A Coalgebraic View of Characteristic Formulas in Equational Modal Fixed Point Logics
Abstract
The literature on process theory and structural operational semantics abounds with various notions of behavioural equivalence and, more generally, simulation preorders. An important problem in this area from the point of view of logic is to find formulas that characterize states in finite transition systems with respect to these various relations. Recent work by Aceto et al. shows how such characterizing formulas in equational modal fixed point logics can be obtained for a wide variety of behavioural preorders using a single method. In this paper, we apply this basic insight from the work by Aceto et al. to Baltag’s “logics for coalgebraic simulation” to obtain a general result that yields characteristic formulas for a wide range of relations, including strong bisimilarity, simulation, as well as bisimulation and simulation on Markov chains and more. Hence this paper both generalizes the work of Aceto et al. and makes explicit the coalgebraic aspects of their work.
Keywords
Label Transition System Kripke Frame Structural Operational Semantic Characteristic Formula Weak Bisimulation1 Introduction

Forth: If \(u Z v\) and \(u \mathop {\longrightarrow }\limits ^{a} u^\prime \) for some action \(a\), then there is \(v^\prime \) with \(v \mathop {\longrightarrow }\limits ^{a} v^\prime \) and \(u^\prime Z v^\prime \).

Back: If \(u Z v\) and \(v \mathop {\longrightarrow }\limits ^{a} v^\prime \), then there is \(u^\prime \) with \(u \mathop {\longrightarrow }\limits ^{a} u^\prime \) and \(u^\prime Z v^\prime \).
The weaker notion of simulation is like bisimulation except that the “Back” condition is dropped. Another way to weaken the notion of strong bisimulation is to “truncate” the silent \(\tau \)transitions, according to the intuition that bisimulation should capture equivalence of observable behaviour. The resulting concept of behavioural equivalence is called weak bisimulation.
 1.
The semantics for the modal operators and the various notions of simulation both arise from the same concept of relation lifting via a lax extension.
 2.
A finite coalgebra can itself be viewed as a system of equations.
2 Basics
2.1 Set Coalgebras and Lax Extensions
In this section we introduce some basic concepts from coalgebra theory that will be used later on. We assume familiarity with basic category theoretic concepts. We fix a functor \(T : \mathbf {Set} \rightarrow \mathbf {Set}\), where \(\mathbf {Set}\) is the category of sets and mappings For simplicity, we assume that \(T\) preserves set inclusions, so that a set inclusion \(\iota : X \rightarrow Y\) is mapped to a set inclusion \(T\iota : TX \rightarrow TY\). This assumption is actually more innocent than it may seem at first, since every set functor is naturally isomorphic “upto\(\emptyset \)” to one that preserves set inclusions. More precisely, for every set functor \(T\) there is a functor \(T^\prime \) such that the restrictions of these two functors to the full subcategory of nonempty sets are naturally isomorphic, see [1] for details.
We will make use of an approach to coalgebraic logic developed by Alexandru Baltag, based on certain methods of extending the signature functor \(T\) to relations [3]. This approach is a generalization of the original formulation of coalgebraic logic due to Moss [13]. While Baltag uses “weak \(T\)relators” (for more on relators and simulations [7, 9, 17]), we shall here use the slightly more general notion of “lax extension” [11], which works just as well. Besides that, our approach is the same as Baltag’s.
Definition 1
Given a function \(f:X\rightarrow Y\), let \(\widehat{f} = \{(x,f(x))\mid x\in X\}\) be the graph of \(f\). Let \(\varDelta _X = \widehat{ Id _X}\) be the graph of the identity map on \(X\).
The concept of a lax extension is defined as follows:
Definition 2

L1: \(R \subseteq S\) implies \(LR \subseteq LS\),

L2: \(LR ; LS \subseteq L(R ; S)\),

L3: \(\widehat{Tf} \subseteq L\widehat{f}\) for any mapping \(f\).
Example 1
Definition 3
\({\varvec{(L}}\)simulation). An \(L\) simulation from a \(T\)coalgebra \((X,\alpha )\) to \((Y,\beta )\) is a binary relation \(Z \subseteq X \times Y\) such that \(u Z v\) implies \(\alpha (u) (LZ) \beta (v)\). Given pointed \(T\)coalgebras \((\mathfrak {A},u)\) and \((\mathfrak {B},v)\), we write \((\mathfrak {A},u)\preceq _L (\mathfrak {B},v)\) to say that there is an \(L\)simulation \(Z\) from \(\mathfrak {A}\) to \(\mathfrak {B}\) with \(u Z v\).
Note that \(L_{ sim }\)simulation (with \(L_{ sim }\) from Example 1) is simulation on Kripke frames.
2.2 Symmetric Lax Extensions and Bisimulation
We write the converse of a relation \(R\) as \(R^\circ \). Given a relation lifting \(L\), let \(L^{\circ }:R\mapsto (L(R^{\circ }))^\circ \). Call a relation lifting \(L\) symmetric if \(L = L^{\circ }\).
Example 2
Definition 4
Given a set functor \(T\), a \(T\) bisimulation is an \(L\)simulation, where \(L = \overline{T}\).
We could also call a \(T\)bisimulation a \(\overline{T}\)bisimulation, and we will more generally define what is meant by \(L\)bisimulation for any lax extension \(L\) (not necessarily symmetric) that extends \(\overline{T}\) in that \(\overline{T}R \subseteq LR\) for each relation \(R\). We first observe the following.
Observation 1
If \(L\) is a lax extension that extends \(\overline{T}\), then \(L^{\circ }\) is a lax extension.
Proof
 By L1 in Definition 2, we have$$ R \subseteq S \; \Rightarrow \; R^\circ \subseteq S^\circ \; \Rightarrow \; L(R^\circ ) \subseteq L(S^\circ )\; \Rightarrow \; L^{\circ }(R) \subseteq L^{\circ }(S). $$
 We reason as follows:$$\begin{aligned} L^{\circ }(R);L^{\circ }(S)&= (L(R^\circ ))^\circ ;(L(S^\circ )^\circ ) = (L(S^\circ );L(R^\circ ))^\circ \\&\subseteq (L(S^\circ ;R^\circ ))^\circ = (L((R;S)^\circ ))^\circ = L^{\circ }(R;S). \end{aligned}$$
 Let \(f:X\rightarrow Y\). For \(a\in TX\), we haveThus \(\widehat{Tf}\subseteq L^\circ (\widehat{f})\). \(\square \)$$\begin{aligned} a (\overline{T}\widehat{f})(Tf(a))&\Rightarrow (Tf(a))(\overline{T}(\widehat{f}^\circ ))a \qquad (\overline{T}~\mathrm{is~a~symmetric ~relation~lifting })\\&\Rightarrow (Tf(a))(L(\widehat{f}^\circ ))a \qquad (\overline{T}R\subseteq LR~\mathrm{for~all~relations~R })\\&\Rightarrow a (L\widehat{f})(Tf(a)). \end{aligned}$$
Definition 5
The reader can easily check that the following holds:
Observation 2
If \(L_1\) and \(L_2\) are lax extensions, then \(L_1\cap L_2\) is a lax extension.
Hence we get:
Observation 3
If \(L\) is a lax extension, such that \(\overline{T}R\subseteq LR\) for each relation \(R\), then its bisimulator \(B(L)\) is a symmetric lax extension.
Proof
First note that as \(L\) extends \(\overline{T}\), we have by Observation 1 that \(L^{\circ }\) is a lax extension. Then by Observation 2, \(B(L)\) is a lax extension. By definition \(B(L)\) is symmetric. \(\square \)
Remark 1
If \(L_{ sim }\) is the lax extension for the finitary power set functor \(\mathcal {P}_\omega \) that was given in Example 1, then \(B(L_{ sim }) = \overline{\mathcal {P}_\omega }\).
Definition 6
If \(L\) is a lax extension, such that \(\overline{T}R\subseteq LR\) for every relation \(R\), then an \(L\) bisimulation is a \(B(L)\)simulation.
By the previous remark, \(Z\) is an \(L_{ sim }\)bisimulation if and only if \(Z\) is a \(\overline{\mathcal {P}_\omega }\)bisimulation.
3 Coalgebraic Logic with Fixed Point Equations
3.1 Basic Coalgebraic Modal Logic
Definition 7

\(p \in \mathcal {L}\) for all \(p \in V\),

\(\varphi ,\psi \in \mathcal {L}\) implies \(\varphi \wedge \psi \in \mathcal {L}\) and \(\varphi \vee \psi \in \mathcal {L}\),

if \(\varPhi \) is a finite subset of \(\mathcal {L}\) then \(\square a \in \mathcal {L}\) and \(\Diamond a \in \mathcal {L}\) for each \(a \in T\varPhi \).
The following observation is made in [18]:
Proposition 1
Let \(T\) be a set functor that preserves inclusions. Then for any \(a \in TX\) where \(X\) is a finite set, there is a unique smallest set \(Y\subseteq X\) with \(a \in T Y\).
In particular, this guarantees that for any formula \(\Box a\) or \(\Diamond a\), there is a unique smallest set of formulas \(\varPhi \) with \(a \in T \varPhi \). We denote this set by \( SPT (a)\), for “support of \(a\)”. When \(X\) is understood by context, we write \( SPT \) for \( SPT _X\).

\((\mathfrak {A},u)\vDash _\upsilon p\) iff \(u \in \upsilon (p)\), for a propositional variable \(p\),

\((\mathfrak {A},u)\vDash _\upsilon \varphi \wedge \psi \) iff \((\mathfrak {A},u)\vDash _\upsilon \varphi \) and \((\mathfrak {A},u)\vDash _\upsilon \psi \),

\((\mathfrak {A},u)\vDash _\upsilon \varphi \vee \psi \) iff \((\mathfrak {A},u)\vDash _\upsilon \varphi \) or \((\mathfrak {A},u)\vDash _\upsilon \psi \),

\((\mathfrak {A},u) \vDash _{\upsilon } \Box a\) iff \(\alpha (u) (L\vDash _\upsilon ) a\),

\((\mathfrak {A},u) \vDash _{\upsilon } \Diamond a\) iff \( \alpha (u) (L^{\circ }(\vDash _\upsilon )) a\).
Observation 4
If \(L\) is a lax extension, then \(\alpha (u) (L\vDash _{\upsilon }\upharpoonright _{X\times SPT (a)}) a\) if and only if \(\alpha (u) (L\vDash _{\upsilon }) a\).
Proof
First note that \({\vDash _{\upsilon }\upharpoonright _{X\times SPT (a)}}\subseteq {\vDash _{\upsilon }}\) and hence by L1, \(\alpha (u) (L\vDash _{\upsilon }\upharpoonright _{X\times SPT (a)}) a\) implies \(\alpha (u) (L\vDash _{\upsilon }) a\). For the other direction, suppose that \(\alpha (u) (L\vDash _{\upsilon }) a\). By definition of \( SPT \), \(a\in T SPT (a)\), and hence \((a,a)\in \varDelta _{T SPT (a)}\). By (1), \((a,a)\in L\varDelta _{ SPT (a)}\). Then \(\alpha (u) (L\vDash _{\upsilon });L(\varDelta _{ SPT (a)})a\). The desired result follows from this, L2, and the fact that \({\vDash _{\upsilon }\upharpoonright _{X\times SPT (a)}} = (\vDash _{\upsilon });(\varDelta _{ SPT (a)})\). \(\square \)
Remark 2
If \(L\) is a symmetric lax extension, then the formulas \(\Box a\) and \(\Diamond a\) are equivalent. In this case, we might write \(\nabla a\) instead of \(\Box a\) to emphasize that \(\Box \) and \(\Diamond \) are the same. If \(L = \overline{T}\), then these modalities are the same as the \(\nabla \)modality from the (finitary version of) Moss’ presentation of coalgebraic logic [13].
3.2 Fixed Point Semantics
In this section we introduce the (greatest) fixed point semantics for the logic \(\mathcal {L}(V)\), relative to a system of equations. First, we have to say more precisely what a system of equations is:
Definition 8
(System of equations). Given a set of variables \(V\), a system of fixedpoint equations is defined to be a mapping \(s : V \rightarrow \mathcal {L}(V)\).
The following observation will be used for proving the correctness of the characteristic formula for mutual simulation in Example 7. It plays a somewhat similar role as [2, Lemma 4.6] toward this goal.
Observation 5
Proof
The proof of this is a straightforward induction. The key observation is that as \(s = t\upharpoonright V_0\), for each \(p \in V_0\), \(t(p)\) is a formula over the variables in \(V_0\), and hence all variables not in \(V_0\) are “unreachable” in \(t\) from variables in \(V_0\). \(\square \)
4 Characteristic Formulas
Definition 9
Note that \(\mathcal {F}_L\) is a monotone increasing function on the complete lattice of relations in \(\mathcal {P}(X\times Y)\), and hence by the KnasterTarski fixed point theorem, \(\mathcal {F}_L\) has a greatest fixed point. It is clear that a relation \(R \subseteq X \times Y\) is a postfixed point of \(\mathcal {F}_L\) iff it is an \(L\)simulation, and so the greatest fixed point of \(\mathcal {F}_L\) is the relation \(\preceq _L\), i.e. we have \((u,v) \in GFP(\mathcal {F}_L)\) iff there is an \(L\)simulation relating \(u\) to \(v\).
We consider the language \(\mathcal {L}(X)\) with \(X\) being the set of variables. Let \(\varPhi \) be a function from relations in \(\mathcal {P}(X\times Y)\) to valuations in \(\mathcal {P}(Y)^X\), such that \(\varPhi (R)(x) = \{y\mid (x,y)\in R\}\). Let \(\varPsi \) be the function from relations in \(\mathcal {P}(Y\times X)\) to valuations in \(\mathcal {P}(Y)^X\), such that \(\varPsi (R)(x) = \varPhi (R^\circ )(x) = \{y\mid (y,x)\in R\}\).
Definition 10
Theorem 1
The idea behind this theorem has been used for some time, and has been given in papers such as [2, 14, 15]. The presentation in this paper is most similar to a formulation given in [15], which addressed probabilistic simulations in a noncoalgebraic setting. The proofs given in those papers apply to this setting as well. However, we provide a sketch of the proof here to emphasize that it applies to our more general (coalgebraic) setting.
Proof
We are now left with the task to find systems of equations that express \(\mathcal {F}_L\) (directly and conversely). The main observation here is that, with the semantics we are using here for the \(\Box \) and \(\Diamond \)operators, this is easy: a finite \(T\)coalgebra almost is a system of equations!
Lemma 1
For any lax extension \(L\), where \(L^\circ \) is also a lax extension, \(s_\Box \) directly expresses \(\mathcal {F}_L\), and \(s_\Diamond \) conversely expresses \(\mathcal {F}_L\).
Proof
From Lemma 1 together with Theorem 1, we immediately get our main result:
Theorem 2
We also get characteristic formulas for various notions of bisimilarity as an easy corollary to this result. Given a lax extension \(L\) that extends \(\overline{T}\), we use \(\sim _L\) as an abbreviation for the simulation relation \(\preceq _{B(L)}\), where \(B(L)\) is the bisimulator of \(L\).
Corollary 1
Proof
Example 3
Remark 3
In the case where \(L\) is the Barr extension of \(T\) (where \(T\) preserves weak pullbacks), the system of equations \(s_\Box \) (equivalently \(s_\Diamond \)) can viewed as a very simple “\(T\)automaton” in the sense of [18]. Hence [18, Proposition 4.9], which shows that any finite \(T\)coalgebra can be characterized up to bisimilarity by a suitable \(T\)automaton, can be seen as a special instance of Theorem 2.
4.1 Predicate Liftings
The \(\Box \) and \(\Diamond \) modalities used to obtained characteristic formulas above have the nice feature that the appropriate connection between the formulas and the lax extension \(L\) is built directly into the semantics. On the other hand, these modalities are rather abstract. By contrast, modalities based on predicate liftings are relatively easy to grasp and are formally closer to the standard modalities used in HennessyMilner logic and other modal logics for specification of various kinds of transition systems. In this section we provide conditions on a lax extension \(L\) that allow us to derive characteristic systems of equations for \(L\)simulation in the language of predicate liftings. This is very closely related to a recent result by Marti and Venema, appearing first in [10] and later in [12]. The result builds on earlier work by A. Kurz and R. Leal [8], and provides a translation of nablastyle coalgebraic logic corresponding to a lax extension into the logic of predicate liftings. The one subtle difference is that, while Marti and Venema restrict attention to symmetric lax extensions, we are interested also in the nonsymmetric case. The nonsymmetric case allows us to characterize simulation preorders whereas the symmetric only allows us to characterize behavioral equivalences.
Definition 11
 A1 Given a mapping \(f : Z \rightarrow X\) and a relation \(R \subseteq X \times Y\), we have$$ \widehat{Tf};LR = L(\widehat{f};R).$$
 A2 Given a relation \(R \subseteq X \times Y\) and a mapping \(f : Z \rightarrow Y\), we have$$ L(R;(\widehat{f})^\circ ) = LR;(\widehat{Tf})^\circ .$$
It is shown in [10, Proposition 3.10] and [12, Proposition 5] that these conditions hold for all symmetric lax extensions.
Example 4
The reader can verify that these conditions hold for the lax extension \(L_{ sim }\) from Example 1.
Observation 6
If A1 and A2 hold for \(L\), then they hold for \(L^\circ \) also.
An immediate consequence of the conditions A1 and A2 is the following:
Lemma 2
The case where \(L\) is symmetric is shown is given in [12, Proposition 19]. Below we verify that the equations A1 and A2 suffice for the proof to go through.
Proof
Lemma 3
Definition 12
We now come to the main lemma of this section:
Lemma 4
Proof
For the second part of the lemma, we make use of Observation 6 and reason exactly the same way using the distributive law determined by \(L^\circ \). \(\square \)
Note that if \(s_1,s_2\) always give rise to the same operators on evaluations as \(s_\Box ,s_\Diamond \), then these systems of equations must be positive! Hence, we get:
Theorem 3
Proof
Easy corollary from the previous lemma and Theorem 2. \(\square \)
5 Applications
In this final section, we provide examples of lax extensions for various functors that give rise to simulations and bisimulations that have been used in the literature. All these examples are taken from the papers [2, 15].
Finitary Power Set Functor.
Example 5
Example 6
Example 7

\(s(w,0) = \square (\mathcal {P}_\omega \iota _1 (\alpha (w)))\) and

\(s(w,1) = \Diamond (\mathcal {P}_\omega \iota _2 (\alpha (w)))\).

\((\mathfrak {B},v)\vDash _{s_\square } u\) iff \((\mathfrak {B},v)\vDash _{t_0} (u,0)\),

\((\mathfrak {B},v)\vDash _{s_\Diamond } u\) iff \((\mathfrak {B},v)\vDash _{t_1} (u,1)\).

\((\mathfrak {B},v)\vDash _{s_\square } u\) iff \((\mathfrak {B},v)\vDash _{s} (u,0)\),

\((\mathfrak {B},v)\vDash _{s_\Diamond } u\) iff \((\mathfrak {B},v)\vDash _{s} (u,1)\).
Given a relation \(R\subseteq X\times Y\) and \(A\subseteq X\), let \(R[A] = \{b\mid \exists a\in A:aRb\}\).
Example 8
The lax extension \(L\) corresponds to both simulation and bisimulation on Markov chains, and so the main theorem gives characteristic formulas for this relation (simulation and bisimulation are distinguished in variations of these Markov chains such as in [6] as well as with the probabilistic automata in Example 9 below). Furthermore, it is immediate from the equivalence of items 1 and 3 in [15, Lemma 1] that \(L\) is in fact just the Barr extension of \(\mathcal {D}\).
Finite Nondeterministic Probability Functor. We call the functor \(\mathcal {P}_\omega \circ \mathcal {D}\) the finite nondeterministic probability functor. A coalgebra for \(\mathcal {P}_\omega \circ \mathcal {D}\) corresponds to a probabilistic automaton (which is essentially a Markov chain with nondeterministic transitions to distributions).
Example 9
Example 10
Labelled Powerset Functor. Let \(A\) be a set of labels, and let \(\mathcal {P}_A\) be the functor that maps each object \(X\) to \( (\mathcal {P}_\omega (X))^A\), and maps each morphism \(f:X\rightarrow Y\) to \(\mathcal {P}_Af: h \mapsto k\), where \(k:a\mapsto f[h(a)]\). Such a functor corresponds to a multimodal Kripke frame.
Example 11
Weak Simulation and Bisimulation. Let \(A\) be a set of labels and designate \(\tau \in A\) to be a “silent action”, not to be counted in a weak simulation. We aim to define a lax extension to capture weak simulation of transition systems. Here, we cannot simply work with the labelled powerset functor; the problem is that this functor only catches the “onestep” behaviours, while weak simulation crucially involves iterated behaviour. We will solve this problem by modelling transition systems as coalgebras for a suitable comonad.

For a set \(X\), \(C_A(X)\) is the set of \((X\times A)\)labelled and finitely branching rooted trees, with \(\pi _2\lambda (\varepsilon ) = \tau \) (as an arbitrary convention).

For a mapping \(h : X \rightarrow Y\), \(C_A h : C_A X \rightarrow C_A Y\) is defined by letting \(C_A h\) map a tree \((t,\lambda )\) to the tree \((t,\lambda ^\prime )\) with labelling \(\lambda ^\prime \) obtained by the assignment \(x\mapsto (h(\pi _1\lambda (x)),\pi _2\lambda (x))\).

\(t_w = \{v \in \mathbb {N}^*\mid w\cdot v\in t\}\) and

\(\lambda _w:t_w\rightarrow X\times A\), where \(\lambda _w(v) = \left\{ \begin{array}{ll} \lambda (w\cdot v) &{} v\ne \varepsilon \\ (\pi _1\lambda (w), \tau )&{} v=\varepsilon \end{array}\right. .\)
Footnotes
 1.
Here, for lax extensions \(L_1\) and \(L_2\) we define \(L_1 \cap L_2\) by \(R \mapsto L_1 R \cap L_2 R\).
 2.
Given a mapping \(h : X \rightarrow Y\), \(Qh : QY \rightarrow QX\) is defined by \(Qh(Z) = h^{1}[Z]\).
 3.To be concrete, we can take \(\varSigma _n = T(n)\), and we can define the action of \(p_X\) on \((\sigma ,u_1,\ldots ,u_n) \in \varSigma _n \times X^n\) bywhere \(h : n \rightarrow X\) is the mapping defined by \(i \mapsto u_i\). These details will not be relevant to us, however. All we need to know is that \(p\) is a natural transformation, and each of its components is surjective.$$p_X(\sigma ,u_1,\ldots ,u_n) = Th(\sigma ),$$
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