Relative Bisimulation: A Unified Perspective

• Relative bisimulation is a framework that tailors back-and-forth matching by using external parameters like policies, action subsets, or logical fragments.
• It is applied in discounted MDPs, labeled transition systems, the calculus of relations, and coalgebra to preserve key observable behaviors while relaxing standard conditions.
• Its variations enable controlled weakening of canonical bisimulation, yielding scalable algorithms and robust approximation guarantees in diverse semantic settings.

Relative bisimulation is a family of parameterized behavioral equivalences or preorders in which the usual bisimulation condition is relativized to an external parameter. In the literature covered here, that parameter is a stationary policy in discounted Markov decision processes, a distinguished subset of actions in labeled transition systems, a logical fragment in the calculus of relations, or a chosen relation lifting or relator in coalgebraic semantics (Kemertas et al., 2022, Markovski, 2012, Fletcher et al., 2012, Staton, 2011, Goncharov et al., 3 Feb 2025). The shared theme is not a single canonical definition but a controlled weakening or specialization of back-and-forth matching that preserves the observables relevant to the ambient theory.

1. Scope and General Form

The term has several established uses, each tied to a different semantic setting. The parameter controls which transitions, observations, or modal resources must be matched symmetrically, and which may be matched only in a weaker or more localized sense.

Context	Relativizing parameter	Canonical effect
Discounted MDPs	stationary policy $\pi$	replace $\max_{a\in A}$ by expectations under $\pi$
Labeled transition systems	action subset $B\subseteq \mathrm{Act}$	actions in $B$ are bisimulated; others are only simulated
Calculus of relations	fragment $F$ and degree $k$	clauses quantify only over $\mathrm{paths}_k^F$ and fragment-specific operators
Coalgebra/relators	relation lifting or relator	step condition is defined relative to the chosen lifting

This variation matters because different communities use “relative bisimulation” for different technical objects. In discounted control, the standard term is $\pi$-bisimulation or policy-relative bisimulation (Kemertas et al., 2022). In concurrency theory, “partial bisimulation,” “relative bisimulation,” and “conditional bisimulation” are used synonymously for relations parameterized by a bisimulated action subset $B$ (Markovski, 2012). In finite-model theory and database semantics, relative bisimulation is fragment-relative: the back-and-forth clauses are tailored to the operators available in a fragment of the calculus of relations (Fletcher et al., 2012). In coalgebra, relative bisimulation typically means bisimulation defined relative to a relation lifting $\max_{a\in A}$0 or, more generally, a relator or lax extension (Staton, 2011, Goncharov et al., 3 Feb 2025).

A recurring misconception is to conflate these notions with weak or branching bisimulation. In the action-set-parametric setting, partial bisimulation is explicitly strong and stepwise, but restricted by $\max_{a\in A}$1; it is distinct from weak or branching bisimulation, which abstract from silent steps and use path-based matching (Markovski, 2012).

2. Policy-Relative Bisimulation in Discounted Markov Decision Processes

In discounted MDPs $\max_{a\in A}$2 with compact state space $\max_{a\in A}$3, bounded reward $\max_{a\in A}$4, and stationary policy $\max_{a\in A}$5, policy-relative bisimulation is defined from the policy-averaged reward and transition kernel

$\max_{a\in A}$6

The $\max_{a\in A}$7-bisimulation metric $\max_{a\in A}$8 is the unique fixed point of

$\max_{a\in A}$9

with $\pi$0 and $\pi$1 (Kemertas et al., 2022).

This construction parallels the standard bisimulation metric

$\pi$2

but replaces the maximization over actions by expectations under $\pi$3. The paper shows that both bisimulation and $\pi$4-bisimulation can be generalized through $\pi$5-Wasserstein and Sinkhorn distances. For $\pi$6 and $\pi$7, the generalized operator uses $\pi$8, and the same value-function-approximation guarantees hold:

$\pi$9

for any $B\subseteq \mathrm{Act}$0-aggregation $B\subseteq \mathrm{Act}$1, provided $B\subseteq \mathrm{Act}$2 (Kemertas et al., 2022).

A central technical point is that policy-relative bisimulation is smooth with respect to policy change. For $B\subseteq \mathrm{Act}$3 and $B\subseteq \mathrm{Act}$4,

$B\subseteq \mathrm{Act}$5

where $B\subseteq \mathrm{Act}$6. Small policy updates therefore induce small changes in the metric. This is the basis for the paper’s conservative API$B\subseteq \mathrm{Act}$7 procedure, which updates

$B\subseteq \mathrm{Act}$8

and obtains a better asymptotic bound than naive API with hard greedy replacement. The same work maps these ideas to actor-critic practice through representation learning objectives that approximate $B\subseteq \mathrm{Act}$9, including the DBC loss with $B$0 and Gaussian latent dynamics (Kemertas et al., 2022).

The policy-relative construction is therefore “relative” in a literal sense: state similarity is evaluated relative to the behavior induced by the current policy through $B$1 and $B$2. This avoids the pessimism of $B$3 and is computationally attractive in large action spaces, while retaining uniform approximation guarantees for $B$4.

3. Partial Bisimulation on Labeled Transition Systems

For a labeled transition system $B$5, relative bisimulation is commonly defined by fixing a subset $B$6 of bisimulated actions. A relation $B$7 is a partial bisimulation with respect to $B$8 if, for all $B$9, actions in $F$0 satisfy both forward and backward matching, while actions in $F$1 satisfy forward simulation only. If a termination predicate is present, one additionally requires termination monotonicity $F$2 (Markovski, 2012).

The induced preorder and equivalence are

$F$3

The two extreme choices recover familiar notions: $F$4 yields simulation preorder and simulation equivalence, while $F$5 yields full bisimulation (Markovski, 2012).

The quotient construction is more delicate than for ordinary bisimulation because simulation-style minimization must track “little brothers.” These are class-level ordering relations $F$6 expressing that $F$7 simulates $F$8. In quotienting and refinement, one must preserve upward closure for simulated actions and additional back-matching constraints for actions in $F$9. The paper emphasizes that this extra structure is what makes simulation-equivalence minimization more expensive than bisimulation minimization (Markovski, 2012).

The algorithmic contribution is a partition-refinement method that maintains a partition–relation pair $k$0 and alternates two phases: partition refinement in Paige–Tarjan style and little-brother updates. Stability requires uniform termination on each block, forward simulation conditions for all labels, and back-matching on bisimulated labels. The paper proves that stable pairs form an upper lattice with a greatest stable pair, and that iterating the refinement operator converges to the coarsest stable pair, hence to the greatest $k$1 and $k$2 (Markovski, 2012).

The worst-case complexity is

$k$3

time and

$k$4

space. These bounds scale with the bisimulated subset because only labels in $k$5 require the more expensive symmetric treatment. The paper’s intended application is supervisory control in discrete-event systems, where $k$6 is chosen as the set of uncontrollable actions: partial bisimulation then enforces bisimulation on uncontrollables and only simulation on controllables (Markovski, 2012).

In this tradition, “relative” means relative to a designated action interface. The relation lies strictly between simulation and bisimulation, and its significance is tied to property preservation and scalable quotienting rather than to metric geometry.

4. Fragment-Relative Bisimulation in the Calculus of Relations

In the calculus of binary relations, relative bisimulation is parameterized by a fragment $k$7 of the language and, in the degree-bounded setting, by a degree $k$8. The semantics are given over relational structures $k$9, and the key auxiliary object is the fragment-relative path predicate $\mathrm{paths}_k^F$0, which captures the pairs reachable by expressions of degree at most $\mathrm{paths}_k^F$1 (Fletcher et al., 2012).

Marked structures are triples $\mathrm{paths}_k^F$2, and indistinguishability is defined by type inclusion:

$\mathrm{paths}_k^F$3

When $\mathrm{paths}_k^F$4 contains complement or difference, one-sided indistinguishability collapses to two-sided indistinguishability except for trivial outside-path cases, and the appropriate notion is an $\mathrm{paths}_k^F$5-bisimulation: a decreasing sequence $\mathrm{paths}_k^F$6 satisfying Atoms Forth/Back together with fragment-specific back-and-forth clauses for composition, projection, and residuals (Fletcher et al., 2012).

The composition clause is representative. At degree $\mathrm{paths}_k^F$7, if $\mathrm{paths}_k^F$8 and $\mathrm{paths}_k^F$9 lie in $\pi$0, then there must exist $\pi$1 such that $\pi$2 and $\pi$3; the back direction is symmetric. Projection and residuals introduce additional node-observational and universal/conditional clauses, and all quantification is explicitly restricted to $\pi$4 (Fletcher et al., 2012).

For positive fragments, where neither complement nor difference is available, the paper replaces bisimulation by a two-relations simulation $\pi$5. This is needed because coprojection and residuals are nonmonotonic in the absence of complement. The adequacy theorem states that, for fragments with complement or difference,

$\pi$6

while for positive fragments, one-sided indistinguishability is characterized by the corresponding $\pi$7-simulation (Fletcher et al., 2012).

For finite structures, the degree-bounded refinement stabilizes, yielding Hennessy–Milner-style characterizations for the full fragment. The paper also gives polynomial-time decidability for indistinguishability of finite marked structures in fixed fragments. Here, “relative” is tied to expressivity: the bisimulation clauses are exactly strong enough to preserve the operators admitted by the chosen fragment.

5. Coalgebraic Relative Bisimulation and Relators

In coalgebraic semantics, relative bisimulation is the relation-lifting-based notion associated with a functor. For a functor $\pi$8, coalgebras $\pi$9 and $B$0, and a relation $B$1, the Hermida–Jacobs condition is

$B$2

equivalently: if $B$3 then $B$4. The paper “Relating coalgebraic notions of bisimulation” identifies this as the notion most commonly called relative bisimulation and denotes it as HJ-bisimulation (Staton, 2011).

That paper studies four coalgebraic generalizations: Aczel–Mendler bisimulation, HJ-bisimulation, AM-precongruence, and kernel bisimulation. It proves the implication chain

$B$5

and then provides structural conditions under which the notions coincide. In particular, if $B$6 preserves weak pullbacks, every kernel bisimulation is an AM-bisimulation; under the standard accompanying hypotheses, the four notions coincide with behavioral equivalence, and the greatest relative bisimulation can be obtained by transfinite refinement

$B$7

starting from $B$8 (Staton, 2011).

A later generalization replaces relation liftings by relators or lax extensions. For a set endofunctor $B$9 and an $\max_{a\in A}$00-relator $\max_{a\in A}$01, a relation $\max_{a\in A}$02 between coalgebras $\max_{a\in A}$03 and $\max_{a\in A}$04 is an $\max_{a\in A}$05-simulation iff

$\max_{a\in A}$06

that is, $\max_{a\in A}$07 implies $\max_{a\in A}$08. When $\max_{a\in A}$09 is symmetric, the resulting greatest fixed point is an $\max_{a\in A}$10-bisimilarity (Goncharov et al., 3 Feb 2025).

Within this framework, soundness and completeness become properties of the chosen relator. The coBarr relator $\max_{a\in A}$11 is sound and complete for behavioral equivalence when the functor preserves $\max_{a\in A}$12-iso pullbacks. The same work shows that the expected closure properties of simulations and bisimulations characterize relator axioms such as extension of functions, laxity, symmetry, and normality, and that for functors preserving inverse images there exists a greatest normal lax extension $\max_{a\in A}$13 (Goncharov et al., 3 Feb 2025). In this sense, coalgebraic relative bisimulation is literally bisimulation relative to a chosen lifting discipline.

6. Distribution-Based and Probabilistic Variants

For finite, image-finite probabilistic automata

$\max_{a\in A}$14

the paper on probabilistic automata introduces a distribution-based bisimulation directly on $\max_{a\in A}$15. For input-enabled automata, a symmetric relation $\max_{a\in A}$16 is a bisimulation if, whenever $\max_{a\in A}$17, two conditions hold: first, label preservation,

$\max_{a\in A}$18

and second, step matching,

$\max_{a\in A}$19

For non-input-enabled automata, the definition is given via the input-enabled extension $\max_{a\in A}$20 (Feng et al., 2013).

This notion is weaker than lifted state-based probabilistic bisimulations because it is defined directly on distributions rather than as a lifting of a state relation. The paper states that the bisimilarity relation is linear and continuous, but not left-decomposable in general. That failure of left-decomposability is exactly what allows the relation to bridge equivalence and bisimulation in Rabin’s reactive automata (Feng et al., 2013).

For reactive automata with the same action alphabet, the paper restates the Doyen–Henzinger–Raskin theorem:

$\max_{a\in A}$21

and shows that, in the direct sum automaton, its own distribution-based bisimulation coincides with $\max_{a\in A}$22 on initial distributions. Thus the probabilistic relation captures Rabin’s language equivalence exactly (Feng et al., 2013).

The same work develops a discounted approximate bisimulation family $\max_{a\in A}$23 and the induced bisimulation distance

$\max_{a\in A}$24

It proves that $\max_{a\in A}$25 is a pseudometric, that $\max_{a\in A}$26 iff $\max_{a\in A}$27, and that $\max_{a\in A}$28 coincides both with a Hennessy–Milner-style logical distance $\max_{a\in A}$29 and with the least fixed point $\max_{a\in A}$30 of a monotone functional on pseudometrics. The distribution-based metric is bounded above by the Kantorovich lifting of the state-based game bisimulation metric, so it is at least as coarse as the lifted state-based metric (Feng et al., 2013).

Although the paper does not define its relation under the label “relative bisimulation,” it explicitly presents the construction through that lens. The relation is relative to the observable interface given by labels and actions and, in reactive automata, to acceptance semantics.

7. Relational Dualities, Modal Invariance, and Conceptual Scope

A recent categorical development reconstructs bisimulation through relational extensions of Tarski and Thomason dualities. For a relation $\max_{a\in A}$31, the lower lifting is

$\max_{a\in A}$32

with dual upper lifting obtained from the converse. On Kripke frames $\max_{a\in A}$33 and $\max_{a\in A}$34, a bisimulation $\max_{a\in A}$35 is characterized by the relational inclusions

$\max_{a\in A}$36

These are exactly the usual forth and back conditions, written diagrammatically (Kozicki et al., 7 May 2026).

The algebraic side is modal: for $\max_{a\in A}$37 and $\max_{a\in A}$38, the induced predicate transformers satisfy modal commutation laws along bisimulations. The paper constructs categories $\max_{a\in A}$39 and $\max_{a\in A}$40 of frames with simulations and bisimulations, together with algebraic categories $\max_{a\in A}$41 and $\max_{a\in A}$42, and proves the relational Thomason duality

$\max_{a\in A}$43

via the lower lifting (Kozicki et al., 7 May 2026).

The logical consequence is formula invariance: if $\max_{a\in A}$44 is a bisimulation and atomic valuations are preserved along $\max_{a\in A}$45, then for every modal formula $\max_{a\in A}$46,

$\max_{a\in A}$47

The paper also presents a proof-theoretic reading through judgments expressing that one predicate simulates another across a relation, together with modal transport rules induced by simulations, cosimulations, and bisimulations (Kozicki et al., 7 May 2026).

This duality-based account sharpens the conceptual scope of relative bisimulation. In some literatures the relativity is to policies, action subsets, logical fragments, or relators; here it is to the relation induced between predicate algebras by the lower lifting. This suggests a broad unifying schema: relative bisimulation is not a single equivalence, but a method for tailoring back-and-forth reasoning to a specified semantic interface while preserving the invariants native to that interface.