choice of Θ it follows that Given an MSO-formula ϕ that is bisimulation-invariant over C, we set ψ := T Θ (S) S ∈ C , S |= ϕ .
(As Θ is finite, this is a finite disjunction of finite conjunctions.) Then ψ ∈ L µ and, for each S ∈ C, it follows that The second observation is much deeper, but fortunately nearly all of the work has already been done by Janin and Walukiewicz. Proof. We adapt the proof of Janin and Walukiewicz [10] which roughly goes as follows. For a transition system M, let M be the tree obtained from the unravelling U(M) by duplicating every subtree infinitely many times. Given an MSO-formula ϕ, one can use automaton-theoretic techniques to construct an L µ -formula ϕ ∨ such that This is the contents of Lemma 12 of [10]. Now the claim follows by bisimulation-invariance since To make this proof work for finite trees, it is sufficient to modify the construction of the system M. A closer look at the proof of Lemma 12 reveals that it does not require infinite branching for M. It is enough if we duplicate each subtree sufficiently often, where the exact number of copies only depends on the formula ϕ. (Note that there is a remark after Corollary 14 of [10] indicating that Janin and Walukiewicz were already aware of this fact.) As a preparation for the more involved characterisation results to follow, we simplify our task by introducing the following property of a class C of transition systems, which will turn out to be equivalent to having a characterisation result for bisimulation-invariant MSO over C. Definition 2.10. We say that a class C of transition systems has the unravelling property if, for every MSO-formula ϕ that is bisimulation-invariant over C, there exists an MSO-formulaφ that is bisimulation-invariant over trees such that Using Theorem 2.5, we can reformulate this definition as follows. This version will be our main tool to prove characterisation results for bisimulation-invariant MSO: it is sufficient to prove that the given class has the unravelling property. Proof. (⇒) Suppose that C has the unravelling property and let ϕ ∈ MSO be bisimulationinvariant over C. Then there exists an MSO-formulaφ that is bisimulation-invariant over trees and satisfies We can use Theorem 2.5 to find an L µ -formula ψ such that For S ∈ C, it follows by bisimulation-invariance of L µ that (⇐) Suppose that, over C, bisimulation-invariant MSO coincides with L µ . To show that C has the unravelling property, consider an MSO-formula ϕ that is bisimulation-invariant over C. By assumption, there exists an L µ -formula ψ such that Letφ be an MSO-formula that is equivalent to ψ over every transition system. As ψ is bisimulation-invariant over all transition systems, the formulaφ is bisimulation-invariant over trees and we have Let us also note the following result, which allows us to extend the unravelling property from a given class to certain superclasses. Lemma 2.12. Let C 0 ⊆ C be classes such that every system in C is bisimilar to one in C 0 . If C 0 has the unravelling property, then so does C.
Proof. Let ϕ be bisimulation-invariant over C. Then it is also bisimulation-invariant over C 0 and we can find a formulaφ that is bisimulation-invariant over trees such that We claim that this formula has the desired properties. Thus, consider a system S ∈ C.
By assumption, we have S ∼ S 0 for some S 0 ∈ C 0 . By Lemma 2.2, it follows that U(S) ∼ U(S 0 ). Consequently, by bisimulation-invariance of ϕ over C and ofφ over trees, we have

Composition lemmas
We have mentioned above that automata-theoretic methods have so far been unsuccessful at attacking the finite version of the Janin-Walukiewicz result. Therefore, we rely on the composition method instead. Let us recall how this method works. Roughly speaking the composition method provides some machinery that allows us to compute the m-theory of a given transition system by breaking it down into several components and looking at the m-theories of these components separately. This approach is based on the realisation that several operations on transition systems are compatible with m-theories in the sense that the m-theory of the result can be computed from the m-theories of the arguments. Statements to that effect are known as composition theorems. For an overview we refer the reader to [3] and [11]. The following basic operations and their composition theorems will be used below. We start with disjoint unions.

Definition 3.2. The disjoint union of two structures
obtained by forming the disjoint union of the universes and relations of A and B and adding two unary predicates Left := A and Right := B that mark whether an element belongs to A or to B. If A and B are transition systems, the initial state of A ⊕ B is that of A.
The corresponding composition theorem looks as follows. It can be proved by a simple induction on m.
A. Blumensath, F. Wolf xx:7 Two other operations we need are interpretations and fusion operations.

Definition 3.4.
An interpretation is an operation τ on structures that is given by a list δ(x), (ϕ R (x)) R∈Σ of MSO-formulae. Given a structure A, it produces the structure τ (A) whose universe consists of all elements of A satisfying the formula δ and whose relations are those defined by the formulae ϕ R . The quantifier-rank of an interpretation is the maximal quantifier-rank of a formula in the list. An interpretation is quantifier-free if its quantifier-rank is 0.

Lemma 3.5. Let τ be an interpretation of quantifier-rank k. Then
Definition 3.6. Let P be a predicate symbol. The fusion operation fuse P merges in a given structure all elements of the set P into a single element, i.e., all elements of P are replaced by a single new element and all edges incident with one of the old elements are attached to the new one instead.
Using the composition theorems for these basic operations we can prove new theorems for derived operations. As an example let us consider pointed paths, i.e., paths where both end-points are marked by special colours. Definition 3.8. We denote the concatenation of two paths A and B by A + B. And we write A • for the expansion of a path A by two new constants for the end-points.
Proof. As the end-points are given by constants, we can construct a quantifier-free inter- Note that, since the concatenation operation is associative, it in particular follows that the set of m-theories of paths forms a semigroup.
Finally let us mention one more involved operation with a composition theorem. Let S be a transition system and C ⊆ S a subsystem. We say that C is attached at the state s ∈ S if there is a unique edge (in either direction) between a state in S \ C and a state in C and this edge leads from s to the initial state of C. For a finite system S this statement can be proved in the same way as Corollary 3.9 by expressing S as a disjoint union followed by a quantifier-free interpretation. For infinite systems, we need a more powerful version of the disjoint union operation called a generalised sum (see [16]).
As presented above these tools work with m-theories, which is not quite what we need since we have to also account for bisimulation-invariance. To do so we modify the definitions as follows. Definition 3.11. Let C be a class of transition systems and m < ω a number. (a) We denote by m C the transitive closure of the union ≡ m ∪ ∼ restricted to the class C. Formally, we define S m C T if there exist systems C 0 , . . . , C n ∈ C such that (b) We denote by Th m C (S) the set of all MSO-formulae of quantifier-rank m that are bisimulation-invariant over C and that are satisfied by S, and we define We also set TH m Note that, up to logical equivalence, there are only finitely many formulae of a given quantifier-rank. Hence, each set TH m C is finite and the relations ≡ m , ≡ m C and m C have finite index.
The relation ≡ m C is what we aim to understand when proving characterisation results. But there is no obvious way to compute it. As an approximation we have introduced the relation m C , which is defined in terms of relations that we hopefully understand much better. Surprisingly, our approximation turns out to be exact. Proof. Clearly S m C T implies S ≡ m C T as no bisimulation-invariant MSO-formula of quantifier rank at most m can distinguish two m C -equivalent transition systems. To prove the converse we consider the formulae (This is well-defined since, up to logical equivalence, there are only finitely many m-theories and each of them only contains finitely many formulae.) We start by showing that Clearly, T m C C implies T |= ψ C by definition of ψ C . Conversely, Furthermore, note that ψ C is bisimulation-invariant over C since Thus, ψ C is an MSO m -formula that is bisimulation-invariant over C, and it follows that Some of the above composition theorems also hold for the relation m C . This is immediate if the operation in question also preserves bisimilarity. We mention only two such results. The second one will be needed below.
A. Blumensath, F. Wolf xx:9 Lemma 3.13. Let C be a class that is closed under disjoint unions.
Proposition 3.14. Let C and D be two classes, S ∈ C a (possibly infinite) transition system and let S be the system obtained from S by replacing an arbitrary number of attached subsystems by subsystems which are m D -equivalent. Then S m C S provided that the class C is closed under the operation of replacing attached subsystems in D.

Types
Our strategy to prove the unravelling property for a class C is as follows. For every quantifier-rank m, we assign to each tree T a so-called m-type τ m (T). We choose the functions τ m such that we can compute the theory Th m C (C) of a system C ∈ C from the m-type τ m (U(C)) of its unravelling. Furthermore, we need to find MSO-formulae checking whether a tree has a given m-type. The formal definition is as follows. (S1) τ m (U(C)) = τ m (U(C )) implies Th m Let us start by showing how to prove the unravelling property using type functions. The following characterisation theorem can be considered to be the main theoretical result of this article. (1) Over C, bisimulation-invariant MSO coincides with L µ .
(2) ⇒ (4) Let m < ω. For every θ ∈ TH m C , we use the unravelling property to find an MSO-formula ϕ θ that is bisimulation-invariant over trees and satisfies Let k be the maximal quantifier-rank of these formulae ϕ θ . Then Th k T (U(C)) = Th k T (U(C )) implies Th m C (C) = Th m C (C ) .
Consequently, there exists a function h m : TH k T → TH m C such that h m Th k T (U(C)) = Th m C (C) .
We start by showing that each ψ θ is bisimulation-invariant over trees: given T ∼ T , (S2) implies that as desired. By the claim we have just proved, it follows that

Lassos
As an application of type functions, we consider a very simple example, the class of lassos. Our proof is based on more or less the same arguments as that by Dawar and Janin [6], just the presentation differs. A lasso is a transition system consisting of a directed path ending in a cycle.
We allow the borderline cases where the initial path has length 0 or the cycle consists of only a single edge.
To define the type of a lasso, note that we can construct every lasso L from two finite paths A and B by identifying three of their end-points.
The paths A and B are uniquely determined by L. We will refer to A as the tail of the lasso and to B as the loop. We introduce two kinds of types for lassos, a strong one and a weak one. The strong m-type of a lasso uniquely determines its m-theory.
Proof. Let A i and B i be the tail and loop of L i . Note that we can write L i in the form is an additional unary predicate marking the vertices to be identified, and fuse Pi is the fusion operation that identifies all vertices in P i . Note that P i is definable by a quantifier-free formula. Hence, there exists a quantifier-free interpretation σ such that As disjoint union, quantifier-free interpretations, and fusion are compatible with m-theories, it follows that

xx:12 Bisimulation Invariant MSO in the Finite
The problem with the strong type of a lasso L is that we cannot recover it from the unravelling of L as the decomposition of U(L) into the parts of L is uncertain. Therefore we introduce another notion of a type where this recovery is possible. For this we recall some facts from the theory of ω-semigroups.
Recall that we have noted in Corollary 3.9 that the m-theories of pointed paths form a finite semigroup with respect to concatenation. Furthermore, every element a of a finite semigroup has an idempotent power a π , which is defined as the value a n where n is the least natural number such that a n · a n = a n . Since the pairs α 0 , β 0 and α 1 , β 1 are conjugate, there exist m-theories ξ and η such that To show that the functions (τ m ) m form a family of type functions, we need the following standard facts about factorisations and their types (see, e.g., Section II.2 of [14]).

Proposition 5.5. Let A be an infinite path.
(a) A has a factorisation of type α, β , for some α and β. A has factorisations of type α, β and γ, δ , then α, β and γ, δ are conjugate.
Note that these two statements imply in particular that the type τ m (T) of a tree T is well-defined. Proof. (S1) Suppose that τ m (U(L 0 )) = τ m (U(L 1 )), for two lassos L 0 and L 1 . By Proposition 5.5 (b), it follows that Hence, the claim follows by Lemma 5.4.
(S2) Suppose that T ∼ T and that every branch of T has a factorisation of type α, β . Then so does every branch of T . Hence, τ m (T) = τ m (T ).
(S3) Given two m-theories α and β, it is straightforward to write down an MSOformula ψ α,β stating that every branch of a tree has a factorisation of type α, β .

Hierarchical Lassos
After the simple example in the previous section, let us give a more substantial application of the type machinery. We consider hierarchical (or nested) lassos. These are obtained from a lasso by repeatedly attaching sublassos to some states. More precisely, a 1-lasso is just an ordinary lasso, while inductively a (k + 1)-lasso is obtained from a k-lasso by attaching one or more lassos to some of the states. (Each state may have several sublassos attached.) Alternatively, we can obtain a (k + 1)-lasso M from a 1-lasso L by attaching k-lassos. We will call this lasso L the main lasso of M. The types we use for k-lassos are based on the same principles as those for simple lassos, but we have to nest them in order to take the branching of a hierarchical lasso into account.
(t(v) is the label of the vertex v and t| u denotes the subtree attached to u.) We will prove that the functions tp k m form a family of type functions. Note that it follows immediately from the definition that they satisfy Properties (S2) and (S3). Thus, to prove that the class of k-lassos has the unravelling property it is sufficient to show that tp k m also satisfies Property (S1). We will do so by induction on k. The base case of this induction rests on the following lemma. Proof. We prove the claim by induction on k. For k = 1, we can take N := M. Hence, suppose that k > 1. By inductive hypothesis, every sublasso attached to the main lasso is equivalent to some 1-lasso. Replacing them by these 1-lassos, we may assume that k = 2. We start by getting rid of the sublassos attached to the main loop of M. Fix a vertex v on the main loop of M and let P be the cycle from v back to v. Let L be a sublasso attached to v. By Lemma 5.4, we have L m L1 P. Hence, we can replace L by P. Let M be the 2-lasso obtained by these substitutions, let K be the main loop of M (including all the sublassos), and let K be the loop obtained from K by removing the sublassos. As every sublasso attached to the main loop K is isomorphic to K , it follows that K ∼ K . Let M be the 2-lasso obtained from M by replacing the loop K by K . Then It remains to remove the sublassos of M attached to the tail. We prove the claim by induction on the number of vertices of M that have sublassos attached. If there are none, we are done. Otherwise, let v be the last such vertex, let L be the part of the main lasso that is attached to v and let K be some sublasso attached to v. By Lemma 5.4, we have K m L1 L. Let M be the 2-lasso obtained from M by replacing all sublassos attached to v by a copy of L and let M (4) be the 2-lasso obtained by removing all these sublassos. Then As M (4) has one less vertex with sublassos attached, we can use the inductive hypothesis to find an 1-lasso N with  N)) that correspond to their main lassos. We first consider the case where wtp m (β) = wtp m (γ). For every tp k−1 m -type σ, we pick a representative C σ . Let M and N be the k-lassos obtained by replacing every sublasso of type σ by its representative C σ . By inductive hypothesis and Proposition 3.14, it follows that M m L K M and N m L K N . As the m-types of β and γ are conjugate (including all the information about attached sublassos), it follows by Lemma 5.4 that the two lassos A and B that correspond to the branches β and γ are m L -equivalent, even with the additional labelling provided by TP k−1 m . Note that M is the k-lasso obtained from A by attaching all representatives C σ as indicated by this labelling, and N is obtained from B in the same way. By Proposition 3.14 it therefore follows that M m L K N . Consequently, It remains to consider the case where β and γ have different m-types. As M and N have the same type, there exists a branch γ of TP k−1 m (U(N)) whose m-type is conjugate to that of β. We will construct a (k − 1)-lasso N m L K N such that tp k m (N ) = tp k m (M) and the main lasso of N has the same type as γ . Then the claim follows from the special case proved above.
We construct N by choosing a copy of γ as its main lasso. For every successor u of a vertex v of γ that does not itself belong to γ , we attach a copy of C σ to the corresponding vertex of N , where σ is the type of the sublasso of N rooted at u. By the definition of tp k m it follows that as desired. Furthermore, Proposition 3.14 implies that N m L K N.
Using Theorem 4.2 we now immediately obtain the following statement.
Theorem 6.5. For every k, the class of all k-lassos has the unravelling property.

Reductions
We would like to define reductions that allow us to prove that a certain class has the unravelling property when we already know that some other class has this property. To do so, we encode every transition system of the first class by some system in the second one.
The main example we will be working with is a function that removes certain attached subsystems and uses additional vertex labels to remember the m-theories of all deleted system. Up to equivalence of m-theories, we can undo this operation by a function η that attaches to each vertex labelled by some m-theory θ some fixed system with theory θ. Let us give a general definition of such pairs of maps.
In this case, we call the function η a (k, m)-decoding map for .
Example. Let T be the class of all trees and C ⊇ T any class containing it. The unravelling operation U : C → T is an (m, m)-encoding map and the identity function id : T → C the corresponding (m, m)-decoding map. For (E1), it is sufficient to note that U(id(T)) = T, for every tree T. For (E2), consider two systems S, S ∈ C. Then Let us note that the two axioms of an encoding map imply dual axioms with the functions and η exchanged. (η( (C))) k D (C) implies η( (C)) m C C .
(E4) By (E1) and (E2), The axioms of an encoding map were chosen to guarantee the property stated in the following lemma. It will be used below to prove that encoding maps can be used to transfer the unravelling property from one class to another. Proof. By (E2) and Proposition 3.12, Hence, there exists a function h on MSO-theories such that We set where Θ ϕ is the set of all MSO m -theories containing ϕ. Note thatφ is bisimulation-invariant over D since bisimulation-invariant formulae are closed under boolean operations. Furthermore, ϕ has quantifier-rank k and It remains to show how to use encoding maps to transfer the unravelling property. Just the existence of such a map is not sufficient. It also has to be what we call definable. (The only reason why we use a family of classes to reduce to, instead of a single one is so that we can have the labellings of systems in D m depend on the quantifier-rank m.) Theorem 7.5. Suppose that C is reducible to (D m ) m<ω . If every class D m has the unravelling property, so does C.
Proof. Let ϕ be bisimulation-invariant over C and let m be its quantifier-rank. By Lemma 7.3, there exists an MSO-formula ψ that is bisimulation-invariant over D m such that Using the unravelling property of D m , we can find an MSO-formulaψ that is bisimulationinvariant over trees such that Finally, definability of m provides an MSO-formulaφ that is bisimulation-invariant over trees such that Consequently, we have C |= ϕ if, and only if, U(C) |=φ, for all C ∈ C.

Finite Cantor-Bendixson rank
One common property of k-lassos is that the trees we obtain by unravelling them all have finite Cantor-Bendixson rank. In this section we will generalise our results to cover transition systems with this more general property. The proof below consists in a two-step reduction to the class of k-lassos.
Definition 8.1. Let T be a finitely branching tree. The Cantor-Bendixson derivative of T is the tree T obtained from T by removing all subtrees that have only finitely many infinite branches. The Cantor-Bendixson rank of a tree T is the least ordinal α such that applying α + 1 Cantor-Bendixson derivatives to T results in an empty tree. The Cantor-Bendixson rank of a transition system S is equal to the Cantor-Bendixson rank of its unravelling.
We can go from the class of k-lassos to that of systems with bounded Cantor-Bendixson rank in two steps. Definition 8.2. (a) A transition system is a generalised k-lasso if it is obtained from a finite tree by attaching (one or several) k-lassos to every leaf.
(b) A transition system T is a tree extension of S if T is obtained from S by attaching an arbitrary number of finite trees to some of the vertices.
With these two notions we can characterise the property of having bounded Cantor-Bendixson rank as follows. (1) S has Cantor-Bendixson rank at most k.
(2) S is bisimilar to a tree extension of a generalised (k + 1)-lasso.
(b) The following statements are equivalent.
(2) S is bisimilar to a tree extension of a generalised k-lasso, for some k < ω.
(3) Every strongly connected component of S is either a singleton or a cycle.
Proof. (a) follows by induction on k. For k = 0, note that a transition system S has Cantor-Bendixson rank 0 if, and only if, its unravelling consists of finitely many infinite branches and attached finite subtrees. This is the case if, and only if, S is bisimilar to a tree extension of a generalised 1-lasso.
For k > 0, note that S has Cantor-Bendixson rank at most k if, and only if, in its unravelling we can choose finitely many branches such that all subtrees that do not contain any of them have Cantor-Bendixson rank at most k − 1. By inductive hypothesis, this is the case if, and only if, the unravelling is bisimilar to a tree with finitely many infinite branches to which tree extensions of generalised k-lassos are attached at arbitrary vertices. Such a structure is bisimilar to a tree extension of a generalised (k + 1)-lasso.
(3) ⇒ (2) Suppose that every strongly connected component of S is either a singleton or a cycle. In the partial order formed by all strongly connected components of S (ordered by the reachability relation), fix a chain of maximal length that consists only of components that are cycles and let k be its length. By induction on k it follows that we can partially unravel S into a tree extension of a generalised k-lasso.
To do so, we present a reduction to the class of generalised k-lassos. Our (m, m)-encoding maps m : E k → H m k map a tree extension M to the generalised k-lasso m (M) obtained by removing all attached finite trees. To remember what was deleted, we label every vertex v with the set of m-theories of the subtrees that were attached to v. The corresponding (m, m)decoding map η m : H m k → E k simply adds a representative of every m-theory to all vertices labelled by this theory.
To see that m and η m form a definable family of encoding and decoding maps, we have to check three conditions.

Conclusion
We have shown in several simple examples how to characterise bisimulation-invariant MSO in the finite. In particular, we have proved that it coincides with L µ over every finite class (Theorem 2.8), the class of all finite trees (Theorem 2.9), the classes of all lassos, k-lassos, and generalised k-lassos (Theorems 5.7, 6.5, and 8.4), the class of all systems of Cantor-Bendixson rank at most k (Theorem 8.5).
Our main tool in these proofs was the unravelling property (Theorem 2.11). It will be interesting to see how far our methods can be extended to more complicated classes. For instance, can they be used to prove the following conjecture?
Conjecture. If a class C of transition systems has the unravelling property, then so does the class of all subdivisions of systems in C.
A good first step seems to be the class of all finite transition systems that have Cantor-Bendixson rank k, for some k < ω that is not fixed. In this paper we have considered only transition systems made out of paths with very limited branching. To extend our techniques to classes allowing for more branching seems to require new ideas. A simple test case that looks promising is the class of systems with a 'lasso-decomposition' of width k, i.e., something like a tree decomposition but where the pieces are indexed by a lasso instead of a tree.